Advances in Boundary Element & Meshless

ISBN: 978-0-9576731-4-4

Advances in Boundary Element & Meshless Techniques XVII

The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques. Previous conferences devoted to Boundary Element and Meshless Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012), Paris, France (2013), Florence, Italy (2014), Valencia, Spain (2015) and Ankara, Turkey (2016)

EC ltd

Advances in Boundary Element & Meshless Techniques XVIII

Edited by L Marin M H Aliabadi

Advances In Boundary Element and Meshless Techniques XVIII

Advances In Boundary Element and Meshless Techniques XVIII

Edited by L Marin M H Aliabadi

EC

ltd

Published by EC, Ltd, UK Copyright © 2017, Published by EC Ltd, 18 Sir Lancelot Close, Eastleigh, SO53 4HJ, UK Phone (+44) 2380 260334

All Rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, except under terms of the Copyright, Designs and Patents Act 1988. Requests to the Publishers should be addressed to the Permission Department, EC, Ltd Publications..

ISBN: 978-0-9576731-4-4

The text of the papers in this book was set individually by the authors or under their supervision. No responsibility is assumed by the editors or the publishers for any injury and/or damage to person or property as a matter of product liability, negligence or otherwise, or from any used or operation of any method, instructions or ideas contained in the material herein.

International Conference on Boundary Element and Meshless Techniques XVIII 11-13 July 2017, Bucharest Romania Organising Committee:

Professor Liviu Marin Department of Mathematics Faculty of Mathematics and Computer Science University of Bucharest 14 Academiei 010014 Bucharest, Romania [email protected] Professor Ferri M H Aliabadi Department of Aeronautics Imperial College, London South Kensington Campus London SW7 2AZ Tel: +44 (0) 20759 45077 [email protected] International Scientific Advisory Committee Alves,C.J.S (Portugal) Benedetti,I (Italy) Blasquez,A (Spain) Chen, Weiqiu (China) Chen, Wen (China) Cisilino,A (Argentina) De Araujo, F C (Brazil) Delvare,F (France) Denda,M (USA) Dong,C (China) Dumont,N (Brazil)

Gao,X.W. (China) Garcia-Sanchez,F (Spain) Johansson,B.T (UK) Lesnic,D (UK) Hematiyan,M.R. (Iran) Mallardo,V (Italy) Martins, N.F.M (Portugal) Mansur, W. J (Brazil) Mantic,V (Spain) Marin, L (Romania) Matsumoto, T (Japan) Mesquita,E (Brazil) Millazo, A (Italy) Minutolo,V (Italy) Ochiai,Y (Japan) Panzeca,T (Italy) Perez Gavilan, J J (Mexico) Pineda,E (Mexico) Qin,Q (Australia) Saez,A (Spain) Sapountzakis E.J. (Greece) Sellier, A (France) Shiah,Y (Taiwan) Sladek,J (Slovakia) Saldek, V (Slovakia) Sollero.P. (Brazil) Taigbenu,A (South Africa) Tan,C.L (Canada) Telles,J.C.F. (Brazil) Wen,P.H. (UK) Wrobel,L.C. (UK) Yao,Z (China) Zhang, Ch (Germany))

PREFACE The Conferences on Boundary Element and Meshless Techniques are devoted to fostering the continued involvement of the research community in identifying new problem areas, mathematical procedures, innovative applications, and novel solution techniques in both boundary element methods (BEM) and boundary integral equation methods (BIEM). Previous successful conferences devoted to Boundary Element Techniques were held in London, UK (1999), New Jersey, USA (2001), Beijing, China (2002), Granada, Spain (2003), Lisbon, Portugal (2004), Montreal, Canada (2005), Paris, France (2006), Naples, Italy (2007), Seville, Spain (2008), Athens, Greece (2009), Berlin, Germany (2010), Brasilia, Brazil (2011), Prague, Czech Republic (2012), Paris, France (2013), Florence, Italy (2014), Valencia, Spain (2015) and Ankara, Turkey (2016). The present volume is a collection of edited papers that were accepted for presentation at the Boundary Element Techniques Conference held at the Radisson Blu Hotel, Bucharest, Romania during 11-13th July 2017. Research papers received from 18 countries formed the basis for the Technical Program. The themes considered for the technical program included solid mechanics, fluid mechanics, potential theory, composite materials, fracture mechanics, damage mechanics, contact and wear, optimization, heat transfer, dynamics and vibrations, acoustics and geomechanics. The conference organizers would also like to express their appreciation to the International Scientific Advisory Board for their assistance in supporting and promoting the objectives of the meeting and for their assistance in the form of reviews of the submitted papers. Editors

CONTENTS Viscous axisymmetric MHD flow about a solid sphere translating in a circular solid tube T. Celik, S. H. Aydin and A. Sellier

1

Inverse simulations of accidental spills in groundwater by the Green element method Akpofure Taigbenu

7

Alternative boundary integral equations for fracture mechanics in 2D anisotropic bodies Vincenzo Gulizzi, Ivano Benedetti, Alberto Milazzo

15

The PWM for the identification of a sound-soft interior acoustic scatterer Andreas Karageorghis, Daniel Lesnic and Liviu Marin

19

Fundamental three-dimensional MHD creeping flow bounded by a plane motionless wall normal to a uniform ambient magnetic field A. Sellier and S. H. Aydin

27

Spherical harmonic expansion of fundamental solutions and their derivatives for homogenous elliptic operators Vincenzo Gulizzi, Alberto Milazzo, Ivano Benedetti

33

Particle-particle interactions in axisymmetric MHD viscous flow S. H. Aydin and A. Sellier

39

2.5D spectral based BEM-FEM formulation to represent waveguide with acoustic and solid interaction F.J. Cruz-Munoz, A. Romero, A. Tadeu and P. Galvın

45

Application of boundary element method to evaluate the seismic site effects of adjacent non-curved valleys Zahra Khakzad, Dana Amini Baneh and Behrouz Gatmiri

50

Identification from partial full-field measurements using a fading regularization MFS algorithm Laëtitia Caillé, Franck Delvare, Nathalie Michaux-Leblond and Jean-Luc Hanus

56

An overview of the peridynamic (PD) formulation with the extended boundary element Method (XBEM) for dynamic fracture G Hattori and J Trevelyan

64

Boundary element crystal plasticity method Ivano Benedetti, Vincenzo Gulizzi, Vincenzo Mallardo

71

A meshless fading regularization algorithm for the Cauchy problem associated with 2D Helmholtz-type equations Laëtitia Caillé, Franck Delvare, Liviu Marin and Nathalie Michaux-Leblond

75

A boundary element formulation for inter- and trans-granular cracking in polycrystalline materials Vincenzo Gulizzi, Chris H. Rycroft, Ivano Benedetti

81

An efficient boundary element procedure to evaluate the design of cathodic protection systems of tank bottoms W. J. Santos, S.L.D.C. Brasil, J. A. F. Santiago and J. C. F. Telles

85

Boundary element formulation for crack surface contact simulation in piezoelectric materials L. Rodrıguez-Tembleque, F. Garcıa-Sanchez, A. Saez and M. Wunsche

91

An efficient hybrid implementation of MLPG method M. Barbosa, E.F. Fontes Jr, J.C.F. Telles, W.J. Santos

97

BEM analysis of crack paths modelled as a sequence of interfacial debonds in unidirectional composites L. Tavara, V. Mantič

104

3D Vortex approach to the unsteady flow generated by a flapping insect wing M. Denda

112

A boundary element implementation for fracture mechanics problems using Generalized Westergaard stress functions Ney Augusto Dumont, Elvis Yuri Mamani and Marilene Lobato Cardoso

120

Superficial 3D mesh generation process using multimedia software for multiscale bone analysis D.M. Prada, A.F. Galvis and P. Sollero

126

Extending the method of fundamental solutions to potential problems with discontinuous boundary conditions Svilen S. Valtchev and Carlos J.S. Alves

132

Dynamic stress intensity factors evaluation with functionally graded materials by finite block method J. Li, C. Shi and P.H. Wen A hierarchic constitutive governed recursive methodology for obtaining threedimensional anisotropic fundamental solution: a theoretical approach T.V. Lisboa and R.J. Marczak

138

A quasi-static model of delamination for structures made of anisotropic layers implemented by SGBEM and nonlinear programming R. Vodička and F. Ksinan

154

Application of SPR for Discontinuous Boundary Elements Results in 2D Elasticity and a New Method to Evaluate Tangential Stress Otávio A. A. da Silveira, Rogério J. Marczak

162

Solution of the Elastodynamic Contact Problem for Cracked Body Using Boundary Integral Equation Method V.V. Zozulya

170

A Boundary Spectral Element Model for Piezoelectric Smart Structures Fangxin Zou, M. H. Aliabadi

178

Multi-Fidelity Modelling for Structural Reliability Analysis Llewellyn Morse, Zahra Sharif Khodaei, M. H. Aliabadi

184

146

Spectral BEM for Wave Propagation and Crack Dynamics Jun Li, Zahra Sharif Khodaei, M. H. Aliabadi

186

Three-dimensional Analysis of Generally Anisotropic Piezoelectric Materials by the BEM Based upon Radon-Stroh Formalism Chung-Lei Hsu, Yui-Chuin Shiah, and Chyanbin Hwu

188

BEM gradient based iterative algorithms for inverse BVPs in steady-state anisotropic heat conduction L. Marin

194

Advances in Boundary Element and Meshless Techniques XVIII

Viscous axisymmetric MHD flow about a solid sphere translating in a circular solid tube T. Celik1 , S. H. Aydin2 and A. Sellier3 Faculty of Education, Bayburt University, 69000 Bayburt, Turkey Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey 3 LadHyX. Ecole polytechnique, 91128 Palaiseau C´ edex, France 1

2

Keywords: MagnetoHydroDynamics, Axisymmetric MHD flow, Particle-wall interactions, Boundary-integral equation.

Abstract. This work looks at the quasi-steady MHD flow about a solid sphere translating in a conducting and quiescent liquid bounded by a solid and motionless tube, of circular cross section, in the presence of a uniform ambient magnetic field. The particle velocity, the tube axis and the magnetic field are parallel. The resulting axisymmetric velocity and pressure fields, in a truncated and bounded liquid domain, are then found to admit boundary integral representations involving the surface traction arising on both the sphere boundary and a closed surface involving a part of the tube. Those representations appeal to fundamental axisymmetric MHD flows previously obtained elsewhere and result in coupled Fredholm boundary-integral equations of the first kind for the surface axial and radial components of the traction exerted by the flow on the boundaries. Numerically inverting those integral equations permits us to give the drag experienced by the sphere and the MHD flow induced in the liquid by its motion for different values of the Hartmann number. This MHD flow is then found to deeply depend upon the Hartmann number. Introduction It is in general of tremendous difficulty to get the unsteady MagnetoHydroDynamic (MHD) flow about a solid body experiencing a prescribed rigid-body migration in a conducting and unbounded quiescent Newtonian liquid subject to a given uniform and steady ambient magnetic field B = Bez [1,2]. Indeed, one has to solve coupled (through the Lorentz body force fL ) unsteady Navier-Stokes and Maxwell equations for the flow and the electric and magnetic fields occurring in the liquid. If the liquid has uniform density ρ, conductivity σ > 0 and viscosity μ while the body has length scale a and the flow has velocity scale V > 0 one introduces the magnetic Reynolds number Rem = μm σV a, the Reynolds number Re = ρV a/μ and the Hartmann number Ha = a/d with d = ( μ/σ)/|B| the so-called Hartmann layer [3]. Then, for a body (and if any, boundaries) having also the magnetic permeability μm and if Rem 1 the magnetic field appears to be B in the liquid [1,2]. When the MHD flow is furthermore without swirl and axisymmetric about an axis parallel with B there is no electric field in the liquid [1,4]. If in addition Re 1 the flow is quasi-steady with velocity u and pressure p governed by the Stokes equations with Lorentz body force fL = σ(u ∧ B) ∧ B. This low magnetic Reynolds and low Reynolds numbers framework for a MDH quasisteady axisymmetric flow without swirl has been used for a sphere translating parallel with B for small [5] or large [6] values of the Hartmann number Ha. Recently [7], the case of arbitrary Ha > 0 has been treated by resorting to a new boundary approach. In the present work, we extent the analysis developed in [7] to the challenging case of a sphere translating in a conducting liquid bounded by a solid and motionless tube of circular cross section and axis of revolution parallel with both the ambient magnetic field and the sphere translational velocity. Governing MHD equations and boundary formulation This section introduces two related MHD problems: the motivating one for the unbounded tube case and another associated one for bounded tube and liquid domain. For this latter case it also presents a suitable boundary approach. Adressed axisymmetric MHD problems As illustrated in Fig. 1, we look at a quasi-steady axisymmetric MHD flow, with velocity u and pressure p, of a conducting and quiescent Newtonian liquid confined in a solid and infinite tube of axis

1

2

Eds L Marin & M H Aliabadi

of revolution (O, ez ) and circular cross-section with radius R. This flow is produced by a solid sphere P, Σ S B = Bez

a

n O

(μ, ρ, σ)

er r z

P

R

M •

U ez D n

Figure 1: A solid sphere, with center O, translating at the velocity U ez in a conducting Newtonian liquid confined by a solid tube having axis of revolution (O, ez ) and circular cross-section with radius R > a. with radius a < R and center O, translating at the velocity U ez parallel to a uniform ambient magnetic field B = Bez . Accordingly, (u, p) is axisymmetric about the (O, ez ) axis and without swirl. This suggests using for any point x the cylindrical coordinates (r, z, θ) such that z = x.ez , r = {|x|2 − z 2 }1/2 ≥ 0 and θ ∈ [0, 2π]. Hence, x = rer + zez with local unit vector er = er (θ) shown in Fig. 1. With these notations, p(x) = p(r, z) and u(x) = ur (r, z)er + uz (r, z)ez . In addition, assuming vanishing Reynolds and magnetic Reynolds numbers and the tube and the sphere to have the same uniform magnetic permeability μm as the liquid implies (see the introduction) that (u, p) satisfies μ∇2 u = ∇p − σB 2 (u ∧ ez ) ∧ ez and ∇.u = 0 in D ,

(1)

u = U ez on S, u = 0 on Σ, (u, p) → (0, 0) as |x| → ∞

(2)

with D the unbounded liquid domain, S the sphere boundary and Σ the tube. The flow (u, p), with stress tensor σ, exerts on the surface S ∪ Σ, with unit n pointing into the liquid, a surface traction f = σ.n = fr (r, z)er + fz (r, z)ez . As a result, the translating sphere experiences a zero torque (about its center O) and a force F given by f dS = [2π fz (P )r(P )dl(P )]ez = −6πμaλU ez (3) F= S

C

with C the half-circle trace of S in the θ = 0 half plane and λ > 0 the so-called drag coefficient. By virtue of the far-field behaviour (2), instead of solving (1)-(2) for the unbounded liquid domain D we solve another MHD problem for a bounded liquid domain DL with prescribed and large enough typical length L > a. More precisely, the boundary ∂DL of DL consists of S, the |z| < L part ΣL of the tube Σ and also the z = −L and z = L circular cross sections D− and D+ , respectively. Accordingly, this second MHD problem depends upon L/a > 1 and reads μ∇2 u = ∇p − σB 2 (u ∧ ez ) ∧ ez and ∇.u = 0 in DL ,

(4)

u = U ez on S, u = 0 on ΣL ∪ D− ∪ D+ .

(5)

Note that (u, p) and the above drag coefficient λ depend upon (L/a, R/a) and the Hartmann number Ha = a/d where d = ( μ/σ)/|B| is the Hartmann layer thickness [3]. Boundary method for the axisymmetric MHD problem in the bounded liquid For convenience for x(r, z) in the bounded liquid domain DL we introduce the related point M (r, z) in the half θ = 0 plane. As for the contour C associated with the sphere (recall (3)) we denote by CL the trace of the boundary ∂DL \ S in the θ = 0 half plane. Appealing to [7,8] and using the usual tensor summation notation for repeated indices α and β in {r, z} then provides the following basic integral representations


3

for the flow velocity and pressure 1 Gαβ (M, P )fβ (P )r(P )dl(P )for α = r, z and x in DL ∪ S ∪ ∂DL , uα (x) = − 8πμ C∪CL 1 Pβ (M, P )fβ (P )r(P )dl(P ) for x in DL p(x) = − 8π C∪CL

(6) (7)

with Green tensor velocity components Gαβ (M, P ) and Green pressure vector components Pβ (M, P ) available in [8] in terms of z − z(P ), r, r(P ) and d. From (6)-(7), calculating the traction f = fr er + fz ez on the entire contour C ∪ CL permits one to gain the flow (u, p) in the liquid domain. In addition, selecting in (6) the point x on C or CL provides for the required radial and axial traction components fr and fz the coupled boundary-integral equations of the first kind [Grr (M, P )fr (P ) + Grz (M, P )fz (P )]r(P )dl(P ) = 0 for M on C ∪ CL , (8) C∪CL [Gzr (M, P )fr (P ) + Gzz (M, P )fz (P )]r(P )dl(P ) = 0 for M on CL , (9) C∪CL [Gzr (M, P )fr (P ) + Gzz (M, P )fz (P )]r(P )dl(P ) = −8πμU for M on C. (10) C∪CL

As briefly described in the next section, we numerically invert in the present work the boundary-integral equations (8)-(10) in a first stage to get the traction f and to calculate the drag coefficient λ. In a second stage we then compute the flow in the bounded liquid domain DL (but not on its boundary S ∪ ∂DL ) by exploiting there the integral representations (6)-(7). Numerical implementation and results The boundary-integral equations (8)-(10) governing the traction f = fr er + fz ez are numerically solved by using a collocation method. This is done by using quadratic boundary elements on the involved contours C and CL . Remind that this latter contour consists in the line Lt , trace of the truncated tube in the θ = 0 half plane (i. e. points M (r, z) such that r = R and |z| < L), and the end lines L− and L+ made of points M (r, z) with 0 ≤ r ≤ R and z = −L or z = L, respectively. In practice we spread Ns nodes on C, Nt nodes on Lt and Ne nodes on each straight end lines L− and L+ so that the typical size of each boundary element is the same on the different curves. In summary, we end up with N = Ns + Nt + 2Ne nodal points on the entire contour C ∪ CL . Each discretized counterpart of (9)-(10) is then enforced at each node while (8) is only imposed at the N − 4 nodes located off the r = 0 axis where we instead of (8) rather require, for symmetry reasons, that fr = 0 (actually (8) is trivially fulfilled on the r = 0 axis since, as shown in [8], Grr (M, P ) = Grz (M, P ) = 0 whenever r(M ) = 0). L/a 5.0 7.5 10.0 12.5 15.0 20.0

Ha = 0.01 3.593 3.595 3.596 3.597 3.599 3.601

Ha = 0.1 3.594 3.595 3.597 3.598 3.599 3.602

Ha = 0.5 3.613 3.615 3.616 3.617 3.619 3.621

Ha = 1.0 3.672 3.673 3.675 3.676 3.678 3.680

Ha = 3.0 4.214 4.213 4.215 4.217 4.219 4.223

Ha = 5.0 5.054 5.027 5.029 5.031 5.034 5.039

Table 1: Computed drag coefficient λ for R = 2.5a and different values of L/a and Ha. Different tube sizes R/a have been numerically addressed with attention paid to the convergence of the drag coefficient versus the domain normalized length L/a at given (R/a, Ha). For a sake of conciseness, we however report here only results for the moderately confined case R = 2.5a.

4


-5

-4

-3

5

-1

0

1

0. 02

00

00

0.2000

0.0143

0.10

1

2

3

4

2

3

z

0

-5

-4

5

50

0 00 .1 -0 85 00 0. -1

0

-3

0. 07

0.1500

29

-0.0

.5

5

00

-0 .0

07 0.01

0 00 0.2

-2

0

79 0.03 07

-3

12

-4

1

03 0.

-0.0 50 -0.01900 -5

00

4

z

5

-2

0.0

07

37

0.0 9

21

z

5

9

07 0.

.5

0 .1 -0

-0.029

0.0312

-0.1500

0 -0.1

50

50 0

2

-0.0

0.0

00

0.1000

0. 10

.5

31

-0.0190 -0.0500

1

-1

-0.0295

2

90

25 0.

0.0219 0.1500

00 .2 0 -0

2

1 .0 -0

65

-2

00

0.0379 5 -0.008 -0.0500

0.0107

.5

0

0

0 .2

r

-0.0500

r

5

-0

65

z

00

0 0.00

4

-0 .1 0

2 43

50

5

3

-0.0

0.0

26

08 0.00

00

0.0

2

5 -0.1

1

.5

1

43

00 1500 0.10 0.

0

2 .0 -0

0

-1

21

-0.050

0.0034

3

0

10

50

000

-2

0.1

9

-3

5 .1

00

0 .0 -0

.5

6

-4

0 .1 -0

0.3000

0.0024

2 10 9 -0.005 -0.0

2

23

-5

20

0.0

00

0

0.0

-0.0 0

-0

4

.5

-0.2500

-0.003

1

50

0.0

.5

0.0

-0 .25 -0.3 00 0 00

0

0. 30

r 50

0

-0.0

50

08 8

0.2

-0.0

6

10

23

.00

0 50 0.1 0.0008-0.0209 -0.0034

-0

2

0.0

r

We first show in Table 1, for different values of the Hartmann number Ha, the drag coefficient λ sensitivity to the mesh truncation normalized length L/a. More precisely we use Ns = 33(65) for Ha ≤ 1(> 1), Ne = 25 and different values of Nt depending upon L. Denoting by (L, Ne ) each adopted pair we made the following choices: (5, 101), (7.5, 151), (10, 201), (12.5, 251), (15, 301) and (20, 401). It is seen that λ increases with Ha, especially for Ha ≥ 1 (note that for Ha = 0.01 our result approaches the prediction λ ∼ 3.596 obtained for Ha = 0 in [9]). Moreover, it appears in Table 1 that at least for Ha ≤ 5 taking L = 7.5a is sufficient to ensure a good accuracy. We thus take L = 7.5a when computing the flow patterns presented below (but we draw these patterns only for |z| ≤ 5a).

0

0.0219

1

2

3

4

Figure 2: Isolevel curves of the normalized radial velocity ur for Ha = 0.1(top left), Ha = 1(top right), Ha = 3(bottom left) and Ha = 5(bottom right).

z

5

-5

-4

2

7 -3

-2

-1

0

1

-1

0

1

-0.0236 0.0470

2

3

4

z

5

0

-5

-4

-3

00

4

z

5

z

5

-0.0236 00

0

00

0.8

.5

47

6 57

0.2000

1

00

.2 0

3

-0.0

.5

00

00

6

0.05 97

0. 20

00

0.6 -2

-0.022

0. 40

0

00

80 0.

3

2

-0

0. 60

-0.065

-0.2000 0.4000

0. 20

7 59

4

0. 00

0

0. 40

00

0

0.0

3

0 0900 800. 0.

00

-3

2

0

0.6

-4

1

0 00 0.6 0.80

-5

0

0

00

r

-1

1

0

0.4

.5

53

0.0000

.5

-0.023

7 00 0

0345

0.7 6 .0

0.0127 1

-0.0657

22

23

0. 01

0. 00

r

-0

.5

0.0420

2

.5

5

0

-2

26

02

000

00

-3

-0.02

-0.0 0.4

0.7

-4

-0.0474 0.2000

0

-5

60 0.

0.3000 0 00 00 0.8

00

0

0

0.9

.5

0

0 .10

0.20 00 0.30 00 0.5 00 0

4

00

18

1

-0.0 -0.0 033 013

-0.2000

2

0 .1 -0

-0.0

.5

000

082

-0.2

-0.0

6

-0.000

r

-0.007

2

r

We plot in Fig. 2 the isolevel curves of the normalized radial velocity ur = ur /U for Ha = 0.1, 1, 3, 5. At given Ha this quantity vanishes on the sphere and on the tube, quickly decays away from the (O, z) axis and remains small especially outside two pockets close the sphere. As Ha increases the previous pockets extends upstream (z > 0) and downstream (z < 0).

-2

-1

0

1

2

3

4

Figure 3: Isolevel curves of the normalized axial velocity uz for Ha = 0.1(top left), Ha = 1(top right), Ha = 3(bottom left) and Ha = 5(bottom right). The isolevel curves of the normalized axial velocity uz = uz /U are drawn in Fig. 3 again for Ha = 0.1, 1, 3, 5. This velocity component takes the unit value on the sphere and vanishes on the tube. Its magnitude is less than unity in the entire liquid domain and this velocity remains negative near the tube while being positive elsewhere. As Ha increases the regions near the sphere where uz exceeds 0.3 extend away and both upstream and downstream the sphere close the r = 0 axis. Finally, we display in Fig. 4 the isolevel curves of the normalized pressure p = ap/(μU ) still for Ha = 0.1, 1, 3, 5. The pressure is large in magnitude close the sphere. It also increases in magnitude with Ha especially near the r = 0 axis both downstream and upstream the sphere.

0 35

5

-2.5

-1.5000

r

-3.1

-0.5000

00 3. 00

0 00

13 .19

-3.5

0

1

.60

65

-7

2

3

4

z

3

4

z

5

0

-5

-4

-3

-2

-1

5

00 2.

00

24

000

3. 63

0

-9

.5

0 .00

00

0

-4.0

-4.6231

.0 0

0

00 2.0

1

2

r

-2.0000

-3

0 00

3.5000

3.1044

r

-1

4.0000

0

-2

1

-1

-3

2

1

00

0

63

-2

-3

3.5460 4.0

00

3.1

-3

-4

3.5

00

00

-4

-5

0

4.50

-5

0

.5

7.0000

.5

00

z

5

0 50 2.

0.0000

0 -4.0

4

00

150

3

0 .0

-3.1

1

0

2

-2

93

0.0000

.5 0

1

3.3198

-3

0

3

2

.5

-1

3.090

-2

.5

07

-3

-3.0000

3.1

-4

00 0

00 3.

.5

1

0

-5

4.5

0

5.000

0

67

1.5

-3.19

.5

2.5

2

90

3.16 35

-1.5000 -1.0000

0 00 -3.0

4 85

-3.0560

-3.0

-3.089

0

00 0

1

3. 08

00

3.1285

-2.5

.5

5

.5000

90

5

5000

-3.0

2

0.5000

r


0

9. 00

4.63

00 1

7. 00

5. 33

00 2

3

73

54 4

z

5

Figure 4: Isolevel curves of the normalized pressure p for Ha = 0.1(top left), Ha = 1(top right), Ha = 3(bottom left) and Ha = 5(bottom right). Conclusions A new accurate and efficient boundary approach has been proposed and implemented to calculate the quasi-steady axisymmetric MHD viscous flow about a solid sphere translating in a tube of circular cross section and of axis of revolution parallel with the sphere translational velocity and a prescribed uniform ambient magnetic field. The advocated method rests on the treatment of coupled boundary-integral equations on the surface of the sphere, on a truncated part of the tube and on two cross sections located sufficiently far from the sphere. It also exploits in the liquid key integral representations for the flow velocity and pressure. As shown by the presented preliminary numerical results, the obtained MHD flow is, for a given tube radius, very sensitive to the magnitude of the imposed ambient uniform magnetic field aligned with the tube axis. In addition, the magnitude of the radial and axial velocity components are quite different in the liquid. References [1] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970). [2] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [3] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [4] K. Gotoh Magnetohydrodynamic flow past a sphere. Journal of the Physical Society of Japan, 15 (1), 189-196 (1960). [5] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech.”, vol 3, 304-308 (1957). [6] W. Chester The effect of a magnetic field on the flow of a conducting fluid past a body of revolution. J. Fluid Mech.”, vol 10, 459-465 (1961). [7] A. Sellier and S. H. Aydin Creeping axisymmetric MHD flow about a sphere translating parallel with a uniform ambient magnetic field. To appear in MagnetoHydrodynamics. [8] A. Sellier and S. H. Aydin Fundamental free-space solutions of a steady axisymmetric MHD viscous

6


flow. European Journal of Computational Mechanics, vol 25, issue 1-2; 194-217 (2016). [9] W. L. Haberman and R. M. Sayre Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes. David W. Taylor Model Basin Report No. 1143, U.S. Navy Dept. (1958).


7

Inverse simulations of accidental spills in groundwater by the Green element method Akpofure Taigbenu School of Civil & Environmental Engineering, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa. [email protected] Keywords: inverse Green element method, inverse simulation, point and distributed instantaneous pollution spills, Tikhonov regularization

Abstract. When accidental spills occur, they impact on the quality of water in underlying aquifers. Such spills can be modelled as instantaneous pollution sources, and estimating their strengths from the concentration plumes they produce is an inverse problem which is addressed in this paper by the Green element method (GEM). Identifying the location and strengths of accidental spills, making use of the concentration data at various locations and times, is an inverse problem whose solution is often associated with non-uniqueness, non-existence and instability. Here GEM is used to predict the strengths of accidental spills from measured concentration data at internal observation points. The performance of the methodology is illustrated using two numerical examples in which the contaminant plumes are from multiple point and distributed pollution sources. It is observed that GEM is more accurate in predicting the strengths of distributed instantaneous pollution sources than point sources because of the discontinuities of the latter in both the spatial and temporal dimensions. Introduction Surface and subsurface pollution spills are a regular occurrence in practice, and their impacts on groundwater systems persist over considerable spatial and temporal scales as the pollutants undergo various chemical and hydrodynamic processes. It is usual to observe these impacts in downstream wells and surface water bodies after months or years of the occurrence of the spills. Forensic or inverse groundwater modelling uses pollution concentration information at different observation points in time to predict the strength of the pollution spill. The numerical challenges presented by such inverse modelling exercises are that many solutions could exist which satisfy the recreated contamination scenario, and the numerical solution could be unstable. There is a wide variety of inverse groundwater contaminant transport problems that present themselves in practice. They range from source characterizations to identification of the historical distribution of a contaminant plumes to aquifer parameters estimations, and the applied numerical techniques to these problems have also been quite varied. The inverse problem addressed in this paper belongs to the first category, and specifically attempts to estimate the source strength of pollutants instantaneously injected into an aquifer at a well or over an area. Geostatistical and backward probability methods are some of the techniques that have been applied in addressing this type of inverse problem [1, 2]. In this paper, the Green element formulation, presented in Taigbenu [3], is used in conjunction with Tikhonov regularization to predict the strengths of point and distributed sources instantaneously introduced into an aquifer from measured data of the concentration at observation points. Two numerical examples are used to demonstrate the capability of the methodology. Contaminant Transport Equation The transient contaminant transport equation in 2-D is addressed in this paper. It is governed by the advection-dispersion differential equation that is given as ( DC ) (VC ) R

wC wt

0

(1)

8


where iw / wx jw / wy is the 2-D gradient operator with spatial variables x and y, C = C(x,y,t) is the concentration in time and space in the domain Ω, V=iu +jv is the velocity vector, D is the hydrodynamic dispersion coefficient and R is the retardation factor. The inverse problem that is addressed solves eq (1) subject to the Dirichlet and Neumann boundary conditions: (2) C ( x, y, t ) f1 on Γ1 D C n

(3)

q1 on Γ2

in which n is the unit outward pointing normal vector, and Ω is the domain with boundary Γ=Γ1 Γ2. The initial condition reflects either point instantaneous contamination that is described as C ( x, y,0) { C (r ,0)

P

¦ S i G (r ri )

(4a)

i 1

or distributed instantaneous contamination that is described as C ( x, y,0) { C (r ,0)

P

(4b)

¦ S i (r )

i 1

where Si is the pollution source strength of the accidental spillage that has to be estimated from the inverse simulation and P is the number of sources (Fig. 1). It is assumed that the spatial characteristics of the spillage are known. There are available measured data on the concentration at B number of observation points, (xb,yb) which, in practice, could have errors. They can be expressed as ~ ~ (3) Cb { C ( xb , yb , t ) C ( xb , yb , t )>1 M u RAN(b)@ where φ is the noise level, and RAN represents random numbers and C~b is the perturbed value of the

observed concentration Cb = C(xb,yb).

D C n q1

C ( x, y, t )

( DC ) (VC ) R

Figure 1: Schematic of the problem statement

wC wt

0

f1 (t )


9

GEM Formulation Eq (1) is solved in a homogeneous aquifer. Green’s second identity is applied to the differential equation and the free space Green’s function of the solution of 2G G r ri in the infinite space is utilized to obtain the integral equation ª§ wC § wG · ·º V C ¸»dA 0 D¨¨ OCi ³ C ds ¸¸ ³ G q ds ³³ G «¨ R ¹¼ * wn ¹ * : ¬© wt ©

(5)

where Ci = C(ri) and ri = (xi,yi) is the source node and λ is the nodal angle at ri, and q

DwC / wn is the

normal contaminant flux. Eq (5) is implemented by discretizing the computational domain into rectangular elements and interpolating the quantities C, V and q by basis functions of the Lagrange family (C ≈ NjCj). The discrete element equation for each element Ωe with boundary Γe is

Vij U ikj

§ · D¨¨ ³ N j Gi G ijO ¸¸, Lij ³ N j Gi ds, Wij *e © *e ¹ wN j wN j dA, Yikj dA ³³ Gi N k ³³ Gi N k e e x w wy : :

³³ N j Gi dA,

:e

(7)

in which Gi=ln(r-ri). The discrete element eq (6) is aggregated for all elements used in discretizing the computational domain, and the temporal term is approximated by a finite difference approximation in time, that is, dC/dt ≈ [C(2)–C(1)]/∆t at the time t=t1 + θ∆t, where 0≤ θ ≤1 is the weighting factor and ∆t is the time step between the current time t2 and the previous time t1. With this approximation eq (6) becomes TEij C (j2) R

where Eij

Wij 't

C (j2) ZEij C (j1) R

Wij 't

C (j1) TLij q ( 2) ZLij q (1) j

j

(8)

0

Vij U ikjuk Yikjvk , ω = 1-θ, and the superscripts represent the times at which the quantities are

evaluated. The instantaneous releases of pollutants into the aquifer, described eqs (4a) and (4b), have to be accounted for in eq (8). Point sources released at t = 0 are accounted in the term R

Wij 't

C (j1) in eq (8) by the

relationship R R P R P ¦ S j ln(ri r j ) ³³ GC (r ,0)dA ³³ ¦ S jG (r r j ) ln(r r j )dA 't : 't : j 1 't j 1

(9)

Eq (9) takes advantage of the property of the Dirac delta function in evaluating the integral. While distributed instantaneous sources released at t = 0 are accounted as R R P R P ¦ Wij S j ³³ GC (r ,0)dA ³³ ¦ S j (r ) ln(r r j )dA 't : 't : j 1 't j 1

(10)

Eq (8) can now be expressed as TEij C (j2) R

Wij 't

C (j2) H ij S j TLij q ( 2) j

ZEij C (j1) ZLij q (1) j

(11)

where Hij denotes the matrix that captures the contribution of the instantaneous pollution sources. In a condensed form, eq (11) is expressed as

Pz f where

(12)

10


Wij º ª C (j2) ½ » «TEij R °° °° 't » « z ® q (j2) ¾ and (13) P « TLij » °S ° « H ij » j ¯° ¿° » « ¼ ¬ where P is an M×N matrix, with M being the number of nodes in the computational domain which is the same as the number of discrete equations generated, and N represents the number of unknowns in the vector z which are Cj and/or qj at external nodes, Cj at internal nodes where measurement data are not available, and the pollution sources, Sj. The vector f represents the known quantities which consist of the terms in the right hand side of eq (11) and as well as the contributions from the available concentration measurements. Eq (12) is an over-determined, ill-conditioned system of equations which is solved by the least squares method and regularized by the Tikhonov regularization technique. The matrix P is decomposed by the singular value decomposition (SVD) technique to P

N

UDVtr

tr

¦V iui vi

(14)

i 1

where U and V are M×M and N×N orthogonal matrices, ui and vi are respectively the column vectors of U and V, and D is an M×N diagonal matrix with N non-negative diagonal elements (D = diag (σ1, σ2, ... σN)) which satisfy the condition: σ1 > σ2 > ···> σN > 0. The least squares solution of eq (12) with Tikhonov regularization minimizes the Euclidean norm ║Pz–f║2+α2║z║2 in computing the solution of z which is given as; z (D )

N

¦

i 1D

Vi 2

V i2

(15)

uitr fvi

where α is the regularization parameter, and the factor σi/(α2+σi2) plays the role of dampening the instability caused by the small singular values which tend to have considerable influence on the quality of the numerical solutions. The choice of the value of α is facilitated by the L-curve technique which is a graphical tool that identifies the suitable compromise of the norms of ║Pz–f║2 and ║z║2 [4].

Numerical Examples and Discussion of Results Two numerical examples are employed to demonstrate the capabilities of the inverse Green element formulation of accidental pollution spills in an aquifer. The first example arises from instantaneous injection of pollutants from four point sources, and it has an exact solution which is used as a benchmark for the numerical solutions. The second example addresses distributed instantaneous pollution sources in an aquifer. Example 1 This example relates to the instantaneous release of contaminants from four point sources (P = 4) in a 2-D homogenous aquifer that is infinitely extensive. The flow in the aquifer is in the x-direction with a uniform velocity u. The exact solution for the spatial and temporal distribution of the concentration plume is given by Bear [5] C ( x, y, t )

ª x x j ut 1 4 ¦ S j exp« 4StD j 1 4 Dt «¬

2 y y j 2 º» 4 Dt

»¼

(16)

The flow and aquifer property values used in the simulations are: u = 0.5, D = 1.0 and R = 1.0. The inverse Green element simulations are carried out on a rectangular domain [50 × 20] with the values of the contaminant flux, obtained from eq (16), prescribed on the top and bottom boundaries, while the left and right boundaries are Dirichlet boundaries with concentration values obtained from eq (16). The analytical solution is generated with pollution source strengths S1 = 85.0 at (x1=4.5, y1=6.0), S2 = 40.0 at (x2=13.0, y2=16.0), S3 = 62.0 at (x3=21.0, y3=8.0) and S4 = 25.0 at (x4=28.0, y4=11.0). The domain is discretized into 160 uniform rectangular elements each [Δx=2.5× Δy=2.5], a uniform time step Δt = 1.5, and a weighting


11

factor, θ = 0.75 are adopted in the simulations. A minimum of 4 observation points is required to solve for the strength s of the pollution sources, and these are placed close to the locations of the pollution sources and on their downstream end. The observation points are at (5.0,5.0), (15.0,15.0), (22.5,7.5) and (30.0,10.0). The inverse GEM predictions of the strengths of the four point sources in comparison with their exact values are presented in Fig. 2. Except for S3 which is over-predicted by GEM, the others are correctly predicted at t = 0. Whereas the source strengths should theoretically become zero for t > 0, the GEM solutions have residual source strengths which only diminish to zero after about four time steps. This is due to the spatial and temporal discontinuities that arise from point sources which are only active at an infinitesimal time.

Figure 2: Exact and inverse GEM solutions for instantaneous point source strengths of Example 1. The mean error between the calculated nodal concentrations and their exact values is evaluated by the relationship M

H

1 M

(e) (n) ¦ Ci Ci

i 1

M

(e) 2

2

u 100

(17)

¦ Ci

i 1

The variation of this error at every simulation time step is presented in Fig. 3. It is observed that the errors are quite small and they decrease exponentially with time, which suggests that the inverse GEM prediction of the contaminant plume is excellent. The values of the regularization parameter used in the simulation ranged from 7.90×10-6 to 3.22×10-5 with an average value of 1.42×10-5.

Figure 3: Variation of error of the GEM predicted contaminant plume with time for Example 1.

12


The contaminant plumes of the exact and the inverse GEM solutions are presented in Figs. 4a and 4b at t = 6 and t = 24, respectively, and there is good agreement in the solutions.

(a)

(b) Figure 4: Contaminant plume of Example 1: (a) t = 6 and (b) t = 24; Exact on the left and GEM on the right. Example 2 This example is a case of instantaneous release of contaminants from three areas shown in the 2-D homogenous aquifer shown in Fig. 5. The flow in the aquifer is in the x-direction with a uniform velocity u. The values of the aquifer and flow parameters used in numerical simulations are: D = 400m2/day, R = 1.0 and u = 6m/day. The computational domain is rectangular [1200m × 800m], and it is assumed initially before the distributed accidental spills that the concentration everywhere in the aquifer is 0.1mg/l. The top and bottom boundaries are considered as no-flux boundaries, while the left boundary has a concentration of 0.1mg/l and right boundary is a Neumann one with a normal contaminant flux, q = 0.15×t m.mg/l. There is no exact solution for this problem, so a direct GEM simulation of the problem is implemented with pollution source strength S1 = 80.0mg/l, S2 = 550.0mg/l and S3 = 180.0mg/l. The domain is discretized into 384 uniform rectangular elements each [Δx=50m× Δy=50m], a uniform time step Δt = 2.0days, and a weighting factor, θ = 1.0 are adopted in the simulations. The results from the direct GEM simulation are used as estimates of the concentration values at six observation points at (200m,600m), (350m,700m), (750m,100m), (800m,250m), (1000m,600m) and (1000m,750m). The results of the inverse GEM simulations in predicting the distributed instantaneous sources are presented in Table 1, and they indicate accurate prediction of their strengths with no residual values at subsequent simulation times. Noise levels of 2% and 5% are introduced into the observed data, and the predicted distributed source strengths are compared to the true values in Table 2. It is observed that the maximum relative error in the prediction of the source strength is 2.8% at noise level of 5%, and this shows that introducing noise in the data has little influence on the source strength prediction.


13

Figure 5: Computational domain of Example 2.

Table 1: GEM solutions of the distributed instantaneous source strengths of Example 2. Time, t (day) 0 2 4 6 8 10

S1 (mg/l) 80.003 -0.006 0.006 -0.006 0.006 -0.005

S2 (mg/l) 550.002 -0.007 0.007 -0.007 0.008 -0.009

S3 (mg/l) 180.000 0.000 0.001 0.000 0.001 0.000

Table 2: Exact and GEM solutions of the distributed various noise levels. φ=0% True values Relative (mg/l) error (%) Sources Strength 0.0 S1 (mg/l) 80.0 80.003 0.0 S2 (mg/l) 550.0 550.002 0.0 S3 (mg/l) 180.0 180.000

instantaneous source strengths of Example 2 for φ=2% Relative error (%) Strength 1.1 79.098 0.0 549.725 0.2 180.429

φ=5% Relative error (%) Strength 2.8 77.741 0.1 549.310 0.6 181.072

The errors of the inverse GEM solutions for the contaminant plume in relation to those from the direct GEM simulations are calculated by eq (17) and presented in Fig. 6. The errors are very small, indicating that the inverse and direct GEM solutions are quite identical. The contaminant plumes of the GEM solutions are presented in Figure 7 at t = 6days and 28days.

14


Figure 6: Error plots of the GEM predicted contaminant plume with time for Example 2.

Figure 7: Contaminant plumes from the GEM simulations: (a) t = 6days, (b) t = 28 days.

Conclusion Inverse Green element solutions of the advection-dispersion equation have been presented for the problem of predicting the source strength when an accidental spill occurs and impacts on the underlying aquifer. The over-determined, ill-conditioned global matrix generated in the numerical formulation is solved by the least squares method and regularized by the Tikhonov technique. Two numerical examples are used to demonstrate the computational capabilities of the GEM formulation. The first example, which has an analytical solution, dealt with the estimation of the strengths of four instantaneous point pollution sources, while the second addressed the problem of estimation of three distributed pollution sources. The current formulation is capable of estimating the multiple instantaneous source strengths from concentration data of contaminant plumes. It is observed GEM gives better prediction of distributed pollution sources than point sources, and this is due to the spatial and temporal discontinuities associated with the latter. Errors associated with measurement data at observation points do not have much influence on the prediction of the distributed instantaneous pollution source strengths.

References [1] A. M. Michalak and P.K. Kitanidis Water Resources Research, 40, W08302, doi:10.1029/2004WR003214 (2004). [2] R. M. Neupaue and R. Lin Water Resources Research, 42, W03424, doi:10.1029/2005WR004115 (2006) [3] A.E.Taigbenu Engineering Analysis Boundary Elements, 36, 125-136 (2012). [4] P.C. Hansen, Numer. Algorithms, 6 1-35, (1994). [5] J. Bear, Dynamics of fluids in porous media, McGraw-Hill, New York, USA (1979).


15

Alternative Boundary Integral Equations for Fracture Mechanics in 2D Anisotropic Bodies Vincenzo Gulizzi1,a, Ivano Benedetti1,b, Alberto Milazzo1,c 1Dipartimento

di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali (DICAM),

Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy [email protected], [email protected], [email protected]

Keywords: Fracture Mechanics, Elasticity, Integral Equations, Dual Boundary Element Method Abstract. An alternative dual boundary element formulation for generally anisotropic linear elastic twodimensional bodies is presented in this contribution. The formulation is based on the decomposition of the displacement field into the sum of a vector field satisfying the anisotropic Laplace equation and the gradient of the classic Airy stress function. By suitable manipulation of the integral representation of the anisotropic Laplace equation, a set of alternative integral equations is obtained, which can be used in combination with the displacement boundary integral equation for the solution of crack problems. Such boundary integral equations have the advantage of avoiding hyper-singular integrals. Introduction The Dual Boundary Element Method (DBEM), as proposed by Aliabadi and co-workers [1,2,3] and involving simultaneously both displacement and traction boundary integral equations, has been successfully employed for modelling the presence of cracks in isotropic and anisotropic materials. However, the classic DBEM formulation involves the evaluation of hyper-singular integrals, which require specific numerical integration treatment or specialized regularization techniques. In this work, an alternative DBEM formulation for generally anisotropic 2D bodies is presented. The formulation revisits and extends previous works of Davì and Milazzo [4,5] valid for isotropic and orthotropic materials and it is based on a decomposition of the displacement field into the sum of two contributions. The first contribution is given in terms of a function solving the anisotropic Laplace equation, whereas the second contribution is given in terms of the stress function gradient. By using the relationship between the gradient of the stress function and the boundary tractions and a suitable manipulation of the integral representation of the anisotropic Laplace equation, it is herein shown that it is possible to obtain additional integral equations that can be used in combination with the displacement boundary integral equation on the crack surfaces. Such an alternative boundary integral equation has the advantage of avoiding the use of hyper-singular integrals. Formulation Simply connected domain. Let us consider a simply connected 2D linear elastic anisotropic body V with boundary S subject to a suitably defined set of kinematical and mechanical boundary conditions. In such a case, the stress field ı ı ([) {V xx ([), V yy ([), V xy ([)}, [ {x, y} V , within the body can be derived from a single function, namely the stress function I I (x) [6]. After introducing the stress function, the displacement field u u(x) {u x (x), u y (x)} can be decomposed into the sum of two contributions depending on a vector field v v (x) {vx (x), v y (x)} and the gradient of the stress function as follows

ªL u(x) Lv (x) ȁI ([), / { « 11 ¬0

0 º ª/ , ȁ { « 11 » L22 ¼ ¬ / 21

/12 º ªw x º , { « ». » / 22 ¼ ¬w y ¼

(1)

The components vx (x) and v y (x) of the vector field v(x) satisfy the same anisotropic Laplace equation

§ w2 w2 w2 · ¨ 2 a1 2 2a2 ¸ vi wy wxwy ¹ © wx

0, i

x, y.

and they are conjugate according to the following relationships

(2)

16


vx , x ( x) v y , y ( x)

(3)

a1vx , y (x) a2 v y , y (x) [v y , x (x) a2 vx , x (x)]

The coefficients ai , Lii and / ij are obtained by suitable manipulations of the elasticity governing equations. Since vx (x) and v y (x) satisfy the anisotropic Laplace equation (3), they have the following boundary integral representation

c(x 0 ) v (x 0 ) ³ p* (x, x 0 ) v (x)dS (x)

w

³ v (x, x ) wn v(x)dS (x) *

(4)

0

S

S

where x0 and x are the source and integration points, respectively; c(x0 ) is the free term and

§ w § w w w · w · { nx (x) ¨ a2 ¸ n y (x) ¨ a2 a1 ¸ , wn w x w y w x w y¹ © ¹ ©

(5)

being n(x) {nx (x), n y (x)} the unit outer normal of the boundary S. Eventually, v* (x, x0 ) and p* (x, x 0 )

wv* (x, x 0 ) / wn are the fundamental solutions of the anisotropic Laplace equation (2).

Denoting by t (x) {t x (x), t y (x)} the boundary tractions, the following relationship holds

1 º ªa w w ªw º , v(x) A v (x) AL1 « u(x) ȁ+ 1W ([) » , $ { « 2 a a2 »¼ s wn ws w ¬ ¼ ¬ 1

(6)

where s(x) {sx (x), s y (x)} is the unit vector tangent to the boundary S at x. The Eq. (6) is obtained by using Eq. (1) and the expression of the boundary tractions in terms of the stress function, i.e.

t ( x)

H

ª 0 1º w I (x), H { « ». ws ¬ 1 0 ¼

(7)

Eq. (7) can also be used to obtain a representation of the tractions resultant R (x, x0 ) over a curve connecting the points x and x0 along the boundary S [7], namely x

R ( x, x 0 )

³ t(x )dS (x )

H > I ( x ) I ( x 0 ) @ .

(8)

x0

Upon combining Eqs. (1), (4) and (8), it is possible to write an alternative boundary integral equation relating the boundary displacements and tractions of the considered mechanical problem. The alternative equation reads

c(x 0 )u(x 0 ) ³ P (x, x 0 )u(x)dS (x) S

³ V(x, x )t(x)dS (x) ³ PI (x, x )R(x, x )dS (x), 0

S

0

0

(9)

S

where the integration kernels are defined as follows

w * v (x, x 0 ), ws V (x, x 0 ) LAL1ȁ+ 1v* ([, [ 0 ), P (x, x 0 ) Ip* (x, x 0 ) LAL1

PI (x, x 0 )

(10)

ȁ+ 1 p* ([, [ 0 ),

being I the 2x2 identity matrix. The key feature of the alternative boundary integral equation (ABIE) given in Eq.(9) is that it involves at most strongly singular integrals as it can be seen by looking at the kernels given in (10). Cracked simply connected domain. Following the procedure used in the dual boundary element approach [1], by writing the ABIE at a point x0 of the boundary crack, one obtains


17

c(x 0 )u(x 0 ) c(x 0 )u(x 0 ) ³ P (x, x 0 )u(x)dS (x) ³ P (x, x 0 )G u(x)dS (x) S

C

³ V (x, x0 )t(x)dS (x) ³ PI (x, x0 )R (x, x0 )dS (x), S

(11)

S

where x and x are opposite points on the crack’s faces, G u(x) is the crack opening, C is the line representing the crack and S is the external boundary. The ABIE (11) can be used as an additional equation for cracked simply connected domains. 0

0

Multiply connected domain. Following similar steps, the alternative boundary integral equations for uncracked and cracked multiply connected domain can be derived. Let us consider for simplicity a multiply connected domain with one traction-free cavity. In such a case, by collocating the alternative boundary integral equation on a point belonging to crack inside the considered domain, one obtains c(x 0 )u(x 0 ) c(x 0 )u(x 0 ) ȁ ª¬I ([ 0 ) I ([ e ) º¼ ³ 3 ([, [ 0 )X([)dS ([) ³ 3 ([, [ 0 )G X([)dS ([) S

C

³ V(x, x0 )t(x)dS (x) ³ PI (x, x0 )R(x, xe )dS (x), S

(12)

S

where xe is an arbitrary point on the external boundary S of the domain for which it is assumed that I (xe ) 0 . This is a legitimate assumption since it is always possible to add a linear function of x to the stress function I (x) without changing the corresponding stress [7]. However, in this case, the value of the gradient of the stress function at the point x0 , namely I (x0 ) , appears explicitly. This is consistent with the fact that is not possible to find the solution of a multiply connected domain in terms of a stress function without introducing additional conditions. For the purpose of this work and following [4,5], to eliminate the presence of I (x0 ) , Eq.(12) is written for a number of collocation points equal to the number of unknowns nodal crack relative displacements plus one. Numerical tests The proposed formulation is employed to compute the linear elastic response of a generic anisotropic domain. The elastic constants of the considered anisotropic material are, in Voigt notation: c11 = 0.5637 GPa, c12 = 0.2963 GPa, c16 = 0.3158 GPa, c22 = 0.5637 GPa, c26 = 0.3158 GPa, c66 =0.3111 GPa. Four tests are considered. Fig (1a) shows a l u w plate, with a circular notch of radius r at the center of its left side, subject to a tensile stress. The figure compares the solution computed by using the classic displacement boundary integral equation (DBIE) only and the solution computed by using the DBIE on the straight edges of the plate and the ABIE (9) on the edges of the notch. Fig. (1b) shows the same plate as in Fig. (1a) with an embedded hole subject to tensile stress. Similarly to the previous case, the solution is obtained by collocating the ABIE (12) on one half of the hole and it is compared to the solution obtained using the DBIE only. Fig. (2a) shows the notched plate of Fig. (1a) with an additional traction-free boundary crack of length a. In this case, the solution is computed by collocating the ABIE (11) on one face of the crack and the DBIE on the remaining part of the domain and it is compared with the solution obtained using the DBIE and a multi-domain approach. Eventually, Fig. (2b) shows the same plate considered in Fig. (1b) with two additional cracks emanating from the hole. The solution in the last case is computed by using the ABIE on one face of the two cracks and the DBIE on the remaining boundaries, including the other crack faces. From the comparisons between the solutions obtained with the two different methods, it is concluded that the ABIE can be employed as an alternative set of equations for the solution of crack problems. Summary An alternative dual boundary element formulation for generally anisotropic linear elastic two-dimensional bodies has been presented. The alternative boundary integral equations employed in the formulation do not include hyper-singular integrals, thus simplifying the integration procedure. From the comparison between the presented solutions and those obtained with the classic displacement boundary integral equations, it is concluded that the formulation can be employed as an alternative single-domain boundary integral approach for fracture problems.

18


(a) (b) Fig 1: Comparison between the classic DBIE and the ABIE in combination with the DBIE for two uncracked domains: (a) plate with notch; (b) plate with hole.

(a) (b) Fig 2: (a) Comparison between the multi-domain DBIE and the single domain ABIE in combination with the DBIE for a cracked plate subject to tensile stress; (b) Solution of a cracked plate with an internal hole computed using the proposed ABIE. References [1] [2] [3] [4] [5] [6] [7]

MH Aliabadi, The boundary Element Method: Application in Solids and Structures, John Wiley & Sons Ltd, England, 2002 P Sollero, MH Aliabadi, International Journal of Fracture, 64(4), 269-284, 1993 A Portela, MH Aliabadi, DP Rooke, International Journal for Numerical Methods in Engineering, 33(6), 12691287, 1992 G Davì, A Milazzo, SDHM Structural Durability & Health Monitoring, 2(3), 177-182, 2006 G Davì, A Milazzo, SDHM Structural Durability & Health Monitoring, 3(4), 229-238, 2007 PC Chou, NJ Pagano, Elasticity: tensor, dyadic, and engineering approaches. Courier Corporation, 1992. C. Truesdell, Mechanics of solids Volume II Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells, Springer, 2013.


19

The PWM for the identification of a sound-soft interior acoustic scatterer Andreas Karageorghis1 , Daniel Lesnic2 and Liviu Marin3,4 of Mathematics and Statistics, University of Cyprus/Panepist mio KÔprou, P.O. Box 20537, 1678 Nicosia/LeukwsÐa, Cyprus/KÔproc 2 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK 3 Department of Matematics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014 Bucharest, Romania 4 Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, 13 Calea 13 Septembrie, 050711 Bucharest, Romania

1 Department

Keywords: Inverse problem; sound-soft scatterer; acoustic scattering; plane waves method; regularization. Abstract. We employ the plane waves method (PWM) for detecting a sound-soft scatterer surrounding a host acoustic homogeneous medium due to a given point source inside it. The measurements are taken inside the medium and, in addition, are contaminated with noise. The PWM discretization yields a nonlinear constrained regularized minimization problem which is solved using standard software. The results of several numerical experiments are presented and discussed. Mathematical Formulation Physically, we consider the interior scattering with a wave number k > 0 due to a given point source z 0 inside the two-dimensional, bounded and simply-connected scatterer domain D with a C 2 boundary ∂D. This means that the incident field is given by i (1) uinc (x) = Φ(x, z 0 ) := H0 (k|x − z 0 |) = 0, 4

x ∈ R2 ,

(1)

(1)

where i2 = −1 and H0 denotes the Hankel function of first kind and of order zero. The scattered field us satisfies the Helmholtz equation Δus + k 2 us = 0 in

D.

(2)

The sound-soft boundary condition on ∂D is of the form i (1) us + H0 (k|x − z 0 |) = 0 4

on

∂D.

(3)

We assume that: (A) k 2 is not a Dirichlet eigenvalue of −Δ in D. The Direct Problem. Under assumption (A), it is well-known, see e.g. [3], that the direct scattering problem which requires finding us satisfying (2) and (3), when D is known, is well-posed. The Inverse Problem. We consider the Helmholtz equation (2) subject to the Dirichlet boundary condition (3) but now the boundary ∂D is unknown and has to be determined from the additional measurement us (x) = f (x), x ∈ Γ = ∂B|z 0 | (0) ⊂ D, (4) of the scattered field us on the circle of radius |z 0 | centred at the origin which is assumed to be contained in the domain D, and f is some given measured data which may be contaminated with noise. Of course, for compatible data, the function f in (4) depends on z 0 . Assuming that: (B) k 2 is not a Dirichlet eigenvalue of −Δ in the interior of Γ, (C) D is contained in a disk of radius t0 /k, where t0 = 2.40482 is the first positive root of the Bessel function J0 , then the boundary ∂D can be uniquely determined from only one measurement (4) for the scattered field us at a single point source z 0 ∈ Γ, see [10]. The purpose of this paper is to solve numerically the nonlinear and ill-posed problem (1)–(4). Since the fundamental solution of the Helmholtz equation (2) is known, one could employ the boundary

20


element method (BEM) or the method of fundamental solutions (MFS) as described in [8, 10, 11]. However, both these methods involve implicitly the discretisation of a single layer integral potential equation which is either singular as in the BEM, or requires the selection of source points outside the domain of the problem as in the MFS. In contrast, these difficulties are not present in the PWM which is employed in this paper as described in the next section. Prior to this study, the PWM has been applied for the solution of both linear and nonlinear inverse problems in [6, 7]. The Plane Waves Method (PWM) In the PWM we seek the solution of the inverse Helmholtz problem (2)–(4) in the form, [1], uN (x) =

N

cn eikx·dn ,

x ∈ D,

(5)

n=1

where the vectors dn are unitary direction vectors and cn ∈ C are unknown complex coefficients to be determined by imposing boundary condition (3) and condition (4). We assume that the unknown boundary ∂D is a smooth, star-like curve with respect to the origin. This means that its equation in polar coordinates can be written as x = r(ϑ) cos ϑ,

y = r(ϑ) sin ϑ,

ϑ ∈ [0, 2π),

(6)

where r is a smooth 2π−periodic function. If we let ϑm = 2π(m − 1)/M for m = 1, M , be a uniform discretization of the interval [0, 2π), then the discretized form of (6) for ∂D becomes rm = r(ϑm ),

m = 1, M .

(7)

On the unknown star-shaped boundary ∂D we consider the points xm = rm (cos ϑm , sin ϑm ) ,

m = 1, M ,

(8)

expressed in polar coordinates, where the radii rm > 0 are unknown. Moreover, the measured data are given at the points of the circle Γ ˜ = |z 0 | (cos ϕ , sin ϕ ) , x

ϕ = 2π( − 1)/L,

= 1, L.

(9)

Finally, the N unitary direction vectors are chosen as dn = (cos φn , sin φn ) ,

φn =

2(n − 1)π , N

n = 1, N .

(10)

We thus have M + 2N unknowns, namely the radii r = (rm )m=1,M and the complex coefficients c = (cn )n=1,N . These are determined by imposing (complex) boundary condition (1) at the M points

(xm )M x )L m=1 which yield 2M equations, and by imposing (complex) condition (4) at the L points (˜ =1 which yield an additional 2L equations. We thus have 2M + 2L equations in M + 2N unknowns and therefore need to take M + 2L ≥ 2N . To obtain a stable approximation to the inverse problem, we minimize the regularized nonlinear leastsquares functional 2 N M i (1) ikxm ·dn Tλ1 ,λ2 (c, r) := cn e + H0 (k|xm − z 0 |) 4 m=1 n=1 2 L N N M + cn eikx˜ ·dn − f ε (˜ x ) + λ1 |cj |2 + λ2 (rm − rm−1 )2 , (11) =1

n=1

j=1

m=2

where λ1 , λ2 ≥ 0 are regularization parameters, subject to the simple bounds on the variables |z 0 | < rm < ζ2 = 2.40482/k,

m = 1, M .

(12)


21

The data (4) come from practical measurements which are inherently contaminated with errors due to noise, and we therefore replace f by f ε , and in computation, the noisy data are generated as f ε (˜ x ) = (1 + ρ p) f (˜ x ) ,

= 1, L ,

(13)

where p represents the percentage of noise added to the data on Γ, and ρ is a pseudo-random noisy variable drawn from a uniform distribution in [−1, 1]. This constrained optimization problem is solved c using the MATLAB [9] toolbox routine lsqnonlin. Numerical Results We take, for simplicity, the wave number k equal to unity. The choice of the regularization parameters λ1 and λ2 in (11) is based on the L-curve criterion, see [5, 4] although the L-surface criterion, see [2], might also be employed. Example 1. We first consider the simple case of a circular scatterer of unit radius. We take z 0 = (0.5, 0) and the measured data on Γ are simulated by solving the direct problem (2) and (3) with M = 60 collocation points and N = 40 terms in the PWM expansion (5). The data were generated at L = 32 points on Γ. In the implementation of the inverse problem we took an initial guess (c0 , r 0 ) = (0, 0.6) and M = N = 50. In Figure 1 we present the reconstructed boundary with no noise and no regularization after 1, 5, 50 and 200 iterations (niter). The corresponding results with niter=200, noise p = 5%, λ2 = 0 and regularization with λ1 , and λ1 = 0 and regularization with λ2 , are presented in Figures 2 and 3, respectively. The L-curves corresponding to the two above cases are presented in Figure 4. From Figure 4(a) it may be seen that the corner of the L-curve corresponds to λ1 between 10−2 and 10−1 which is consistent with the results presented in Figure 2. From Figure 4(b) we observe that the corner of the L-curve corresponds to λ2 between 10−1 and 1 which is consistent with the results presented in Figure 3. niter=1

niter=5

niter=50

niter=200

Figure 1: Example 1: Results for no noise and no regularization for various numbers of iterations. Example 2. We next consider the case of a peanut shape scatterer given by the radial parametrisation, r(ϑ) =

1 1 + 3 cos2 ϑ, 2

ϑ ∈ [0, 2π).

(14)

We take z 0 = (0.25, 0) and the measured data on Γ are simulated by solving the direct problem (2) and (3) with M = 60 collocation points and N = 40 terms in the PWM expansion (5). The data were generated at L = 32 points on Γ. In the implementation of the inverse problem we took an initial guess (c0 , r 0 ) = (0, 0.6) and M = N = 50. In Figure 5 we present the reconstructed boundary

22


λ1=0

−3

λ1=10

λ1=10−5

λ1=10−4

−2

−1

λ1=10

λ1=10

Figure 2: Example 1: Results for noise p = 5%, λ2 = 0 and regularization with λ1 . λ2=0

−2

λ2=10

λ2=10−5

λ2=10−4

−1

0

λ2=10

λ2=10

Figure 3: Example 1: Results for noise p = 5%, λ1 = 0 and regularization with λ2 . with no noise and no regularization after 1, 5, 50 and 200 iterations. The corresponding results with niter=200, noise p = 5%, λ2 = 0 and regularization with λ1 , and λ1 = 0 and regularization with λ2 , are presented in Figures 6 and 7, respectively. Similar conclusions to those obtained for Example 1 can be drawn although, as expected, retrieving a non-convex interior scatterer is more difficult because waves can be trapped in concave regions. Example 3. We next consider the case of a complicated scatterer given by the radial parametrisation, r(ϑ) = 1 + 0.3 cos 3ϑ,

ϑ ∈ [0, 2π).

(15)

We take z 0 = (0.25, 0) and the measured data on Γ are simulated by solving the direct problem (2) and (3) with M = 60 collocation points and N = 40 terms in the PWM expansion (5). The data were generated at L = 32 points on Γ. In the implementation of the inverse problem we took an initial guess (c0 , r 0 ) = (0, 0.4) and M = N = 50. In Figure 8 we present the reconstructed boundary with no noise and no regularization after 1, 5, 50 and 200 iterations. The corresponding results with niter=200, noise p = 5%, λ2 = 0 and regularization with λ1 , and λ1 = 0 and regularization with λ2 , are presented in Figures 9 and 10, respectively. By comparing the current results with the corresponding


23

(a) 0.3

|| c||2

0.2 0.1 λ =10−2 1

λ =10−1 1

0.04

0.05 ||Residual||

2

(b) 0.3 0.2

′

|| r ||2

0.1

−1

λ2=10

λ =1 2

0.05

0.1

0.15

||Residual||2

Figure 4: Example 1: L-curves for p = 5%. (a) Varying λ1 with λ2 = 0; (a) Varying λ2 with λ1 = 0. reconstructions for the circle (Example 1) and the peanut (Example 2), one can observe that the recovery of the three-petal flower (15) is much more challenging. Conclusions In this paper the PWM was used for the numerical solution of inverse acoustic scattering inside a sound-soft obstacle generated from a point source inside the domain. Since this is an ill-posed problem, its discretized version was regularized with respect to not only the magnitude of the PWM coefficients, but also the smoothness of the curve. The values of the regularization parameters were selected based on Hansen’s L-curve criterion. The numerical results retrieved for three examples revealed that the method is well suited for the reconstruction of the unknown boundaries even when the measured data was contaminated with noise. The extension to three dimensions is currently i (1) eik|x−z0 | under investigation and involves replacing throughout the function H0 (k|x − z 0 |) by , 4 4π|x − z 0 | the constant 2.40482 by π and using spherical coordinates instead of polar coordinates. Future work will concern identifying scatterers D subject to a Robin impedance boundary condition, [12], instead of the sound-soft boundary condition (3), using the PWM.

References [1] C.J.S. Alves and S.S. Valtchev, Numerical comparison of two meshfree methods for acoustic wave scattering, Engineering Analysis with Boundary Elements 29, 371–382 (2005). [2] M. Belge, M.E. Kilmer, and E.L. Miller, Efficient determination of multiple regularization parameters in a generalized L-curve framework, Inverse Problems 21, 133–151 (2005). [3] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin (1998). [4] P.C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia (2010). [5] P.C. Hansen and D.P. O’Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM Journal on Scientific Computing 14, 1487–1503 (1993). [6] B. Jin and L. Marin, The plane wave method for inverse problems associated with Helmholtz-type equations, Engineering Analysis with Boundary Elements 32, 223–240 (2008). [7] A. Karageorghis, D. Lesnic, and L. Marin, The plane waves method for numerical boundary identification, submitted. [8] A. Karageorghis, D. Lesnic, and L. Marin, The MFS for the identification of a sound-soft interior acoustic scatterer, submitted.

24


niter=1

niter=5

niter=50

niter=200

Figure 5: Example 2: Results for no noise and no regularization for various numbers of iterations. λ1=10−4

λ1=0

−2

λ1= 10

−1

λ1= 10

λ1= 10−3

0

λ1=10

Figure 6: Example 2: Results for noise p = 5%, λ2 = 0 and regularization with λ1 . [9] The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA, Matlab. [10] H.-H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems 27, 035005 (2011). [11] H.-H. Qin and D. Colton, The inverse scattering problem for cavities, Applied Numerical Mathematics 62, 699–708 (2012). [12] H.-H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Advances in Computational Mathematics 36, 157–174 (2012).


λ2=0

−3

λ2= 10

25

λ2=10−4

−2

λ2= 10

λ2=5 × 10−4

−1

λ2=10

Figure 7: Example 2: Results for noise p = 5%, λ1 = 0 and regularization with λ2 .

niter=1

niter=5

niter=50

niter=200

Figure 8: Example 3: Results for no noise and no regularization for various numbers of iterations.

26


λ1=10−5

λ1=0

−4

λ1= 10

λ1= 5 × 10−5

−4

λ1= 5 × 10

−3

λ1=10


λ2=10−6

λ2=0

−5

λ2= 5 × 10

−4

λ2= 10

λ2=10−5

−3

λ2=10



Fundamental three-dimensional MHD creeping flow bounded by a plane motionless wall normal to a uniform ambient magnetic field A. Sellier1 and S. H. Aydin2 LadHyX. Ecole polytechnique, 91128 Palaiseau Cédex, France Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey 1

2

Keywords: 3D MHD Stokes flow, Fundamental flow, conducting wall, insulating wall.

Abstract. This work investigates the so-called fundamental MagnetoHydroDynamic (MHD) creeping velocity and pressure fields produced by a source point with arbitrary unit strength e located in a conducting Newtonian liquid above a solid, motionless and plane wall normal to a uniform magnetic field imposed in the entire liquid. This is done both for insulating and conducting walls by computing the analytical solutions recently established in [1] for e either parallel with or normal to the wall. Each fundamental flow is then found to deeply depend upon the source-wall gap, normalized with the Hartmann layer thickness, and also when e is parallel with the wall to the wall nature (insulating or conducting). Introduction MagnetoHydrodynamics (MHD) deals with flows of conducting Newtonian liquids [2,3] in which act in general coupled electric field E and magnetic field B . It addresses very involved problems since the liquid flow velocity u and pressure p are coupled with (E , B ) through the Lorentz body force fL = j ∧ B which, adopting for the current density j the usual Ohm’s law with σ > 0 the liquid uniform conductivity, reads fL = σ(E + u ∧ B ) ∧ B . More precisely, (E , B , u, p) are governed by unsteady and coupled Maxwell and non-linear incompressible Navier-Stokes equations. For a magnetic field with typical magnitude B > 0, a liquid with velocity scale V > 0, length scale a and uniform viscosity μ, magnetic permeability μm and density ρl three dimensionless numbers arise for the MHD problem: the Reynolds magnetic number Rem = μm σV a, the Reynolds number Re = ρl V a/μ and the Hartmann number Ha = a/d where d = ( μ/σ)/B is the Hartmann layer thickness [4]. Some applications pay attention to the MDH flow about a solid body moving with a rigid-body velocity field of typical magnitude V in an unbounded liquid in which prevails far from the body a prescribed ambient, uniform and steady magnetic field B. For a slow body motion one gets a quasi-steady solution (E , B , u, p). Assuming furthermore a slow body (and thus fluid) motion yields Re 1 wich for most applications also results in Rem 1. This latter property yields B = B in the entire liquid and one then ends up with unknown steady and coupled electric field E and Stokes flow (u, p). For a solid axisymmetric body translating parallel with both its axis of revolution and the ambient magnetic field B the flow (u, p) is axisymmetric and without swirl. In such circumstances, one gets E = 0 [5] and (u, p) is a Stokes flow driven by the body force fL = σ(u ∧ B) ∧ B. This pleasant framework has been employed for a solid sphere, with radius a, translating parallel with B either at small Harmann number Ha = a/d [6] or large Ha [7]. Recently [8] dealt with arbitrary Ha for the translating sphere by resorting to two axisymmetric fundamental MHD Stokes flows without swirl due to a distribution of axial or radial forces on a circular ring. Those fundamental axisymmetric flows have been obtained in [9] from the coupled fully three-dimensional MHD Stokes flow and electrical potential produced by a point force with arbitrary strength and analytically determined in [10]. Sometimes boundaries can’t be ignored and it is then necessary to see to which extent a boundary can affect the solution prevailing for the case of a body moving in an unbounded liquid. Of course, the answer will depend upon the boundary shape and nature (slip, no-slip, insulating, conducting,...). A simple case is the one of a sphere translating parallel to a uniform ambient magnetic field and normal to a plane solid no-slip wall. In order to mimick the treatment adopted in [9] it is then necessary to get the fundamental MHD Stokes flow and electrical potential produced by a point force with arbitrary strength g located in a conducting liquid bounded by a plane solid wall Σ normal to B. Such a task has been done for the axisymmetric case of g normal to Σ in [11] and also recently for the asisymmetric case of g parallell with a

27

28


conducting or insulating wall Σ in [1]. Unfortunately, neither [11] nor [1] report and discuss the associated flow patterns. The aim of the present work is to numerically address this issue. Fundamental MHD coupled Stokes flow and electrical potential This section gives the MHD problem governing the coupled fundamental Stokes flow and electrical potential produced by a point force with arbitrary unit strength e located in a conducting liquid above a solid, motionless and either conducting or insulating plane wall. Governing equations As shown in Fig. 1, we consider a point force, with arbitrary unit strength e, located at point x0 in a conducting Newtonian liquid with uniform viscosity μ and conductivity σ > 0. The liquid occupies the z > 0 domain D above the z = 0 plane solid and motionless wall Σ normal to the prescribed ambient uniform magnetic field B = Bez . z D

e r •x

0

B = Bez O

•

x(x, y, z)

z0 > 0 Σ(z = 0)

• •

x x0

Figure 1: A concentrated force, with unit strength e, located at point x0 in the liquid domain z > 0 liquid domain bounded by the solid, motionless z = 0 plane wall Σ normal to the magnetic field B. The point x0 is the symmetric of x0 with respect to the wall. Any point x in the liquid admits Cartesian coordinates (O, x, y, z) with origin O on the plane wall Σ. For the source x0 located on the (O, z) axis we have x0 = y0 = 0 and z0 > 0. The unit force placed at x0 produces in the liquid so-called fundamental coupled MHD steady electrical potential φ and flow with velocity and pressure p. Assuming (see the introduction) vanishing Reynolds and magnetic numbers shows that the magnetic field is B = Bez in the entire liquid whereas (u, p, φ) satifies μ∇2 u = ∇p + σB∇φ ∧ ez − σB 2 (u ∧ ez ) ∧ ez − δ(x − x0 )e for x = x0 ∈ D,

(1)

∇.u = 0 and Δφ = B∇.(u ∧ ez ) for x = x0 ∈ D

(2)

with Δ and δ the three-dimensional Laplacian operator and Dirac delta pseudo-function, respectively. For a conducting or insulating wall the boundary and far-field boundary conditions supplementing (1)-(2) read (u, ∇φ, p) → (0, 0, 0)far from x0 ,u = 0 on Σ, φ = 0 (conducting) or ∇φ.ez = 0 (insulating) on Σ. (3) Analytical solution The solution (u, p, φ) to (1)-(3) has been recently obtained in [1] by using the following decomposition u(x) = v0 (x) − v0 (x) + U(x), p(x) = q0 (x) − q0 (x) + P (x), φ(x) = ψ0 (x) − ψ0 (x) + Φ(x)

(4)

in which the auxiliary regular electric potential Φ and flow (U, P ) obey (1)-(2) for e = 0 while (v0 , q0 , ψ0 ) and (v0 , q0 , ψ0 ) are the solutions to (1)-(2) and the far-field behaviour (3) produced in an unbounded liquid by a point force, with strength e, located at x0 or at the symmetric x0 of x0 with respect to the wall (see Fig. 1), respectively. Using the analytical solution (v0 , q0 , ψ0 ) derived in [10], imposing on the wall relevant


29

boundary conditions (dictated by (3)) to U and Φ made it possible to analytically determine in [1] the quantities U, P and Φ for a conducting or insulating plane wall Σ (the treatment appeals to direct and inverse two-dimensional Fourier transform with respect to the variables (x, y)). The resulting fundamental flow (u, p) is given, without explanations, in the next sections. For the associated electrical potential φ and further details the reader is directed to [1]. Axisymmetric case of a point force normal to the wall For a point x(x, y, z) in the liquid we shall note r = {x2 + y 2 }1/2 (see Fig. 1) and introduce the local unit vector er = er (x) such that x = rer + zez . We also set R = |x − x0 | and R = |x − x0 | so that here R = [r 2 + (z − z0 )2 ]1/2 and R = [r 2 + (z + z0 )2 ]1/2 . For e = ez one gets, whatever the wall nature (conducting or insulating), φ = 0 and a fundamental axisymmetric flow without swirl with u(x) = ur (r, z)er + uz (r, z)ez and p(x) = p(r, z). Curtailing the details (see [1]), (ur , uz , p) obeys z − z0 z0 A 8πμur = r H(R) sinh( ) + 2[ ] sinh( ) , (5) 2d R 2d z z0 (6) 8πμuz = F (R, z − z0 ) − F (R , z + z0 ) + 2B sinh( ) sinh( ), 2d 2d z z 0 (7) 8πdp = G(R, z − z0 ) − G(R , z + z0 ) + 2B sinh( ) cosh( ), 2d 2d −R /(2d) 2 z z + z0 6d 12d e z 2d ](1 + + 2 ) , (8) A = cosh( )[1 + ] − sinh( )[ 2d R 2d R R R R 2d 4d2 z + z0 2 6d 12d2 e−R /(2d) e−s/(2d) 2d B = 1 + + 2 − [ (9) ] (1 + + 2 ) , H(s) = [1 + ], R R R R R R s2 s e−s/(2d) t t 2d t [cosh( ) + sinh( )(1 + ) ], s 2d 2d s s t t 2d t e−s/(2d) [sinh( ) + cosh( )(1 + ) ] G(s, t) = s 2d 2d s s where it is recalled that d = ( μ/σ)/|B| is the Hartmann layer thickness [4]. F (s, t) =

0.4

0

5

7.5

r

0

02

3 .0

75

r

1

-0

-0.1

75

40

8

-0.01

0.08

7

0.75

2.9

0

3

00 .0 3

367 6

30

-0.0408 -0 0083

30

-0 73

.20

-0.5

.0 3

00

0.1

0. -025 0.4 0.8470 .0 09 74 01 9 9

0.2

10

-0.0 8 .067

8

04

02

-0

-0

0 -0.0

35

0.5

47

02

-0.0530 -0.0 -0.0175 9518.819 5

-0 25 00 -00. 17 -0.0

87

8 83

-0 0089

z 1 01 -0.0

.0 0

07

-0 0001

0 .0

.0 2

-0

03 .0 0 -0

87

0.25

1.3

0.4

-0

.0 7

-0 .0 5

1.20.2413 0.14 74 31 4

3

08

.02

-0.1

0

0.3

143 3 21 2.5

0

-0.0

-0.0

066

0.0712

-0.0

0

-0.00

6

0

-0.0066

4

-0.003

00 -0.0

09

00

.0

1 01

10

-0.3211

-0

1

r

7.5

14.3963

0.28 0.148 3 6 .1734 -0. 00 00 36 0 .0 -0-.0 343 .0 88 0.0 004 5 -70 .0

600

-0.001

8

5

-0.0

-0

0.9

2

75 -0

-0

-0

0.5

0.00 2 -0.00.0 20517

z 4

53

9

13 24 0. 08 5.11 -0.9767 .03-70

7 80 72 0592 1.3 1232 0050

5

-0.001

-0.0047

0 0000 2.5

3

0.1

8.3301

38

-0 -0 1108 0089

0 0658

287

0.2

08

1

-0 . 05

-0 12 0. 1 00 37

0.0

18

27

-0.0

735

75

3

0.3

-0.0

0

.53

00 8 -0.0 650513 502 .0.0 15-0.0 0 -0 .0 -0

-0.18

0

-0

74

4 077 1470.3

1

13

-0.00

00 -0.

14 0.

0.24

2

-0.0

2 01 0.

22

-

3

0.1474

z

z

02 08

02 -0.

00 0.

0.3593 0.5296

.0 0

00 -0 .0 0

59

0 0121

2 0.0

4

(11)

0.5

0.0037 0.0 121

5

(10)

-0.0 0.25

958 0.5

0.75

r

1

Figure 2: Isolevel curves of the normalized radial velocity ur for z0 = d (top left) and z0 = 0.1d (top right) and axial velocity uz for z0 = d (bottom left) and z0 = 0.1d (bottom right). In Fig. 2 we plot, versus the normalized coordinates r = r/d and z = z/d, the isolevel curves of the components of the normalized velocity u = 8πμdu for two locations of the source x0 : z0 = d and (point

30


close to the wall) z0 = d/10. As expected both velocities vanish on the z = 0 plane no-slip wall Σ. Those quantities are also seen to quickly decay away from the (O, z) axis and to be positive near this axis and negative elsewhere. Not surprisingly, the magnitude of uz is larger than the magnitude of ur in the entire normalized liquid domain whatever z0 . Moreover, at any given location (r/z0 , z/z0 ) one gets a larger value of both |ur | and |uz | for z0 = 0.1d than for z0 = d.

9 7.5

00 -0 .

155 -1.0

5

-0.0

14 -0.0

00

02

r

0

10

-0.1498

6

05 -3 .2 8

058

89 0. 13

3 593

.0 -0 5

3

2.5

-0.607

-0.0

-0.0089

0.7452

4 00

015

.0 05-0 8.00 -0

09

z

-0.0025

37 0.0 -00.01 010 1

z

-0 . 00

-0.0

-6.9747

.3865 3.5

0

968

0.2

-7 2

0

0.3

0.1

6

-0.0

1

3

26

7

25 4

.0 -0

2

-0.2

67

-0.0

4 00

45

.3 -0

6

8.9

8 96 98 .0 .14 -0 -0

528

0.0579 92

54

0.4

2.8

0.08

0.3

3

992

4

0.5

1.5

5

0

0.25

0.5

r

0.75

1

Figure 3: Isolevel curves of the normalized pressure p for z0 = d (left) and z0 = 0.1d (right). Isolevel curves of the normalized pressure p = 8πd2 p are given in Fig. 3, again for z0 /d = 0, 0.1. These quantities exhibit the same trends as the previous velocity components ur and uz . Asymmetric case of a point force parallel with the wall For this case we take e = ex . The electrical potential φ is non-zero, depends upon the wall nature [1] and is discarded in the present work. For point x we introduce the angle θ in [0, 2π] about the (O, z) axis, i. e. such that x = r cos θ and y = r sin θ. Then, both the fundamental velocity component uz (x) and pressure p(x) do not depend upon the wall nature (conducting or insulating) and are given by E1 z − z0 z + z0 z0 ) − H(R ) sinh( ) + 2[ ] sinh( ) , (12) 8πμuz = Vz cos θ, Vz = r H(R) sinh( 2d 2d R 2d z − z0 E z + z z 0 2 0 8πdp = Q cos θ, Q = r H(R) cosh( ) − H(R ) cosh( ) + 2[ ] sinh( ) , (13) 2d 2d R 2d 2d z z z + z0 6d 12d2 e−R /(2d) ](1 + + 2 ) , (14) E1 = cosh( )[1 + ] + sinh( )[ 2d R 2d R R R R 2d z z z + z0 6d 12d2 e−R /(2d) E2 = sinh( )[1 + ] + cosh( )[ ](1 + + 2 ) . (15) 2d R 2d R R R R

0

0

-0

7 27

3 04

60

2.5

9 27 0.8

1

5

0.0004 7.5

64

22

0.1

8 0.000

4

32 3.4

8 13 0.0 .0080 0

0.0000

r

10

0

1.7

8. 48

0.06

67

1.1068

2

0.0

02 0.0

4 22

33

0.2

12 0.

77 0.0037

4 9832184 00002

00 8 0 00 000008

4039 0 .1726 .03 1 .05

0.3

0.0

0.2814 0.22 13

2

1.4559

3

1

0.4

0.8279

0.0473

4

0.2

z

0 0008

z

0.5

0.04730.02240.0004 0.008 0 0.0138 0.0 91 3

5

0

0.3

245

93

2

16188 0.60.46

92 0.14510 0.0 0.25

0.0983 0.5

0.75

r

1

Figure 4: Isolevel curves of the normalized velocity V z for z0 = d (left) and z0 = 0.1d (right). Isolevel curves of the normalized velocity V z = dVz are plotted in Fig. 4 for z0 /d = 0, 0.1. This quantity vanishes on the plane wall z = 0, remains positive in the entire liquid and is small except close to the source point z 0 . Moreover, V z is at given normalized location (r, z) larger for z0 = 0.1d than for z0 = d.


53

0.2

808

13.3829

84.5

1.0844

78

8 93

36

9

0.0

44 7 0

72 59 67 1. 92 22 13 38

0.1

2.5

5

r

7.5

10

0

0

37

0.25

1.3985

.3 3

55

2.0117

20

49.0489

55

1

0

8

0.1.80 0879 2.0 442.7 11083 7 9. 59 5.3 74 7 2

0.0

01

03

7

4

0.0230

0.3

0.0

00.2.0337 036

2 0.9

0.0583

0.0 8 0.93 13 09

0 0017

2

6. 86

3.9267

z

0.00

0.0

0.4

2.7083

0.0034

62

0.011

58

3

z 0 0008

4

3

1.398

0.5

7

5

31

0.8

28

0.5

07

9

r

0.75

1

Figure 5: Isolevel curves of the normalized pressure Q for z0 = d (left) and z0 = 0.1d (right). Isolevel curves of the normalized pressure Q = dQ are drawn in Fig. 5. This quantity is positive everywhere and quickly decays away from the source point z = 0. At any point (r, z) it is smaller for the z0 = d case than for the z0 = 0.1d case. Finally, the velocities ux and uy , analytically given in [1], depend upon the wall nature. For a sake of conciseness, the formulae for ux and uy are not given here (see [1]). For comparison purpose, we give in Fig. 6 (only in the y = 0 plane and for z0 /d = 0, 0.1) the associated isolevel curves for the normalized quantities ux = 8πμdux (while uy = 0 in the y = 0 plane).

20

0.0 3

0.0493

10

0

0

5 0. 0 0040

4 0.01 0 0011

2.5

00 0.

34

49

0 0.0

4

65

5

17

x

7.5

10

0.5

0

19

0.4

21 5

80

0.25

6 66

26 0. 44

37 0.6

0.2

33 0.

5 0.34

0.6382 0

58

0.5

82

0.1

0.1376 0.75

2831

x

1

0

90

0.1

81

0

31

75

3 13 1.

0.1

0

28 0.

0

5 9.

5 63 1.1169

2. 44

0.2

0.5

3.32

18

3

6

0.4

1.87

82 2

169 0.3

0.8

1.4926

247

0.2

56

z

z

0.7 0. 89

x

7.5

03 0.

6 72

0. 02

04

0.0011

5

0.4

1.5

0.0

2

1

0.0022

0.5

0.3

7 64

23

0.0

0.0

51

7

0.3253 95 4 .1726 00062 0 0097 00.0 0 0004 2.5

0

9

0.0

2

0

0.01

2

1

34

34

0.2352

45

0. 11

0.0347

3

0

02 0.

4

27

4

0.0040

5

0.00

2

z

z

23

0.0098

0.0

0.00

5

0.1101 0.25

0.5

0.75

x

1

Figure 6: Isolevel curves of the normalized velocity ux in the y = 0 plane for the conducting wall (top left) and the insulating wall (top right) for z0 = d and for the conducting wall (bottom left) and the insulating wall (bottom right) for a close point force (z0 = 0.1d). As revealed by Fig. 6, the velocity ux depends upon the wall nature. It is positive in the liquid and of course vanishes on the wall. Conclusions The fundamental steady velocity u and pressure p induced by a point force, with unit stregth e, located in a conducting liquid above the z = 0 plane solid wall have been numerically investigated when the wall is normal to the uniform magnetic field prevailing in the liquid. This has been done for both insulating and conducting wall. The quantities u and p do not depend upon e and the wall nature except u − (u.ez )ez in the case of e tangent to the wall. Our computations for two source point locations reveal that (u, p) quickly decays away from the source point and are sensitive to the source point location.

32


[1] A. Sellier Fundamental MHD creeping flow bounded by a motionless plane solid wall. To appear in The European Journal of Computational Mechanics. [2] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970). [3] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Publisher. (1990). [4] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [5] H. Yosinobu and T. Kakutani Two-dimensional Stokes flow of an electrically conducting fluid in a uniform magnetic field. Journal of the Physical Society of Japan, 14 (10), 1433-1444 (1959). [6] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech.”, vol 3, 304-308 (1957). [7] W. Chester The effect of a magnetic field on the flow of a conducting fluid past a body of revolution. J. Fluid Mech.”, vol 10, 459-465 (1961). [8] A. Sellier and S. H. Aydin Creeping axisymmetric MHD flow about a sphere translating parallel with a uniform ambient magnetic field. To appear in MagnetoHydrodynamics. [9] A. Sellier and S. H. Aydin Fundamental free-space solutions of a steady axisymmetric MHD viscous flow. European Journal of Computational Mechanics, vol 25, issue 1-2; 194-217 (2016). [10] J. Priede Fundamental solutions of MHD Stokes flow arXiv: 1309.3886v1. Physics. fluid. Dynamics, (2013). [11] A. B. Tsinober Green’s function for axisymmetric MHD Stokes flow in a half-space. MagnetoHydrodynamics, 4, 559-562 (1973).


33

Spherical harmonics expansion of fundamental solutions and their derivatives for homogenous elliptic operators Vincenzo Gulizzi1,a, Alberto Milazzo1,b, Ivano Benedetti1,c 1Dipartimento

di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali (DICAM),

Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy [email protected], [email protected], [email protected],

Keywords: fundamental solutions, spherical harmonics, elliptic operators, integral equations, boundary element method

Abstract. In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material. Introduction Fundamental solutions are necessary for solving many boundary value problems in engineering and represent an essential item in boundary integral formulations [1]. Closed form expressions can be obtained only for simple cases, such as potential problems or isotropic elasticity. Therefore, efficient schemes for computing the fundamental solutions of generic linear systems of partial differential equations (PDEs) are still of engineering and scientific interest. In the literature, several numerical approaches have been used to compute the fundamental solutions of linear second-order partial differential operators. The formal expression has been classically obtained using either Fourier or Radon transforms leading to a formula involving a line integral over the unit circle [2]. Several researchers transformed the unit circle integral into the integral over the infinite line and obtained its solution in terms of Stroh eigenvalues [3]. However, in such a case, the problem of mathematically degenerate materials needs to be robustly addressed [3,4]. In the present work, considered a linear system of PDEs defined by a second-order homogeneous elliptic partial differential operator with constant coefficients, the corresponding fundamental solutions and their derivatives are obtained in a unified fashion in terms of spherical harmonics expansions [5]. The peculiarities of the developed spherical harmonics representation are: a) the derivatives of the fundamental solutions are obtained without any term-by-term differentiation; b) the coefficients of the expansions depend on the material properties and need to be computed only once; c) mathematically degenerate materials do not need any specific treatment. The mathematical steps for obtaining the spherical harmonics expansions are presented and discussed. It is shown that in case of isotropic system of PDEs, such as isotropic Laplace or isotropic elasticity equations, the spherical harmonics expansion leads to the exact solutions. Then, the developed expansion is employed to compute the fundamental solution of generally anisotropic operators and two tests involving isotropic elastic and general magneto-electro-elastic materials are presented. Problem statement Let us consider a linear system of partial differential equations (PDEs) represented by a homogeneous elliptic partial differential operator with constant coefficients. The system of equations can be written as follows

Lij (w x )I j (x) f i (x) 0,

(1)

34


where x {x1 , x2 , x3 } R 3 is the spatial variable, Ii (x) are the unknown functions, fi (x) are the volume forces, i, j 1,, N where N is the number of equations as well as the number of unknown functions; Lij (w x ) is the homogeneous partial differential operator with constant coefficients and can generally be written as Lij (w x ) cijkl w 2 () / wxk wxl where cijkl are the constant coefficients. Moreover, Lij (w x ) is supposed to be elliptic, i.e. its symbol Lij (ȟ ) defined as Lij (ȟ ) cijkl [ k [l can be inverted for any value ȟ {[1 , [ 2 , [3 } R 3 . The system of PDEs (1) can be specialized to different specific problems such as the classical Laplace equation or the governing equations of anisotropic materials with linear magneto-electro-elastic coupling. The fundamental solutions ) ij (x, y ) of the system of PDEs (1) are then obtained by solving the adjoint problem [6] L*ij (w r )) jp (r ) G piG (r ) 0,

(2)

where r x y {r1 , r2 , r3 } , G pi is the Kronecker delta function, G (r ) is the Dirac delta function, L*ij (w x )

L ji (w x ) is the adjoint differential operator. Upon applying the Fourier transform to Eq.(2), the fundamental solutions ) ij (x, y ) can be formally written as follows

) ij (r ) where i

1 8S 3

³ )

ij

(ȟ ) exp(i rk [ k ) dȟ,

(3)

R3

1 and

(ȟ ) [ L* (ȟ )]1 . ) ij ij

(4)

Closed form expressions of the integral representation of the fundamental solutions as given in Eq.(3) can be obtained for simple operators only such as the Laplace or the isotropic elasticity operators. For general anisotropic materials, the three-dimensional integral given in Eq.(3) can be reduced to a one dimensional unit circle integral [2]. The unit circle integration has been successfully used in the numerical implementation of the boundary element method [7,8,9]. More recently, several authors have proposed different expressions of the fundamental solutions based on the Stroh’s eigenvalues [3,4,10,11] or Fourier series [12]. In this work, the fundamental solutions and their derivatives are computed in terms of spherical harmonics expansions. Spherical harmonics expansions of the fundamental solutions and their derivatives The spherical harmonics expansions of the fundamental solutions and their derivatives are obtained by considering the homogeneous nature of the function ) ij (ȟ ) given in Eqs. (3) and (4) and by using the Rayleigh expansion in combination with the spherical harmonics addition theorem [13]. More specifically, since ) ij (ȟ ) is homogeneous of order n 2 , is it possible to write

(ȟ ) [ n ) (ȟ ), ) ij ij

(5)

where [ [ k [ k and ȟ ȟ / [ . Moreover, the Rayleigh expansion and spherical harmonics addition theorem permit to express the quantity exp(i rk [ k ) in terms of spherical harmonics as follows f

exp(i rk [ k ) 4S ¦

"

¦i J

" 0 m "

where r

rk rk , r

"

"

(r[ )Y "m (r )Y "m (ȟ ),

(6)

r / r , J " () is the spherical Bessel function of order " , Y "m () is the spherical harmonic

of order " and degree m and () denotes the complex conjugate. It is easy to see that Eqs. (5) and (6) allow to write the integral given Eq.(3) as the product of an integral with respect to [ and an integral with respect to ȟ . The former integral can be evaluated analytically whereas the latter provides the coefficients of the expansions as shown in [5]. The fundamental solutions ) ij (r ) can then be written as follows


³ )

"m ) ij

ij

35

(ȟ )Y "m (ȟ )dS (ȟ ).

(8)

S1

Similarly, upon noting that the derivatives of order I of the fundamental solutions has a Fourier transform that is homogeneous of order n I 2 , it is possible to show that the derivatives of the fundamental solutions can be written by means of the following compact and unified expression w ( I ) ) ij (D ) 1

wr

(E ) 2

wr

(J ) 3

wr

(r )

f " 1 "m "m (r ), P I (0) ¦ ) ij ,(D , E ,J )Y I 1 ¦ " 4S r " 0 m "

(9)

where I D E J , P"I (0) is the associated Legendre function of order " and degree I, and "m ) ij ,(D , E ,J )

³ ([ )

D

1

(ȟ )Y "m (ȟ )dS (ȟ ). ([ 2 ) E ([3 )J ) ij

(10)

S1

It is worth noting that the expression of the fundamental solutions derivatives given in Eq.(9) and of the expansion coefficients given in Eq.(10) coincides to the expression of the fundamental solutions given in Eq.(7) and their coefficients given in Eq.(8) for I D E J 0 . Eq.(9) represents the unified expression of the fundamental solutions of a homogeneous elliptic second order operator and their derivatives presented in this work. The expression permits to compute the derivatives of the fundamental solutions up to the desired order in an extremely compact fashion and shows how the fundamental solutions and their derivatives can be written as the product of a regular part depending on r and a singular part depending on r . Furthermore, from a numerical perspective, the coefficients of the expansions do not depend on the vector r and therefore need to be computed only once. Results In this section, the proposed spherical harmonics expansions are employed to compute the fundamental solutions of isotropic elastic materials and of an anisotropic magneto-electro-elastic material. The partial differential operator describing the governing equations of three-dimensional isotropic elasticity can be written as Lij (w x ) (O P )

w2 w2 P G ij wxi wx j wxk wxk

(11)

where O and P are the Lamé constants. Since Lij (w x ) is symmetric, the adjoint operator L*ij (w x ) coincides with Lij (w x ) and its symbol reads L*ij (ȟ ) (O P )[i[ j P[ k [ k G ij . Using Eqs.(4) and (10), the coefficients of the expansions are obtained as "m ) ij ,(D , E ,J )

³ ([ )

D

1

S1

§1 · 1 ([ 2 ) E ([3 )J ¨ G ij [î[ˆ j ¸ Y "m (ȟ )dS (ȟ ) 2 P (1 Q ) ©P ¹

(12)

being Q the Poisson’s coefficients. Using the orthogonality properties of the spherical harmonics, it is possible to show that the expansions coefficients are identically zero for " ! D E J 2 and the expansions involve only a finite number of terms, thus providing the exact expressions of the fundamental solutions. As an example, let us consider the fundamental solution )11,1 (r ) . Using Eq.(12), the non-zero coefficients for m ! 0 are computed as 11 ) 11,(1,0,0)

S / 6 (7 10Q ) 31 , ) 11,(1,0,0) 5P (1 Q )

which, using Eq.(9), provide

3S / 7 33 , ) 11,(1,0,0) 10 P (1 Q )

S / 35 2 P (1 Q )

(13)

36


)11,1 (r )

(1 4Q 3rˆ12 ) rˆ1 . 16 S P (1 Q ) r 2

(14)

The partial differential operator describing the governing equations of three-dimensional anisotropic magneto-electro-elasticity can be written as [11,14,15] LJK (w x ) ciJKl

w2 wxl wxi

(15)

where ciJKl are the constants of the magneto-electro-elastic coupling that are defined as

ciJKl

cijkl , °e , ° lij °eikl , ° °qlij , ® °qikl , °Oil , ° °il , °N , ¯ il

J,K d 3 J d 3, K 4 J 4, K d 3 J d 3, K 5

(16)

5, K d 3

J J

4, K

J

K

4

5 or J

J

K

5

5, K

4

being cijkl the elasticity stiffness tensor, ij the dieletric permittivity tensor, N ij the magnetic permeability tensor, elij the piezoelectric coupling tensor, qlij the piezo-magnetic coupling tensor and Oil the magnetoelectric coupling tensor. These satisfy certain symmetry identities that ensure that the symbol of the magneto-electro-elastic operator is symmetric and can always be inverted. In this case, the operator is generally anisotropic and the equality of the spherical harmonics expansions given in Eq.(9) is valid for an infinite number of terms. Therefore, the expansions are truncated and the coefficients are computed for " 0, }, L with L finite. The accuracy of the expansions is assessed by defining the following measures of error: e ª) ij ,(D , E ,J ) (r ) º ¬ ¼

) ij ,(D , E ,J ) (r ) ) ijR,(D , E ,J ) (r ) ) ijR,(D , E ,J ) (r )

S1

and ' ª) ij ,(D , E ,J ) (r ) º ¬ ¼

) ij ,(D , E ,J ) (r ) ) ijR,(D , E ,J ) (r ) )R (r )

S1

ij ,(D , E ,J )

S1

(17) where S

1

³

dS , ) ij ,(D , E ,J ) (r ) is the fundamental solution computed using the present scheme and

S1

) ijR,(D , E ,J ) (r ) is a reference value of the considered fundamental solution.

Figs.(1a) and (1b) show the error e ª¬) ij ,(D , E ,J ) (r ) º¼ as defined in Eq.(17) of the components of two selected second derivatives of the fundamental solutions of a magneto-electro-elastic material, whose constants are taken from [15]. The unit circle integration has been used as reference. Fig.(2a) shows the plot of the selected fundamental solution )11,(2,0,0) (r ) computed using L 32 whereas Fig.(2b) shows the error ' ª¬)11,(2,0,0) (r ) º¼ as defined in Eq.(17) of the selected fundamental solution )11,(2,0,0) (r ) computed using the presented scheme and the unit circle integration. Summary This work presents a novel unified and compact scheme to compute the fundamental solutions of a threedimensional homogeneous second order elliptic partial differential operator. The scheme has been obtained by using the Rayleigh expansion and the Fourier representation of a homogeneous function, and provides a representation of the fundamental solutions and their derivatives up to the desired order in terms of spherical harmonics. The scheme is easy to implement as it does not require any term by term differentiation to obtain the derivatives of the fundamental solutions. Furthermore, the coefficients of the expansions depend on the material properties only and need to be computed only once. Two operators have been considered: first, the

scheme has been employed to compute the fundamental solution of the differential operator of threedimensional isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme has been assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

(a)

(b)

Fig 1: Error as a function of the series truncation number L of the spherical harmonics expansions of the components of two selected second derivatives of the fundamental solutions of a magneto-electro-elastic material: (a) ) ij ,(2,0,0) (r ) ; (b) ) ij ,(1,1,0) (r ) .

(a) )11,(2,0,0) (r )[m 1 ]

(b) '[)11,(2,0,0) (r )]

Fig 2: (a) Selected second derivative )11,(1,1,0) (r ) of the fundamental solutions of a magneto-electro-elastic material; (b) plot over the unit sphere of the difference '[)11,(2,0,0) (r )] of the second derivative )11,(1,1,0) (r ) computed using the spherical harmonics expansion and the unit circle integration. References [1] [2] [3] [4] [5] [6]

MH Aliabadi, The boundary element method: applications in solids and structures. John Wiley & Sons Ltd, England, 2002. JL Synge, The Hypercircle in mathematical physics. University Press Cambridge, 1957. TCT Ting, VG Lee, The three-dimensional elastostatic green’s function for general anisotropic linear elastic solids. Quarterly Journal of Mechanics & Applied Mathematics, 50(3), pp. 407-426, 1997. L Xie, C Zhang, J Sladek, V Sladek, Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials. Proceedings of the Royal Society A, 472, pp. 20150272, 2016. V Gulizzi, A Milazzo, I Benedetti, Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions. International Journal of Solids and Structures, 100, pp. 169-186, 2016. PK Banerjee, R Butterfield, Boundary element methods in engineering science. McGraw-Hill London, 1981

38


[7] [8] [9] [10] [11] [12] [13] [14] [15]

I Benedetti, MH Aliabadi. Multiscale modeling of polycrystalline materials: A boundary element approach to material degradation and fracture. Computer Methods in Applied Mechanics and Engineering, 289, pp.429-453, 2015. V Gulizzi, A Milazzo, I Benedetti. An enhanced grain-boundary framework for computational homogenization and micro-cracking simulations of polycrystalline materials. Computational Mechanics, 56(4), pp.631-651, 2015. I Benedetti, V Gulizzi, V Mallardo, A grain boundary formulation for crystal plasticity, International Journal of Plasticity, 83, pp. 202-224, 2016. VG Lee, Derivatives of the three-dimensional Green’s functions for anisotropic materials. International Journal of Solids and Structures, 46(18), pp. 3471-3479, 2009. FC Buroni, A Saez, Three-dimensional Green's function and its derivative for materials with general anisotropic magneto-electro-elastic coupling. Proceedings of the Royal Society A, 466, pp. 515-537, 2010. YC Shiah, CL Tan, CY Wang, Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis. Engineering Analysis with Boundary Elements, 36(12), pp. 1746-1755, 2012. GB Arfken, Mathematical methods for physicists: A comprehensive guide. Academic press, 2005. E Pan, F Tonon, Three-dimensional Green’s functions in anisotropic piezoelectric solids, International Journal of Solids and Structures, 37(6), pp. 943-958, 2000. MM Muñoz-Reja, FC Buroni, A Saez, 3D explicit-BEM fracture analysis for materials with anisotropic multifield coupling. Applied Mathematical Modelling, 40(4), pp. 2897-2912, 2016.


Particle-particle interactions in axisymmetric MHD viscous flow 1

S. H. Aydin1 and A. Sellier2 Department of Mathematics, Karadeniz Technical University, 61080 Trabzon, Turkey 2 LadHyX. Ecole polytechnique, 91128 Palaiseau C´ edex, France

Keywords: Particle-particle interactions, MagnetoHydroDynamics, Axisymmetric MHD flow, Boundary-integral equation.

Abstract. This work investigates the interactions between two solid spheres translating, in an unbounded conducting Newtonian liquid, parallel with both their line of centers and a prescribed steady and uniform ambient magnetic field. The procedure actually extents to this cluster the new boundary formulation recently proposed and successfully worked-out by the authors for a single sphere. The resulting coupled boundary-integral equations, which govern the traction arising on each sphere boundary, are numerically inverted and the liquid velocity and pressure fields are then subsequently computed in the liquid domain by exploiting the relevant integral representations prevailing in the liquid for those quantities. From the preliminary numerical results reported and discussed for two equal spheres (with one translating sphere and another motionless sphere) the particle-particle interactions are found to deeply depend upon the particles spacing and the Hartmann number. Introduction For some applications it is useful to determine the so-called MagnetoHydroDynamic (MHD) flow about a solid particle experiencing a given rigid-body motion in a conducting and unbounded Newtonian liquid subject to a prescribed ambient uniform and steady magnetic field B = Bez with magnitude |B|. Depending upon the solid migration this problem is cumbersome since in general coupled electrical field E and magnetic field B take place in the liquid and one has to simultaneously get those fieds (E , B ) and the liquid flow velocity u and pressure p by solving coupled unsteady Navier-Stokes and Maxwell equations [1,2]. Actually, this coupling is due to the Lorentz body force which, using the usual Ohm’s law, reads In addition, near the body fL = σ(E + u ∧ B ) ∧ B where σ > 0 denotes the liquid uniform conductivity. surface arises a so-called Hartmann layer [3], with typical thickness d = ( μ/σ)/|B| (with μ the uniform liquid viscosity) in which the viscous term μ∇2 u magnitude becomes comparable with the magnitude of σ(u ∧ B ) ∧ B (here B has typical magnitude |B|). Assuming a sufficiently slow body motion however ensures that E , B , u and p are quasi-steady fields. For a particle with typical length scale a and a liquid with uniform magnetic permeability μm > 0, density ρ and velocity typical magnitude V > 0 it appears that (E , B , u, p) depends upon three so-called magnetic Reynolds number Rem , Reynolds number Re and Hartmann number Ha defined as follows (1) Rem = μm σV a, Re = ρV a/μ, Ha = a/d = a|B|/ μ/σ. Fortunately, under some additional assumptions one ends up with a more tractable problem. First, when the liquid and the body admit identical magnetic permeability and Rem 1 the ambient magnetic field B turns out [1,2] to be undisturbed by the particle, i. e. B = B in the entire liquid domain. Second, as soon as the flow (u, p) is axisymmetric about the direction of B and without swirl one also gets E = 0 [1,4]. Finally, when Re 1, our steady MHD flow (u, p) is then governed by the linear Stokes equation with the non-uniform Lorentz body force fL = σ(u ∧ B) ∧ B. This case of a creeping steady MHD flow with axis of revolution parallel with B and without swirl has been investigated for a sphere translating parallel with B for either small [5] or large [6] values of the Hartmann number Ha. It has also recently been addressed still for the translating sphere but for arbitrary Ha > 0 in [7] by using a new boundary approach. This approach appeals to two fundamental axisymmetric MHD flows, produced by distributing on a ring either axial or radial points forces, obtained in [8]. Since collections of particles are also encountered in applications, it is also worth examining to which extent particle-particle interactions might affect the results obtained for a single sphere translating parallel with B. This is done in this work by addressing the case of two translating spheres having line of centers and translational velocities parallel with B.

39

40


Governing axisymmetric MHD problem and advocated boundary approach This section presents the addressed MHD problem for two translating spheres and also the advocated boundary approach. Governing equations We consider two solid spheres P1 and P2 immersed in a conducting unbounded Newtonian liquid with uniform viscosity μ, conductivity σ > 0 and density ρ. As sketched in Fig. 1, for n = 1, 2 the sphere Pn has center On , radius an , surface Sn with unit normal n directed into the liquid and translational velocity U (n) ez where ez = O1 O2 /|O1 O2 |. er M• B = Bez r S1 a1

ez z

n O1 P1

(μ, ρ, σ) n

S2 U (1) ez

a2

O D

O2

U (2) ez

P2

Figure 1: Two solid spheres having ligne of centers parallel with ez and translating in a conducting Newtonian liquid at the velocities U (1) ez and U (2) ez parallel with the ambient magnetic field B = Bez . For such a geometry the liquid MHD viscous flow around the spheres is axisymmetric about the (O1 , ez ) axis and without swirl. Assuming, as in the introduction, vanishing Reynolds magnetic and Reynolds numbers its axisymmetric velocity field u and pressure field p obey μ∇2 u = ∇p − σB 2 (u ∧ ez ) ∧ ez and ∇.u = 0 in D ,

(u, p) → (0, 0) as |x| → ∞ , u = U (n) ez on Sn for n = 1, 2

(2) (3)

with D the unbounded liquid domain and x = OM where O designates a given origin (see Fig. 1) taken on the (O1 , ez ) axis. For convenience, each point x in D ∪ S1 ∪ S2 is located by its cylindrical coordinates (r, z, θ) with θ ∈ [0, 2π], z = x.ez and r = {|x|2 − z 2 }1/2 ≥ 0. Thus, x = rer + zez with local unit vector er = er (θ) shown in Fig. 1 and u(x) = ur (r, z)er + uz (r, z)ez while p(x) = p(r, z). On the boundary Sn , with unit n pointing into the liquid, the flow (u, p), with stress tensor σ, exerts a surface traction f = σ.n = fr (r, z)er + fz (r, z)ez . For symmetry reasons, each sphere experiences a zero torque about its center and a force parallel with ez . More precisely, the force F(n) acting on Pn reads F(n) = f dS = [2π fz (P )r(P )dl(P )]ez (4) Sn

Cn

where Cn is the half-circle trace of Sn in the θ = 0 half plane. Inspecting (2)-(3) shows that the MHD flow (u, p) linearly depends upon U (1) and U (2) . Accordingly, one can write F(1) = −6πμa1 [λ11 U (1) + λ12 U (2) ]ez , F(2) = −6πμa2 [λ21 U (1) λ22 U (2) ]ez

(5)

where the coefficients λ11 , λ12 , λ21 and λ22 are drag coefficients. Both the flow (u, p) and these coefficients depend upon the spheres spacing |O1 O2 | and the Hartmann number Ha = a/d with a = M ax(a1 , a2 ) and d = ( μ/σ)/|B| the Hartmann layer thickness [3]. For widely-separated spheres the coupling coefficients λ12 and λ21 vanish while each coefficient λnn tends to the value prevailing for a single sphere Pn , i. e. to


41

the drag coefficient recently computed in [7]. Relevant integral representations and coupled boundary-integral equations To a point x(r, z) in the liquid domain we associate the point M (r, z) in the half θ = 0 plane. Using the material obtained in [8] it is then possible to extend [7] to the present problem thereby arriving at integral representations for the MHD flow velocity and pressure in the liquid domain D. Adopting henceforth the usual tensor summation notation for repeated indices and taking α and β in {r, z}, we get the key relations 1 Gαβ (M, P )fβ (P )r(P )dl(P ) for α = r, z and x in D ∪ S1 ∪ S2 , (6) uα (x) = − 8πμ C1 ∪C2 1 p(x) = −

Ha 0.01 0.1 0.5 1.0 5.0 10.0

N = 16 2.503 2.521 2.643 2.858 5.626 10.691

λ11 N = 32 2.499 2.517 2.639 2.853 5.616 10.656

N = 64 2.498 2.516 2.637 2.852 5.613 10.644

N = 16 -1.842 -1.831 -1.832 -1.908 -3.725 -7.7191

λ21 N = 32 -1.838 -1.827 -1.828 -1.904 -3.715 -7.685

N = 64 -1.837 -1.826 -1.827 -1.903 -3.712 -7.674

Table 1: Computed drag coefficients λ11 and λ21 for several Hartmann numbers Ha and different meshes (N1 = N2 = N ) for close equal spheres with β = 0.9.

42


Clearly (recall (5)), under our choice U (2) = 0 we gain the drag coefficients λ11 for the translating sphere P1 and λ21 for the motionless sphere P2 . One should note that for symmetry reasons (equal spheres) here we actually also have λ22 = λ11 and λ12 = λ21 . Several meshes have been tested putting the same number N/2 (here N = N1 = N2 ) of elements on each half-circle Cn . The results for different (Ha, β) shown a nice convergence even, as illustrated in Table 1, for close spheres (here β = 0.9). It is actually found that spreading N/2 = 16 quadratic 3-node boundary elements (i. e. N + 1 = 33 nodes) on each sphere half-circle contour Cn is quite sufficient to ascertain a very good accuracy for Ha in the range [0, 10] when β ≤ 0.9. Numerical results For a sake of conciseness we present in this subsection results for Ha = 10 (taking N = 32). First, we plot in Fig. 2 both drag coefficients λ11 (= λ22) and λ21 (= λ12) versus the separation parameter β. 12 10 8 6 4 2 0 -2 -4 -6 -8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

β

Figure 2: Drag coefficients λ11 (solid line) and λ21 (dashed line) versus the cluster separation parameter β for Ha = 10. Remind that under our choice a = a1 = a2 the force F(n) exerted here on the sphere Pn is given by F(n) = −6πμaλn1 U ez . Not surprisingly, for the translating sphere P1 one has λ11 > 0 while for the motionless sphere P2 it appears that λ21 < 0 because when U > 0 the second sphere is pushed by the first one (the force F(2) excerted on P2 is then directed in the z > 0 downstream direction). Both drag coefficients increase in magnitude as spheres approach each other (i. e. as β increases). For close spheres (β = 0.9) the quantity −λ21 , which is zero for widely-separated spheres (β = 0), becomes large and almost as big as λ11 therefore indicating strong sphere-sphere interactions (from Table 1 we have for β = 0.9 the values λ11 ∼ 10.6 and −λ21 ∼ 7.67.) For comparison, note that for a single sphere P1 (case β = 0) we obtain (see [7]) the value λ11 ∼ 4.72. We now pay attention to the normalized radial and axial velocity components ur = ur /U and uz = uz /U in the normalized liquid domain with coordinates z = z/a and r = r/a. The isolevel curves of those quantities are given in Fig. 3 again for close spheres (β = 0.9). The normalized axial velocity uz quickly decays aways from the r = 0 axis and is positive except in a pocket located close above the normalized contour C 1 in which it takes negative values of small magnitude. Outside this pocket 0 < uz ≤ 1 and it takes values above 0.3 only close C 1 and upstream z < 0 near the (O1 , ez ) axis. The normalized radial velocity ur remains small in magnitude in the entire liquid domain and is negative solely upstream the first sphere.


3

43

3

-0.0125 2.5

0.0082

2.5

r

44

0.5

z

-0

0.3

0

2

4

0

0.00

82

-0 .0 1

-0.0

1 44

44

0

-2

0.85

0

1 0 00

500

0.500

000

1

0 0.10044 82.01 0-0

1.5

0.1000

0.0

0.0

0 00

0.050 0.30000 0.8500

-0.1

125

0.5

2

0.3000

r

500

5

1

125

1.5

-0.106

-0.0

0.0

.0 1

-0.0

2

0.0082

-2

0

z

2

4

Figure 3: Isolevel curves of the normalized axial and radial velocity components uz (left) and ur (right) for Ha = 10 and close spheres (β = 0.9). Finally, the associated streamlines and also the normalized pressure p = ap/(μU ) are displayed in Fig. 4 still for close spheres (β = 0.9). The computed streamlines show that the normalized flow around the 2-sphere cluster flows downstream (in the z > 0 direction) except above and upstream the moving first sphere where there is a reverse flow. The normalized pressure also strongly decays aways from the r = 0 axis. It is negative upstream the first sphere and positive elsewhere. In adddition, it reaches large positive values in the narrow gap between the close spheres.

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0.2

4

0

10.0

33

-2

1.9

1.9801 4

0

801

000 25.0000

z

2

74

z

2

74

28 0.

0

78 9.

87

.0

-2

0.28

100

0

0.2874

r

3

r

3

4

Figure 4: Flow streamlines (left) and isolevel curves of the normalized pressure p (right) for Ha = 10 and close spheres (β = 0.9). Conclusions It is possible to efficiently and accurately investigate the particle-particle interactions taking place for two solid spheres translating, in a conducting Newtonian liquid, parallel with their line of centers and a given ambient uniform magnetic field. The advocated procedure is a boundary approach which reduces the task to the (numerical) determination of the traction arising on each sphere surface and permits one, in a second step, to calculate the flow velocity and pressure about the cluster in the entire liquid domain by appealing to basic integral representations of those quantities. The numerical preliminary results for two equal spheres reveal that the flow and the forces experienced by the spheres deeply depend upon the Hartmann number and the spheres separation. Additional numerical results will be presented and discussed at the oral presentation. References [1] A. B. Tsinober MHD flow around bodies. Fluid Mechanics and its Applications (Kluwer Academic Publisher, 1970). [2] R. J. Moreau Magnetohydrodynamics, Fluid Mechanics and its applications. Kluwer Academic Pub-

44


lisher. (1990). [3] J. Hartmann Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl . Danske Videnskabernes Selskab . Mathematisk-fysiske Meddelelser, XV (6), 1-28 (1937). [4] K. Gotoh Magnetohydrodynamic flow past a sphere. Journal of the Physical Society of Japan, 15 (1), 189-196 (1960). [5] W. Chester The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech.”, vol 3, 304-308 (1957). [6] W. Chester The effect of a magnetic field on the flow of a conducting fluid past a body of revolution. J. Fluid Mech.”, vol 10, 459-465 (1961). [7] A. Sellier and S. H. Aydin Creeping axisymmetric MHD flow about a sphere translating parallel with a uniform ambient magnetic field. To appear in MagnetoHydrodynamics. [8] A. Sellier and S. H. Aydin Fundamental free-space solutions of a steady axisymmetric MHD viscous flow. European Journal of Computational Mechanics, vol 25, issue 1-2; 194-217 (2016).


2.5D spectral based BEM-FEM formulation to represent waveguide with acoustic and solid interaction F.J. Cruz-Muñoz1 , A. Romero1∗ , A. Tadeu2 and P. Galv´ın1 1

Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain. [email protected], [email protected], [email protected] 2 Department

of Civil Engineering, University of Coimbra, Pólo II, Rua Lu´ıs Reis Santos, 3030-788 Coimbra, Portugal. [email protected]

Keywords: BEM, wave propagation, spectral, fluid acoustics.

Abstract. This work presents a two-and-a-half (2.5D) spectral formulation based on the finite element method (FEM) and the boundary element method (BEM) to study three dimensional (3D) wave propagation within acoustic and elastic regions. The numerical method is based on the domain decomposition to study the fluid-solid coupled problem. The BEM is used to analyse the acoustic field in unbounded regions, whereas the FEM allows to represent the solid waveguide with arbitrary cross-section. Both approaches are extended to its spectral formulation using Lagrange interpolant polynomials as element shape functions at the Legendre-Gauss-Lobatto (LGL) points. The proposed method is verified from a benchmark problem: the acoustic scattered wave in an unbounded medium by either rigid or elastic solid cavities. An h-p analysis is carried out to assess the accuracy of the method. The results show a high accuracy of the proposed method to represent this kind of problem. Introduction Time-harmonic wave propagation, such as fluid acoustics and solid scattering, is a common phenomenon that appears in many engineering fields. The propagation of acoustic waves triggered by static and moving pressure sources, the vibration assessment and the acoustic insulation, involve fluid and solid interaction and must be considered rigorously. The finite element method (FEM) has been used in several works to predict the response in fluid-structure interaction problems. For the low frequency range, the conventional finite elements with linear shape functions represent accurately the fluid and solid scattering waves. However, at high frequencies, these shape functions do not provide reliable results due to so-called pollution effects [1, 2]: the accuracy of the numerical solution deteriorates with increasing non-dimensional wave number and it is not sufficient the commonly employed rules of n elements per wavelength [3]. High element resolutions are required in order to obtain results with reasonable accuracy. In this work, a two-and-a-half dimensional (2.5D) approach to represent scattered waves in fluid media is proposed. This approach is useful for problems where the material and geometric properties are uniform along one direction, and the source exhibits 3D behaviour. Numerical Model The 2.5D formulation computes the problem solution as the superposition of two-dimensional (2D) problems with a different longitudinal wavenumber, kz , in the z direction. An inverse Fourier transform is used to compute the 3D solution: +∞ a(x, ω) = a( x, kz , ω)e−ikz z dkz (1) −∞

where a(x, ω) is an unknown variable (e.g., displacement or pressure), a( x, kz , ω) is its representation √ = x(x, y, 0), ω is the angular frequency, and i = −1. in the frequency-wavenumber domain, x

45

46


The boundary element formulation presented in this work considers an arbitrary boundary submerged in an unbounded fluid medium. The integral representation of the pressure pi for a point i located at the fluid subdomain Ωf ∞ , with zero body forces and zero initial conditions may be written as [4]: ci pi (xi , ω) =

Γf

pi∗ (x, ω; xi )ui (x, ω)dΓ −

Γf

ui∗ (x, ω; xi )pi (x, ω)dΓ

(2)

where ui (x, ω) and pi (x, ω) are respectively the normal displacement and the pressure at point i. ui∗ (x, ω; xi ) and pi∗ (x, ω, xi ) are respectively the fluid full-space fundamental solution for normal displacement and pressure at point x due to a point load at xi . The integral-free term ci depends only on the boundary geometry at point i. The integration boundary Γf represents the boundary between the unbounded fluid medium (Ωf ∞ ) and the solid subdomain (Ωs ). Assuming that the unbounded medium is invariant in the longitudinal direction z, eq. (2) is expressed in terms of integrals in this direction and over the cross-section boundary, Σf : +∞ +∞ pi∗ (x, ω; xi )ui (x, ω)dSdz − ui∗ (x, ω; xi )pi (x, ω)dSdz (3) ci pi (x, ω) = −∞

−∞

Σf

Σf

Eq. (3) is then transformed to the wavenumber domain as: i ) i ) x, ω, kz ) = pi∗ ( x, ω, kz ; x ui ( x, ω, kz )dS − u i∗ ( x, ω, kz ; x pi ( x, ω, kz )dS ci pi ( Σf

(4)

Σf

where a hat above a variable denotes its representation in the frequency-wavenumber domain. The problem is discretised into elements, leading to a boundary approximation of the normal displacement and pressure using the interpolation shape functions φj . Then, eq. (4) is written as:

Q Q i i i∗ j i i∗ j ij pi ij u p φ dΣ u − u φ dΣ pi = i − G H (5) c p = j=1

Σjf

Σjf

j=1

Σjf

are the elements which contains where Q is the number of boundary nodes at the boundary Σf and the node j. After interpolating the boundary variables, the integral representation defined by eq. (5) yields a system of equations that is solved for each frequency. The system of equations for all the boundary elements becomes: H(ω, kz ) u( x, ω, kz ) = G(ω, kz ) p( x, ω, kz )

(6)

The proposed spectral boundary element method for the 2.5D fluid element uses Legendre polynomials of order p as interpolation shape functions. The shape interpolation functions φ are given by: ξ − ξi (7) φi = ξ j − ξi j=i

where the local nodal coordinates ξ are found at the LGL integration points: (1 − ξ 2 )

∂φ(ξ) =0 ∂ξ

(8)

The boundary Γf is discretized into elements that are generated from a base mesh. The element definition is done by a polynomial interpolation from the element base mesh and its representation in the natural coordinates system. As an example, Fig. 1.(a) shows an element base mesh defined by four geometrical points used to generate an element with order p = 6 (Fig. 1.(c)). The seven spectral element nodes are defined in natural coordinates in ascending order (Fig. 1.(b)) with ξ ∈ [−1, 1]. The shape function φ and its derivatives are symbolically computed in a straightforward procedure for an arbitrary element order. Fig. 2 shows the interpolation shape functions for the p = 6 spectral element presented in Fig. 1.


1 2

y

47

3

4

5

6 7

ξ

x (a)

y x

(b)

(c)

Figure 1: (a) Element mesh seed representation, (b) element definition and (c) physical representation of spectral fluid boundary element of order p = 6 1

φ1 φ2

0.8

φ3

0.6

φ4 φ5

0.4

φ6

0.2

φ7

0 -0.2 -1

-0.5

0 ξ

0.5

1

Figure 2: Element shape functions for an element with order p = 6 Numerical verification The BEM model was verified with a benchmark problem. The model was validated by applying it to a fixed cylindrical circular cavity submerged in a homogeneous unboundend fluid medium. The cavity is subjected to a harmonic point pressure load. The analytical solution to this problem can be found in Reference [5]. The cavity had a radius r = 5m , located at the origin (x, y) = (0, 0). The unbounded fluid medium properties were: pressure wave velocity α = 1500 m/s and density ρ = 1000 kg/m3 . The problem 0 = (x0 , y0 ) = (0, 15) solution was computed for a dilatational point source placed at the fluid medium x i.e, 15 m away from the cavity centre. This loads emits a harmonic incident field pinc at a point x given by: −iA (2) H0 (kα (x − x0 )2 + (y − y0 )2 ) x, ω, kz ) = (9) pinc = ( 2 (2) where A is the source amplitude, H0 is the Hankel function of the second kind, and kα = ω 2 /α2 − kz2 . In this problem, the longitudinal wavenumber was set to kz = 0. The problem solution was computed over a grid of 1376 receivers regularly spaced in a outer region defined by −10m ≤ x ≤ 10m and −10m ≤ y ≤ 10m, at a frequency f = 200 Hz. Four different h − p strategies were investigated to get the optimal discretisation with the lowest computational effort. The characteristic element sizes were 1/h = {0.4/π, 1/π, 2/π, 4/π} m−1 . The nodal density per wavelength was used to describe the mesh density: dλ =

2πp kh

(10)

The numerical results were compared with the analytical solution and L2 scaled error, 2 , was used to asses the accuracy:

48


1/h [m−1 ] 0.4 1 2 4

p [-] 9 5 3 2

L2 scaled error 2 6.52 × 10− 5 1.50 × 10− 5 4.15 × 10− 5 4.99 × 10− 5

CPU time [s] 1.796 1.133 0.886 1.256

dλ [-] 8.59 11.94 14.32 19.10

Table 1: Summary of L2 scaled error 2 , CPU time and nodal density (dλ ) for different spectral h − p discretisation to approximate the problem solution at excitation frequency 200Hz with the accepted accuracy 2 ≤ 10−4

2 =

||fex − fh || ||fex ||

(11)

where fex denotes the reference solution and fh is the results computed by the proposed methodology.

(a)

(b)

(c)

Figure 3: (a) L2 scaled error 2 , (b) nodal density per wavelegth dλ and (c) CPU time for different discretisations 1/h and element polynomial order p. Fig. 3 shows the error 2 , the nodal density per wavelength dλ , and the CPU time for different h − p configurations. All the error curves have an initial value 2 = 1 for p = 1 and start to decrease depending on the mesh discretisation. The error curves show a monotonic convergence with the element order p. Denser meshes tend to a minimum error with a lower element order p. Table 1 summarizes the nodal density per wavelength and the CPU time for the optimal h − p discretisation to reach the solution with the acceptable error 2 ≤ 10−4 . This analysis shows that the configuration (1/h, p) = (2, 3) allows the minimum computational effort for the given accuracy. Other h − p pairs involve higher computational effort due to the high order interpolation functions or the


49

higher number of nodes. ×10 -4

3.5

0.12

0.1

0.08

0.05

0.06

0.04

0 10

-10 0 x [m]

10

(a)

2

2

1

1.5

0 10

1

-10 0.5

0 0

y [m]

2.5

3

0.02

0 -10

3

4

L2 scaled error 2

psca [P a]

0.1

×10-4

x [m]

0 -10

10

y [m]

(b)

Figure 4: (a) Scattered pressure field and (b) L2 scaled error 2 for the discretisation (1/h, p) = (2, 3) Conclusions This work proposes a spectral formulation based on the BEM to study acoustic wave propagation. This method looks at 3D problems whose materials and geometric properties remain homogeneous in one direction. The developed methodology avoids the pollution effect at high frequencies. The approach has been verified with a benchmark problem with known analytical solution. The numerical results were in good agreement with the reference solution. Acknowledgments The authors would like to acknowledge the financial support provided by the Spanish Ministry of Economy and Competitiveness under research projects [BIA2013-43085-P and BIA2016-75042-C21-R]. Additionally, the authors also wish to acknowledge the support provided by the Andalusian Scientific Computing Centre (CICA). This research was also supported by the project POCI-01-0247FEDER-003474 (coMMUTe - Multifunctional Modular Train Floor), funded by Portugal 2020 through the Operational Programme for Competitiveness Factors (COMPETE 2020). References [1] I. Babuska, F. Ihlenburg, E.T. Paik and S.A. Sauter A generalizaed finite element method for solving the helmholtz equation in two dimensions with minimal pollution, Computer Methods in Applied Mechanics and Engineering, Vol. 128, no. 3-4, pp. 325–359 (1995). [2] F. Ihlenburg, I. Babuska and S.A. Sauter Reliability of finite element methods for the numerical computation of waves, Advances in Engineering Software, Vol. 28, no. 7, pp. 417–424 (1997). [3] F. Ihlenburg (2003): The medium-frequency range in computational acoustics: Practical and numerical aspects, Journal of Computational Acoustics, no. 2, pp. 175–193 (2003). [4] J. Dom´ınguez Boundary elements in dynamics, Computational Mechanics Publications and Elsevier Aplied Science, Southampton (1993). [5] A.J.B. Tadeu and L.M.C. Godinho Three-dimensional wave scattering by a fixed cylindrical inclusion submerged in a fluid medium, Engineering Analysis with Boundary Elements, Vol. 23 no. 9, pp. 745–755 (1999).

50


Application of Boundary Element Method to evaluate the Seismic Site Effects of Adjacent Non-curved Valleys Zahra Khakzad2, Dana Amini Baneh1 and Behrouz Gatmiri1 2

1 Faculty of Civil Engineering, University of Tehran, Tehran, Iran, [email protected] Department of Structural Civil Engineering, Islamic Azad University of Roudehen, Tehran, Iran, [email protected]

Keywords: Non-curved Adjacent Valleys, Site Effect, spectral response, Hybrid Numerical Method.

Abstract. An extensive numerical analysis on the seismic site effects due to local topographical and geotechnical characteristics is performed. Two dimensional (2D) non-curved configurations under incidence of vertically propagating SV waves is modelled with the aid of HYBRID program prepared by Gatmiri and his collaborators. The HYBRID software combines the finite elements in the near field and the boundary elements in the far field (FE/BE). This paper focuses on the modelling of non-curved adjacent valleys. A parametric study is conducted to examine the effects of topography and adjacency on the amplification of the response spectrum at various points across the valleys. Valleys are characterized by their depth (H) and their half width at the surface (L) and the calculations are made for different depth ratios H/L= 0.2, 0.4, 0.6, 1. Finally, some practical graphs are proposed in terms of engineering applications to assess the spectral response at the surface of non-curved adjacent valleys.

1. Introduction It has been recognized that effects of geometrical of a site can significantly affect the nature of strong ground motion during earthquakes. The modification of the seismic movement due to local topographical and geotechnical conditions is called site effect. Certainly in the recent past, there have been numerous cases of recorded motions and observed earthquake damage pointing towards geometrical and geotechnical amplification as an important effect. Thus study of site effects is one of the most important topics in earthquake engineering. Geometrical of a site modify the nature of seismic waves in transition from depth to the surface. The majority of seismic codes rest on seismic site effects by using one-dimensional (1D) model. The purpose of this paper is study of site effects in two-dimensional (2D) non-curve adjacent valleys in a building code. The 2D wave scattering is studied with a hybrid numerical method, combining finite elements in the near field and boundary elements in the far field (FEM/BEM). This program has been developed by Gatmiri and his coworkers (Gatmiri, B. & Kamalian, M, 2002; Gatmiri, B. & Arson, C. Nguyen, K.H, 2007 and 2008; Gatmiri, B. & Dehghan, K, 2005).

2. Summary of previous works Gatmiri et al have performed several parametric analyses of site effects. In order to better clarify the usage of HYBRID program, some of these studies are mentioned in the following. It should be noted, that sediments are modeled by finite elements and substratum is represented by boundary elements, which is adapted to the study in the far field. Gatmiri et al., (2007, 2008) studied various configurations and considered the influence of configuration of irregularities, slope angle of irregularities and dimensionless frequency of incident wave. The several salient features of topographic effects obtained are as follows: The seismic ground motion was amplified at the crest of ridges, at the upper corner of slopes and at the edges of canyons; it was systematically attenuated at the base of these reliefs. This conclusion was normally verified for the cases of low dimensionless frequency. The ground motion as not homogeneous as in case of the half-space, but it strongly varied on the free field. There were successive regions that movements of round ere attenuated. The magnitude of response at a


location on the top surface was dependent on the distance from this location on the relief. This distance was a function of the frequency content of the relief itself. The effects of topography were also influenced by the slope angle of the relief. Generally, the stiffer the slope of the relief was, the more the effects of topography due to this relief were accentuated. The topographic effects of a relief on the seismic response of that relief strongly depended on the frequency content of the excitation. In general, the higher the excited frequency was the more significant and complex were the site effects due to relief, and the wider the region influenced by the presence of the relief was, especially for the wavelengths comparable to or lower than the characteristic dimension of the relief. Gatmiri and Arson, (2008) studied several parametric analysis in order to characterize the combined effects of topographical irregularities and sedimentary filling on ground motion under seismic solicitation due to vertically incident SV Ricker wave. Indeed, the horizontal displacement in a canyon tend to be attenuated at the centre and slightly amplified at the edge but in an alluvial basin, horizontal displacements are amplified at the centre and can be locally attenuated near the edge if depth is large enough. A qualitative comparison between seismic response of the filled and empty was carried out suggesting that 2D geotechnical effects increase with depth and steep sidedness. Gatmiri et al., (2009, 2011) studied acceleration response spectra of different empty valleys. Curves were collected on a unique figure, which characterized topographical effects in a quantitative and qualitative way in the spectral domain. The results showed that maximum amplification was reached at the edge point of valleys. The spectral acceleration responses were classified according to a unique geometrical criterion except for elliptical valleys: the “S/A” ratio (where S is the area of the valley opening, and A indicates the angle between the horizontal line and slope in the above corner) (Fig. 1). The spectral response increased by increasing the parameter of S/A, in elliptical valleys for each depth ratio.

Figure 1. Definition of parameters S, A Sedimentary aspect of alluvial valleys was underlined by Gatmiri and Foroutan, (2012). New criteria were offered in order to develop simple methods to incorporate 2D combined site effects in building codes. Filling ratio effects of Non-curved alluvial valleys and the influence of the changes in impedance ratio between sediments and the bedrock were investigated. The derived conclusions are presented briefly as follows: Existence of sediments could smooth valley’s response at the edge and amplify it at the centre. When combining the depth and shape effects, two geometrical parameters S/A and sin (A) were presented; by increasing S/A, SR*sin (A) increased (S and A are similar to prior work).In order to combine filling ratio and depth ratio effects, the two geometrical parameters S1/A and H/L were considered. As increasing the S1/A, SR*H/L increased (S1, the area which was occupied by sediment, and H/L was the valley’s depth ratio) (Fig. 2). Spectral ratio had an inverse relation to impedance ratio. By sediment softening in comparison to rocky bed, the spectral ratio increased and the seismic response of a configuration became more and more complicated and the data sequencing became more and more difficult. Finally, variation S1/A n1/ß as a function of dimensionless parameter SR*sin (A)* H1/L (H1 was sediments depth) was plotted as a linear trend.

51

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Figure 2. Definition of parameters S1, H/L

3. PROBLEM PARAMETERS 3.1. Geometrical parameters In order to evaluation of influence of shapes of empty non-curved adjacent valleys on the site effects the different shapes of valleys include rectangular, triangular and trapezoidal was modelled. Valleys are characterized by their depth, H and their half width at the surface, L (Fig. 3). Simulations are carried out with depth ratios, H/L, equal to 0.2, 0.4, 0.6, 1. The value of L for all of the valleys is kept equal to100m.

Figure 3. Adjacent trapezoidal valley

3.2. Mechanical parameters of the materials In adopted models, the rocky bed are assumed to be homogeneous linear elastic materials. The main parameters of the bedrock are given in Table 1. The data used as input in this research are the digits and numbers considered for simulation in this program and their practical application calls for assessment of the extent by which they are factual and statistical as well as their sensitivity of results to these parameters, an assessment which is beyond the scope of objectives of this study. E(MPa)

ν

ρ(kg/m³)

C(m/s)

6720

0.4

2450

1000

Table 1. Mechanical characteristics of bedrock

3.3 Incident wave characteristics The main focus of this work is the study of the effect of 2D geometrical irregularities on modification of seismic response and this study relies on simplified geometrical conditions as seismic loading is considered to be the simplest one; vertically incident SV Ricker wave. Imposed displacements are therefore expressed as;

Where


Amplification A0 is constant value of 1; predominant frequency (f) is thus equal to 2 Hz; and TP = TS = 0.5s. The incident signal lasts 3s, but it can be seen from Fig. 2 that amplitude is nearly zero as soon as it reaches t = 1s. That is why the window has been defined from t=0 to t=3s (Fig. 4). It should be noted, in all the models above, vibration is applied to the base of the left valley (Fig. 5).

Figure 4. Incident Ricker signal

Figure 5. Point of the wave diffusion

4. 2D Site effects in non-curved adjacent valleys The aim of this section is compare of influence of non-curved adjacent valleys on the seismic response of valleys with different ratios of H/L. The geometrical characteristics of valleys are displayed in Figure 1. In all of the shapes of valley, L is half of the width of the valley and is equals to 100 m for all the valleys, as well as the distance between the adjacent valleys was determined 3L and the depth of valleys are H. In the present work, simulations are carried out with a depth (H) equal to 20, 40, 60, and 100 m and for different ratios (H/L) equal to 0.2, 0.4, 0.6, and 1. According to the following graph, the results of different models show a general trend that spectral ratio is increased with increasing H/L ratio, and this increase is more evident in the inner edge of the valleys. The spectral ratio at the inner edge of rectangular valleys are more critical than spectral ratio at the inner edge of trapezoidal and triangular valleys (Fig. 6).

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a) Rectangular

b) Trapezoidal

c) Triangular

Figure 6. The results of Different models

5. Conclusion Site effects in non-curve adjacent valleys are studied by means of a hybrid numerical technique. The main results of this study are: • Spectral ratio is increased with increasing H/L ratio. • In the constant H/L ratio, spectral ratio at the adjacent rectangular valleys is more than spectral ratio at the trapezoidal and triangular valleys. • According to the results, the spectral ratio at the inner edge of all valleys is more critical and the value of that is nearly uniform, between two adjacent valleys.


6. References B.Gatmiri and M. Kamalian, Two-dimensional transient wave propagation in an elastic saturated porous media by a hybrid FE/BE method, Proceedings of the fifth European conference of numerical methods in geotechnical engineering, Paris, France, 947–56 (2002). B.Gatmiri and K.Dehghan, Applying a new fast numerical method to elasto-dynamic transient kernels in HYBRID wave propagation analysis, Proceedings of the sixth conference on structural dynamics (EURODYN2005), Paris, France, p.1879–84 (2005). KV. Nguyen and B. Gatmiri, Evaluation of seismic ground motion induced by topographic irregularity. Int J Soil Dyn Earthquake Eng 27:183–8 (2007). B. Gatmiri, C. Arson and KV. Nguyen, Seismic site effects by an optimized 2D BE/FE method I. Theory, numerical optimization and application to topographical irregularities. Int J Soil Dyn Earthquake Eng 28:632–45 (2008). B. Gatmiri and C. Arson, Seismic site effects by an optimized 2D BE/FE method II. Quantification of site effects in two-dimensional sedimentary valleys. Int J Soil Dyn Earthquake Eng 28:646–61 (2008). B.Gatmiri and C.Arson and K.H.Nguyen, Seismic site effects by an optimized 2D BE/FE method I. Theory, numerical optimization and application to topographical irregularities, Int J Soil Dyn Earthquake Eng 28, 632–645 (2008). B. Gatmiri, P. Maghoul and C. Arson, Site-specific spectral response of seismic movement due to geometrical and geotechnical characteristics of sites. Int J Soil Dyn Earthquake Eng 29:51–70 (2009). B.Gatmiri and P.Maghoul, Site-specific spectral response of seismic movement due to geometrical and geotechnical characteristics of sites, Int J Soil Dyn Earthquake Eng 29, 51–70 (2009). B. Gatmiri, S. Lepense and P. Maghul, A multi-scale seismic response of two dimensional sedimentary valleys due to the combined effects of topography and geology. J Multi scale Model 03:133–49 (2011). B. Gatmiri and T. Foroutan, New criteria on the filling ratio and impedance ratio effects in seismic response evaluation of the partial filled alluvial valleys. Int J Soil Dyn Earthquake Eng 41:89–101 (2012). B.Gatmiri and D.Amini, Practical Recommendations of Spectral Response Analysis in NonCurved Alluvial Valleys Using Hybrid FE/BE Method, Journal of Multiscale Modelling Vol. 5, No. 2 (2013) 1350006 (2013). B.Gatmiri and D.Amini, Impact of geometrical and mechanical characteristics on the spectral response of sediment-filled valleys, Int J Soil Dyn Earthquake Eng 67, 233-250 (2014).

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Identification from partial full-field measurements using a fading regularization MFS algorithm Laëtitia Caillé1, Franck Delvare1, Nathalie Michaux-Leblond1 and Jean-Luc Hanus2 2 INSA

1 Normandie Univ, UNICAEN, CNRS, LMNO, 14000 Caen, France Centre Val de Loire, Université d’Orléans, PRISME, 18022 Bourges Cedex, France

Keywords : Identification ; Partial full-field measurements ; Inverse problem ; Method of fundamental solutions (MFS) ; Fading regularization method.

Abstract. We investigate the application of the fading regularization method [1, 2] combined with the method of fundamental solutions (MFS) [3] to a boundary condition identification problem. From the knowledge of partial full-field measurements in a part of the domain, we present a numerical reconstruction of missing data and boundary conditions. The performance and robustness of the method are highlighted by two numerical compression tests. Introduction The limits of the current experimental setup generate several difficulties, such as measurements carried out only on a part of the sample and possibly incomplete or noisy boundary conditions. These difficulties give rise to identification problems which can be considered as a class of inverse problems. An inverse method (fading regularization method) was previously introduced to solve Cauchy problems associated with the Laplace equation [1, 2] or the Lamé equation [4]. The fading regularization method is based on the idea of seeking, among all solutions of the equilibrium equation, those that fits the best the boundary conditions available on a part of the boundary. The resolution of the inverse problem is reduced to a sequence (fixed point algorithm) of constraint optimization problems. Each functional involves several terms. The first one is a relaxation term which represents the gap between the computed optimal element and the data. The second one acts on the whole domain and expresses the distance between the actual optimal solution and the previous one. This second term acts as a regularization term and tends to zero as the iterative procedure is continued. The solution thus computed does not depend on a regularization coefficient, verifies the equilibrium equation and is stable with respect to the noise on the data since these are recomputed to be compatible. This method is, in particular, able to deblur the given noisy data and can be implemented by employing the method of fundamental solutions (MFS) [5]. We present the extension of the combination of the fading regularization method with the MFS to identification problems from partial full-field measurements associated with heat conduction or linear elasticity problems (elliptic problems). The measurements are only available on a central zone of the specimen. With our algorithm, we reconstruct the field in the entire domain and the boundary conditions inaccessible to measurements. The method is numerically validated with synthetic data for linear elasticity problems. The data completion problem Let Ω be an open bounded domain Ω ⊂ R2 which is bounded by a piecewise smooth boundary Γ ≡ ∂Ω, such that Γ = Γd ∪ Γu , where Γd , Γu = ∅ and Γd ∩ Γu = ∅. We denote by Γu the boundary where no information is specified. We define the domain Ωd ⊂ Ω (Fig. 1) where the measurements are available. Γd Ωd L(u) = 0 Ω Γu

Figure 1 – Domain We assume that u satisfies the following elliptic equation L(u) = 0,

x ∈ Ω,

(1)


57

where L is either the Laplace or the Lamé operator. If it is possible to measure u in Ωd , then this leads to the mathematical formulation of a data completion problem consisting of Eq. (1) and the measured field (2) u = φd ∀x ∈ Ωd The data, evaluated for example by DIC [6], in Ωd are considered unreliable due to noisy images or DIC errors. This problem is difficult to solve since it is ill-posed. If data (2) are compatible, the data completion problem admits a unique solution, but this is known to be very sensitive to small perturbations of data [7]. Therefore, a regularization method is required to solve accurately this problem. The fading regularization-MFS algorithm Continuous formulation. Let us introduce the space H(Ω) of solutions of the equilibrium equation (1) : H(Ω) = v ∈ H 1 (Ω) / L(v) = 0 in Ω An equivalent formulation of problem (1) and its associated data (2) reads as

Find u ∈ H(Ω) such as :

(3)

u = φd in Ωd

Problem (3) remains also ill-posed even if it admits a unique solution. Therefore, an iterative regularizing method is introduced to solve it. The method chosen is a generalization of the inverse technique introduced by Cimetière et al. [1, 2] to solve the Cauchy problem for the Laplace equation. It can be considered as an iterative Tikhonov-type regularization method. Given c > 0 and u0 ∈ H(Ω), the iterative algorithm is given by : ⎧ Find uk+1 ∈ H(Ω) such as : ⎪ ⎪ ⎨ Jck (uk+1 ) ≤ Jck (v) ∀v ∈ H(Ω) with (4) ⎪ ⎪ ⎩ J k (v) = v − φ 2 + cv − uk 2 c

|Ωd

d Ω d

Ω

The fading regularization method is based on the idea of seeking, among all solutions of the equilibrium equation, those that fits the best the measured data in a part of the domain (Ωd ). Therefore, in this iterative process (4), Eq. (1) is taken into account exactly since at each step the search for the optimal element is performed in space H(Ω) and the measurements φd ∈ Ωd are considered as unreliable. The functional in (4) is composed of two terms which play different roles. The first one acts only in Ωd and represents the gap between the optimal element and the data. It relaxes the data which can be possibly perturbed by noise (i.e. a relaxation term). The second term acts in the entire domain Ω, is a regularization term and controls the distance between the new optimal element and the previous one, and tends to zero as the number of iterations increases. Therefore, at each step of the iterative procedure, the optimal element obtained is an exact solution of Eq. (1) and is close to the data φd . It is important to mention that the proposed fading regularization-MFS algorithm can be generalized to any linear elliptic operator, provided that we know a corresponding fundamental solution. Discrete formulation in the case of the linear elasticity. Herein, we present the discrete formulation of the algorithm in the case of linear elasticity. We consider an isotropic linear elastic material. The displacement vector u satisfies the Lamé equation : L u(x) =

E E Δ u(x) + ∇(divu(x)) = 0, 2(1 + ν) 2(1 + ν)(1 − 2ν)

∀x ∈ Ω

(5)

Let n(x) be the outward unit normal vector at a point x ∈ Γ and T (x) = σ(x).n(x) be the stress vector at a point x ∈ Γ. Solving (3) for any geometry and boundary conditions requires to discretize the space H(Ω) of solutions. In this paper this is accomplished in finite dimension by a meshless method, namely the method of

58


fundamental solutions (MFS). The main idea of the MFS consists of approximating the displacement vector solution in Ω by a linear combination of fundamental solutions of the Lamé equations (5). In the two-dimensional case, the fundamental solution uj , j = 1, 2, is given [8], at a point x ∈ Ω, as a function of a singular point Q, outside the domain, as : uj (x, Q) = Uij (x, Q) ei , with Uij (x, Q) = −

1 8πG(1 − ν)

x ∈ Ω,

Q ∈ R2 \ Ω,

(3 − 4ν) ln r(x, Q)δij −

j = 1, 2

(xi − xiQ )(xj − xjQ ) r 2 (x, Q)

(6) ,

i, j = 1, 2

where ei , i = 1, 2, is the unit vector along the xi -axis and and δij is the Kronecker delta tensor. r(x, Q) = (x1 − x1Q )2 + (x2 − x2Q )2 represents the Euclidean distance between the point x = (x1 , x2 ) and the E source point Q of coordinates (x1Q , x2Q ), whilst G = is the shear modulus and ν is introduced 2(1 + ν) ν . to distinguish the plane strain state (ν = ν) and the plane stress state ν = 1+ν With respect to N source points located outside the physical domain, Ql ∈ R2 \Ω, 1 ≤ l ≤ N , the displacement vector at a point x is approximated as u(x) ≈ u(a, b, Q; x) =

N

al u1 (x, Ql ) + bl u2 (x, Ql ),

x ∈ Ω,

(7)

l=1

where a = (a1 , ..., aN ), b = (b1 , ..., bN ) and Q is the 2N -vector containing the coordinates of the source points Ql , 1 ≤ l ≤ N . Taking into account the definitions of the components of the stress vector T (x) and the fundamental solutions (6), the traction vector is approximated on Γ as T (x) ≈ T (a, b, Q; x) =

N

al T 1 (x, Ql ) + bl T 2 (x, Ql ),

x ∈ Γ,

(8)

l=1

where with

T j (x, Q) = Tij (x, Q) ei ,

x ∈ Γ,

Q ∈ R2 \ Ω,

j = 1, 2

1 − ν ∂u1j (x, Q) ν ∂u2j (x, Q) n1 (x) T1j (x, Q) = 2G + 1 − 2ν ∂x1 1 − 2ν ∂x2 ∂u1j (x, Q) ∂u2j (x, Q) + n2 (x), j = 1, 2, +G ∂x2 ∂x1 ∂u1j (x, Q) ∂u2j (x, Q) T2j (x, Q) = G + n1 (x) ∂x2 ∂x1 ν ∂u1j (x, Q) 1 − ν ∂u2j (x, Q) n2 (x), + 2G + 1 − 2ν ∂x1 1 − 2ν ∂x2

j = 1, 2.

Further, approximations (7) and (8) can be written as u(x) ≈ A(x)X,

x ∈ Ω,

T (x) ≈ B(x)X,

x ∈ Γ,

where A(x) and B(x) are matrices containing the fundamental solutions and their derivatives, respectively, and X is the unknown 2N -vector containing a and b . According to the fading regularization algorithm described previously, at each step k ≥ 0 of the sequence of minimization problems (4), one has to approximate both the unknown data u(k) Ω\Ω d


59

and the unknown boundary conditions u(k) Γ and φ(k) Γ , at the same time accounting for the noisy measurements u ε Ω . d To do this, we collocate the data at a set of Md points located inside the domain (xl ∈ Ωd , 1 ≤ l ≤ Md ). We also express the MFS approximations (7) and (8) for the unknown data at a set of Mu points, namely MΩ\Ωd points inside the domain where the displacement vector is unknown and MΓ points on the boundary where the displacement vectors and the stress vectors have to be determined, such that MΓ = MΓd + MΓu . Then, we express the MFS approximations for the displacement field in Ω and its associated stress vector on Γ at these M collocation points. Consequently, at each step k ≥ 0, the minimization problem (4) is reduced to a sequence of linear minimization problems with respect to the corresponding unknown MFS constants X, namely J k (X) = A(x)X − u ε Ω 2Ωd + cA(x)X − A(x)X k−1 2Ω + cA(x)X − A(x)X k−1 2Γ d

+ cB(x)X − B(x)X k−1 2Γ (9)

k Further, we denote by JΩk d and Jreg. the first term and the sum of the following terms, respectively, of the functionals which define the sequence (9).

Numerical results Herein we present the numerical results obtained using the fading regularization method and the MFS. We consider two examples for which an analytical solution uan is known and two loading conditions are investigated. In all examples considered, we have taken Md , MΩ\Ωd and MΓ uniformly distributed collocation points in Ωd , in Ω \ Ωd and on Γ, respectively, as well as N uniformly distributed sources. It should be mentioned that the sources are preassigned and kept fixed throughout the solution process on a pseudo boundary Γ. Example 1. As a first example, an unconfined compression test is selected (Fig. 2). For a force an imposed experiment in the linear regime, the analytical solution for the displacements uan = (uan 1 , u2 ) is the following : ν σ0 uan x1 , (x1 , x2 ) ∈ Ω, 1 (x1 , x2 ) = E (10) σ0 an u2 (x1 , x2 ) = − x2 , (x1 , x2 ) ∈ Ω, E where E = 1 Pa, ν = 0.2 and σ0 = 0.1 N.m−2 . The domain Ω, the data grid Ωd ∈ Ω and Γ (the boundary defined the by Γ = Γd ∪ Γu) are defined on = x = (x1 , x2 ) ∈ R2 x2 + x2 = d2 and Fig. 2. The MFS parameters have been set as N = 90 on Γ 1 2 d = 10, whilst MΓd = 100 and MΓu = 50. The data grid Ωd is four times smaller than Ω. The domain Ω is discretize with (30 × 60) points. Γu (1, 2)

(0, 2)

Γd

Γd Ωd

x2 (0, 0)

(1, 0) Γu x1

Figure 2 – Compressed pavement

60


It is necessary to investigate the influence of parameter c, which defines the relative weight of the reguk larization term Jreg. compared to the relaxation term JΩk d , Figs. 3(a) and (b) show the reconstructions of the displacement component u2 and the stress vector component T2 on Γ, for various values of c, respectively. The reconstructions of the displacement component u1 and the T1 component are not presented because the results are similar to those obtained for the u2 and T2 components. 0.15

0.15

Analytical c = 1.10−4 c = 1.10−2 c = 1.10−1 c=1

0.1

0.05

u2

0.05

T2

0

0

-0.05

-0.05

-0.1

-0.1

-0.15

Analytical Data c = 1.10−4 c = 1.10−2 c = 1.10−1 c=1

0.1

0

1

2

3

4

5

-0.15

6

0

1

2

3

Γ

4

5

6

Γ

(a)

(b)

an Figure 3 – (a) The analytical, uan 2 , and the numerical displacements, u2 , and (b) the analytical, T2 , and the numerical stress vectors, T2 , retrieved on the boundary Γ for various values of c.

Next, we investigate how the reconstructions are influenced by noisy data. The reconstructions of the components u2 and T2 on Γ, obtained with the noise level of 1%, 3%, 5% and 10%, are represented in Figs. 4(a) and (b), respectively. 0.15

Analytical δ = 1% δ = 3% δ = 5% δ = 10%

0.1

0.05

u2

Analytical δ = 1% δ = 3% δ = 5% δ = 10%

0.1

0.05

T2

0

0

-0.05

-0.05

-0.1 -0.1 0

1

2

3

4

5

6

-0.15

0

1

2

3

Γ (a)

4

5

6

Γ (b)

an Figure 4 – (a) The analytical, uan 2 , and the numerical displacements, u2 , and (b) the analytical, T2 , and the numerical, T2 , retrieved on Γ, for various levels of noise and c = 1.

The proposed algorithm allows to reconstruct the displacements in the data grid Ωd and to retrieve the displacements in the whole domain Ω. Figs. 5(a), (b), (c) and (d) show the data u1 d , the reconstructions of the displacement component u1 in Ωd , the residual |ud1 − u1 | and the displacement component uΩ 1 in Ω, respectively, for c = 1.10−1 and the noise level δ = 10%.


61

u1 d

u1

2

2

0.004 0.004 1.5

1.5 0.002

0.002

0

1

1

0

-0.002 -0.002 0.5

0.5 -0.004 -0.004

0 -0.5

0

0.5

1

0 -0.5

1.5

0

(a)

0.5

1

1.5

(b)

|ud1 − u1 |

uΩ 1

2

2

0.01

0.001

1.5

1.5 0.0008

0.005

0.0006 1

1

0

0.0004

-0.005 0.5

0.5

0.0002

-0.01 0 -0.5

0

0.5

1

0 -0.5

1.5

0

(c)

0.5

1

1.5

(d)

Figure 5 – (a) The noisy data, u1 d , (b) the reconstruction of the displacement, u1 in Ωd , (c) the −1 and 10% residual |ud1 − u1 | and (d) the reconstruction of the displacement, uΩ 1 in Ω, for c = 1.10 noise level. The reconstructions obtained are very accurate. This shows that even for noisy data, the numerical reconstructions are also independent of c and, therefore, the combined fading regularization methodMFS algorithm is, at the same time, robust. Example 2. A similar study is further performed for the same analytical displacement solution as in the first example but with a different loading condition. The solution domain represented in Fig. 6 is a subdomain of the initial one (Fig. 2). This particular configuration allows to apply the proposed algorithm to a geometry that causes a non-constant stress vector component T2 on Γu . Γu

(0, 2)

Γd

(1, 2)

Γd Ωd

x2

(0, 0)

(1, 0) x1

Γu

Figure 6 – Compressed pavement with rounded shape

62


Figs. 7(a) and (b) show the reconstructions of the displacement component u2 and the stress vector component T2 on Γ, respectively, with the noise level of 1%, 3%, 5% and 10% and c = 1.10−1 . 0.1

0.15

Analytical δ = 1% δ = 3% δ = 5% δ = 10%

0.08 0.06 0.04

Analytical δ = 1% δ = 3% δ = 5% δ = 10%

0.1

0.05

0.02

u2

T2

0

0

-0.02 -0.05

-0.04 -0.06

-0.1

-0.08 -0.1

0

1

2

3

4

5

-0.15

0

1

2

Γ

3

4

5

Γ

(a)

(b)

an Figure 7 – (a) The analytical, uan 2 , and the numerical displacement, u2 , and (b) the analytical, T2 , −1 and the numerical, T2 , retrieved on Γ, for various levels of noise and c = 1.10 .

It may be noted the accuracy of the reconstructions on Γ and the fading regularization-MFS algorithm ability to deblur the Ωd noisy data and to retrieve the displacement field in Ω (see also Figs. 8(a) and (b)). We note the accurate reconstructions when we search non-constant solutions on Γu . uΩ 1

uΩ 2

2

2

0.01

1.5

1.5

0.05

0.005

1

0

0

1

-0.005 0.5

0 -0.5

-0.05

0.5

0

0.5

1

(a)

1.5

0 -0.5

0

0.5

1

1.5

(b)

Figure 8 – (a) The reconstruction of the displacement component, uΩ 1 , and (b) the reconstruction of −1 the displacement component, uΩ 2 , in Ω for δ = 10% and c = 1.10 . Conclusions The application of the fading regularization MFS-algorithm to partial full-field measurements was investigated. We have studied the reconstruction of missing data and boundary conditions, from only partial full-field measurements in a part of the domain. In the case of the two-dimensional linear elasticity, the problem is to reconstruct the missing data, such that both displacements and stress vectors on the boundary where measurements are inaccessible, from noisy displacement field measured only in a part of the domain. In addition to computing the missing data and the unknown boundary conditions, the procedure was also used to recompute the given noisy measurements, in the sense


that the latter was denoising. Two numerical compression examples have highlighted the accuracy, convergence, stability, efficiency of the proposed inverse method, as well as its capability to deblur noisy measurements. The solution was shown to be independent of the regularization parameter c. Acknowledgements. L. Caillé and J.L. Hanus would like to acknowledge the financial support received from Région Centre (IMFRA2 project). The financial support received by L. Caillé from Région Normandie is also gratefully acknowledged.

References [1] A. Cimetière, F. Delvare, and F. Pons, Comptes Rendus de l’Académie des Sciences-Series IIBMechanics, 328, 639–644 (2000). [2] A. Cimetière, F. Delvare, M. Jaoua, and F. Pons, Inverse Problems, 17, 553–570 (2001). [3] R. Mathon and R.L. Johnston, SIAM Journal on Numerical Analysis, 14, 638–650 (1977). [4] F. Delvare, A. Cimetière, J.-L. Hanus, and P. Bailly, Computer Methods in Applied Mechanics and Engineering, 199, 3336–3344 (2010). [5] L. Marin, F. Delvare, and A. Cimetière, International Journal of Solids and Structures, 78–79, 9–20 (2016). [6] M. Grediac and F. Hild Full-Field Measurements and Identification in Solid Mechanics, Wiley (2012). [7] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press (1923). [8] J.R. Berger and A. Karageorghis Engineering Analysis with Boundary Elements, 25, 877–886 (2001).

63

64


1∗ 1 1

! " ! ! ! # ! ! $

%& ! ! ' ( # ! %& $ %& ) * ! # !

* %& ) ! ! + ! +

%& $

+ + * ! * ,- ./ 0# * ' 0# " % %&

+

* 0 ,1/ + !

+ %& !

! ! 2 %& ' (

( # %& ! # %& ) ! + %& ! ! ! +

%&

+ ( b * σij,j + bi = ρ¨ ui

-


65

ρ uï σij = σji ;

εij =

εij = εji

1 (ui,j + uj,i ) 2

ui i !"# σij = Cijkl εkl

$

Cijkl % Cijkl = Cjikl = Cijlk = Cklij

&

F(x) x % '$( ∂y F(x) = ) ∂x x % y = x + u % % J = (F(x)) J > 0 % * +, P(x) = JσF(x)T

-

*. " % '(

ρ¨ u(x, t) =

H

f (u(x , t) − u(x, t), x − x)dVx + b(x, t)

/

f x 0 x H x % " *. 1 2 3 x 3 δ , 4 % % , *.

*. '( * %0 " 3 *. " '&( *. 5 %

H

T

{T[x, t]x − x − T[x , t]x − x }dVx + b(x, t) = ρ¨ u(x, t)

6

66


x

δ x H Ω

y t t

u δ

u

x x

δ

y z

y

x

! u u x x ! "# $ #

# # % # #!

! &#

&"! # " ' % ! ( "# ! $

# # & (" ) # (" ! * +, -./ % # #! # !

#! & ) #

# " 0 # % &

# $

! & 0 -1/ ! " Y[x, t]ξ = y(x + ξ, t) − y(x, t)

2

$ F(x) " # B(x) = H F(x) = H

ω(|ξ|)(ξ ⊗ ξ)dVξ

−1

ω(|ξ|)(Y(ξ) ⊗ ξ)dVξ .B(x)


67

B(x) ⊗

ω(|ξ|) x ω(|ξ|) = 1 ! "#$ ∂W P(x)T = %&'( ∂F W ) t "#$ T[x, t]x − x = ω(|x − x|)P(x)T .B(x).(x − x) %&*(

)

) ) ) ) "+$ , -

E(x) 1 F(x)T F(x) − I E(x) = %&.( 2 /0 !

Eeq (x, x ) =

Eeq (x, x ) ≥ Ecrit

4 I = 3 2

2 (x, x ) E (x, x )Eij 3 ij

1 Eij (x, x ) = Eij (x, x ) − Ekk (x, x ) 3 1 E(x) + E(x ) Eij (x, x ) = 2 )

) x

x ) 1 Eeq (x, x ) < Ecrit μ(x, x ) = 0

%&1( %&#( %&+(

, μ(x, x ) %&2(

0 ϕ ) 0

!

μ(x, x )dVx ! H dVx

H

ϕ =1−

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) )

) 0

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)

! %67( 67

67 5678 cij (ξ)uj (ξ) + cij (ξ)pj (ξ) + Nk

Γ

Γ

p∗ij (x, ξ)uj (x)dΓ(x) +

s∗kij (x, ξ)uj (x)dΓ(x) + Nk

Γc

Γc

˜ l ψlj (ξ)dΓ = p∗ij (x, ξ)K

Γ

˜ l ψlj (ξ)dΓ = Nk s∗kij (x, ξ)K

u∗ij (x, ξ)pj (x)dΓ(x)

%3&(

Γ

d∗kij (x, ξ)pj (x)dΓ(x)

%33(

68


Γ Γc = Γ+ ∪ Γ− Γ+ Γ− Nk K˜ l KI KII

ψlj

u∗ij p∗ij d∗kij

s∗kij

u∗ij p∗ij p∗ij d∗rij s∗rij !"#$ "#$ %&'

("$) *#+ ,- . ("$) /! uXBEM = uP D

0 1 2 $3 &2 cij (ξ) = 1 & KI KII -

- %2 &' -

4 /!

5 3 w/h = 0.04 2a = 0.02 /! . 3 . 0.02a + 6 /! . E = 69 7/ ν = 0.33 ("$) 8

σ = 1 )/ σ

0.1a

0.1a 2a 2h

2w

σ

+ 61 9 ("$) /! 5 ("$) 0 /!

1 Einc = 10E ν = 0.33 /!

+ : - ϕ /! + ; 0. In the above formulation of boundary conditions (2), it can be seen that the boundary Γd is overspecified by prescribing both

76


the solution u and its normal derivative Φ, whilst the boundary Γu is underspecified since both the solution u and its normal derivative Φ are unknown and have to be determined. This problem, referred to as the Cauchy problem, is much more difficult to solve than the direct problem since its solution does not satisfy the general conditions of well-posedness, see [5]. Whilst the Dirichlet, Neumann or mixed direct problems associated with Eq. (1) do not always have a unique solution due to the eigensolutions of the Laplacian [6], the solution of the Cauchy problem (1) and (2) is unique if it exists, based on the analytical continuation property [7]. However, it is well-known that if this solution exists, then it is unstable with respect to small perturbations in the data on Γd [5]. Thus the problem under investigation is ill-posed. Therefore, regularization methods are required in order to solve accurately the Cauchy problem associated with the two-dimensional Helmholtz-type equations. Fading regularization method Next, we introduce the space H(Ω) of solutions of the equation (1) H(Ω) = v ∈ H 1 (Ω)v satisfies (1) in a weak sense , where H 1 (Ω) is the Sobolev space. We denote by H(Γ) the space composed of couples of restrictions on Γ of elements v ∈ H(Ω) and their associated normal derivatives Φ. An equivalent formulation of problem (1) and (2) reads as Find U = (u, Φ) ∈ H(Γ) such that U = ψd on Γd ,

(3)

where ψd =(ud , Φd ). Problem (3) is also ill-posed even if it admits a unique solution and hence an iterative regularizing method has to be introduced to solve it in a stable manner. This method is a generalization of the inverse technique introduced by Cimetière et al. [1, 2] to solve the Cauchy problem for the Laplace equation and can be considered as an iterative Tikhonov-type method. Given c > 0 and U 0 ∈ H(Γ), the iterative algorithm is given by Find U k+1 ∈ H(Γ) such that k k+1 Jc (U ) ≤ Jck (V ), ∀V ∈ H(Γ) with (4) " " " "2 k Jc (V ) = "V |Γ − ψd "2 + c "V − U k " . d

Γ

Γd

The functional in (4) is composed of two terms which play different roles. The first one acts only on Γd and represents the gap between the optimal element and the overspecified boundary data. It relaxes the overspecified data which can be possibly perturbed by noisy measurements (i.e. a relaxation term). The second term in (4) acts on the entire boundary Γ and not only on Γu , where the boundary conditions are to be computed. This term is a regularization term, controls the distance between the new optimal element and the previous one and tends to zero as the number of iterations increases. Therefore, at each step of the iterative procedure, the optimal element obtained is an exact solution of Eq. (1) and is close to the overspecified data ψd =(ud , Φd ). This unique optimal element U k+1 is characterized by $ $ # # U k+1 |Γd − ψd , V + c U k+1 − U k , V = 0, ∀ V ∈ H(Γ). (5) Γd

Γ

It can be proved that the sequence produced by the iterative scheme (4) strongly converges to ψd on Γd and weakly converges to ψe on Γ. We note that this proof is based on the same arguments that are used in [1] to prove the convergence of the sequence when solving the Cauchy problem associated to the Laplace equation. It is important to mention that the proposed fading regularization algorithm allows for both the reconstruction of the missing boundary data on the underspecified boundary Γu and the denoising of the perturbed overspecified boundary data on Γd .


77

Method of fundamental solutions (MFS) The main idea of the MFS consists of approximating the solution in the domain Ω and on its boundary ∂Ω by a linear combination of fundamental solutions with respect to N source points y j ∈ R2 \ Ω, j = 1, N , in the form N u(x) ≈ uN (a, Y ) = aj F (x, y j ), x ∈ Ω, (6) j=1

where F = FH and F = FMH for the Helmholtz and the modified Helmholtz equations, respectively. FH and FMH in the two-dimensional case are given by [8] : FH (x, y) =

i (1) H (kr(x, y)), 4 0

FMH (x, y) = Here i =

√

x ∈ Ω,

1 K0 (kr(x, y)), 2π

x ∈ Ω,

y ∈ R2 \Ω. y ∈ R2 \Ω.

(7) (8)

−1, r(x, y) is the distance between the point x = (x1 , x2 ) ∈ Ω and the source point (1)

y = (y1 , y2 ) ∈ R2 \ Ω, H0 is the zeroth-order Hankel function of the first kind, K0 is the zeroth-order modified Bessel function of the second kind, a = (a1 , ..., aN ) and Y is the 2N -vector containing the coordinates of the source points y j , j = 1, N . Then, the normal derivative on Γ can be approximated by Φ(x) ≈ ΦN (a, Y , n; x) =

N

aj G (x, y j ; n),

x ∈ Γ,

(9)

j=1

where G (x, y j ; n) = ∇x F (x, y).n(x), G = GH and G = GMH for the Helmholtz and the modified Helmholtz equations, respectively, and these are given by GH (x, y; n) = −

((x − y)n(x))ki (1) H1 (kr(x, y)), 4r(x, y)

x ∈ Ω,

y ∈ R2 \Ω.

(10)

((x − y)n(x))k K1 (kr(x, y)), 2πr(x, y)

x ∈ Ω,

y ∈ R2 \Ω.

(11)

GMH (x, y; n) = − (1)

Here H1 is the first-order Hankel function of the first kind and K1 is the first-order modified Bessel function of the second kind. Further, approximations (6) and (9) can be written as u(x) ≈ A(x)a,

x ∈ Ω, and Φ(x) ≈ B(x)a,

x ∈ Γ,

where A(x) and B(x) are matrices containing the fundamental solutions and their associated normal derivatives, respectively. Discrete formulation According to the fading regularization algorithm previously described, at each step k ≥ 0 of the minimization problem (4) or, equivalently, Eq. (5), one has to approximate both the known boundary data u(k) Γ and Φ(k) Γ and the unknown boundary data u(k) Γu and Φ(k) Γu , at the same time d d ε . accounting for the given perturbed boundary conditions u ε Γ and Φ Γd d To do this, Md and Mu collocation points are chosen on the overspecified boundary Γd and the underspecified boundary Γu , respectively, and we express the MFS approximations for the solution and its normal derivative at these M = Md + Mu collocation points. Consequently, at each step k ≥ 0, the minimization problem (4) or, equivalently, Eq. (5) is reduced to a linear minimization problem with respect to the corresponding unknown MFS constants a : ε 2Γ + cS(x)a − S(x)ak−1 2Γ ∪Γ J k (a) = S(x)a − U (12) u d d Γ d

78


where the matrix S is defined by Sij (x) = Aij (x) and S(M +i)j (x) = Bij (x) with i = 1, ..., M and ε = ( ε ). ak−1 represents ε is the given perturbed boundary conditions U uε Γ , Φ j = 1, ..., N . U Γd Γd Γd d the MFS constants associated with the numerical solution of the discretised minimization problem (5) at the previous step (k − 1). Further, we denote by JΓkd and JΓk the first and the second terms, respectively, of the functionals defining sequence (12). Numerical results Herein we present the numerical results obtained using the fading regularization method and the MFS. Two examples, for which an analytical solution uan is known, are considered and, furthermore, we take Md and Mu uniformly distributed collocation points on Γd and Γu , respectively, as well as N uniformly distributed sources. It should be mentioned that the sources are and kept fixed preassigned such that dist Γ, Γ is a fixed constant (i.e. throughout the solution process on a pseudo-boundary Γ the so-called static [9]). The MFS parameters have been set as been employed MFS approach has 2 2 2 2 N = 100 on Γ = x = (x1 , x2 ) ∈ R x1 + x2 = d and d = 10, whilst Md = Mu = 150. Note that 1 meas(Γd ) = meas(Γu ) = meas(Γ). 2 Example 1. We consider the Cauchy problem associated with the modified Helmholtz equation in the unit disk and the following analytical solution √ (13) uan (x) = exp(x1 + 3x2 ), x = (x1 , x2 ) ∈ Ω, in the unit disk Ω = {x = (x1 , x2 )|x21 + x22 < 1}, where k 2 = −4. Here Γd = {x ∈ Γ|0 ≤ θ(x) ≤ π} and Γu = {x ∈ Γ|π ≤ θ(x) ≤ 2π}, where θ(x) is the angular polar coordinate of x. 8

6 5

u

16


7


14 12 10

4

Φ

3

8 6 4

2

2

1

0

0

-2

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Γ

1

Γ (b)

(a)

Figure 1 – (a) The analytical, uan , and numerical solutions, u, and (b) the analytical, φan , and numerical normal derivatives, φ, retrieved on the boundary Γ, for various values of c. It is necessary to investigate the influence of parameter c, which defines the relative weight of the regularization term JΓk compared to the relaxation term JΓkd , on the accuracy of the numerical solution. Figs. 1(a) and (b) show the reconstructions of the solution and its normal derivative on Γ, for various values of c, respectively. We observe in the previous figures that the algorithm converges to the same solution, regardless the value of c. Next, we investigate influence of noisy data in the boundary data. We assume numerically reconstructed = Φan have been perturbed as that the given exact boundary data u Γ = uan Γ and Φ Γ Γ d

d

d

d

u ε (x) = uan (x) + δuan (x).ρ, x ∈ Γd , ε (x) = Φan (x) + δΦan (x).ρ, x ∈ Γd , Φ

(14)


79

where δ is the level of percentage noise added and ρ is a pseudorandom number drawn from the standard uniform distribution in [−1, 1]. The reconstructions of u and Φ on Γ, obtained with the noise level of 1%, 3%, 5% and 10%, are represented in Figs. 2(a) and (b), respectively. It can be seen from these figures that these reconstructions are very accurate. 8

6 5

u

16

Analytical Noisy data 10% δ = 1% δ = 3% δ = 5% δ = 10%

7


14 12 10

4

Φ

3

8 6 4

2

2

1

0

0

-2

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Γ

1

Γ

(a)

(b)

Figure 2 – The numerical (a) solution, u, and (b) normal derivative, Φ, retrieved on the boundary Γ, for various levels of noise and c = 1.10−1 . Example 2. A similar study is further performed for the Cauchy problem associated with the Helmholtz equation in the unit square. We consider the following analytical solution in Ω = (0, 1)2 with k2 = 4 √ (15) uan (x) = cos(x1 + 3x2 ), x = (x1 , x2 ) ∈ Ω. Here Γd = {x ∈ Γ|0 ≤ x1 ≤ 1, x2 = 0} ∪ {x ∈ Γ|x1 = 1, 0 ≤ x2 ≤ 1} and Γu = {x ∈ Γ|0 ≤ x1 ≤ 1, x2 = 1} ∪ {x ∈ Γ|x1 = 0, 0 ≤ x2 ≤ 1}. 1.5

1.5 Analytical Noisy data 10% δ = 1% δ = 3% δ = 5% δ = 10%

1 0.5


1 0.5

u

Φ 0

0 -0.5 -1

-0.5 -1

-1.5 0

0.5

1

1.5

2

2.5

3

3.5

4

-2

0

0.5

1

1.5

(a)

2

2.5

3

3.5

4

Γ

Γ (b)

Figure 3 – The numerical (a) solution, u, and (b) normal derivative, Φ, retrieved on the boundary Γ, for various levels of noise and c = 1. Figs. 3(a) and (b) show the analytical solution and the numerical reconstructions for u and the normal derivatives Φ, respectively, for δ ∈ {1%, 3%, 5%, 10%} and c = 1. It can be remarked from these figures the very good accuracy of the reconstructions on Γu and the capability of the fading regularizationMFS algorithm to deblur the noisy data on Γd . We also notice the accurate reconstructions when the boundary Γu is piecewise smooth.

80


Conclusions In this paper, we have studied the numerical reconstructions of the missing solution and the associated normal derivative on an inaccessible part of the boundary, for two-dimensional Helmholtz-type equations, from overprescribed noisy data taken on the remaining accessible boundary part. Two examples were considered and thoroughly investigated by combining the fading regularization method and the MFS. The numerical simulations performed have highlighted the efficiency, accuracy, convergence, stability and robustness of the proposed algorithm for noisy data, as well as its ability to deblur noisy data. For all situations analysed, it was observed that the errors in the numerical normal derivatives, obtained using the fading regularization-MFS algorithm, were higher than those corresponding to the reconstructed solutions. In addition to computing the solution on the underspecified boundary, the algorithm was also used to recompute the given noisy data, in the sense that the latter was denoised. Moreover, the solution was shown to be independent of the regularization parameter c. Acknowledgements. L. Caillé, F. Delvare and L. Marin would like to acknowledge the financial support received from L.E.A. Math Mode. The financial support received by L. Caillé from Région Normandie is also gratefully acknowledged.

References [1] A. Cimetière, F. Delvare, and F. Pons, Comptes Rendus de l’Académie des Sciences-Series IIBMechanics, 328, 639–644 (2000). [2] A. Cimetière, F. Delvare, M. Jaoua, and F. Pons, Inverse Problems, 17, 553–570 (2001). [3] R. Mathon and R.L. Johnston, SIAM Journal on Numerical Analysis, 14, 638–650 (1977). [4] F. Delvare, A. Cimetière, J.-L. Hanus, and P. Bailly, Computer Methods in Applied Mechanics and Engineering, 199, 3336–3344 (2010). [5] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press (1923). [6] G. Chen and J. Zhou, Boundary Element Methods, Academic Press (1992). [7] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell’s Equations, Applied Mathematical Sciences (2015). [8] G. Fairweather and A. Karageorghis, Advances in Computational Mathematics, 9, 69–95 (1998). [9] P. Gorzelańczyk and J.A. Kołodziej, Engineering Analysis with Boundary Elements, 32, 64–75 (2008).


81

A boundary element formulation for inter- and trans-granular cracking in polycrystalline materials Vincenzo Gulizzi1,2,a, Chris H. Rycroft2,b,Ivano Benedetti 1,c 1

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei Materiali Università degli Studi di Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy 2

Paulson School of Engineering and Applied Sciences, Harvard University 29 Oxford Street, Cambridge, MA 02138, USA

a

[email protected]

, [email protected]

Keywords: polycrystalline materials, inter-granular cracking, trans-granular cracking, boundary element method

Abstract. A boundary element formulation for inter- and trans-granular cracking of three-dimensional polycrystalline materials is presented. Inter-granular cracking occurs at the grain boundaries whereas transgranular cracking takes place along specific cleavage planes, whose orientation depends on the grains crystallographic orientation. The evolution of inter- and trans-granular cracks is then governed by suitably defined cohesive laws, whose parameters characterize the behaviour of the two fracture modes. Results show how the model captures the inter/trans-granular cracking competition. Introduction At the grain scale, the micro-mechanical behavior of polycrystalline materials, which include metals and ceramics, is characterized by different deformation and damage mechanisms. Two of the most common micro-cracking failure processes are: a) inter-granular failure, which represents the micro-cracking formation at the interface between two neighboring grains and b) trans-granular failure, which represents the failure of the bulk crystals. The main difference between the two mechanisms is represented by the fact that intergranular damage initiates and evolves over the grain-boundaries, which are well-defined when the polycrystalline morphology is generated; on the other hand, at any point of the considered bulk grain, transgranular failure may potentially occur along different cleavage planes, whose actual activation is not a priori known and depends on the comparison between the local stress acting on the cleavage plane and the corresponding strength. In this study, a novel boundary element model for the analysis of inter- and trans-granular micro-cracking for three-dimensional (3D) polycrystalline materials is presented. The model is based on a multi-domain grainboundary formulation developed by Sfantos and Aliabadi [2,3] and Benedetti and Aliabadi [4-8] for 2D and 3D polycrystalline morphologies respectively, and it has the advantage of expressing the polycrystalline problem in terms of boundary displacements and tractions, thus reducing the computational cost with respect to other numerical techniques. The first BEM model for inter- and trans-granular failures in 2D polycrystals has been recently presented by Geraci and Aliabadi [9]. Here, we discuss a formulation for 3D polycrystals, which present specific features [10]. In the present model, trans-granular cleavage planes are dynamically introduced into the morphology during the load history upon the fulfilment of a suitably defined stress threshold condition and the 3D morphology is re-meshed accordingly. The evolution of inter- and trans-granular cracking is then governed by means of suitably defined cohesive traction-separation laws, whose parameters are chosen so to represent the fracture energies of the two mechanisms. The iterative incremental algorithm based on the grain-boundary model and involving the on-the-fly re-meshing is presented and discussed. The activation condition for the introduction of trans-granular cleavage planes and the effect of the cohesive law parameters on the macroscopic response of polycrystalline aggregates are investigated. A few numerical tests are presented to show the capabilities of the proposed model. 3D boundary element formulation for inter/trans-granular polycrystalline failures

82


Polycrystalline morphologies. In the present study, the polycrystalline micro-morphologies are obtained as 3D Voronoi tessellations, in which each grain is represented as a convex polyhedron bounded by flat convex polygonal faces. From the crystallographic point of view, each grain is an anisotropic elastic domain with the principal directions randomly oriented in the 3D space. Their constitutive behavior is expressed as V ij Cijkl H kl where V ij , H ij and Cijkl are the components of the stress, strain and elasticity tensors. Boundary integral equations. The presented formulation is based on the use of a multi-region boundary integral approach for 3D anisotropic solids [4,11,12]. To model polycrystalline inter/trans-granular cracking, the displacement boundary integral equations (DBIEs) and the stress boundary integral equations (SBIEs) are used during the analysis [11,12]. The DBIEs read

cij ( y )u j ( y ) ³ Tij ( x, y )u j ( x) dS ( x)

³U

S

ij

( x, y )t j ( x) dS ( x),

(1)

S

where ui and ti are the grain boundary displacements and tractions; Tij and U ij are the fundamental solutions for 3D anisotropic elasticity [13], x is the boundary integration point, y is the collocation point and cij are the free-terms stemming from the boundary limiting process. The SBIEs, at an internal point y, are written as

V ij ( y ) ³ Sijk ( x, y )uk ( x) dS ( x) S

³D

ijk

( x, y )t k ( x) dS ( x )

(2)

S

where Sijk and Dijk are derived from the fundamental solutions in Eq.(1) [12]. The polycrystalline problem is closed by means of a properly defined set of boundary and interface conditions. The boundary conditions (BCs) are enforced over the external faces of the aggregate by prescribing values of displacements and/or tractions. The interface conditions (ICs) are enforced on the boundaries shared between contiguous grains and, as long as the generic interface is intact, are defined by displacements continuity and tractions equilibrium. As damage starts, inter/trans-granular cohesive laws are introduced in the model. Inter/trans-granular cohesive zone modelling. Inter- and trans-granular cracks are modelled with cohesive laws [2,5]. When a suitably defined effective traction overcomes the interface or cleavage strength, damage arises over either inter- or trans-granular interfaces. Cohesive laws of the form ti K ij ( d * )G u j are introduced when the threshold criterion is fulfilled, to link the interface traction components ti with the interface displacements jump G ui , expressed in a grain boundary local reference system. The cohesive laws are given in terms of K ij ( d * ) , which are function of the irreversible damage parameter d*, defined in terms of an effective opening displacement [2,5]. Although similar cohesive approaches are used to model inter- and trans-granular processes, it is worth highlighting the different nature of such processes and their numerical treatment. Inter-granular failures occur at the interfaces between contiguous grains; such interfaces are natural material discontinuities within the original polycrystalline morphology. On the other hand, at a generic point within the bulk crystallographic lattice, trans-granular failures may potentially occur along different planes and and may or may not activate depending on the fulfillment of a threshold cleavage condition. Thus, trans-granular cracks are introduced dynamically during the analysis as soon as the cleavage stress overcomes the strength of the potential cleavage planes. Moreover, the evolution of inter- and trans-granular cracking is governed by different cohesive parameters, selected to represent the different properties of the two types of interfaces. The interplay between the two mechanisms is governed by the parameter J G GIg / GIgh , expressing the ratio between the mode I cleavage fracture toughness and the mode I inter-granular fracture toughness. In turn, for each failure mechanism, i.e. inter- or trans-granular failure, the work of separation GI, the interface strength Tmax, the ratio GI/GII, where GII is the work of separation in pure sliding mode and two parameters weighing the normal and sliding modes are subsequently used to completely define each cohesive law [2,5].


83

Discretization, integration and numerical solution. The 3D polycrystalline problem with inter- and transgranular cracking is solved by discretising Eqs.(1-2) according with the strategy developed in Ref.[6]. The equations for the whole aggregate are retrieved by discretising Eq.(1) for each grain and by enforcing the suitable boundary and interface equations [2,5]. A non-linear system of equations of the form ( , ) O is the load factor [5,6]. During the generic load step, Eq.(2) is used to assess the stress based threshold at the selected control points. If the trans-granular trigger condition is fulfilled, a trans-granular cleavage plane is introduced and a re-meshing of the aggregate is performed. Then, the solution with the new morphology is computed for the same load factor. Numerical test The micro-cracking behavior of 10 SiC grains with ASTM grain size G = 12 is shown. For hexagonal 6H SiC, the favorite cleavage plane is the basal plane identified by the (0001) Miller indices. The non-zero elastic constants in the grain’s local reference system are C1111 = C2222 = 502 GPa, C3333 = 565 GPa, C1122 = 95 GPa, C1133 = C2233 = 96 GPa, C2323 = C1313 = 169 GPa, C1212 = (C1111-C1122)/2. The non-zero inter-granular cohesive coefficients are K11

K 22

D Tmax 1 d * G u scr d *

and K 33

Tmax 1 d * , G u ncr d *

(3)

whose parameters are Tmax 500MPa , G uncr , gh 7.80 102 , G uscr , gh 1.56 101 , D gh 1 . To enforce the ratio g gh , G u ncr , g J G GIg / GIgh , the parameters of the trans-granular cohesive law are Tmax J G Tmax J G G u ncr , gh , G u scr , g

J G G u scr , gh , D gh

Dg .

The aggregate shown in Fig.( 1a) is subject to prescribed macro-strain * 33 along the x3 direction. Two values of J G are considered, namely J G 1 and J G 1/ 4 . The curves showing the macro-stress 6 33 as a function of the load factor for the two values of J G are given in Fig.( 1b). Fig.(XX) show the crack pattern at the last computed step for J G 1 and J G 1/ 4 , respectively. In Fig.(2a), although one grain gets cut during the analysis, the fracture is predominantly inter-granular, whereas, in Fig.(2b), it is possible to see that reducing the trans-granular fracture energy favors the trans-granular mechanism itself.

(a)

(b)

Fig. 1: a) Mesh of a 10-grain polycrystalline morphology with ASTM grain size G = 12; b) Volume stress average 6 33 vs. load factor O for two values of the ratio J G .

84


(a)

(b)

Fig. 2: Crack patterns at the last computed step of the considered morphology for the two values of the ratio J G between the inter- and trans-granular fracture toughness: a) J G 1 b) J G 1 / 4 . Conclusions A novel numerical scheme for addressing the competition between inter- and trans-granular crack propagation in three-dimensional polycrystalline micro-morphologies has been developed and tested. Transgranular failures are modelled by suitably oriented trans-granular cracks introduced upon the fulfillment of a threshold cleavage condition. Cohesive traction-separation laws with different parameters are used to model both inter- and trans-granular processes. Results show that the method captures the behavior of polycrystalline aggregates when the two mechanisms are active. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

N Sukumar, DJ Srolovitz, TJ Baker, JH Prévost. Brittle fracture in polycrystalline microstructures with the extended finite element method. International Journal for Numerical Methods in Engineering, 56(14), pp.20152037, 2003 GK Sfantos, MH Aliabadi. A boundary cohesive grain element formulation for modelling intergranular microfracture in polycrystalline brittle materials. International journal for numerical methods in engineering, 69(8), pp.1590-1626, 2007 GK Sfantos, MH Aliabadi. Multi-scale boundary element modelling of material degradation and fracture. Computer Methods in Applied Mechanics and Engineering, 196(7), pp.1310-1329, 2007 I Benedetti, MH Aliabadi. A three-dimensional grain boundary formulation for microstructural modeling of polycrystalline materials. Computational Materials Science, 67, pp.249-260, 2013 I Benedetti, MH Aliabadi. A three-dimensional cohesive-frictional grain-boundary micromechanical model for intergranular degradation and failure in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering, 265, pp.36-62, 2013 V Gulizzi, A Milazzo, I Benedetti. An enhanced grain-boundary framework for computational homogenization and micro-cracking simulations of polycrystalline materials. Computational Mechanics, 56(4), pp.631-651, 2015 I Benedetti, MH Aliabadi. Multiscale modeling of polycrystalline materials: A boundary element approach to material degradation and fracture. Computer Methods in Applied Mechanics and Engineering, 289, pp.429-453, 2015 I Benedetti, V Gulizzi, V Mallardo, A grain boundary formulation for crystal plasticity, International Journal of Plasticity, 83, pp. 202-224, 2016 G Geraci, MH Aliabadi. Micromechanical boundary element modelling of transgranular and intergranular cohesive cracking in polycrystalline materials. Engineering Fracture Mechanics, In Press , 2017, http://dx.doi.org/10.1016/j.engfracmech.2017.03.016 I Benedetti and F Barbe. Modelling polycrystalline materials: an overview of three-dimensional grain-scale mechanical models. Journal of Multiscale Modelling 5.01: 1350002, 2013 P.K. Banerjee & R. Butterfield: Boundary element methods in engineering science (McGraw-Hill, 1981) M.H. Aliabadi, The boundary element method: applications in solids and structures. John Wiley & Sons Ltd, England, 2002 V Gulizzi, A Milazzo and I Benedetti, Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions. International Journal of Solids and Structures, 100, pp. 169-186, 2016


An efficient boundary element procedure to evaluate the design of cathodic protection systems of tank bottoms W. J. Santos1, S.L.D.C. Brasil2, J. A. F. Santiago3 and J. C. F. Telles4 1

Department of Mathematics, UFRRJ, Seropedica/RJ – Brazil, [email protected] 2

School of Chemistry, UFRJ, Rio de Janeiro – Brazil, [email protected]

3

Department of Civil Engineering, COPPE/UFRJ, Rio de Janeiro – Brazil, [email protected] 4

Department of Civil Engineering, COPPE/UFRJ, Rio de Janeiro – Brazil, [email protected]

Keywords

85

86


2. Mathematical Model As can be seen in Fig. 1, the tank bottom analysed is in electrical contact with a slender conductive concrete layer, of low resistivity ߩଵ , which in turn is in direct contact with a homogeneous deep soil region of resistivity ߩଶ . Furthermore, the CP is achieved using concentric ring anodes placed between the liner and the slender conductive layer. The conductive concrete layer has length represented by ݄ଵ , whereas the length of the soil region is denoted by ݄ଶ .

Figure 1: Hypothesis of the potential problem. Considering that the CP technique is developed within each homogeneous subregion presented in Fig. 1, the mathematical model of the problem is based on a Laplace equation for the electrochemical potential ሺ߶ሻ: ‫׏‬ଶ ߶ ൌ ͲǤ

(1)

Over the metal surface in direct contact with the conductive concrete, the boundary condition is given by the polarization curve, which describes a nonlinear relationship between the electrochemical potential and current density ሺ݅ሻ on the metal, ߶ ൌ ݂ሺ݅ሻ. All the other boundaries are insulated ሺ݅ ൌ Ͳ ‫ܣ‬Τ݉ ଶ ሻ, excepted the interface. In addition, from Ohm's law డథ

݅ ൌ ݇ , డ௡ in which ݇ is the conductivity of the electrolyte and ࢔ is the outward normal to the boundary Ȟ.

(2)

The familiar BEM algebraic system, obtained after discretizing the starting equation that represents the electrical field problem using boundary elements [10], can be written as (3)

ࡴࣘ ൌ ࡳ࢏.

For nonlinear boundary conditions, the system (3) is usually solved by a tangent Newton-Raphson method, which can be formulated considering only the first order terms in a Taylor series expansion: ࢏௞ ൌ ࢏௞ିଵ ൅ ࡶ௞ିଵ ࣘ࢑ ǡ

(4)

where the Jacobian matrix ࡶ is given by ࡶ௞ିଵ ൌ ቀ

ࣔ࢏ ௞ିଵ ቁ Ǥ ࣔࣘ

(5)

The axisymmetric fundamental solution for Laplace's equation is considered to simulate the CP systems of tank bottoms using concentric rings. In this case, the fundamental potential can be calculated explicitly in terms of the complete elliptic integral of the first kind ‫ܭ‬ሺ݉ሻ as [10] ଶగ

‫כ‬ ߶௔௫௜௦ ൌ ‫׬‬଴ ߶ ‫ כ‬ሺߦǡ ‫ݔ‬ሻ݀ߠ ൌ

ସ௄ሺ௠ሻ భ , ሺ௔ା௕ሻ ൗమ

(6)


87

where ߶ ‫ כ‬is the three-dimensional fundamental solution in cylindrical polar coordinates ሺܴǡ ߠǡ ܼሻ and ݉ൌ

ଶ௕ ௔ା௕

ǡ

ܽ ൌ ܴଶ ሺߦ ሻ ൅ ܴଶ ሺ‫ݔ‬ሻ ൅ ሾܼሺߦሻ െ ܼሺ‫ݔ‬ሻሿଶ ǡ ܾ ൌ ʹܴ ሺߦ ሻܴ ሺ‫ݔ‬ሻǤ

(7)

The parameter ݉ has range of [0,1]. The normal derivative of the axisymmetric fundamental solution is given by ‫כ‬ ݅௔௫௜௦ ൌ ଵ

Ͷ ଵൗ ଶ

ሺܽ ൅ ܾሻ

ோ మሺకሻିோ మሺ௫ሻାሾ௓ሺకሻି௓ሺ௫ሻሿమ

‫ܧ‬ሺ݉ ሻ െ ‫ܭ‬ሺ݉ ሻቃ ቑǡ (8) ݊ோ ሺ‫ݔ‬ሻ ൅ ௔ି௕ ‫ܧ‬ሺ݉ ሻ݊௭ ሺ‫ݔ‬ሻ where ‫ܧ‬ሺ݉ሻ is the complete elliptic integral of the second kind. For convenience of numerical implementation, the complete elliptic integrals is approximated by polynomial expressions [11]. Ǥ ቐଶோሺ௫ሻ

ቂ

௔ି௕ ௓ሺకሻି௓ሺ௫ሻ

3. Traditional solution technique The potential problem cited above and illustrated in Fig. 1 is defined over a piecewise homogeneous body consisting of two subregions and, therefore, the boundary element procedure can be applied to each homogeneous subregion (see Fig. 2). The final set of equations for the whole region can then be obtained by assembling the set of equations (3) for each subregion using compatibility of potentials and fluxes between the common interfaces [10].

Figure 2: Domain divided into two subregions. 4. A new solution technique In order to create the theoretical polarization curve in the interface, the current flowing through the slender conductive concrete was considered like a linear circuit (conductive wires with resistivity ߩଵ ). According to Ohm's law, there is a linear relationship between the voltage drop across a circuit element and the current flowing through it. In equation form, this relation can be written as follows: ܸ ൌ ‫ܴܫ‬ǡ (9) where ܸ is the potential difference ሺ߶ଵ െ ߶ଶ ሻ between any two points in a circuit in volts ሺܸሻ, ‫ ܫ‬is the current flowing through the circuit in units of amperes ሺ‫ܣ‬ሻ and ሺܴሻ is the resistance of the circuit with unit ohm ሺȳሻ, as schematically shown in Fig. 3. Considering a homogeneous and isotropic material with resistivity ߩଵ , a cross-sectional area ‫ ܣ‬and the length ݄ଵ , the current density and the resistance are given in a circuit, respectively, by the following equations:

88


ூ

(10)

݅ ൌ ஺ǡ ܴൌ

ఘభ௛భ ஺

.

(11)

Figure 3: Potential difference between any two points in a circuit. Using the eqs (10) and (11), it is possible to rewrite the relation (9) for any two points of the circuit presented in Fig. 3 as ߶ଵ െ ߶ଶ ൌ ݅ߩଵ ݄ଵ Ǥ

(12)

After experimentally determining the polarization curve of the tank bottom, which is in electric contact with the conductive concrete, the eq (12) can be used to create a theoretical polarization curve for the concrete/soil interface. For each pair of points ሺ݅ ௞ ǡ ߶ଵ௞ ሻ given by experimental polarization curve over the tank bottom, the corresponding theoretical polarization curve in the concrete/soil interface is approximated by the point pairs ൫݅௞ ǡ ߶ଶ௞ ൯ ‫ ؠ‬൫݅ ௞ ǡ ߶ଵ௞ െ ݅ ௞ ߩଵ ݄ଵ ൯. Thus, it is only considered the soil layer of resistivity ߩଶ for modelling CP system and hence a standard BEM procedure can be easily applied. Finally, after solving the potential in the interface, the inversion relation ߶ଵ௞ ൌ ߶ଶ௞ ൅ ݅ ௞ ߩଵ ݄ଵ is used to determine the potential distribution over the tank bottom. 5. Numerical results Example 1 For the purpose of testing the proposed methodology, an angled anode cathodic protection system, shown in Fig. 4, was initially analysed. The sacrificial anodes are mathematically represented by linear sources and each anode has a current intensity of െͲǤͲ͸ ‫ܣ‬Τ݉ , where this value was chosen with the goal of providing a potential distribution over the metal surface below the critical potential: ߶ ൑ ߶௖ ൌ െͲǤͺͷͲܸ (vs. SCE). The two zones considered have resistivities ߩଵ ൌ ͺͲȳǤ ݉ and ߩଶ ൌ ͷͲͲȳǤ ݉. The height of the first subregion is ͲǤͷ݉.

Figure 4: Two-dimensional potential problem. The proposed formulation is compared with the alternative subregion technique. Figure 5 shows the potential distribution obtained on the metal surface using the subregion technique and the proposed methodology, where the similarity of the results can also be seen.


Figure 5: Potential distribution on the metal. The CPU time for the numerical calculation of the example 1 using BEM with the usual subregion technique was approximately 4.3 seconds. Using BEM and the tested methodology, the process time was approximately 1.4 seconds. In applications involving optimization problems and inverse problems, this computational time difference will be a lot more significant, due to necessity of solving a numerical problem many times. In addition, for this example, the geometry is most simple considering only the soil layer, avoiding the reentrances of the original problem. Example 2 In this application an external tank bottom cathodic protection (see Fig. 1) is analysed. It is considered a conductive concrete with height ݄ଵ ൌ ͲǤͷ݉ and resistivity ߩଵ ൌ ͺ͹ȳǤ ݉, whereas the resistivity in the soil layer is ߩଶ ൌ ͵͵ͳǤ͵ͶȳǤ ݉. Here, for reducing corrosion and extending the service life of the tank bottom is proposed a system of anodes in concentric rings, where the number of rings varies with tank diameter. The diameter of the tank bottom is 84 m and the fixed distance between the 12 anodes is 3 m. In addition, these anodes are localized over the liner and each anode has a prescribed potential of -1.6 V. The problem is solved considering the soil layer and the polarization curve in the interface. All boundary values have axial symmetry and consequently all domain values are also axisymmetric. Thus, an axisymmetric analysis ሺܴǡ ܼሻ can be considered in this example, using the axisymmetric fundamental solutions, eqs (6) and (8). The potential values on the metal solved by the standard BEM solution procedure are presented in Fig. 6.

Figure 6: Potential distribution on the metal.

89

90


6. Conclusions The main goal of this paper is to evaluate the design of CP systems of tank bottoms using an efficient boundary element method. Example 1 shows a complete satisfactory equivalence between the proposed methodology and the alternative subregion technique, where the process time for the numerical calculation using the tested methodology was significantly less. In example 2, a newer design to protect tank bottoms was considered, the system of linear anodes in concentric rings. Most of the applications involving CP systems comprises optimization and inverse problems. In these cases, the computational time difference between the proposed methodology and the subregion technique should be a lot more significant, due to necessity of solving the numerical problem many times. This analysis will be the subject of future a work. 7. Acknowledgements The CAPES-Brazil and CNPq-Brazil. References [1] US EPA. Title 40 code federal regulations, parts 280 and 281, September (1988). [2] L. Koszewski Retrofitting asphalt storage tanks with an improved cathodic protection system, Materials Perfomance, 38, 20-24 (1999). [3] D. P. Riemer and M. E. Orazem L. Koszewski A mathematical model for the cathodic protection of thank bottoms, Corrosion Science, 47, 849-868 (2005). [4] J. C. F. Telles, W. J. Mansur, L. C. Wrobel and M. G. Marinho Numerical Simulation of a Cathodically Protected Semisubmersible Platform using PROCAT System, Corrosion, 46, 513-518 (1990). [5] J. A. F. Santiago and J. C. F. Telles On Boundary Elements for Simulation of Cathodic Protection Systems with Dynamic Polarization Curves, International Journal for Numerical Methods in Engineering, 40, 2611-2622 (1997). [6] BEASY User Guide Beasy, Computational Mechanics BEASY Ltd, Ashurst, Southampton, UK (2000). [7] W. J. Santos, J. A. F. Santiago and J. C. F Telles An Application of Genetic Algorithms and the Method of Fundamental Solutions to Simulate Cathodic Protection Systems, Computer Modeling in Engineering & Sciences, 87, 23-40 (2012). [8] W. J. Santos, J. A. F. Santiago and J. C. F Telles Optimal positioning of anodes and virtual sources in the design of cathodic protection systems using the method of fundamental solutions, Engineering Analysis with Boundary Elements, 46, 67-74 (2014). [9] W. J. Santos, J. A. F. Santiago and J. C. F Telles Using the Gaussian function to simulate constant potential anodes in multiobjective optimization of cathodic protection systems, Engineering Analysis with Boundary Elements, 73, 35-41 (2016). [10] C. A. Brebbia., J. C. F. Telles and L. C. Wrobel Boundary Element Techniques: Theory and Applications in Engineering, Springer, Berlin (1984). [11] M. Abromowitz and I. A. Stegun Handbook of Mathematical Functions, Dover, New York (1965).


91

Boundary element formulation for crack surface contact simulation in piezoelectric materials L. Rodr´ıguez-Tembleque1,∗ , F. Garc´ıa-Sánchez2 , A. Sáez1 and M. Wünsche3 1

3

Escuela Técnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los descubrimientos s/n, E41092 Sevilla, SPAIN 2 Escuela de Ingenier´ ıas Industriales, Universidad de M´ alaga, Doctor Ortiz Ramos s/n, 29071 M´ alaga, SPAIN Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava, SLOVAKIA ∗ [email protected]

Keywords: Piezoelectric Materials, Fracture Mechanics, Contact Mechanics, Boundary Element Method.

Abstract. Piezoelectric materials exhibit a multifield coupling which allows for their use as actuators and sensors in many technological sectors of current interest such as the aerospace and automotive industries, or the biomedical and the electronics industries. Actuators, sensors, micro- and nanoelectromechanical systems and other PE components are generally constructed in block form or in a thin laminated composite. The study of the integrity of such materials in their various forms and small sizes is still a challenge nowadays. To gain a better understanding of these systems, this work presents a crack surface contact formulation which makes it possible to study the integrity of these advanced materials under more realistic crack surface multifield operational conditions. The boundary element method (BEM) is used for modeling frictional crack surface contact on piezoelectric solids, in the presence of electric fields. The formulation, based on previous works [1, 2, 3, 4], uses the BEM for computing the elastic influence coefficients and contact operators over the augmented Lagrangian to enforce contact constraints on the crack surface. The BEM reveals to be a very suitable methodology for these interface interaction problems because it considers only the boundary degrees of freedom what makes it possible to reduce the number of unknowns and to obtain a very good accuracy with much less number of elements than finite element formulations. The capabities of this methodology are illustrated solving some benchmark problems. Introduction Let us consider a two-dimensional, homogeneous, anisotropic and linear piezoelectric (PE) cracked solid Ω ⊂ R2 with boundary ∂Ω, in a Cartesian coordinate system (xi ) (i = 1, 2). The mechanical equilibrium equations of this problem, in the absence of body forces, and the electric equilibrium equations under free electrical charge are σij,j = 0 Di,i = 0

in Ω, in Ω,

(1)

where σij are the components of Cauchy stress tensor and Di are the electric displacements. The infinitesimal strain tensor γij and the electric field Ei are defined as γij = (ui,j + uj,i )/2 Ei = −ϕ,i

in in

Ω, Ω,

(2)

with ui being the elastic displacement and ϕ being the electric potential. The elastic and electric fields are coupled through the linear constitutive law σij = cijkl γkl − elij El Di = eikl γkl + il El

in in

Ω, Ω,

(3)

92


where cijkl and il denote the components of the elastic stiffness tensor and the dielectric permittivity tensor, respectively; and eijk are the PE coupling coefficients. These tensors satisfy the following symmetries: cijkl = cjikl = cijlk = cklij , ekij = ekji , kl = lk , with the elastic constant and dielectric permittivity tensors being positive definite. The boundary ∂Ω is divided in two disjoint parts: ∂Ω = ∂Ωe ∪∂Ωc , where ∂Ωe denotes the external boundary and ∂Ωc is the crack surface. Two partitions of the boundary ∂Ωe are considered to define the mechanical and the electrical boundary conditions. The first partition is: ∂Ωe = ∂Ωu ∪ ∂Ωp , i.e., ∂Ωu being the external boundary on which diplacements u ˜i are prescribed and ∂Ωp with imposed ˜ prescribed on tractions p˜i . The second partition is: ∂Ωe = ∂Ωϕ ∪ ∂Ωq , being the electrical potential ϕ ∂Ωϕ , and the electrical charges q˜ assumed on ∂Ωq . Consequently, the Dirichlet boundary conditions are ˜i on ∂Ωu , ui = u (4) ϕ = ϕ˜ on ∂Ωϕ , and the Neumann boundary conditions are given by σij νj = p˜i Di νi = q˜

on on

∂Ωp , ∂Ωq ,

(5)

with νi being the outward unit normal to the boundary. − Finally, on the upper and lower crack faces (i.e. ∂Ωc = ∂Ω+ c ∪ ∂Ωc ) self equilibrated tractions and + − + electric charges are considered: Δpi = pi + pi = 0 and Δq = q + q − = 0. However, aditional crack surface contact conditions have to be considered, as follows, on ∂Ωc . Crack face mechanical contact conditions In order to avoid material interpepenetration between crack-faces, the unilateral contact law involves Signorini’s contact conditions on ∂Ωc : Δuν ≥ 0,

p+ ν ≥ 0,

Δuν p+ ν = 0,

(6)

− + + + + + where Δuν = u+ ν − uν and pν = p · ν c , with ν c being the unit normal on ∂Ωc . The normal contact constraints presented in (6) can be formulated as:

p+ p+ ν − PR+ ( ν ) = 0,

(7) p+ ν

p+ ν

= − rν Δuν is the where PR+ (•) is the normal projection function (PR+ (•) = max(0, •)) and augmented normal traction. The parameters rν is the normal dimensional penalization parameter (rν ∈ R+ ). In general, frictional contact condition on crack surfaces should be considered. So the Coulomb friction restriction can be summarized as: Δuτ = −λp+ τ ,

λ ≥ 0,

+ |p+ τ | ≤ μpν ,

u+ τ

+ λ(p+ τ − μpν ) = 0,

u− τ

p+ τ

p+

(8) τ+ c ,

τ+ c

− and = · with being where λ is an scalar, μ is the friction coefficient, Δuτ = the unit tangential vector on ∂Ω+ c . The frictional contact constraints (8) can be also formulated using a contact operators as: pτ − PEρ ( p+ τ ) = 0,

(9)

+ where p+ τ = pτ − rτ Δuτ (rτ ∈ R ) is the augmented tangential traction and PEρ (•) : R −→ R is the tangential projection function defined as + pτ if | p+ τ | < ρ, (10) p+ ) = PEρ ( τ + |) if | /| p p+ ρ( p+ τ τ τ | ≥ ρ,

with ρ = μpν , as it was defined in Eq. (7).


93

Crack face electrical contact conditions The electrical boundary conditions on the crack-faces ∂Ωc can be defined, in the general form as q + = κc Δϕ/Δuν ,

(11)

where Δϕ = ϕ+ −ϕ− . In Eq. (11), the electrical permittivity κc = κr κo is defined by the product of the relative permittivity of the considered medium κr and the permittivity of the vacuum κo = 8.85 · 103 C/(GV m). In contrast to the impermeable and permeable crack-face boundary conditions, a nonlinear relation between mechanical displacements, electrical potentials and electrical charges is now present. Boundary integral equations The mixed formulation for the BE solution of crack problems considers both the extended displacement (EDBIE) and the extended traction (ETBIE) boundary integral representations to overcome the − difficulty of having two coincident boundaries ∂Ω+ c and ∂Ωc . In this way, the EDBIE is applied for collocation points ξ on ∂Ωe and on either of the crack faces, say ∂Ω− c , to yield cIJ (ξ)uJ (ξ) + − p∗IJ (x, ξ)uJ (x)dS(x) = u∗IJ (x, ξ)pJ (x)dS(x), (12) ∂Ω

∂Ω

where x is a boundary point, uJ is the extended displacement vector (see Barnett & Lothe representation [5]) uj J 2 uJ = (13) ϕ J = 3, pJ is the extended tractions vector pJ =

pj q

J 2 J = 3,

(14)

cJK depends on the local geometry of the boundary ∂Ω at the collocation point ξ; u∗IJ and p∗IJ are the extended displacement fundamental solution [6] and the extended traction fundamental solution at a boundary point x due to a unit extended source applied at point ξ, respectively. Consequently, the ETBIE is applied for collocation points ξ on the other crack surface, ∂Ω+ c , (15) cIJ (ξ)pJ (ξ)+ = s∗IJ (x, ξ)uJ (x)dS(x) = − d∗IJ (x, ξ)pJ (x)dS(x), ∂Ω

∂Ω

to complete the set of equations to compute the extended displacements and tractions on ∂Ω. In Eq. (15) s∗IJ and d∗IJ are obtained by differentiation of u∗IJ and p∗IJ [6], as d∗IJ = −Ns CsIKr u∗KJ,r ,

(16)

s∗IJ = −Ns CsIKr p∗KJ,r ,

(17)

with N being the outward unit normal to the boundary at the source point and ⎧ Cijkl , J, K = 1, 2 ⎪ ⎪ ⎨ J = 1, 2; K = 3 elij , CiJKl = J = 3; K = 1, 2 eikl , ⎪ ⎪ ⎩ J, K = 3, −il ,

(18)

where the lowercase (elastic) and (extended) subscripts take values 1, 2 and 1, 2, 3, respec! uppercase ! tively. Furthermore, Symbols − and = in Eqs. (12) and (15) stand for the Cauchy Principal Value (CPV) and the Hadamard Finite Part (HFP) of the integral, respectively.

94


− When the cracks are mechanically and electrically selfequilibrated, i.e., ΔpI = p+ I + pI = 0 on ∂Ωc (the superscripts + and − stand for the upper and lower crack surfaces), it would be enough to apply the EDBIE for collocation points ξ on ∂Ωe and the ETBIE for collocation points ξ on either side of the crack, say ∂Ω+ c cIJ (ξ)uJ (ξ) + − p∗IJ (x, ξ)uJ (x)dS(x) + − p∗IJ (x, ξ)ΔuJ (x)dS(x) = u∗IJ (x, ξ)pJ (x)dS(x) (19) ∂Ωe

pJ (ξ)+ = s∗IJ (x, ξ)uJ (x)dS(x)+ = ∂Ωe

∂Ω+ c

∂Ωe

s∗IJ (x, ξ)ΔuJ (x)dS(x) = − d∗IJ (x, ξ)pJ (x)dS(x)

∂Ω+ c

(20)

∂Ωe

Eqs. (19) and (20) yield a complete set of equations to compute the extended displacements and − tractions on ∂Ωe and the extended crack opening displacements ΔuI = u+ I − uI on ∂Ωc . In Eq. (20) the free term has been set to 1 because of the additional singularity arising from the coincidence of the two crack surfaces. Crack surface contact discrete equations Numerical evaluation of the ETBIE requires C 1 continuity of the displacements. As in previous works [1], discontinuous quadratic elements with the two extreme collocation nodes shifted towards the element interior are used to mesh the cracks. The asymptotic behavior of the extended displacements near the crack tip is modelled via discontinuous quarter-point elements. For the rest of the boundaries, continuous quadratic elements are employed. A detailed justification of the discretization procedure can be found in [1]. A collocation procedure on boundary integral equations (19) and (20) leads to the following system of equations: Ax = F, where the boundary conditions have been imposed and all the unknowns have been passed to vector x, to yield ⎧ ⎫ xe ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ & ⎨ Δuc ⎬ % Axe AΔuc AΔϕc Apc Aqc Δϕc = F. (21) ⎪ ⎪ ⎪ ⎪ p ⎪ ⎪ c ⎪ ⎪ ⎩ ⎭ qc In expression (21), xe collects the nodal external unknowns (i.e. the nodal unknowns on ∂Ωe ), Δuc and Δϕc collect the nodal crack opening displacements and electric potentials, respectively, on xc , pc contains the normal and tangential nodal contact tractions (i.e. pν and pτ ) and qc contains the nodal electric charges. Matrices Axe , AΔuc , AΔϕc , Apc and Aqc are constructed with the columns of matrices yielded from the numerical integration of Eqs. (19) and (20). The electric charge on every contact node i can be expressed in terms of the electric potential according to the electrical contact condition (11), as: (qc )i = −κ((Δuν )i )(Δϕc )i . So equation (21) can be written as ⎧ ⎫ xe ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ & % Δuc ˜ = F, (22) Axe AΔuc AΔϕc Apc ⎪ ⎪ Δϕc ⎪ ⎪ ⎩ ⎭ pc ˜ ϕc = Aϕc − κ(Δuν )i )Aqc and κ(Δuν ) a diagonal matrix, i.e.: being A κ(Δuν ) = diag ( κ((Δuν )1 ), · · · , κ((Δuν )i ), · · · , κ((Δuν )Nc ) ) .

(23)

Finally, the mechanical contact restrictions (7) and (9) are defined on every contact node i as: (pν )i − PR+ ( (pν )i − rν (Δuν )i ) = 0,

(24)

(pτ )i − PEρ ( (pτ )i − rτ (Δuτ )i ) = 0,

(25)

where pν and pτ contain the normal and tangential contact tractions of every contact node i and Δuν and Δuτ contain the normal and tangential nodal crack opening displacements, respectively.


95

ߪ

ߪ ‫ݔ‬ଶ

ܾ

ܾ

ߙ

‫ݔ‬ଵ

Figure 1: A crack under compression in an unbounded domain. Solution method The nonlinear equations set (22–25) is solved using the iterative Uzawa scheme proposed in [3, 4] for multifield PE materials in contact. Numerical example In order to validate this crack surface frictional contact formulation, a benchmark problem is solved. In this example, the formulation is applied for a mathematical degenerate case, i.e., elastic and isotropic material. Fig. 1 shows a single crack of length 2b in an unbounded domain and subject to a compressive remote stress σ. The analytical solution of this plane strain state is available in [7] for comparison. The mode-I stress intensity factor (SIF) KI = 0, as the crack surfaces remain closed under compression. However, the analytical solution for the mode-II SIF is √ (26) KII = σ πb sin α(cos α − μ sin α) where μ can be written as a function of the friction angle (φ): μ = tan(φ). The material constants employed are: Youngs modulus E =√70 GP a and Poissons ratio ν = 0.2. Results are presented in Fig. 2, where the normalized KII (KII /σ πb) is showed for various inclination angles (α) of the crack and different friction angles (φ = 0o , 15o , 30o , 45o ). An excellent agreement between analytical and numerical solutions can be observed. Conclusions A boundary element formulation has been applied to study crack surface frictional contact problems in PE materials. The proposed formulation has been applied to analyze a classical benchmark problems. The results present an excellent agreement with the analytical solution presented in [7]. So this boundary element formulation proves to be a very interesting numerical methodology to study the integrity of multifield materials such as PE materials. Acknowledgments This work is supported by the Ministerio de Ciencia e Innovaci´ on (Spain) through the research projects: DPI2013-43267-P and DPI2014-53947-R.

96


Ϭ͘ϱ

ʔсϬΣ ʔсϭϱΣ ʔсϯϬΣ ʔсϰϱΣ ŶĂůǇƚŝĐĂůƌĞƐƵůƚƐ

Ϭ͘ϰϱ Ϭ͘ϰ

EŽƌŵĂůŝǌĞĚ −1 ∧ (x1 + 1)2 + (x2 + 1)2 < 9 ∧ x21 + (x2 + 1)2 > 1 ∧ (x1 + 2.5)2 + x22 > 0.25} and bounded by a (piecewise) smooth curve Γ = ∂Ω, see Fig. 1-left. Here x = (x1 , x2 ) ∈ R2 . We will assume that the boundary condition G is a piecewise continuously differentiable function √ with two jump discontinuities, located at the points A = (0, 8 − 1) and B = (0, 0). Note that point A belongs to a convex boundary segment, while point B belongs to a non-convex one, with respect to the interior of the domain. Also, in order to simplify the notation and facilitate the error analysis,


A

2

2

ΓA

Ω

1

1

Γ0

0

133

ΓB

0

B

−1

−1 −4

−3

−2

−1

0

1

2

−4

−3

−2

−1

0

1

2

Figure 1: Domain (left) and a sample distribution of collocation (blue) and source (red) points (right). let Γ0 denote the interior boundary of the domain and ΓA and ΓB , the circular arcs containing the points A and B, respectively. Numerical method As we will illustrate further on, the classical MFS shows poor accuracy when applied to problem (1). Here we propose to augment the MFS approximation basis with a set singular particular solutions, while preserving all the meshfree characteristics of the original method. Classical MFS solution. In the classical MFS, the unknown solution u of problem (1) is approximated in terms of a linear combination u ˜(x) = α0 +

m

αj Φ(x − yj ),

x ∈ Ω,

(2)

j=1 1 ln |x| of the Laplace operator, with singularities Y = {y1 , . . . , ym } of fundamental solutions Φ(x) = − 2π selected in the exterior of the domain, see Fig. 1-right. Since, by its definition, u ˜ is a particular solution of the Laplace PDE, the unknown coefficients a = {α0 , . . . , αm } can be calculated by collocating the boundary condition G on a set X = {x1 , . . . , xn } of n boundary collocation points. Explicitly, the n × m + 1 collocation linear system may be written in matrix form as

Fa = G, with G = G(X ) and

⎡

1 Φ(x1 − y1 ) . . . ⎢ .. .. F(X , Y) = ⎣ ... . . 1 Φ(xn − y1 ) . . .

(3) ⎤ Φ(x1 − ym ) ⎥ .. ⎦ .

(4)

Φ(xn − ym )

and it can be solved in the least squares sense. In most cases, particularly when the domain and/or the boundary data are not sufficiently regular, a regularization technique, e.g. Tikhonov regularization or TSVD, e.g. [7], is also required. From a theoretical point of view, the applicability of the MFS is justified in terms of density results for linear combinations of fundamental solutions in an appropriate functional space on Γ, e.g. [8]. Note that, the approximate solution u ˜, defined by (2), is analytic in Ω and therefore it cannot recreate accurately the singular behaviour of the exact solution u in the neighborhood of points A and B. It becomes necessary to augment the MFS approximation basis with a set particular solutions of the PDE with discontinuous boundary traces. Special harmonic functions with discontinuous boundary traces. Motivated by the mathematical formulation of the boundary element method (BEM), we define a set of harmonic functions {Ψ0 , Ψ1 , Ψ2 , . . . } by analytically evaluating a double layer potential for each element of the polynomial canonical basis {1, y, y 2 , . . . }, Ψk (x) = ∂νy Φ(x − y)y k dsy , x ∈ R2 \γ, k = 0, 1, 2, . . . , (5) γ

134


where γ is an arbitrary linear segment and νy denotes the unit normal vector at y ∈ γ. These functions, also known as cracklets [4], are analytic and harmonic in R2 \γ and the trace of Ψk (x) on γ is the polynomial function xk . Also, Ψk (x) = 0 for x ∈ L\γ, where L is the infinite straight line defined by γ and, from the theory of the boundary layer potentials, the normal derivative of Ψk is continuous across γ. The analytic evaluation of the integral in the definition of Ψk may be carried out using a symbolic computation softwares, e.g. Mathematica. In particular, for γ =] − a, a[×{0} with a > 0, the first two such functions are given by: x1 − a x1 + a Ψ0 (x) = tan−1 − tan−1 x2 x2 2ax1 . Ψ1 (x) = x1 Ψ0 (x) + x2 tanh−1 x21 + x22 + a2 Remark 1 In the definition of Ψk , the parameter a > 0 allows us to vary the length of the segment γ. Also, the position of the segment can be changed by translation of the coordinate axes and since the Laplace operator is rotationally invariant we can change the orientation of γ, using an appropriate change of variables, without affecting the harmonic properties of Ψk . Geometry considerations. Depending on the geometry of the domain Ω we may distinguish between three cases. The difference consists in the relative position of the cracklets Ψk , i.e. of γ, with respect to the domain and its boundary. 2

γ

2

2

A

Ω

1

Ω

1

0

0

−4

−3

B

0

γ−

γ

−1

Ω

1

−2

−1

0

1

2

γ+

−1

−1 −4

−3

−2

−1

0

1

2

−4

−3

−2

−1

0

1

2

Figure 2: Position of γ for a discontinuity point located on a linear (left), convex (center) and nonconvex (right) boundary segment. Case 1. Linear boundary. When the discontinuity points of the boundary condition are located on a linear segment of the boundary we consider a set of cracklets with γ ⊂ Γ, see Fig. 2-left. This case has been extensively analyzed and illustrated in [6]. An improvement of up to 5 orders of magnitude of the absolute error was observed, in comparison with the results from the classical MFS. Also, numerical tests indicate that the accuracy of the enriched MFS increases when higher order Ψk functions are added to the MFS approximation basis. From a theoretical point of view, the enriched MFS is related to the classical subtraction of singularity approach. However, we avoid the splitting of the BVP into a continuous and a discontinuous subproblems, where, in general, the exact solution of the latter cannot be calculated with sufficiently high precision. For further details we refer the reader to [6]. Case 2. Convex boundary. Since γ is a linear segment, the previous approach is not possible when the discontinuity point lies on a curvilinear boundary segment. Instead, we will consider cracklets Ψk with γ that is tangential to the boundary at the discontinuity point, see Fig. 2-center. Another possibility for selecting the position of the segment γ would be to consider a local linear approximation of ΓA by γ. However, since the traces of the cracklets Ψk are singular at both tips of γ we would be including a second discontinuity on the boundary of the domain which is not present in the original problem (1). On the other hand, even for reasonably small values of |γ|, the geometric approximation of Ω will lead to an increase in the approximation error of the BVP’s solution.


135

Remark 2 We require that γ ∩ Ω = ∅ because otherwise the approximation basis would include functions which are discontinuous in the interior of the domain Ω. Note that Ψk (x) are discontinuous in the normal direction, with respect to γ, at any point x ∈ γ. Case 3. Non-convex boundary. In the non-convex case it is not possible to consider γ that is tangential to the boundary at the discontinuity point, without having a non-empty intersection between γ and Ω, see Remark 2. Therefore, and also in order to avoid linearization of the boundary and inclusion of more singularities in the approximate solution, it remains to consider γ starting at the singularity point and laying in the exterior of the domain. Recall that, it is possible to vary the length and the orientation of γ without destroying the harmonic properties of the cracklets, see Remark 1. Since the discontinuity of the trace of Ψk is not in the same direction as the discontinuity of the boundary condition, adding just one set of cracklets, i.e. with the same γ, to the MFS basis is not sufficient in order to solve accurately the BVP. Numerical tests indicate that two sets of cracklets, i.e. with γ − and γ + , as shown in Fig. 2-right, lead to a significant improvement of the numerical results. Enriched MFS. For each segment γ we will append a set {Ψ0 , . . . , Ψp } of p + 1 singular shape functions to the approximation basis of the classical MFS. In particular, denoting by γA , γB − and γB + the three segments required for the analysis of problem (1), we seek an approximate solution of the following form pB − pB + pA m − − + + A A B B u ˜(x) = α0 + αj Φ(x − yj ) + βk Ψk (x) + βk Ψk (x) + βkB ΨB x ∈ Ω. (6) k (x), j=1

k=0

k=0

k=0

In matrix form, the unknown coefficients of the basis functions in (6) can be calculated by solving the following block linear system ⎤ ⎡ a ⎢ % % & & bA ⎥ ⎥ F(X , Y) P(X , A) P(X , B − ) P(X , B + ) ⎢ (7) ⎣b B − ⎦ = G , bB + where P(X , ) denotes the block corresponding to the collocation of the cracklets associated with γ ⎤ ⎡ Ψ0 (x1 ) . . . Ψp (x1 ) ⎥ ⎢ .. .. P(X , ) = ⎣ ... ⎦ . . Ψ0 (xn ) . . .

Ψp (xn )

and b = {β0 , . . . , βp } are the corresponding coefficients in the linear combination (6). Numerical Results For a given discontinuous boundary function G, the exact solution of problem (1) is not known, and due to the presence of singularities, the accuracy of the approximate solution u ˜ will be measured in terms of the boundary RMS error ε2,Γ := G − u ˜2,Γ . Since the maximum principle holds for the exact and approximate solutions of the BVP, the approximation error of u ˜ in Ω is bounded by ε2,Γ . Example 1. For problem (1), we will consider the following Dirichlet boundary condition ⎧ 1 if x ∈ Γ0 ⎪ ⎪ ⎨ 0.5x1 + 0.5 if x ∈ Γ\Γ0 and x1 < 0 , (8) G1 (x) = ⎪ ⎪ ⎩ −0.5x1 − 0.5 if x ∈ Γ\Γ0 and x1 > 0 which has two jump discontinuities at the boundary points A and B. Note that, even though G1 is an analytic function on Γ\{A, B}, the domain Ω has four corners which will also induce singularities on the solution of the BVP. The local accuracy of the approximate solution in the neighborhood of these geometric singularities may be improved as shown in [5] for the Helmholtz BVP, however, this analysis is out of the scope of the present work.

136


We start by illustrating the Gibbs phenomenon that affects the classical MFS solution in the neighborhood of the discontinuity points. In particular, for n = 795 boundary collocation points and m = 399 source points, distributed as shown in Fig. 1-right, the collocation linear system was solved with a residual error of 0.559009, measured on the 2-norm. This low precision solution led to a relatively high boundary error of ε2,Γ = 0.0264156, measured on 5075 error test points on Γ. The maxima of the absolute error were observed in the neighborhood of the points A and B. The exact and approximate solutions on ΓA and ΓB , as functions of the corresponding arc-lengths, are shown in Fig. 3. The incorrect oscillations of the approximate solution do not attenuate even if the number of collocation and source points is increased. These oscillations are due to the impossibility to approximate accurately a discontinuous function by a linear combination of analytic basis functions. Also, this lack of precision of the boundary approximation affects the global accuracy of u ˜ in Ω. 0.5

0.5

0

0

−0.5 −0.5

−1 −1.5 0

2

4

6

8

−1 0

0.5

1

1.5

2

2.5

3

Figure 3: The exact and approximate solution on ΓA (left) and ΓB (right). Boundary condition G1 . Keeping the above knot configuration, we added one extra shape function Ψ0 , for each of the segments γA , γB − and γB + , to the MFS approximation basis.We considered |γA | = |γB − | = |γB + | = 2 in order to move the second discontinuity of Ψ0 away from the boundary of the domain. The angle between γB − and γB + was θ = π4 . The corresponding linear system was solved with a residual error of 4.02777 × 10−3 and we measured a boundary RMS error of ε2,Γ = 2.72276 × 10−4 . These results correspond to a decrease of two orders of magnitude of the boundary error, in comparison with the results from the classical MFS. Note that, the increase in the computational cost of the method was neglectable, corresponding to the addition of 3 extra columns to the linear system. In terms of the maximum absolute error we measured ε∞,Γ = 9.4154 × 10−3 . Again, the maxima of the absolute error occur in the neighborhood of the discontinuity points but this time no incorrect oscillations of the approximate solution were visible. Also, the absolute error rapidly decreases away from the discontinuity points, and, given that maxΩ |u| = maxΓ |G1 | = 1.5, these numerical results correspond to a maximum relative error in Ω of approximately 0.63%. The above numerical results may be improved significantly if more cracklets are included in the B − and ΨB + to the basis and keeping the rest approximation basis. For example, by adding ΨA 1 , Ψ1 1 of the parameters as before, we were able to decrease the RMS error to ε2,Γ = 9.41392 × 10−7 and the absolute error to ε∞,Γ = 3.48793 × 10−5 . Further improvement may be achieved by increasing the number of cracklets and simultaneously increasing the number of collocation and source points. For example, for p = 3, i.e. 4 cracklets per γ , n = 2175 and m = 1124, the corresponding linear system was solved with a residual error of 1.21716 × 10−7 and we measured ε2,Γ = 9.45897 × 10−9 and ε∞,Γ = 8.55777 × 10−7 for the boundary errors. These results correspond to a decrease of more than 7 orders of magnitude in comparison with the corresponding results from the classical MFS. Graphical results for the absolute error of the approximate solution on ΓA and ΓB are shown in Fig. 4. Error analysis was also performed for several values of the angle θ between γB − and γB + . Numerical results indicate that better accuracy can be achieved if θ < π2 . However, in general, poor numerical results were observed for θ ≈ 0 or γ ⊥ Γ. Example 2. We considered the following non-polynomial boundary function ⎧ 0 if x ∈ Γ0 ⎪ ⎪ ⎨ x1 sin(x1 ) + cos(x2 ) if x ∈ Γ\Γ0 and x1 < 0 , G2 (x) = (9) ⎪ ⎪ ⎩ x1 cos(x1 ) + sin(x2 ) if x ∈ Γ\Γ0 and x1 > 0


137

−5

10

−5

−8

10

−11

10

10

−8

10

−11

10

−14

10

−14

0

2

4

6

10

8

0

0.5

1

1.5

2

2.5

3

Figure 4: Absolute error on ΓA (left) and ΓB (right), semi-log scale. Boundary condition G1 . and took n = 2175, m = 1124, p = 3, |γA | = |γB − | = |γB + | = 2 and θ = π4 for the enriched MFS. The corresponding linear system was solved with a residual error of magnitude 1.79835 × 10−6 and we measured ε2,Γ = 1.1152 × 10−7 and ε∞,Γ = 4.8341 × 10−6 for the boundary error. These results correspond to a decrease of the RMS error of approximately 5 orders of magnitude in comparison with the results from the classical MFS. Also, no oscillations were present in the neighborhood of the discontinuity points, see Fig. 5. 2

2 1

1

0 0 −1 −4

−1 −2 −2

0

2

Figure 5: Contour plot of the approximate solution in Ω. Boundary condition G2 . Conclusions A meshfree variant of the classical MFS has been proposed for the numerical solution of BVP with discontinuous Dirichlet boundary conditions. The method has been tested for 2D Laplace BVPs and convex and non-convex boundaries, but the approach can be extended to other PDEs, provided double layer potentials with polynomial densities can be analytically evaluated on a reference segment. Numerical results indicate that the approximate solution is no longer affected by the Gibbs phenomenon, as in the case of the classical MFS. Furthermore, highly accurate numerical solutions can be derived without a significant increase in the computational cost of the method. Acknowledgements. The authors gratefully acknowledge the financial support from CEMAT, through Funda¸c˜so para a Ciência e a Tecnologia (FCT) project UID/Multi/04621/2013.

References [1] G. Fairweather and A. Karageorghis, Adv. Comput. Math. 9, 69–95 (1998). [2] M. A. Golberg and C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, WIT Press (1998). [3] Jae-Hun Jung, Appl. Numer. Math. 57, 213–229 (2007). [4] Carlos J.S. Alves and Victor M.A. Leitão, Eng. Anal. Bound. Elem. 30, 160–166 (2006). [5] Pedro R.S. Antunes and Svilen S. Valtchev, J. Comput. Appl. Math. 234, 2646–2662 (2010). [6] Carlos J.S. Alves and Svilen S. Valtchev, submitted (2017). [7] P.C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM (1998). [8] A. Bobomolny, SIAM J. Numer. Anal. 22, 644–669 (1985).

138


Dynamic stress intensity factors evaluation with functionally graded materials by finite block method J. Li1,a, C. Shi2,b and P.H. Wen2,c 1 2

School of Mathematics and Statistics, Changsha University of Science and Technology, China School of Engineering and Materials Science, Queen Mary University of London, London, UK a

[email protected], b [email protected], c [email protected]

Keywords: functionally graded materials, finite block method, dynamic stress intensity factors, Laplace transform method, Durbin inversion method.

Abstract. The dynamic stress intensity factors with orthotropic functionally graded materials under dynamic load are investigated by using the finite block method in this paper. The higher order derivative matrix for two and three dimensional problems can be constructed directly. For linear elastic fracture mechanics, the COD techniques to determine the stress intensity factors is used. Several examples are given and comparisons have been made with analytical and numerical solutions in order to demonstrate the accuracy and convergence of the finite block method. Introduction For exponentially graded non-homogeneous, isotropic and linear elastic solids, the fundamental solutions have been derived by Martin et al [1] for three dimensions statics and Chan et al [2] for two dimensions statics even the analytical solutions are expressed in some complicated finite integrals. The Fourier-integral representations of the elastodynamic fundamental solutions have been recently derived by Zhang et al. [3] for fracture analysis in FGMs. Due to the mathematical complexities for the non-homogeneous nature of FGMs, only a few investigations on the transient dynamic responses of cracked FGMs can be found in journals including the dynamic responses under impact loading investigated by Babaei and Lukasiwicz [4], Li and Zou [5]. In addition, FGMs may exhibit isotropic or anisotropic material properties depending on the processing technique and the practical engineering requirements. In recent years, meshless formulations are becoming popular due to their high adaptively and low cost to prepare input and output data for numerical analysis [6]. Sladek et al [7] extended the meshless method based on the local Petrov-Galerkin approach for stress analysis in two-dimensional (2D), anisotropic and linear elastic/viscoelastic solids with continuously varying material properties. Jin and Paulino [8] investigated a crack in a viscoelastic strip of FGM under tensile load. The stress intensity factors of mixed modes are obtained in viscoelastic FGMs with correspondence principle. Kim and Paulino [9] presented a general purpose FEM formulation and implementation for linear FGMs and fracture of FGMs for mixed mode cracks with COD and J-integral techniques. The Finite Block Method, based on the point collocation method was developed firstly to solve the heat conduction problem in the functionally graded media and anisotropic materials by Li and Wen [10]. This method has been applied to elasticity, contact and fracture mechanics, see [11]. The essential feature of the FBM is that the physical domain is divided into few blocks and for each block the partial differential matrices are obtained in terms of nodal values using mapping technique. The decretization of domain is similar to FEM, i.e. the domain is divided into several blocks with continuity conditions of stress and displacement on the bounded surfaces. It is easy to prove that all stress components are continuous along the interface between two blocks. In the normalised coordinate system, the first order derivative matrix is constructed with nodes collocated on a straight line. Then higher orders of partial differential matrix can be obtained straight forwards. With quadratic type of block to transfer real domain to normalised domain with 8 seeds for 2D, the partial differential matrices in physical domain are obtained by the differential matrices in the normalised domain. A set of linear equations from equilibrium equations in strong form is


139

formulated to determine all nodal values of displacement. For linear elastic fracture mechanics, the dynamic stress intensity factors are evaluated by crack opening displacement (COD) technique for both isotropic and orthotropic FGMs. To demonstrate the accuracy and efficiency of the FBM, several numerical examples are given with ccomparisons made with the finite element method and local Petrov-Galerkin approach. Two dimension differential matrices Consider a set of nodes shown in Figure 1 with the nodes collocated at [D(k ) , D 1,2 , k 1,2,..., ND , where ND are numbers of nodes along two axes. By two dimension Lagrange interpolation polynomials, the function u ([1 , [ 2 ) can be approximated by u ([1 , [ 2 )

N1 N 2

¦¦ F ([ , [ 1

1

(i ) 1

) F2 ([ 2 , [ 2( j ) )u ( k )

(1)

i 1 j 1

where ND

([ [D( m ) ) (i ) (m) D [D )

([ D

FD ([D , [D(i ) )

m 1 m zi

(2)

and the superscript k in (1) equals to ( j 1) u N1 i . The number of nodes in total is M Then the first order partial differential is determined easily with respects to [1 wu w[1

wF1 ([1 , [1( i ) ) F2 ([ 2 , [ 2( j ) )u ( k ) ¦¦ w[1 i 1 j 1 N1 N 2

N1 u N 2 . (3)

(a)

(b)

Figure 1. Two-dimensional node distribution in mapping domain: (a) the local number system of node; (b) square domain with 8 seeds for the mapping geometry. where wF ([1 , [1( i ) ) w[1

w N1 ([1 [1( m ) ) w[1 m 1, ([1(i ) [1( m ) ) m zi

N1

N1

¦ ([

1

l 1 k 1,k zi ,k zl

[1( k ) ) /

N1

([

(i ) 1

[1( m ) )

(4)

m 1,m zi

and wu w[ 2 where

wF2 ([ 2 , [ 2( j ) ) F1 ([1 , [1( i ) )u ( k ) w[ 2 1

N1 N 2

¦¦ i 1 j

(5)

140


wF2 ([ 2 , [ 2( j ) ) w[ 2

w N 2 ([ 2 [ 2( m ) ) w[ 2 m 1, ([ 2( j ) [ 2( m ) ) mzi

N2

N2

¦ ([

N2

(k ) 2 [2 ) /

l 1 k 1,k zi ,k zl

([

( j) 2

[ 2( m ) )

(6)

m 1,m z j

For a block in the real domain, the mapping technique is introduced. In general, for twoGLPHQVLRQDO DUHD ȍ LQ WKH &DUWHVLDQ FRRUGLQDWH ( x1 , x2 ) can be mapped into a square :' in the domain ([1 , [ 2 ) [1 d 1; [ 2 d 1 by using a set of quadratic shape functions with 8 seeds. The quadratic shape functions are defined below 1 N i ([1 , [ 2 ) (1 [1( i )[1 )(1 [ 2( i )[ 2 )([1( i )[1 [ 2(i )[ 2 1) for i 1,2,3,4 4 1 N i ([1 , [ 2 ) (1 [12 )(1 [ 2(i )[ 2 ) for i 5,7 2 1 N i ([1 , [ 2 ) (1 [ 22 )(1 [1(i )[1 ) for i 6,8 2 The coordinate transform can be written as 8

xD

¦N

k

(7) (8) (8)

([1 , [ 2 )xD( k )

(9)

k 1

where ( x1( k ) , x2( k ) ) denotes the coordinate of seed k in the real domain. The first order partial differentials of function u ( x1 , x2 ) in the Cartesian coordinate system gives wu wx1 where

1§ wu wu · wu ¨¨ E11 ¸, E12 J © w[1 w[ 2 ¸¹ wx2

1§ wu wu · ¨¨ E 21 ¸ E 22 J© w[1 w[ 2 ¸¹

,

wx2 wx wx wx1 , J E 22 E 11 E 21 E12 , E12 2 , E 21 1 , E 22 w[ 2 w[1 w[ 2 w[1 Then, they can be expressed in terms of the nodal values as ( j) wu 1 N1 N 2 ª wF1 ([1 , [1(i ) ) ( j) ( i ) wF2 ([ 2 , [ 2 ) º ( k ) E ( [ , [ ) E ( [ , [ ) F F ¦¦ 11 2 2 2 12 1 1 1 « »u wx1 J i 1 j 1 ¬ w[1 w[ 2 ¼

(10)

E11

wu wx 2

wF ([ , [ ( j ) ) º 1 M N ª wF1 ([1 , [1(i ) ) F2 ([ 2 , [ 2( j ) ) E 22 F1 ([1 , [1(i ) ) 2 2 2 »u ( k ) ¦¦ « E 21 w[1 w[ 2 J i 1 j 1¬ ¼

(11)

M

¦D

1k

([1 , [ 2 )u ( k )

k 1

(12)

M

¦D

2k

([1 , [ 2 )u

(k )

k 1

where again k ( j 1) u N1 i . We can also evaluate the first order partial differentials at each node in form of vector as (13) u,D DD u , where T

wu ( x1(1) , x2(1) ) wu ( x1( 2 ) , x2( 2 ) ) wu ( x1( M ) , x2( M ) ) ½ (1) ( 2) (M ) T , ,..., ® ¾ , u {u , u ,..., u } , DD ^DDlk `M uM , w w w x x x D D D ¯ ¿ (k , l 1,2,..., M ) , (14) DDlk DDk ([1(l ) , [ 2(l ) ) where DD is defined as the first order differential matrix. Furthermore, the L-th order partial differentials in two dimensional problems with respect to both coordinates [1 and [ 2 can be obtained approximately by w m nu ([1 , [ 2 ) ) (15) uˈ(1mn , mn L ...2... ([1 , [ 2 ) w[1mw[ 2n The vectors of the higher order partial differentials can be written, in terms of the first order partial differential matrices D1 and D2 , as u,D

( mn ) uˈ 1...2...

D1m Dn2u.

(16)


141

Alternatively, the real domain of one block can be mapped into a square domain, by using Lagrange series rather than using quadratic shape functions with 8 seeds, as following

xD ([1 , [ 2 )

N1' N 2'

¦¦ F ([ , [ ' 1

1

(i ) 1

) F2' ([1 , [1( i ) ) xD( k ')

(17)

i 1 j 1

in which ND'

([ [D( m ) ) (i ) (m) D [D )

([ D

FD' ([1 , [1(i ) )

m 1 m zi

(18)

and the subscript k ' ( j 1) u N1' i and ND' is number of seeds of mapping along axis [D . The number of seeds in total for mapping is M ' N1' u N 2' . In this case, two sets of Lagrange series are applied: one set for interpolation for the functions of displacements and one set for block mapping. Finite block method with FGMs Assumed that the material properties are dependent on the spatial coordinates in a nonhomogeneous material. The relationship between stress and strain anisotropic materials gives ª H11 º «H » « 22 » ¬«2H12 ¼»

ª E11 «E « 12 ¬« E16

E12 E16 º ªV 11 º E 22 E 26 »» ««V 22 »» E 26 E 66 ¼» «¬V 12 ¼»

(19)

where Eij are the elastic compliances of the FGMs. The compliance coefficients can be written in terms of the engineering constants as

E11 1 / E1 , E 22 1 / E2 , E12

Q 12 / E1

Q 21 / E2 ,

E16 K12,1 / E1 K1,12G12 , E 26 K12, 2 / E2 K2,12G12 , E 66 1 / G12

(20)

where E1 and E2 are the Young's moduli along two axes of coordinate, Q 12 and Q 21 are Poisson's ratios, G12 is the shear modulus and K jk ,l and Kl , jk are the mutual coefficients of first and second. The inverse form of the relationship in (19) yields ªV 11 º «V » « 22 » ¬«V 12 ¼»

ªQ11 Q12 Q16 º ª H11 º «Q Q Q26 »» «« H 22 »» . 22 « 12 «¬Q16 Q26 Q66 ¼» ¬«2H12 ¼» For plane stress orthotropic elasticity, material mechanical constants give E1 E2 Q 12 E1 Q11 , Q12 , Q22 , Q16 Q26 0, Q66 G12 . 1 Q 12Q 21 1 Q 12Q 21 1 Q 12Q 21 The equilibrium equations give V DE ,E bD 0 D , E 1,2, x :

(21)

(22)

(23)

where bD are body forces. Applying the differential matrices over (16) for each block, and substituting (21) into equilibrium equation in (23) results, in matrix form, as D1Q11D1 D2Q66D2 u1 D1Q12D2 D2Q66D1 u 2 b1 0 (24) D2Q12D1 D1Q66D2 u1 D1Q66D1 D2Q 22D2 u 2 b 2 0 where uD

{uD(1) , uD( 2 ) ,..., uD( M ) }T , bD

{bD(1) , bD( 2 ) ,..., bD( M ) }T are nodal value vectors and

142


§ Qij(1) ¨ ¨ 0 ¨ ... ¨ ¨ 0 ©

Qij

0 Qij( 2 ) ... 0

... 0 · ¸ ... 0 ¸ . ... ... ¸¸ ... Qij( M ) ¸¹

(25)

in which Qij(k ) indicates the elasticity coefficient at node k . The boundary conditions give

uD (x) uD0 (x)

x *u

V DE nE

x *q

0

tD (x)

(26)

where uD0 and tD0 are specified displacements and tractions on the boundary. Obviously there are 2 M linear algebraic equations from (24) and (26) for each block, and therefore, all nodal values of displacement should be determined. In the case of more than one blocks, the continuous condition on the interface between blocks I and II gives (27) uDI (x) uDII (x) 0, tDI (x) tDII (x) 0. x *int In addition, it can be proved easily that all components of stress long the bounded surface are continuous and thus the numerical accuracy can be improved as expected. For two-dimensional dynamic problems, the equilibrium equations yield w 2u (28) V DE ,E bD U 2D wt where U indicates the mass density of the media, w 2 u i / wt 2 is accelerations along xD axis. By applying the Laplace transformation over both sides of equations (28) with the initial condition, one obtains ~ (29) x: V~DE ,E b U s 2u~D sUD VD

LQZKLFKWKHLQLWLDOFRQGLWLRQVDUHJLYHQLQWKHGRPDLQȍDV wuD (x,0) x: VD (x), uD (x,0) U D (x), wt and the Laplace transformation is defined as f ~ f ( s ) ³ f (t )e st dt

(30)

(31)

0

where s is Laplace transform parameter. Considering the boundary conditions, we have (32) u~D (x, s ) u~D0 (x, s ) x *u , V~DE nE ~ tD0 (x, s ) x *V ~ 0 0 ~ where uD and tD indicate the transformed boundary displacements and tractions on *u and *V . Following the same way as static case, we can express two equilibrium equations in matrix form by applying partial differential matrices in (16) for each block. Substituting (21) into equilibrium equation in (29) results, in matrix form, as D1Q11D1 D2Q66D2 u~1 D1Q12D2 D2Q66D1 u~2 b~1 U s 2u~1 sV1 U1 (33) x: D Q D D Q D u~ D Q D D Q D u~ b~ U s 2u~ sV U 2

12

1

1

66

2

1

1

66

1

2

22

2

2 T

2

2

2

2

where the vectors of initial conditions V {VD } and U {UD } . Again, there are 2 N1 u N 2 linear algebraic equations in total from (33) and from boundary conditions from (32) for each block. By solving a set of linear algebraic equations, all nodal values of displacements can be determined. In the case of two and more blocks, the continuities of displacements and tractions on the interface *int( I , II ) between blocks I and II yield ~ I (x, s ) u ~ II (x, s ), ~ (34) t I (x, s ) ~ t II (x, s ) 0 u x * ( I , II ) D

D

D

D

T

int


143

In the case of two blocks, the number of node in total is 2 M ( 2 M I 2 M II ) which is the same of equation number from the equilibrium equations in the doPDLQ ȍ ERXQGDU\ FRQGLWLRQV DQG connection conditions. Select (L+1) samples in the transformation space sl , l 0,1,..., L for transformed values. Then the displacements in time domain can be obtained by the inversion technique. A simple and accurate inverse method proposed by Durbin [12] is adopted as follows L ~ º 2eKt ª 1 ~ (35) f (K ) ¦ Re f K 2Sik / T e 2Stil / T » , f (t ) « T ¬ 2 l 0 ¼ ~ where f ( sl ) denotes the transformed variable in the Laplace domain, the transformation parameter is

^

`

and T which chosen as sl K 2S ik / T depend on the observing period in the time domain. In the following examples, all variables are normalized with unit dimensions for the convenience of the analysis, i.e. T / t0 20, K 5 , where t0 is time for a specified elastic wave. Stress intensity factors with FGM

In general case, four blocks at least are needed to model mixed mode crack problem as shown in Figure 2. Uniformly distributed or irregular distributed nodes can be used for each block. The stress intensity factors are computed from the asymptotic expansion of the displacements near the crack-tip. For non-homogeneous linear elastic solids, Eischen [13] showed that the asymptotic crack-tip stress and displacement fields have the same form as those in homogeneous linear elastic materials. Even the structure of the asymptotic crack-tip fields are not influenced by the material gradient parameters in FGMs, the stress intensity factors are dependent on the material gradation. Therefore, the simplest and most direct formulations to determine the stress intensity factors are

KI K II

S 2(D 3D 2 D 1D 4 ) 2r

(D 2 'u 2 v D 4 'u1 ) (36)

S (D 3 'u1 D 1 'u 2 ) 2(D 3D 2 D 1D 4 ) 2r

where 'ui ui ui , r is the distance of the evaluation point to the crack tip. The coefficients in (36) are defined § q q · §P q Pq · §p p · §P p P P · (37) D1 Im¨¨ 2 1 1 2 ¸¸, D 2 Im¨¨ 1 2 ¸¸, D 3 Im¨¨ 2 1 1 2 ¸¸, D 4 Im¨¨ 1 2 ¸¸ © P1 P2 ¹ © P1 P2 ¹ © P1 P2 ¹ © P1 P 2 ¹

E11Pk2 E12 E16 Pk , qk E12 Pk E 22 / Pk E 26 in which P k are the roots of the following characteristic equation

(38)

E11P 4 2E16 P 3 (2 E12 E 66 ) P 2 2E 26 P E 22

(39)

pk

0

where all material property parameters Eij are specified at the crack tip. For orthotropic FGM, we have D1 D 4 KI

0 and stress intensity factors can be simplified as

S 'u 2 ˈK II 2D 3 2r

Sa 'u1 . 2D 2 2r

For isotropic FGM, they become Sa 'u 2 Sa 'u1 ˈK II KI 4 E tip 2r 4 E tip 2r

(40)

(41)

For dynamic problems, the mixed mode stress intensity factors are determined, in the Laplace transformed domain, as

144


~ K I ( s)

S (D 2 'u~2 D 4 'u~1 ) 2(D 3D 2 D 1D 4 ) 2r

~ K II ( s )

S (D 3 'u~1 D 1 'u~2 ) 2(D 3D 2 D 1D 4 ) 2r

(42)

Example: Edge crack in a plate under dynamic load An orthotropic FGM plate of w u 2h containing an edge crack of length a shown in Figure 6 is analysed. Heaviside uniform tensile load V 0 H (t ) is applied on the top of the plate and the bottom is fixed. The dimensions is selected w 2a , h w and the shear modulus is an exponential function of x1 , i.e. isotropic material is considered and the elastic moduli have an exponential variation in the x1 direction as i.e. E1 E10 f ( x1 ) , E2 E20 f ( x1 ) , G12 G120 f ( x1 ) and U U0 f ( x1 ) , where f ( x1 )

exp(Dx1 / w) and D is dimensionless constant and E10 , E20 , G120 and U0 are elastic moduli

and mass density at origin, D ln ( E1w / E10 ) , in which E10 and E1w are the Young's moduli of the lefthand and right-hand edges respectively. For the sake of simplicity of the numerical investigation, it is assumed that E1w / E10 E2w / E20 U w / U0 G12w / G120 5 . Firstly, consider a orthotropic material, E-glass-epoxy (A), with Young's moduli E10 8.26G120 , E20 2.26G120 , Q 12 0.227 , Q 21 0.062 and G120 5.5 GPa. Again four blocks are used for the discretization of the considered problem. The stress intensity factor is computed from the normal crack displacements near the crack-tip. Because the structure of the asymptotic crack-tip fields for non-homogeneous solids under dynamic load is the same as that for homogenous material, the dynamic stress intensity factors can be evaluated by (40). Mode I normalized time dependent stress intensity factors K I (t ) /

V 0 Sa versus the normalized time cst / a and solutions by FEM(ABAQUS) are shown Figure 3, where cs Figure 2. Edge crack under dynamic loads.

G120 / U0 . Excellent

agreement compared with the results obtained by FEM has been achieved.

Second rotate the composite with 900, then the material constants become, E-glass-epoxy (B), E10 2.26G120 , E20 8.26G120 , Q 21 0.227 , Q 12 0.062 and G120 5.5 GPa. Mode I normalized time dependent stress intensity factors K I (t ) / V 0 Sa versus the normalized time cst / a for an orthotropic FGMs with different angles are shown Figure 4. The difference between these two results are huge. It can be seen that the arrival time of longitudinal wave travelling from the top to the crack surface for composite (B) is much shorter than that for composite (A).

145

Summary The Finite Block Method was proposed for general linear elastic fracture mechanics for two dimensional problems with anisotropic functionally graded materials. This method considered the governing equations in strong form and is of all advantages of meshless methods. The system equations are formulated with partial differential matrices from the equilibrium equations, boundary conditions and continuous conditions for all blocks in functionally graded media. As the order of the partial differentials is evaluated by Lagrange series in the mapping domain, the computational effort is reduced significantly compared with RBM and MLS interpolations. The dynamic mixed mode stress intensity factors are obtained by COD techniques.

146


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Application of SPR for Discontinuous Boundary Elements Results in 2D Elasticity and a New Method to Evaluate Tangential Stress Otávio A. A. da Silveira1, Rogério J. Marczak2 1

Civil Engineering Department, UFSC

Campus Reitor João David Ferreira Lima, Rua João Pio Duarte da Silva, 205, Bairro Córrego Grande 88037-000, Florianópolis, Brazil [email protected] 2

Mechanical Engineering Department, Universidade Federal do Rio Grande do Sul Rua Sarmento Leite, 435, Centro, 90050-170, Porto Alegre, Brazil [email protected]

Keywords: Boundary element methods, boundary stress, stress recovery, stress smoothing.

Abstract. The use of discontinuous elements in the boundary element method (BEM) does not provide continuous results across de boundary mesh, i.e. variables are not single valued across element interfaces. The implementation of a smoothing technique, able to retrieve continuous results for isoparametric discontinuous boundary elements in two-dimensional elasticity is proposed. The methodology is based on the recovery of smoothed values at the geometric nodes shared by two elements, using least squares fit of the physical nodes values in the neighbourhood. New solutions with the same degree of interpolation of the original ones are obtained in each element from the recovered values and, consequently, a continuous solution can be achieved. Moreover, continuous as well as discontinuous boundary elements generate discontinuous, low accuracy results for the tangential component of stress, which is usually obtained by post processing. This paper presents a new proposal for computing that stress component based on the use of a lower number of sampling for the evaluation of displacement derivatives prior the application of the Hooke’s law. The efficiency of the proposed techniques is verified by solving static elasticity problems using linear and quadratic elements.

Introduction The relaxation of continuity conditions in discretization-based methods has gained impulse among several numerical methods in the last decade. The use of discontinuous elements in boundary element methods (BEM) is somewhat old, but discontinuous Galerkin finite element methods (FEM) are prime examples of the developments in this field. These approaches greatly simplify the computational implementation of the solution methods, and may increase their efficiency, particularly in nonlinear problems or problems containing discontinuous fields. In the BEM context, there are a number of advantages in the use of discontinuous boundary elements in spite of the characteristic interelement discontinuities. Discontinuous interpolation presents C1 continuity on all physical nodes, which simplifies the computation of strongly singular integrals. It also avoids the need of double nodes in cases containing corners and discontinuities in the boundary conditions. In addition, the use of discontinuous elements has already proved its efficiency in the solution of multidomain BEM formulations and FEM-BEM couplings (Zhang and Zhang, 2002). On the other hand, the


recovery of variables at the ends of the elements by simple extrapolation or by averaging the extrapolated results of two or more elements is usually inadequate. The issue approached by the present work is related to the boundary stress components not directly evaluated from BEM boundary solution. It is well known that in 2D elasticity problems only of two stress components are directly given by the traction components along the boundary. The remaining component of the stress tensor must be computed by mixing the known stress components and another term, evaluated by differentiating the shape functions in order to estimate the normal strain in the tangential direction. Regardless the elements are continuous or not, lower accuracy is generally found for these post-processed stress components due to the reduction by one degree in the approximation polynomial. Therefore, one can expect problems similar to those found in FEM for Mindlin plates, where the shear strain is evaluated by mixing p polynomials for the plate rotations with p-1 polynomials for the derivatives of the transverse displacement. This is not a robust method because of two reasons: (a) mixing primal variables with dual ones (obtained by numerical differentiation) may lead to ill-conditioned equations (Guiggiani, 1994); (b) nodes are not the optimal ordinates to recover derivative (dual) variables. Although it is a viable technique for many applications, the tangential stress component may present significant errors when coarse meshes are employed. Aiming the evaluation of more reliable values for the stress components on the boundary, a low-cost alternative technique for computing the normal tangential stress component is presented and tested in this work. The proposed alternative technique for evaluation of the tangential stress is implemented for linear and quadratic discontinuous boundary elements, and used to solve 2-D elasticity benchmarks. The results obtained are compared with the conventional BEM results.

Alternative Tangencial Stress Calculation in Discontinuous Elements In numerical analysis, the computation of quantities by combining interpolated values and its derivatives must be done with care, as the optimal sampling points of the derivatives are not coincident with the interpolation points themselves. This issue is relatively common in many branches of computational mechanics. Prime examples can be found in FEM, for instance, in the calculation of stress in twodimensional elasticity elements, in the evaluation of shear strains in structural elements (plate/shell), or in the pressure-velocity coupling in fluid mechanics. This is essentially the very same problem that causes the locking phenomenon of in low order thick plate finite elements (Oñate et al. 1992, Zienkiewicz et al. 1993). A similar problem occurs in the standard evaluation of the tangential boundary stress components for elasticity in BEM, although not characterized by the same consequences as in FEM. As aforementioned, the missing boundary stress components in the conventional BEM are obtained using shape functions derivatives (tangential strain) and boundary tractions (Brebbia et al., 1984); however, it is known that this technique not necessarily provides good results along the whole element (Guiggiani, 1994). This work suggests a small change in the use of Hooke’s law in order to obtain a more reliable estimate of the tangential stress component for boundary elements without any significant increase in the computational cost. Basically, the tangential strain is sampled at optimal locations, instead of the nodes. It is important to note that the ideas presented herein are implemented and tested for 2D elasticity discontinuous boundary elements, but they can be used in a fairly broad class of problems, regardless the continuity of the interpolation. Standard technique for tangential stress calculation: In 2D elasticity problems, the normal (σnn) and shear (σnt) boundary stress are directly related to the boundary tractions (pn, pt) in a local coordinate system (n, t). Assuming that the tractions are written in

163

164


the global coordinate system, the boundary stress components are easily obtained by rotating the tractions according to the local system: σ nn ½ p1 ½ ® ¾ = R (α ) ® ¾ ¯σ nt ¿ ¯ p2 ¿

(1)

where R(α) is a rotation marix and α is the angle between global and the local coordinate systems (Fig. 1). The tangential strain (εtt) is obtained by using the interpolated displacements (Zhao, 1996): ε tt =

dut du1 du = t1 + 2 t2 dt dt dt

ε tt =

∴

dφ (ξ ) · 1 § dφi (ξ ) i ¦ ¨ dξ u1t1 + di ξ u2i t2 ¸ J © ¹

(2)

where t1 and t2 are components of the unit tangential vector in x1 and x2 directions, respectively, u1i and u2i are the nodal displacements in the i-th node in x1 and x2 directions, respectively; and φi(ξ) are the physical interpolation functions (J is the Jacobian of the element transformation to the normalized space).

n

t

P

x2

α

Ω

x1

Γ

Figure 1. Coordinate system over the boundary. The tangential stress component (σtt), can be obtained by Hooke’s law for plane-strain: σ tt (ξ ) =

1 ªν σ nn (ξ ) + 2Gε tt (ξ )º¼ 1 −ν ¬

(3)

where ν is the Poisson’s ratio, and G is the shear modulus. Therefore, when Eq.(3) is used in the standard BEM, it sums two polynomial terms of different orders. Depending on where σtt is evaluated, this procedure process may lead to unreliable results unless the ξ coordinate is known to be an optimal point to retrieve derivative quantities (present in the εtt term). Alternative technique for tangential stress calculation: The existence of points able to represent optimally the derivative of an interpolated function is well known and can be proved mathematically. In the FEM, these points are known as Barlow points, and they are used to evaluate stress fields from differentiation of interpolated displacements (Barlow, 1976 and Prathap, 1996). When the interpolation function is of the polynomial type, these points are located at the Gauss-Legendre stations corresponding to one order less than the minimum order necessary to integrate the interpolation function exactly. The underlying idea of the scheme proposed here is to use these points


165

to evaluate Eq.(3). To the best of the authors’ knowledge, there are no similar studies correlating these aspects in the BEM context. In the case of linear boundary elements, the normal stress is obtained directly from the traction forces, and therefore it is a linear function (as well as the displacements). The tangential deformation is represented by a constant function in each element since it is obtained by the displacement derivative. The combined use of these two functions, through the Hooke’s law, is the origin of often unsatisfactory results (Guiggiani, 1994). The present work suggests the use of the central point of the element – Gauss point for a linear function integration (ξ = 0) – to sample the differentiation of the interpolated displacement. The coordinate ξ = 0 delivers the best estimate for the tangential strain along the element. It is worth to note that the evaluation of this strain at nodal locations (ξ = ±1) will overestimates or underestimates the strain value. In summary, it is proposed that both, the normal stress and tangential strain should be evaluated at the center of the element, thus obtaining a constant function for tangential stress over each element. For clarity, Table 1 compares both ways for the evaluation of the tangential stress in linear elements. For quadratic discontinuous elements where, a priori, the normal stress and tangential strain are represented by quadratic and linear functions respectively, it is suggested that the Gauss points for a cubic quadrature (ξ = ±1/√3) should be used to represent the tangential strain field along the element. Therefore, replacing the standard technique, the calculation of the tangential stress is performed using the values for normal stress and tangential strain just at two points. A linear interpolation of the values obtained in the Gauss points is made in order to obtain the nodal stress values of the quadratic element. Table 2 shows the two methods for tangential stress calculation on quadratic elements. In summary, the tangential stress calculation is made with one degree less than the other stress components. This method may initially seems less sound than the conventional procedure, but later it will be shown that when used with the smoothing technique described by Silveira (2007), the proposed scheme leads to better results. In many cases, it was found that the conventional scheme will produce wrong signs to the εtt term in Eq.(3), a direct consequence of it being sampled at non-optimal points. Table 1. Tangential stress calculation for linear discontinuous boundary elements*. STANDARD TECHNIQUE

1 σ = (νσ 1nn + 2Gε tt1 ) 1 −ν 1 σ tt2 = (νσ nn2 + 2Gε tt2 ) 1 −ν 1 tt

ALTERNATIVE TECHNIQUE 1 σ tt (ξ1 ) = (νσ nn (ξ1 ) + 2Gε tt (ξ1 ) ) 1 −ν 1 2 σ tt = σ tt refers to constant interpolation of σ tt ( ξ1 )

* ξ1 = 0 , and the superscript represent the associated nodal value of the variable.

Table 2. Tangential stress calculation for quadratic discontinuous boundary elements*. STANDARD TECHNIQUE 1 σ = (νσ nn1 + 2Gε tt1 ) 1 −ν 1 tt

σ tt3 = * ξ1 = −1

1 (νσ nn3 + 2Gε tt3 ) 1 −ν

3 , ξ2 = 1

ALTERNATIVE TECHNIQUE 1 (νσ nn (ξ1 ) + 2Gε tt (ξ1 ) ) 1 −ν 1 σ tt (ξ 2 ) = (νσ nn (ξ 2 ) + 2Gε tt (ξ 2 ) ) 1 −ν 1 2 3 σ tt , σ tt e σ tt are obtained by linear interpolation

σ tt (ξ1 ) =

of σ tt ( ξ1 ) e σ tt (ξ 2 )

3 , and the superscript represent the associated nodal value of the variable.

166


Numerical Results In order to investigate the performance of the proposed technique, this section shows some results. Numerical integration was carried out using 16 Gauss points, in order to minimize the influence of quadrature errors. Dimensions, material properties, and other physical data are given without units, but they were specified to represent a compatible system of units. The material properties used in all case are: E = 210e9 and ν = 0.3 Plane-stress condition is assumed throughout this section. Square-plate with a central hole under traction: A 100×100 square plate with a central hole of radius R = 5 was analyzed. Due to symmetry, only one quarter of the plate was considered (Fig. 2). The traction loading along the upper side was set to P = 1. The offset of all boundary elements used in the mesh is 15% of the element length. Linear and quadratic elements were used with two different meshes for each type of element. Mesh 1 used an element size of 2.5 along the straight boundaries and four elements along the quarter-circle. Mesh 2 used an element size of 1.25 and eight elements along the quarter-circle. P

50

r

θ B

A

50

Figure 2. Squared plate with a central hole under uniform traction. Regarding the application of the alternative method for tangential stress calculation, it can be used with or without a smoothing procedure (Silveira, 2007), leading to four possibilities for postprocessing the results: • Method A: Discontinuous BEM without smoothing – the raw results of discontinuous elements are considered with standard tangential stress calculation (section 2.1). • Method B: Discontinuous BEM with smoothing – same as Method A, but the results are smoothed. • Method C: Modified discontinuous BEM without smoothing – raw results of discontinuous elements with alternative tangential stress calculation as outlined in section 2.2. • Method D: Modified discontinuous BEM with smoothing – same as Method C, but the results are smoothed. These methods were used to post-process the normal radial stress along the edge AB, which is the tangential component along that piece of boundary. Figure 3 compares graphically these results for linear elements with meshes 1 and 2.


The graphs depicted in Fig. 3 shows that the response which agrees more closely to the analytical solution is the smoothed solution considering the alternative tangential stress calculation. As expected, it can be seen that the alternative tangential solution without smoothing is simply an element average value from discontinuous BEM without smoothing. Figure 4 shows the recovery of radial stress on the same edge, this time using quadratic elements. As in Fig. 3, these graphs show the four types of post-processed results against the analytical solution. Although the differences between the four methods are not as drastic as in the case of linear elements, it is evident that the smoothed results obtained with alternative tangential stress calculation agree more closely to the analytical solution, particularly at point A. Another important aspect is that the differences between all methods tend to vanish where analytical solution is less oscillatory (away from stress concentration areas). Interestingly, it is also evident from Figs.3-4 that none of the methods provided very good results near the hole, although the modified stress calculation seems to recover the better ones. This is direct consequence of the different signals of the terms in Eq. (3), i.e. the high gradients of the tangential strain near the hole are miscalculated when the displacements are differentiated at nonoptimal locations. Of course, this effect becomes more conspicuous when coarse meshes are used.

(a)

(b) Figure 3. Radial stress recovery. Linear elements: (a) mesh 1, (b) mesh 2.

167

168


(a)

(b) Figure 4. Radial stress recovery. Quadratic elements: (a) mesh 1, (b) mesh 2.

Conclusions This work presented an alternative technique for tangential stress component calculation in BEM methods, which estimates more reliable results when compared to the standard boundary stress technique. Moreover, it was used a method to obtain smoothed results from the results of tangential stress component, since it is now evaluated at different positions than the nodes. This method of smoothing appears to work very well with the alternative technique for tangential stress calculation. The increase in the computational cost is negligible. The scheme proposed presents potential to be used with other types of elements or different governing equations without hurdles.


References [1] Barlow, J., Optimal Stress Locations in Finite Element Models, International Journal For Numerical Methods in Engineering, vol. 10, pp. 243-251, (1976). [2] Brebbia, C.A., Telles, J.C.F., Wrobel, L.C., Boundary Element Techniques, Springer Verlag, Berlin Heidelberg (1984). [3] Guiggiani, M., Hypersingular Formulation for Boundary Stress Evaluation, Engineering Analysis with Boundary Elements, vol. 13, pp. 169-179 (1994). [4] Silveira, O.A.A., Implementação de Técnicas de Suavização de Resultados para Elementos de Contorno Descontínuos. Dissertação de Mestrado, UFRGS/Porto Alegre. (2007) [5] Prathap, G., Barlow Points and Gauss Points and the Aliasing and Best Fit Paradigms, Computer & Structures, vol. 58, No. 2, pp. 321-325 (1996) [6] Zhang, Xiaosong, Zhang, Xiaoxian, Coupling FEM and Discontinuous BEM for Elastostatics and Fluid–structure Interaction, Engineering Analysis with Boundary Elements, vol. 26, pp. 719-725 (2002). [7] Zhao, Z.Y., Interelement Stress Evaluation by Boundary Elements, International Journal for Numerical Methods in Engineering, vol. 39, pp. 2399-2415 (1996). [8] Zienkiewicz, O.C., Zhu, J.Z., Wu, J., Superconvergent Patch Recovery Techniques – Some Further Tests, vol. 9, pp. 251-258 (1993).

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Solution of the Elastodynamic Contact Problem for Cracked Body Using Boundary Integral Equation Method V.V. Zozulya Centro de Investigacion Cientifica de Yucatan A.C., Calle 43, No 130, Colonia Chuburná de Hidalgo, C.P. 97200, Mérida, Yucatán, México. E-mail: [email protected] Keywords: crack, elastodynamics, contact problem, boundary integral equations, divergent integrals . Abstract. This paper considers the contact interaction of the crack edges in 2-D and 3-D spaces is studied for the case of normal incidence of a periodic tension–compression and shear- dilatational waves respectively. The problem is solved by the method of boundary integral equations (BIE). The dependence of the stress intensity factor on (SIF) the wave number is studied. 1. Introduction. Let us consider an infinite linearly elastic isotropic body referred to Cartesian coordinates x( x1 , x2 , x3 ) . Assume that the body contains a plane crack finite (of length 2l ) along the x1 -axis and infinite along x3 -axis, (1) {x : x2 0, l x1 l} Let a harmonic tension-compression wave that does not depend on the coordinate x3 propagate perpendicularly to the surface . Thus, we deal with a plane problem for an infinite body with a crack of finite length 2l under a harmonic load. The analysis fulfilled in [1-3] without considering the contact interaction of crack edges shows that all the quantities (stresses and displacements) have the factor eiωt everywhere including the neighborhood of the crack tips. Let us consider phenomena occurring in this case near the right or left tip of the crack. When loaded, the body goes through three phases near the crack tip: (i) the initial undeformed state (Fig. 1a), (ii) the tensile phase, corresponding to the maximum crack opening (Fig. 1b), and (iii) the compressive phase, corresponding to the maximum crack closure (Fig. 1c). Phase (iii) begins a half-period after the onset of phase (ii). From Fig. 1c follows that the displacement discontinuities u2 and u2 at some point near the right tip of the crack in the tensile and compression phases are equals. Therefore neglecting of the contact interaction of crack edges led to erroneous results. Researches carried out by the author of this article were addressed this shortcoming. Upper edge

Upper edge

A

Lower edge

a

Upper edge

B

Lower edge

b

Lower edge

c

Fig. 1. Crack tip under harmonic loading The correct formulation of the elastodynamic problem for a cracked body, that takes into account the possibility of crack edge contact interaction and the formation of areas with close contact, adhesion and sliding, was presented first in [4]. Important for applications case of harmonic loading was studied in [6]. In particular, it was shown that harmonic loading results in a steady-state periodic process, but not a harmonic one, if crack edge contact interaction is taken into account. The algorithm for the solution of this problem


171

was elaborated in [5] and in [14] that algorithm is based on a theory of subdifferentional functionals and the finding of their saddle points of convex functional. Mathematical aspects of the problem and a convergence of this algorithm were investigated in [26, 37, 38, 48, 53, 57, 60]. It has been shown that the algorithm may be considered as a compressive operator, acting on special Sobolev`s functional spaces introduced in [48, 53]. The algorithm comprises two parts. The first part is a solution of an elastodynamic problem for bodies with cracks, without taking into account unilateral restrictions. In this stage the elastodynamic problem for cracked body is solved in the space-time, frequency or Laplace transform domains [30]. In [16] for solution of the elastodynamic problem for cracked body without considering crack edges contact interaction the BIE method is proposed. The singularities of the integral operator kernels were studied in for the first time in [7]. Then methods of the divergent integrals regularization have been developed in numerous our publications (see [22, 24, 25, 27, 42, 43, 49, 52, 51, 62]). The second part is a projection on a set of unilateral restrictions. The projection operators were constructed in [5, 14, 48, 57], and several different algorithms were developed. Convergence of the proposed algorithms have been studied analytically in [38, 48, 53] and numerically in [35, 53]. The crack edge contact interaction for a plane with a finite length crack under harmonic loading was studied for the first time in [8] for one crack and in [9] for two cracks. Then influence of the crack edge contact interaction on the stress intensity factor was studied in numerous our publications ( see [10-13, 15, 17, 19-21, 23, 28, 31-34, 36, 39-41, 44-47, 50, 51, 54-56, 58]. For more information see extended reviews [20, 29, 30] and book [18]. In this paper elastodynamic contact problem for cracked body is considered for the case of arbitrary periodical loading. Influence of the crack edges contact interaction on the stress intensity factor is considered. 2. Statement of the problem. Let an elastic body in 3-D Euclidean space R3 occupy a volume V. The body’s boundary V is piece-wise smooth and consists of sections Vp and Vu, to which the vectors of surface load p(x,t) and displacements u(x,t), respectively, are assigned. There is an arbitrary oriented crack with surfaces , where and are the opposite edges. The body may be subjected to volume forces b(x,t). The stress-strain state of the body is described by the displacement equations of the linear dynamic theory of elasticity (2) Aij u j bi ρ t2ui , x V , t [to , t1 ] The operator Aij for an isotropic body has the form Aij μδ ij k k (λ μ ) i j ,

(3)

where i and t are derivatives with respect to a coordinate and time, respectively, λ and μ are the Lame constants, and ρ is the density of the material. We consider the problem defined in an infinite region, then the solution of the Eqs. (2) is uniquely determined by assigning displacements and velocity vectors in the initial instant of time. Then the initial conditions are ui (x, to ) ui0 (x) , t ui (x, to ) vi0 (x) , x V

(4)

The conditions at infinity must be satisfied too. These conditions were considered in [18, 29, 30]. Let us formulate crack-edge conditions. On the crack edges, the vectors of contact forces and displacement discontinuity must satisfy unilateral contact constraints with friction. In [18, 29, 30] it was shown that unilateral boundary conditions with friction have the form u n ho , qn 0 , (u n ho )qn 0 qτ kτ qn t uτ 0 , qτ kτ qn t uτ λt qτ , x , t

(5)

where qn , qτ and u n , uτ are the normal and tangential components of the vectors of contact forces and displacement discontinuity, respectively, h0 is the initial crack opening, and kτ and λτ are coefficients depending on the properties of the contacting surfaces and . In the case of periodical loading the problem is formulated in frequency domain. The stress-strain components have to be expanded into Fourier series, which depend on the loading parameter ,

172


p

pi (x, t ) Re{

σ ij (x, t ) Re{

n iωnt } , u i ( x, t ) i ( x )e

u

Re{

σ ijn (x)e iωnt } , ε ij (x, t ) Re{

n iωnt , i ( x )e

(6)

ε ijn (x)e iωnt },

where qin (x)

σ ijn (x)

ω 2π

T

q (x, t )e i

iωnt

dt , uin (x)

0 T

ω 2π

T

u (x, t )e i

iωnt

dt ,

0 T

(7)

ω ω σ ij (x, t )e iωnt dt , ε ijn (x) ε ij (x, t )e iωnt dt. 2π 2π

0

0

Substituting (6) into (2) we obtain a set of differential equations for steady-state elastodynamics in the form (8) Aij u nj ω 2 n 2ui* 0, x V , n 0, 1,..., . In view of (1.9), the boundary conditions have the form q n (x, t ) , pin (x, t ) i * pi (x, t ) qin (x, t ),

x e , n 1 , x e , n 1

(9)

where e is the close-contact region. Thus, the problem under consideration reduces to finding the Fourier coefficients (7) of the stress-strain components. These components must satisfy additional conditions. After calculations the physical components of the vectors of contact-interaction forces and displacement discontinuity using Eqs.(6), they must satisfy the unilateral constraints (5). With such an approach, the initial-boundary-value problem (2), (4) with the unilateral constraints (5) is reduces to the boundary-value problems (8) with the parameter ω 2 n 2 and the unilateral constraints (5). 3. Transforming the problem to the BIE and unilateral constrains. In [16, 29, 30] it has been shown that the elastodynamic boundary-initial problem for a body with cracks and with allowance for their edge contact interaction may be transformed into the BIEs and the unilateral constrains (5). In the space-time domain for finite length crack in an unbounded body with homogeneous initial data the BIE has the form (10) p j (y , t ) dτ ui (x,τ ) Fij (x, y , t τ )d

In the frequency domain the BIE for the Fourier coefficients for the traction pin (x,ω ) and the displacement discontinuity uin (x,ω ) may be presented in the form p nj (y , ωn ) uin ( x, ωn ) Fij ( x, y , ωn ) d

(11)

The kernels in the integral equations are calculated in the following way. Fist fundamental solutions for the differential equations (2) or (8) are calculated if the problem is solved in space-time or frequency domains respectively. In [20] it is shown that corresponding fundamental solutions have the form ψδ ij χ i r j r (12) U ij (y x, ) , r 2 ( yi xi )( yi xi ) , απμ Here “ ” stands for t in the time domain formulation, and for ω in the frequency domain formulation, α 4 in the 3-D case and α 2 in the 2-D case and the functions ψ and χ are introduced in [30]. The kennels Fij ( y x, ) can be obtained by applying to U ij (y x, ) the differential operator Pij λδ ij n μ ni j n j i

(13)

with respect to y and x . Exactly form of the corresponding fundamental solutions were presented in [12, 29] for the 2-D case and in [25, 27] for the 3-D case respectively. Obtained kernels are hypersingular and corresponding integrals are


173

divergent. In above mentioned our publications special regularization methods have been developed. The main results in simplest firm can be expressed as a

J 2 F .S .

( x y) dy

a

2

dy

2 , ax

(14)

in the 2-D case and as J 30,0 F .P. S

J

2,0 5

r dS n3 dl , r3 r S

( x1 y1 )( x2 y2 ) 1 3( x y )( x y2 )rn r* 3 ]dl , dS [ 1 1 5 2 5 r 3 S r r S

J 51,1 F .P.

( xα yα ) 2 ( x y ) 2 r 2( xα yα ) nα 2rn 3 ]dl dS [ α 5α n 5 r r 3r 3 3r S S

(15)

F .P.

in the 2-D case. For more information see [22, 24, 25, 27, 42, 43, 49, 52, 51, 62]. The algorithm combines either the solution of the unconstrained problem and the projection operators introduced in [14]. In simplest form the algorithm consists of the following steps: (a) specify an initial distribution of contact forces qi0 (x) , x on the crack edges, (b) solve the problem without constraints and determine the unknown quantities ui (x) over the entire region and on its boundary and ui (x) on the crack edges, (c) correct the normal and tangential components of the vector of contact forces to satisfy the unilateral constraints with friction

qnk 1 (x) Pn [qnk (x) ρ n (unk 1 (x) h0 (x))], qτk 1 (x) Pτ [qτk (x) ρτ uτk 1 (x)] ,

(16)

where Pn [qn ] and Pτ [qτ ] are the operators of projection on the sets K cn (qn ) and Kτc (qτ ) defined in (5.7), and the parameters ρ n and ρτ are selected so as to provide the best convergence for the algorithm, (d) proceed to the next step of iteration. The convergence of this algorithm has been proved in above mentioned our publications. 4. Dependence of the SIF on wave number in 2-D case. We consider a crack of finite length be located in the plane R 2 x : x3 0 . The surface of the crack is described by the Eq. (1). Harmonic waves propagate in the plane perpendicular to the surface of crack. The vector of traction on the crack edges has the form pn (x, t ) Re{μ k12φ0 ei ( k1 x2 ωt ) } or pτ ( x, t ) Re{μ k22ψ 0 ei ( k2 x2 ωt ) }

(17)

for tension-compression P-wave or dilatational SV-wave respectively. For information on the distribution of contact forces in this problem refer to mentioned above our publications. Fig. 2 shows the dimensionless SIF K Imax p πl versus the wave number k1. Curves 1 and 2 represent the cases, respectively, without and with regard for the contact interaction. From the figure, it follows that the maximum SIF corresponds to k2 = 0.8. In the problem accounting for the contact interaction of crack edges, the maximum SIF exceeds the corresponding static values by 15%. In the problem neglecting the contact interaction, the dynamic SIF exceeds the static SIF by 30%. Fig. 3 shows the dimensionless SIF K IImax p πl versus the wave number k1. Curve 1 corresponds to the case where friction is not taken into account, while curves 2 and 3 to the cases where the friction force is 20% and 60%, respectively, of the maximum induced force. From the figure, it follows that the SIF decreases as the coefficient of friction increases. The maximum SIF in the dynamic problem without friction exceeds the corresponding static SIF by 20%.

174


Fig. 2. Dependence of K Imax K Istat on the wave number: 1 – without contact, 2 – with contact

Fig. 3. Dependence of K IImax K IIstat on the wave number: 1 – without contact, 2, 3 – with contact

5. Dependence of the SIF on wave number in 3-D case. In the 3-D case we consider unbounded elastic bodywith a penny-shaped crack in the plane R 2 x : x3 0 . The surface of the crack is described by its Cartesian coordinates: (18) x12 x22 1, x3 0

Harmonic waves propagate in the plane perpendicular to the surface of crack. The vector of traction on the crack edges has the form

pn (x, t ) Re k12φ0 ei ( k1 x3 ωt )

or pτ* (x, t ) ( p1* , p2* ) Re μ k22ψ 0 ei ( k2 x3 ωt ) ,0

(19)

for tension-compression P-wave or dilatational SV-wave respectively. For information on the distribution of contact forces in this problem refer to mentioned above our publications. Fig. 4 shows the dimensionless SIF K Imax p πl versus the wave number k1. Curves 1 and 2 represent the cases, respectively, without and with regard for the contact interaction. From the figure, it follows that if the contact interaction is taken into account, the maximum SIF exceeds its static value by 20%. If the contact interaction is neglected, the dynamic SIF exceeds the static SIF by 55%. However, the contact interaction changes the solution not only quantitatively but also qualitatively - the maximums of curves 1 and 2 do not coincide.

Fig. 4. Dependence of K Imax K Istat on the wave

Fig. 5. K IImax K IIstat on the wave

number: 1 - without contact; 2 - with contact

number: 1 - without contact; 2,3 - with contact


Fig. 5 shows the dimensionless SIF K IImax p πl versus the wave number k1. Curve 1 corresponds to the case where friction is not taken into account, while curves 2 and 3 to the cases where the friction force is 20% and 60%, respectively, of the maximum induced force. From the figures, it follows that if the contact interaction is neglected, the dynamic SIF exceeds the static SIF by 30%. However, the contact interaction changes the solution not only quantitatively but also qualitatively—the maximums of curves 1, 2, and 3 do not coincide. Conclusions. We have considered solution problems of fracture dynamics where the unilateral contact interaction of crack edges and friction are taken into account by BIE methods. The results presented here and in the previous publications show how the contact interaction of the crack edges influences the criteria of fracture mechanics. This should be taken into account in strength analyses of structures by methods of fracture mechanics.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14]

[15] [16]

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[40] Guz A. N., Zozulya V.V., Men'shikov A.V., Three-dimensional dynamic contact problem for an elliptic crack interacting with a normally incident harmonic compression–expansion wave, International Applied Mechanics, 39(12), 1425-1428, 2003. [41] Guz A. N., Zozulya V.V., Men'shikov A.V., General spatial dynamic problem for an elliptic crack under the action of a normal shear wave, with consideration for the contact interaction of the crack faces, International Applied Mechanics, 40(2), 156-159, 2004. [42] Zozulya V.V. Regularization of the divergent integrals. I. General consideration. Electronic Journal of Boundary Elements, 4(2), 49-57, 2006. [43] Zozulya V.V. Regularization of the divergent integrals. II. Application in fracture mechanics. Electronic Journal of Boundary Elements, 4(2), 58-56, 2006. [44] Zozulya V.V. Contact problem for the flat crack under normally incident antiplane shear wave, International Applied Mechanics, 43(5), 532-537, 2007. [45] Guz A.N., Menshykov O.V., Zozulya V.V., Guz I.A. Contact problem for the flat elliptical crack under normally incident shear wave computer. CMES. Modeling in Engineering & Science, 17(3), 205-214, 2007. [46] Zozulya V.V. Stress intensity factor in contact problem for the flat crack under antiplane shear wave. International Applied Mechanics, 43(9), 1043-1047, 2007. [47] Guz A.N., Zozulya V.V. Investigation of the effect of frictional contact in III-mode crack under action of the SH-wave harmonic load. CMES Modeling in Engineering & Science, 22(2), 119-128, 2007. [48] Zozulya V.V. Variational formulation and nonsmooth optimization algorithms in elastostatic contact problems for cracked body. CMES Computer Modeling in Engineering & Science, 42(3), 187-215, 2009. [49] Zozulya V.V. The Regularization of the divergent Integrals in 2-D elastostatics. Electronic Journal of Boundary Elements, 7(2), 50-88, 2009. [50] Guz A.N., Zozulya V.V. On dynamical fracture mechanics in the case of polyharmonic loading by P– waves, International Applied Mechanics, 45(9), 1033-1039, 2009. [51] Guz A.N., Zozulya V.V. On dynamical fracture mechanics in the case of polyharmonic loading by H– waves, International Applied Mechanics, 46(1), 138-144, 2010. [52] Zozulya V.V. Regularization of hypersingular integrals in 3-D fracture mechanics: Triangular BE, and piecewise-constant and piecewise-linear approximations, Engineering Analysis with Boundary Elements, 34(2), 105-113, 2010. [53] Zozulya V.V. Nonsmooth optimization algorithms in some problems of fracture dynamics, Intelligent Information Management, 2, 637-646, 2010. [54] Guz A.N., Zozulya V.V. Contact problem for the flat crack under two normally incident shear Hwaves with wave mode-shifting, Theoretical and Applied Fracture Mechanics, 54(3), 189-195, 2010. [55] Zozulya V.V. Divergent integrals in elastostatics: regularization in 3-D case. CMES. Computer Modeling in Engineering & Science, 70(3), 253-349, 2010. [56] Guz A.N., Zozulya V.V. Contact problem for the flat crack under two normally incident tensioncompression waves with wave mode-shifting, Engineering Analysis with Boundary Elements, 35(1), 34-41, 2011. [57] Zozulya V.V. Variational formulation and nonsmooth optimization algorithms in elastodynamic contact problems for cracked body, Computer Methods in Applied Mechanics and Engineering, 200(58), 525-539, 2011. [58] Guz A.N., Zozulya V.V. Contact problem for the mode III crack under two normally incident shear HS-waves with wave mode-shifting, Theoretical and Applied Fracture Mechanics, 35(1), 34-41, 2012. [59] Zozulya V.V. Comparative study of time and frequency domain BEM approaches in frictional contact problem for antiplane crack under harmonic loading, Engineering Analysis with Boundary Elements, 37, 1499-1513, 2013. [60] Zozulya V.V. Variational formulation and nonsmooth optimization algorithms in elastostatic contact problems for cracked body. In Hetnarski R.B. (Ed.) Encyclopedia of Thermal Stresses, 6327 – 6341. Springer, New York, 2014. [61] Zozulya V.V. An approach based on generalized functions to regularize divergent integrals, Engineering Analysis with Boundary Elements, 40, 62-80, 2014. [62] Zozulya V.V. Regularization of the divergent Integrals. Comparison of classical and generalized functions approaches. Advances in Computational Mathematics, 41, 727-780, 2015.

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A Boundary Spectral Element Model for Piezoelectric Smart Structures Fangxin Zou1, M. H. Aliabadi2 1

Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK, [email protected] 2

Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK, [email protected]

Keywords: boundary integral equation, boundary spectral element method, structural health monitoring, damage detection, smart structure, piezoelectric transducer

Abstract. It has been demonstrated in previous work that the computational efficiency of the boundary element method for elastodynamic analysis can be significantly improved through the employment of high-order spectral elements for boundary discretization. In this work, a boundary spectral element model of the piezoelectric smart structures that are commonly seen in structural health monitoring applications is formulated. The new formulation has been validated by both the finite element method and physical experiments. A considerable reduction in computational expenses over conventional boundary element models has been achieved. The new formulation aims at providing an efficient, stable and accurate numerical tool for the development of SHM applications. Introduction Structural health monitoring (SHM) has long become a highly sought-after approach for ensuring the integrity of in-service equipment and structures. The realization of SHM depends on transducers that are permanently installed on the objects to be monitored. Piezoelectric transducers have been widely used to excite and receive ultrasonic waves for non-destructive testing purpose. Bare piezoelectric patches are particularly favored by SHM applications due their small form factor, low power consumption, and thus the ability to form transducer networks. Structures that are instrumented by piezoelectric transducers for self-monitoring purpose are often referred to as piezoelectric smart structures. If the dynamic response of a piezoelectric smart structure can be predicted accurately and efficiently using mathematical models, the development process will be significantly simplified since experimental studies will not heavily needed. While the responses of simple structures can certainly be determined analytically, the formulation of structures with complex geometries and/or material properties demands numerical approaches. Until now, the finite element method (FEM) has no doubt been the most widely used tool for modelling SHM applications. Lately, the spectral element method (SEM) [1] has gained much interest from the SHM community since it requires much less computational resources than the FEM for resolving high-frequency wave propagation. The boundary element method (BEM) has been used by Zou et al [2] and Zou and Aliabadi [3] for modelling the behaviors of piezoelectric smart structures in damage detection applications. Although the boundary element models demonstrate much higher efficiency and stability than equivalent finite element models, they still struggle in large-scale and high-frequency applications. While the incorporation of spectral elements in the BEM has been briefly investigated in the scope of fluid mechanics [4], it has not been paid much attention to due to the nature of the problems that are of greater interest to the BEM community. Recently, Zou [5] exploited features of the SEM to improve the computational efficiency of a 3D BEM. The resultant formulation is referred to as the Boundary Spectral Element Method (BSEM). In this paper, the conventional boundary element models of the 3D piezoelectric smart structures in ultrasonic wave (UW) based and electrotechnical impedance (EMI) based damaged detection applications are reformulated by the BSEM. Elastic substrates are modelled by a frequency domain transformed BEM, and piezoelectric patches by a semi-analytical finite element approach. The two parts are coupled via boundary displacements and tractions. The results of the new formulation show


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excellent agreement with both the results computed by the FEM and the signals obtained from physical experiments. Boundary Spectral Element Method Transformed Boundary Element Method. For elastodynamic analysis in frequency domain, the displacement of a point in the domain along the boundary of the body by Somigliana identity as (1)

where and are referred to as the source and the field point, and are the fundamental solutions of elastodynamic displacement and traction in frequency domain, and is the generalized frequency term. By imposing the limit , i.e. moving the source point to the boundary, eq (1) can be rewritten as (2)

in which the value of depends on the location of the source point. In this work, the model of the piezoelectric smart structures in UW based damage detection applications is formulated in the Laplace domain such that (3) In contrast, EMI based damage detection applications are simulated in the Fourier domain in which (4) Boundary Discretization. By employing high-order interpolating polynomials as their shape functions, isoparametric spectral elements enable more efficient and accurate approximation of variables. In this work, 36-node Lobatto elements (an example of which is illustrated in Fig. 1) are used for boundary discretization. The nodes of these elements are distributed according to six Lobatto quadrature points. The shape functions comprise 5th-order Lagrange polynomials and are given by (5) For each of the 36 shape functions attainable from eq (5), and are the coordinates of the node that the shape function is associated with, and and represent the coordinates of the remaining nodes. Using 36-node Lobatto elements for boundary discretization, eq (2) can be discretized as (6)

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where is the total number of elements, and can be obtained by the consideration of rigid body motion. By evaluating eq (6), a linear system of equations can be obtained as (7)

Figure 1 A 36-node Lobatto element Model of Piezoelectric Transducers Elemental State-Space Equation. The coupling of the mechanical, the electrical and the piezoelectric behaviors of a piezoelectric patch can be represented by an elemental state-space equation as [3] (8)

where is the electric potential, is the stress, is the electric displacement, is the thickness of patch, and the subscripts and refer to the top and the bottom surface of the patch. Eq (10) is essentially finite element based in the in-plane direction and analytical across the thickness. Model of Piezoelectric Actuators. In practice, the bottom surface of a piezoelectric patch is often made the reference surface with zero electric potential. Also, in this work, the top surfaces of piezoelectric patches are free from attachment and are therefore traction-free. By considering these two boundary conditions, the model of piezoelectric actuators can be derived from eq (10) as [3] (9) Model of Piezoelectric Sensors. In addition to the two boundary conditions used to derive the model of actuators, additional boundary conditions are needed in order to arrive at the model of sensors. Firstly, the top and the bottom surface of a piezoelectric patch are always coated with thin metallic films so that on each surface a single electric potential is measured. Also, as piezoelectric sensors are not subjected to external electric loading, there are no free electric charges. By considering all four boundary conditions, the model of piezoelectric sensors can be found to be [3] (10) (11)


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Coupling of Substrates and Piezoelectric Transducers For a substrate on which piezoelectric transducers are bonded, eq (7) can be expanded into (12)

where the subscripts indicate the interfaces between the transducers and the substrate, the subscripts denote the remaining boundary of the substrate, and are the number of actuators and that of sensors, and and refer to the -th actuator and the -th sensor. By considering the continuity of displacements and tractions at the interfaces between the transducers and the substrate and by substituting eq (9) and eq (10) into eq (12), the following equation can be obtained (13)

By acknowledging that the terms , , and together resemble the boundary of the substrate and by separating known boundary conditions from unknown nodal values, eq (13) can be rearranged into (14)

The solution of eq (14) are essentially the nodal displacements of the boundary of the substrate. Solution of System of Equations For modelling UW based damage detection applications, eq (14) is to be solved for a certain number of Laplace terms, and the nodal displacements of the bottom surfaces of piezoelectric sensors are converted into electric potential via eq (11). On the other hand, the simulation of EMI based damage detection applications depends on the solutions of eq (14) at the frequencies of interest. EMI values can be calculated from the nodal displacements at the bottom surfaces of piezoelectric actuators using the following equations (15) (16) (17) (18) (19) Numerical and Experimental Validations Fig. 2 shows the experimental specimen that was used to validate the boundary spectral element models introduced in this work. Equivalent finite element models were established using Abaqus®.

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Figure 2 The experimental specimen and the schematic diagram of its top view Mesh Convergence. The mesh convergence study of the BSEM was carried out based on the Laplace domain impulse response of a 1D bar structure [3]. The parameters used in the simulations are and . From Fig. 3, it is can be seen that the BSEM with 36-node Lobatto elements converges much more quickly that the conventional BEM with 8-node serendipity elements.

Figure 3 Comparison of the mesh convergences of the BSEM and the conventional BEM UW based Damage Detection. The sensor signals of the pristine and the damaged state of the specimen, computed by the BSEM and the FEM and obtained from physical experiments, are shown in Fig. 4. The excitation frequency is 100 kHz, and the simulation parameters are and . The three sets of results show excellent agreement though after a certain time period, the results of the FEM start to demonstrate numerical instability. EMI based Damage Detection. The EMI values of the pristine and the damaged state of the specimen, computed by the BSEM and obtained from physical experiments, are displayed in Fig. 5. An outstanding agreement between the two sets of results is observed. Conclusion In this paper, the boundary spectral element formulations of the piezoelectric smart structures used in UW and EMI based damage detection applications are presented. Comparing to conventional boundary


element models, the new formulations have achieved a significant improvement in computational efficiency through the use of high-order spectral boundary discretisation. The results computed by the BSEM have been validated by physical experiments, and where possible, by the FEM. In summary, the BSEM has shown great potential to become an efficient, accurate and robust numerical tool for the development of piezoelectric transducer based damage detection applications.

Figure 4 Sensor signals of the pristine and the damage state of the specimen

Figure 5 EMI values of the pristine and the damage state of the specimen References 1. 2. 3. 4. 5.

Patera, A.T., A spectral element method for fluid dynamics: laminar flow in a channel expansion. Journal of computational Physics, 1984. 54(3): p. 468-488. Zou, F., I. Benedetti, and M. Aliabadi, A boundary element model for structural health monitoring using piezoelectric transducers. Smart Materials and Structures, 2013. 23(1): p. 015022. Zou, F. and M. Aliabadi, A boundary element method for detection of damages and selfdiagnosis of transducers using electro-mechanical impedance. Smart Materials and Structures, 2015. 24(9): p. 095015. Muldowney, G. and J. Higdon, A spectral boundary element approach to three-dimensional Stokes flow. Journal of Fluid Mechanics, 1995. 298: p. 167-192. Zou, F., A boundary element method for modelling piezoelectric transducer based structural health monitoring. 2015, Imperial College London.

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Multi-Fidelity Modelling Structural Reliability Analysis Llewellyn Morse 1, a, Zahra Sharif Khodaei 1, M. H. Aliabadi 1 1

Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW7 2AZ, London, UK. a

[email protected]

Keywords: Multi-fidelity modelling; Boundary Element Method (BEM); Reliability analysis; Monte Carlo Simulation (MCS); First Order Reliability Method (FORM); Second Order Reliability Method (SORM)

Abstract In this work, a method for the application of multi-fidelity modelling to the reliability analysis of 2D elastostatic structures using the Boundary Element Method (BEM) is proposed. Reliability analyses were carried out on a rectangular plate with a centre circular hole subjected to uniaxial tension using Monte Carlo Simulations (MCS), the First Order Reliability Method (FORM), and the Second Order Reliability Method (SORM). Two BEM models were investigated, a low-fidelity model (LFM) of 20 elements and a high-fidelity model (HFM) of 100 elements. The response of these models at several design points was used to create multi-fidelity models (MFMs) utilising 2nd order polynomial response surfaces and their reliability, alongside that of the LFM and the HFM, was evaluated. Results show that the MFMs that directly called the LFM were significantly superior in terms of accuracy to the LFM, achieving very similar levels of accuracy to the HFM, while also being of similar computational cost to the LFM. These direct MFMs were found to provide good substitutes for the HFM for MCS, FORM, and SORM. References [1] Huang X, Aliabadi F. A boundary element method for structural reliability. Key Engineering Materials. 2015;627:453-6. [2] Haldar A, Mahadevan S. Probability, Reliability and Statistical Methods in Engineering Design: John Wiley & Sons; 1999. [3] Du X. Probabilistic Engineering Design 2005 [14/02/17]. Available from: http://web.mst.edu/~dux/repository/me360/me360_presentation.html. [4] Su C, Zhao S, Ma H. Reliability analysis of plane elasticity problems by stochastic spline fictitious boundary element method. Engineering Analysis with Boundary Elements. 2012;36(2):118-24. [5] Leonel ED, Venturini WS. Probabilistic fatigue crack growth using BEM and reliability algorithms. 2011;1:3-14. [6] Huang X, Aliabadi MH. Probabilistic fracture mechanics by the boundary element method. International Journal of Fracture. 2011;171(1):51-64. [7] Su C, Xu J. Reliability analysis of Reissner plate bending problems by stochastic spline fictitious boundary element method. Engineering Analysis with Boundary Elements. 2015;51:37-43. [8] Aliabadi MH. The Boundary Element Method: Applications in solids and structures: John Wiley & Sons; 2002. [9] Yu L, Das PK, Zheng Y. A response surface approach to fatigue reliability of ship structures. Ships and Offshore Structures. 2009;4(3):253-9. [10] Hassanien S, Kainat M, Adeeb S, Langer D, editors. On the use of surrogate models in reliabilitybased analysis of dented pipes. 11th International Pipeline Conference; 2016; Calgary, Alberta, Canada. [11] Lee OS, Kim DH, editors. Reliability of Fatigue Damaged Structure Using FORM, SORM and Fatigue model. World Congress on Engineering; 2007; London, U.K. [12] Zhao W, Wang W. Application of Cokriging Technique to Structural Reliability Analysis. 2011:170-4. [13] Sun G, Li G, Zhou S, Xu W, Yang X, Li Q. Multi-fidelity optimization for sheet metal forming process. Structural and Multidisciplinary Optimization. 2010;44(1):111-24. [14] Vitali R, Haftka RT, Sankar BV. Multi-fidelity design of stiffened composite panel with a crack. Structural and Multidisciplinary Optimization. 2002;23(5):347-56.


[15] Perdikaris P, Venturi D, Royset JO, Karniadakis GE. Multi-fidelity modelling via recursive cokriging and Gaussian-Markov random fields. Proceedings Mathematical, physical, and engineering sciences. 2015;471(2179):20150018. [16] Keane AJ, Sóbester A, Forrester AIJ. Multi-fidelity optimization via surrogate modelling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2007;463(2088):3251-69. [17] Glaz B, Goel T, Liu L, Friedmann P, Haftka R. Application of a Weighted Average Surrogate Approach to Helicopter Rotor Blade Vibration Reduction. 2007. [18] Simpson T, Toropov V, Balabanov V, Viana F. Design and Analysis of Computer Experiments in Multidisciplinary Design Optimization: A Review of How Far We Have Come - Or Not. 2008. [19] Khuri AI, Mukhopadhyay S. Response surface methodology. Wiley Interdisciplinary Reviews: Computational Statistics. 2010;2(2):128-49. [20] Rakwitz R, Fiessler B. Structural Reliability Under Combined Random Load Sequences. Computers and Structures. 1978;9(5):484-94. [21] Breitung K. Asymptotic Approximations for Multinormal Integrals Journal of Engineering Mechanics 1984;110(3):357-66. [22] Rackwitz R. Reliability analysis - a review and some perspectives. Structural Safety. 2001. [23] Won Kim D, Man Kwak B. Reliability-based shape optimization of two-dimensional elastic problems using BEM. Computers and Structures. 1995;60(5):743-50.

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Spectral BEM for Wave Propagation and Crack Dynamics Jun Li 1, a, Zahra Sharif Khodaei 1, M. H. Aliabadi 1 1

Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW7 2AZ, London, UK. a

[email protected]

Keywords: Boundary Element Method (BEM); Dual Boundary Element Method (DBEM); Spectral elements; High-frequency wave propagation; Dynamic Stress Intensity Factor (DSIF)

Abstract The aim of this paper was to investigate the computational efficiency of the spectral BEM, which involves the use of spectral elements. The types of spectral elements adopted in this study were based on Lobatto polynomials and Legendre polynomials. Two-dimensional analyses of wave propagation and fracture mechanics were carried out with the BEM and the spectral BEM on the basis of the Laplace transform method. Under the same level of accuracy, it was found that the use of spectral elements, compared with conventional quadratic elements, reduced the total number of nodes required for modelling both high-frequency wave propagation and cracks. Although more integration points were used for the integrals associated with spectral elements than the conventional quadratic elements, shorter computation times were achieved through the application of the spectral BEM. This indicates that the spectral BEM is a more efficient method for the numerical modelling of structural health monitoring (SHM) processes, in which high-frequency waves are commonly used to detect damage, such as cracks, in structures. References [1] Kudela P, Żak A, Krawczuk M, Ostachowicz W. Modelling of wave propagation in composite plates using the time domain spectral element method. Journal of Sound and Vibration. 2007;302:728-45. [2] Dauksher W, Emery AF. The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements. Computer methods in applied mechanics and engineering. 2000;188:217-33. [3] Żak A. A novel formulation of a spectral plate element for wave propagation in isotropic structures. Finite Elements in Analysis and Design. 2009;45:650-8. [4] Schulte R, Fritzen C, Moll J. Spectral element modelling of wave propagation in isotropic and anisotropic shell-structures including different types of damage. IOP Conference Series: Materials Science and Engineering: IOP Publishing; 2010. p. 012065. [5] Ostachowicz W, Kudela P. Wave propagation numerical models in damage detection based on the time domain spectral element method. IOP Conference Series: Materials Science and Engineering: IOP Publishing; 2010. p. 012068. [6] Peng H, Meng G, Li F. Modeling of wave propagation in plate structures using three-dimensional spectral element method for damage detection. Journal of Sound and Vibration. 2009;320:942-54. [7] Zou F, Benedetti I, Aliabadi M. A boundary element model for structural health monitoring using piezoelectric transducers. Smart Materials and Structures. 2013;23:015022. [8] Portela A, Aliabadi M, Rooke D. The dual boundary element method: effective implementation for crack problems. International journal for numerical methods in engineering. 1992;33:1269-87. [9] Rezayat M, Shippy D, Rizzo F. On time-harmonic elastic-wave analysis by the boundary element method for moderate to high frequencies. Computer methods in applied mechanics and engineering. 1986;55:349-67. [10] Zou F, Aliabadi M. A boundary element method for detection of damages and self-diagnosis of transducers using electro-mechanical impedance. Smart Materials and Structures. 2015;24:095015. [11] Zou F. A Boundary Element Method for Modelling Piezoelectric Transducer based Structural Health Monitoring: Imperial College London; 2015. [12] Zou F, Aliabadi M. On modelling three-dimensional piezoelectric smart structures with boundary spectral element method. Smart Materials and Structures. 2017;26:055015. [13] Durbin F. Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate's method. The Computer Journal. 1974;17:371-6. [14] Fedelinski P, Aliabadi M, Rooke D. The Laplace transform DBEM for mixed-mode dynamic crack analysis. Computers & structures. 1996;59:1021-31.


[15] Aliabadi M. The Boundary Element Method: Applications in Solids and Structures, Vol. 2. Chicester: Wiley. 2002. [16] Telles J. A selfϋadaptive coϋordinate transformation for efficient numerical evaluation of general boundary element integrals. International journal for numerical methods in engineering. 1987;24:95973. [17] Martínez J, Domínguez J. On the use of quarterϋpoint boundary elements for stress intensity factor computations. International journal for numerical methods in engineering. 1984;20:1941-50. [18] Fedelinski P, Aliabadi M, Rooke D. A single-region time domain BEM for dynamic crack problems. International journal of solids and structures. 1995;32:3555-71. [19] Eringen A, Suhubi E. Elstodynamics, Vol. II.(Linear Theory). Academic Press: New York; 1975.

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Three-dimensional Analysis of Generally Anisotropic Piezoelectric Materials by the BEM Based upon Radon-Stroh Formalism Chung-Lei Hsu, Yui-Chuin Shiah, and Chyanbin Hwu* Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan, R.O.C. (* Corresponding to: [email protected]) Keywords: Boundary element method, Radon-Stroh formalism, anisotropic piezoelectricity.

Abstract. This paper presents a piezoelectric study for three-dimensional (3D) anisotropic structures by the boundary element method (BEM) with fundamental solutions formulated using real variable Radon-Stroh formalism. As in 2D cases, pertinent equations of the 3D cases are formulated by simply combining matrices of elasticity as well as those of piezoelectricity into extended forms. Also, evaluations of the fundamental solutions by the Radon-Stroh formalism are discussed. In the end, an illustrative example is presented. Introduction In advanced structures, piezoelectric materials have been widely applied for various purposes. A great amount of literatures of BEM have been reported in this regard, they are mostly for two-dimensional (2D) problems though. For successful implementation of BEM to practical applications, the key step is to evaluate the fundamental solutions in the boundary integral equation. Due to mathematical complexity involved, evaluations of the fundamental solutions of 3D anisotropic elasticity have been a focus in the BEM community (e.g [1]-[5]); however, studies on its derivatives have remained very scarce indeed. Among these, the approach by Radon transform [2,6] based on the Stroh formalism has revealed itself usefulness when applied to construct complicated 3D Green’s functions from corresponding 2D solutions to particular problems, such as those of half-space [2] and bi-materials [7]. The well-known Stroh formalism can be used to obtain solutions of the particular 2D problems, by which the corresponding 3D solutions can thus be constructed simply by integrating those of the 2D cases. For saving computation costs in calculating Stroh’s eigenvalues and eigenvectors for the complex solutions, real variable solutions can be formulated using 2D Stroh’s identities as presented in [8,9]. It is well known that the classical solutions for piezoelectric effects can be obtained by simply combining all related matrices of elasticity as well as piezoelectricity into the same mathematical forms but with expanded dimensions in the Stroh formalism. Among the very scarce works to treat 3D problems, Xie et al. [10,11] presented this extended approach for piezoelectric and magnetoelectroelastic materials, where the Green’s function was obtained by both residue calculus and Stroh eigenvector method but not the Radon-Stroh formalism; however, no further implementation has been reported. Previously, the present authors had demonstrated a BEM analysis for pure elastic analysis based upon real variable Radon-Stroh formalism [9]. As in 2D cases, the 3D fundamental solutions for piezoelectric solids can also be obtained using the Radon-Stroh formalism and the expansion of matrices. In this paper, this methodology is implemented in BEM to study the piezoelectric effects in 3D generally anisotropic materials, where the Green’s function is derived based upon the real variable RadonStroh formalism. For verification, analyses by the commercial software ANSYS were also performed to make comparison with our BEM results. Extended Stroh formalism Before presenting the extended Stroh formalism for piezoelectric anisotropic elasticity, some basic equations are reviewed here, which include the constitutive law of piezoelectric solids, strain-displacement relationship, equilibrium equation and electrostatic equation, given as follows [12]:

°V ij ® ¯° D j

E Cijkl H kl ekij Ek ,

e jkl H kl Z jk Ek ,

H ij

1 °V ij , j 0, i, j , k , l 1,2,3, (ui , j u j ,i ), ® 2 ¯° Di ,i 0,

(1)


189

BEM gradient based iterative algorithms for inverse BVPs in steady-state anisotropic heat conduction Liviu Marin1,2 1 Department

of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014 Bucharest, Romania 2 Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, 13 Calea 13 Septembrie, 050711 Bucharest, Romania Keywords: Cauchy problem; anisotropic heat conduction; variational formulation; gradient methods; regularization; boundary element method (BEM). Abstract. We investigate the numerical reconstruction of the missing thermal boundary conditions on an inaccessible part of the boundary in the case of steady-state heat conduction in anisotropic solids from the knowledge of over-prescribed noisy data on the remaining accessible boundary. This inverse problem is tackled by employing a variational formulation which transforms it into an equivalent control problem. Four such approaches are presented and, consequently, a gradient based algorithm is obtained for each of these formulations. The numerical implementation is realized for the 2D case via the boundary element method (BEM), for both exact and perturbed data. Mathematical Formulation by a We consider an anisotropic solid which occupies a domain Ω ⊂ Rd , usually d = 2 or d = %3, bounded & surface ∂Ω of class C2 , and is characterised by the thermal conductivity tensor K(x) = kij (x) i,j=1,d ∈

Rd×d , x ∈ Ω, where

K(x) = K(x) , ∀ x ∈ Ω ; ξ · K(x)ξ ≥ 0 , ∀ x ∈ Ω , ∀ ξ ∈ Rd ; ξ · K(x)ξ = 0 , ∀ x ∈ Ω ⇐⇒ ξ = 0 ∈ Rd .

(1a) (1b)

In the absence of heat sources, the temperature distribution, u, in the domain Ω satisfies the so-called anisotropic heat conduction equation, also referred to as the Laplace-Beltrami equation [4] L u(x) ≡ −∇ · K(x)∇u(x) = 0 , x ∈ Ω , (2) We now let n(x) be the unit outward normal vector at x ∈ ∂Ω and q(x) be the normal heat flux at a point x ∈ ∂Ω defined by, see e.g. [4], q(x) = n(x) · K(x)∇u(x) , x ∈ ∂Ω . (3) We further consider the so-called Cauchy problem for the anisotropic heat conduction equation (2), given by the aforementioned equation and the following over-prescribed boundary conditions on Γ0 ∂Ω, Γ0 = ∅: u(x) = u∗ (x) , x ∈ Γ0 , and q(x) = q ∗ (x) , x ∈ Γ0 , (4) where u∗ and q ∗ are the prescribed temperature and normal heat flux on Γ0 , respectively. Moreover, in the sequel we use the following notation Γ1 := ∂Ω \ Γ0 and further assume that Γ0 ∩ Γ1 = ∅ and kij ∈ L∞ (Ω), i, j = 1, d. We also assume that the prescribed Cauchy data in (4) belongs to L2 (Γ0 ) and we seek the solution u of the inverse problem (2) and (4) in L2 (Ω). We also note that the Cauchy problem (2) and (4) is much more difficult to solve both analytically and numerically than direct problems since its solution does not satisfy the general conditions of well-posedness [2]. Variational Formulations Consider the following direct mixed problem: ⎧ = 0 in Ω , ⎨ −∇· K∇u) n · K∇u = q ∗ on Γ0 , ⎩ u=ϕ on Γ1 ,

(5)

190


for some ϕ ∈ L2 (Γ1 ), and denote its solution by u = u(q ∗ , ϕ). Next, we define the following functional related to problem (5): "2 1" J1 (ϕ) = "u(q ∗ , ϕ) − u∗ "L2 (Γ0 ) , J1 : L2 (Γ1 ) −→ [0, ∞) , 2 and aim to minimise it over L2 (Γ1 ), that is Given u∗ ∈ L2 (Γ0 ) and q ∗ ∈ L2 (Γ0 ), find ϕ∗ ∈ L2 (Γ1 ) :

ϕ∗ = arg

min

ϕ∈L2 (Γ1 )

(6)

J1 (ϕ) .

(7)

Consider the direct mixed problem (5) with non-homogeneous Neumann data on Γ0 , p ∈ L2 (Γ0 ), and homogeneous Dirichlet data on Γ1 , also referred to as the adjoint problem to problem (5), i.e. ⎧ ⎨ −∇· K∇s 1 = 0 in Ω , n · K∇s1 = p on Γ0 , (8) ⎩ on Γ1 , s1 = 0 and, analogously to problem (5), denote its solution by s1 = s1 (p, 0). Under these assumptions, the following results can be obtained: Lemma 1 (Green’s formula) Let u = u(q ∗ , ϕ) and s1 = s1 (p, 0) be the unique solutions of problems (5) and (8), respectively. Then % & p u dΓ + s1 q ∗ dΓ . (9) n · K∇s1 ϕ dΓ = Γ0

Γ1

Theorem 1 (Solvability) Let q ∗ ∈ L2 (Γ0 ). Then the set L2 (Γ0 ).

Γ0

u(q ∗ , ϕ)Γ0 ϕ ∈ L2 (Γ0 ) is dense in

Remark 1 Theorem 1 states that the Cauchy problem (2) is solvable for almost all compatible Cauchy data u∗ ∈ L2 (Γ0 ) and q ∗ ∈ L2 (Γ0 ). Corollary 1 inf

ϕ∈L2 (Γ1 )

J1 (ϕ) = 0 .

(10)

Theorem 2 (Gradient formula) The functional J1 , defined by relation (6), is twice Fréchet differentiable and strictly convex. Moreover, its first gradient is given by (11) J1 (ϕ) = −n · K∇s1 (p, 0) Γ1 . Remark 2 Instead of the minimisation approach (6) adopted above, the Cauchy problem (2) can also be formulated equivalently as an operator equation. More precisely, by defining the following Dirichlet-to-Dirichlet and Neumann-to-Dirichlet operators associated with the mixed problem (5) with homogeneous Neumann data on Γ0 and non-homogeneous Dirichlet data on Γ1 , ϕ, and the same problem (5) with non-homogeneous Neumann data on Γ0 , Φ, and homogeneous Dirichlet data on Γ1 , respectively, i.e. ϕ ∈ L2 (Γ1 ) −→ K ϕ = u(0, ϕ)Γ0 , (12a) K : L2 (Γ1 ) −→ L2 (Γ0 ) , Φ ∈ L2 (Γ0 ) −→ K0 Φ = u(Φ, 0)Γ0 , (12b) K0 : L2 (Γ0 ) −→ L2 (Γ0 ) , the Cauchy problem (2) reduces to solving the following operator equation: Given u∗ ∈ L2 (Γ0 ) and q ∗ ∈ L2 (Γ0 ), find ϕ ∈ L2 (Γ1 ) : K ϕ = u∗ − K0 q ∗ .

(13)

Furthermore, one can prove the following results which relate (13) to the minimisation problem (6):


191

Theorem 3 (Properties of operators K and K0 ) jective and compact operator.

(i) K is a well-defined, linear, bounded, in-

(ii) K0 is a well-defined, linear and bounded operator. (iii) The adjoint operator to operator K regularizes the operator equation (13) and is given by p ∈ L2 (Γ0 ) −→ K ∗ p = −n · K∇s1 (p, 0) Γ1 (14) K ∗ : L2 (Γ0 ) −→ L2 (Γ1 ) , Instead of considering the mixed problem (5) associated with the minimisation problem (6) and referred to as the variational formulation I, one can also consider the following formulations: (i) Variational formulation II: Consider q ∗ ∈ L2 (Γ0 ) and some Neumann data on the underprescribed boundary Γ1 , i.e. Φ ∈ L2 (Γ1 ), and denote by u = u(u∗ , Φ) the solution of the Laplace-Beltrami equation (2) with these conditions. Define the functional J2 : L2 (Γ1 ) −→ [0, ∞) ,

J2 (Φ) =

" 1" "n · K∇u(u∗ , Φ) − q ∗ "2 2 , L (Γ0 ) 2

which we aim to minimise over L2 (Γ1 ), namely Given u∗ ∈ L2 (Γ0 ) and q ∗ ∈ L2 (Γ0 ), find Φ∗ ∈ L2 (Γ1 ) :

Φ∗ = arg

min

Φ∈L2 (Γ1 )

J2 (Φ) .

(15)

(16)

(ii) Variational formulation III: Consider u∗ ∈ L2 (Γ0 ) and some Dirichlet data on the underprescribed boundary Γ1 , i.e. ϕ ∈ L2 (Γ1 ), and denote by u = u(u∗ , ϕ) the solution of the Laplace-Beltrami equation (2) with these conditions. Define the functional J3 : L2 (Γ1 ) −→ [0, ∞) ,

J3 (ϕ) =

" 1" "n · K∇u(u∗ , ϕ) − q ∗ "2 2 , L (Γ0 ) 2

which we aim to minimise over L2 (Γ1 ), namely Given u∗ ∈ L2 (Γ0 ) and q ∗ ∈ L2 (Γ0 ), find ϕ∗ ∈ L2 (Γ1 ) :

ϕ∗ = arg

min

ϕ∈L2 (Γ1 )

J3 (ϕ) .

(17)

(18)

(iii) Variational formulation IV: Consider q ∗ ∈ L2 (Γ0 ) and some Neumann data on the underprescribed boundary Γ1 , i.e. Φ ∈ L2 (Γ1 ). Denote by u = u(q ∗ , Φ) the solution of the LaplaceBeltrami equation (2) with these conditions, with the mention that we may select x0 ∈ Γ0 and prescribe u at x0 in order to obtain a unique solution. Define the functional J4 : L2 (Γ1 ) −→ [0, ∞) ,

J4 (Φ) =

" 1" "u(q ∗ , Φ) − u∗ "2 2 , L (Γ0 ) 2

and aim to minimise it over L2 (Γ1 ), namely Given u∗ ∈ L2 (Γ0 ) and q ∗ ∈ L2 (Γ0 ), find Φ∗ ∈ L2 (Γ1 ) :

Φ∗ = arg

min

Φ∈L2 (Γ1 )

(19)

J4 (Φ) .

(20)

Algorithm Suppose that instead of the exact Dirichlet data on the over-specified boundary Γ0 , a perturbation of it is available, say u∗ε ∈ L2 (Γ0 ) such that " " ∗ "uε − u∗ " 2 ≤ ε, (21) L (Γ0 )

192


where ε > 0 is an a priori known measure in the perturbed Dirichlet data on Γ0 . To solve stably the Cauchy problem (2) and (4) with noisy data u∗ε , one needs to invert the corresponding perturbed operator equation (13) via the adjoint operator (14), that is ϕ = K ∗ u∗ε − K0 q ∗ . (22) However, from Theorems 2 and 3 we observe that this means computing the gradient (11) of functional (6). Herein we present only the algorithm resulting from the variational formulation I, namely Step 1. Set k = 1 and choose ϕk ∈ L2 (Γ1 ). Step 2. Solve the direct problem

to determine qk Γ1

⎧ ⎨ −∇· K∇u k =∗ 0 n · K∇uk = q ⎩ uk = ϕk := n · K∇uk Γ1 and the residual

in Ω on Γ0 on Γ1

(23)

rk := uk Γ0 − u∗ .

Step 3. Solve the adjoint problem ⎧ ⎨ −∇· K∇s k = 0 in Ω n · K∇sk = rk on Γ0 (24) ⎩ sk = 0 on Γ1 to determine the gradient dk := n · K∇sk Γ1 = −J1 (ϕk ). Update the Dirichlet data on Γ1 as ϕk := ϕk + γ dk , where γ > 0 is a constant to be prescribed. Step 4. Set k = k + 1 and repeat steps 2 and 3 until a prescribed stopping criterion is satisfied. As a stopping criterion, we have employed Morozov’s discrepancy principle [3], whilst γ > 0 is chosen to be a small number. One can also prove the following result: Theorem 4 (convergence) Consider the inverse problem (2) and (4) with u∗ ∈ L2 (Γ0 ) and q ∗ ∈ L2 (Γ0 ). Assume that the solution u ∈ L2 (Ω) to problem (2) and (4) exists and choose 0 < γ < K −2 . Let uk be the k−th approximate solution to the algorithm described above. Then lim uk − uL2 (Ω) = 0,

k−→∞

(25)

for any initial data ϕ1 ∈ L2 (Γ1 ). Numerical Results We further consider the annular domain Ω = x ∈ R2 Rint < |x| < Rout , Rint = Rout /2 = 1.0, occupied by an anisotropic material characterised by k11 = 5, k12 = k21 = 2 and k22 = 1. We also consider the following analytical solution: 2k12 − k22 3 2k12 − k11 3 x1 − x21 x2 + x1 x22 − x2 , x = (x1 , x2 ) ∈ Ω, 3k11 3k22 2 k12 (an) (x) = k12 − k22 x1 + 2 k12 − k11 x1 x2 + 2k12 − k11 − k11 x22 n1 (x) q k22 k12 2 2k12 − k22 − k22 x1 + 2 k12 − k22 x1 x2 + k12 − k11 x22 n2 (x), + k11 x = (x1 , x2 ) ∈ ∂Ω.

u(an) (x) =

(26)


N 120

180

240

pu 1% 3% 5% 1% 3% 5% 1% 3% 5%

eu (kopt ) 1.07(−1) 2.02(−1) 2.70(−1) 8.57(−2) 1.51(−1) 2.04(−1) 5.69(−2) 1.23(−1) 1.84(−1)

eq (kopt ) 6.28(−1) 9.20(−1) 1.06( 0) 5.55(−1) 7.23(−1) 8.23(−1) 3.69(−1) 5.77(−1) 7.37(−1)

193

Eu (kopt ) 6.79(−2) 2.04(−1) 3.39(−1) 6.27(−2) 1.88(−1) 3.13(−1) 5.84(−2) 1.75(−1) 2.92(−1)

τε 6.79(−2) 2.04(−1) 3.39(−1) 6.28(−2) 1.88(−1) 3.14(−1) 5.84(−2) 1.75(−1) 2.92(−1)

kopt 212 81 54 234 93 61 288 111 71

Table 1: The accuracy errors eu (k) = uk − u(an) L2 (Γ1 ) and eq (k) = qk − q (an) L2 (Γ1 ) , and the residual error Eu (k) = uk − u(an) L2 (Γ0 ) , obtained using the algorithm I, γ = 10−1 , k = kopt , N ∈ 120, 180, 240 and various levels of noise in the Dirichlet data on Γ0 .

N 120

180

240

pq 1% 3% 5% 1% 3% 5% 1% 3% 5%

eu (kopt ) 6.09(−2) 1.39(−1) 2.03(−1) 5.29(−2) 1.11(−1) 1.60(−1) 3.88(−2) 8.89(−2) 1.36(−1)

eq (kopt ) 3.56(−1) 6.74(−1) 8.98(−1) 3.12(−1) 5.38(−1) 6.88(−1) 2.36(−1) 4.44(−1) 6.26(−1)

Eq (kopt ) 2.00(−1) 5.99(−1) 9.99(−1) 1.81(−1) 5.43(−1) 9.05(−1) 1.72(−1) 5.15(−1) 8.58(−1)

τε 2.00(−1) 6.00(−1) 9.99(−1) 1.81(−1) 5.43(−1) 9.05(−1) 1.72(−1) 5.15(−1) 8.58(−1)

kopt 240 89 59 276 110 73 359 137 86

Table 2: The accuracy errors eu (k) = uk − u(an) L2 (Γ1 ) and eq (k) = qk − q (an) L2 (Γ1 ) , and the residual error Eq (k) = qk − q (an) L2 (Γ0 ) , obtained using the algorithm II, γ = 10−1 , k = kopt , N ∈ 120, 180, 240 and various levels of noise in the Neumann data on Γ0 .

N 120

180

240

pq 1% 3% 5% 1% 3% 5% 1% 3% 5%

eu (kopt ) 8.89(−2) 1.47(−1) 1.87(−1) 8.29(−2) 1.43(−1) 1.86(−1) 1.10(−1) 1.94(−1) 2.38(−1)

eq (kopt ) 7.45(−1) 1.02( 0) 1.15( 0) 6.71(−1) 9.34(−1) 1.13( 0) 8.25(−1) 1.24( 0) 1.45( 0)

Eq (kopt ) 2.00(−1) 5.99(−1) 9.99(−1) 1.81(−1) 5.41(−1) 9.02(−1) 1.71(−1) 5.14(−1) 8.51(−1)

τε 2.00(−1) 6.00(−1) 9.99(−1) 1.81(−1) 5.43(−1) 9.05(−1) 1.72(−1) 5.15(−1) 8.58(−1)

kopt 43 17 11 43 18 12 36 14 10

Table 3: The accuracy errors eu (k) = uk − u(an) L2 (Γ1 ) and eq (k) = qk − q (an) L2 (Γ1 ) , and the residual error Eq (k) = qk − q (an) L2 (Γ0 ) , obtained using the algorithm III, γ = 5 × 10−2 , k = kopt , N ∈ 120, 180, 240 and various levels of noise in the Neumann data on Γ0 .

194


N 120

180

240

pu 1% 3% 5% 1% 3% 5% 1% 3% 5%

eu (kopt ) 5.82(−2) 1.08(−1) 1.50(−1) 6.22(−2) 1.46(−1) 2.30(−1) 7.12(−2) 1.36(−1) 1.92(−1)

eq (kopt ) 3.61(−1) 5.49(−1) 6.87(−1) 3.52(−1) 6.35(−1) 9.17(−1) 3.99(−1) 6.42(−1) 8.50(−1)

Eu (kopt ) 6.79(−2) 2.04(−1) 3.39(−1) 6.27(−2) 1.88(−1) 3.13(−1) 5.84(−2) 1.75(−1) 2.92(−1)

τε 6.79(−2) 2.04(−1) 3.39(−1) 6.28(−2) 1.88(−1) 3.14(−1) 5.84(−2) 1.75(−1) 2.92(−1)

kopt 345, 685 161, 151 110, 846 310, 278 144, 206 88, 979 284, 624 146, 702 97, 908

Table 4: The accuracy errors eu (k) = uk − u(an) L2 (Γ1 ) and eq (k) = qk − q (an) L2 (Γ1 ) , and the residual error Eu (k) = uk − u(an) L2 (Γ0 ) , obtained using the algorithm IV, γ = 10−3 , k = kopt , N ∈ 120, 180, 240 and various levels of noise in the Dirichlet data on Γ0 .

In this study, we consider the (2) and (4) inverse problem with the over- and under-specified boundaries Γ0 = Γout = x ∈ R2 |x| = Rout and Γ1 = Γint = x ∈ R2 |x| = Rint , respectively, and perturbed boundary temperature (pu= 1%, 3%, 5%) or normal heat flux (pq = 1%, 3%, 5%). We have considered N0 = Nout ∈ 80, 120, 160 and N1 = Nint = N0 /2 constant boundary elements, such that N = N0 + N1 , see e.g. [1], to discretise the over- and under-specified boundaries, respectively. Tables 1–4 present the numerical results obtained using the algorithms corresponding to the variational methods I–IV, respectively, in terms of the accuracy errors eu (k) = uk − u(an) L2 (Γ1 ) and eq (k) = qk − q (an) L2 (Γ1 ) , as well as the residual error Eu (k) = uk − u(an) L2 (Γ0 ) , corresponding to k = kopt obtained using the Discrepancy principle of Morozov [3], N ∈ 120, 180, 240 and various levels of noise in the data on Γ0 . It can be seen from these tables that the numerical solutions for the temperature and the normal heat flux on Γ1 , retrieved by any of the algorithms I–IV, are stable approximations to their corresponding exact solutions and they converge to the exact solutions as the level of noise decreases. However, the number of iterations required to obtain a stable solution to inverse problem (2) and (4) differs for these algorithms. Conclusions In this paper, the solution of the Cauchy problem for two-dimensional anisotropic heat conduction was investigated using the BEM in conjunction with four stable and convergent gradient based algorithms derived from four variational approaches. The numerical results obtained show the numerical stability, convergence and accuracy of the proposed BEM–gradient based algorithms. The main drawback of these algorithms is given by the fact that they depend on the parameter γ > 0. Therefore, further investigations are related to overcoming this inconvenient.

References [1] G. Chen and J. Zhou J, Boundary Element Methods, Academic Press, London (1992). [2] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press (1923). [3] V.A. Morozov, On the solution of functional equations by the method of regularization, Soviet Mathematics Doklady 7 (1966) 414–417. ¨ sik, Heat Conduction, John Wiley & Sons, New York, (1993). [4] M.N. Ozi¸


195

where ui is the displacement component in xi -axis, V ij and H ij are respectively the stress and strain tensors, Cijkl is the elastic stiffness tensor, eijk and Z jk are piezoelectric constants and dielectric permittivity, Di and Ei are electric displacements and electric field, respectively. In eq (1), by following tensor notations, the repeated indices imply summation and a comma stands for differentiation. By letting

D j V 4 j , E j u4, j ° E °Cijkl Cijkl ® °Cij 4 l Ci 4 lij elij , °C ¯ 4 j 4 l Z jk ,

2H 4 j , i, j , k , l 1,2,3,

(2)

the basic equations (1) can be rewritten in an expanded notation as

V IJ

1 (uI , J uJ , I ), V IJ , J 2

C IJKLH KL , H IJ

in which the assumptions of uJ ,4

0 , CIJKL

CJIKL

C IJLK

0, I , J , K , L 1,2,3,4,

CKLIJ , V IJ

V JI , and H IJ

(3)

H JI have been made, and

the components of V 44 , H 44 , C IJ 44 , and C44 IJ are undefined. To avoid the undefined symbols, sometimes eq (3) was written as

V Ij

1 (uI , j u j , I ), V Ij , j 2

C IjKl H Kl , H Ij

0,

I,K

It should be careful that if eq (4) is considered, the relation E j

E j

u4, j

H 4 j , and the assumption u j ,4

0 should also be changed to u j ,4

1,2,3,4, j , l 1,2,3.

u4, j

(4)

2H 4 j should be changed to

u4, j . To be consistent with all the

basic equations used for the anisotropic elastic materials, in this study we will follow the expressions given in eq (2) and (3) instead of eq (4). Boundary integral equation for 3D anisotropic piezoelectric analysis

With the foregoing illustrations for dimension expansions, the boundary integral equation (BIE) for 3D electroelastic analysis can now be written in the same form as the conventional one for elastostatics [14,15], namely cIJ (xˆ )uJ (xˆ ) ³ t IJ* ( xˆ , x ) uJ ( x )d *( x ) *

³

*

u*IJ ( xˆ , x ) t J ( x )d *( x ),

(5)

where all notations follow the same definitions for elastostatic analysis except u4 , being the electric potential, and t4 , being the electric displacement along the normal direction v ( v1 , v2 , v3 ) , i.e., t4

V 4 jv j

D jv j

Dv .

(6)

In eq (5), u*IJ ( xˆ , x ) and t IJ* ( xˆ , x ) are the generalized fundamental solutions of displacements and tractions in xJ direction at the field point x with point force/charge acting in the x I -direction applied at point xˆ ; cIJ ( xˆ ) are the free term coefficients of the source point xˆ , which can be determined from the relation of rigid body motion as usual. By using the expanded matrix form, the mathematical expression of the fundamental solutions for 3D electroelastic analysis remain the same as those for 3D anisotropic elasticity [7,13,14]. With reference to [8,9], the solutions formulated using the real variable Radon-Stroh formalism can be written in the following matrix forms:

196


sgn( 'x3 ) 4S 2

[u*IJ ]

³

S

0

sgn( 'x3 ) 4S 2

1ˆ N 2 (\ ,T )dT , [t IJ* ] r

³

S

0

1 ˆT M1 (\ ,T )dT , r2

(7a)

where

ˆ (\ ,T ) N 2

ˆ (\ ,T ) sin\ N 2 (\ ,T ), M 1

M1 (\ ,T )

[sin 2 \ N1 (\ ,T )]c cos 2\ I,

cos\ s M1 (\ ,T ) sin\ s M1* (\ ,T ), (7b)

\ ,T ) [sin 2 \ N1* (\ ,T )]c cos 2\ N1 (T ),

M1* (

N1* (\ ,T )

N1 (T ) N1 (\ ,T ) N 2 (T ) N 3 (\ ,T ).

In eq (7a), sgn( 'x3 ) 1 for 'x3 ! 0 , sgn( 'x3 ) 1 for 'x3 0 , and 'xi xi xî , i 1,2,3 ; ( x1 , x2 , x3 ) and ( xˆ1 , xˆ 2 , xˆ3 ) are, respectively, the locations of field point x and source point xˆ ; Ni (T ) and Ni (\ ,T ), i 1,2,3 are the Stroh’s fundamental elasticity matrices on the 2D Radon plane whose normal is related to the angle T , r and \ are the distance and polar angle related to the coordinates of Radon plane by

U 2 ( 'x3 )2 , \

r

U

tan 1

'x3

U

,

(8)

'x1 cos T 'x2 sin T .

cos\ s wU / ws and sin\ s wx3 / ws where s is a parameter denoting the tangential direction of the boundary surface, and U and x3 are the coordinates used in 2D Radon-domain. The superscript T represents the transpose of a matrix; the prime xc denotes the derivative with respect to \ , and the derivative of the generalized fundamental matrices Ni (\ ,T ) can be calculated by using the identities obtained for 2D anisotropic elasticity [9,14]. With the free term coefficient set to be unity, the corresponding BIE for calculating displacement gradients at internal points can be obtained by directly differentiating eq (5) with respect to xˆ K , yielding u J , K (xˆ ) ³ t IJ* , K (xˆ , x) u J ( x) d *(x) *

³u *

* IJ , K

(xˆ , x) t J (x)d *(x).

(9)

In eq (9), the derivatives of fundamental solutions, [u*IJ ]T,K and [t IJ* ]T,K , can be obtained directly from eq (7). Numerical examples All the presented formulations have been implemented in an existing BEM code. To demonstrate our successful implementation, an analysis was performed for a PZT cube excavated with a cylindrical through hole when uniform pressure is applied on top, and the bottom is totally fixed (Fig. 1(a)). Also, internal analyses were conducted for giving solutions at points along r = R2. For treating general anisotropy, the principal axes are rotated counter-clockwise about the x1 , x2 , x3 -axis successively by 30°, 45°, and 60°, respectively; the following coefficients are obtained:


197

p

x2

R2 x3

R1 x1

(a) (b) (c) Figure 1. Example problem: (a) a cube with central through hole, (b) BEM mesh, (c) ANSYS mesh.

ª¬Cij º¼

ª¬ eij º¼

5.37 2.46º 1.48 2.75» » 5.60 0.49 » » , (unit: GPa) 0.80 2.16» 26.61 0.58» » 0.58 27.60 ¼ ª16.32 2.88 0.50 1.95 2.62 1.20 º « 0.93 8.21 0.19 7.04 1.95 8.49 » , (unit: C/m) « » «¬ 2.03 2.38 15.14 1.94 5.12 1.95»¼ ª 6.001 0.174 0.381º ª¬Zij º¼ « 0.174 6.395 0.144 » . (unit: 109C/(Vm)) « » ¬« 0.381 0.144 6.146 ¼» ª 125.1 « 75.74 « « 75.19 « « 0.32 « 5.37 « ¬ 2.46

75.74 75.19 0.32 136.9 76.3 2.28 76.3 129.3 2.12 2.28 2.12 28.42 1.48 5.60 0.80 2.75 0.49 2.16

(10)

Uniform pressure p = 1 (N/mm2) is assumed to be applied on top of the cube with side length S = 2 m. For the BEM mesh modeling, multiple nodes are applied along edges and at corners. As shown in Fig. 1(b), the BEM modeling employed 136 quadratic quadrilateral elements with total 540 nodes. For numerically integrating the integrals in eq (7a), the Gauss quadrature rule with up to 64 Gaussian points was employed for the highly anisotropic properties. For independent verification, the analysis was also carried out using the finite element software ANSYS, where 57,600 SOLID226 elements were applied (Fig. 1(c)). Fig. 2 shows the displacements and electric potentials calculated at the boundary points along R1 = 0.5 m on the plane x3 = 0. For internal solutions, Fig. 3 plots the von Mises stresses and total electric displacements computed at points along R2 = 0.75 m on the plane x3 = 0. As can be clearly seen from the plots, our BEM results are in quite satisfactory agreements with the FEM analysis.

198


-5 4 x10

-5 BEM u1 BEM u2

-10

BEM u3

Electric potential (kV)

Displacement (m)

2

0

-2 ANSYS u1

-4

ANSYS u2

-15 -20 -25 ANSYS BEM

-30

ANSYS u3 -6

-35 0

90

180

270

360

0

90

T (degrees)

180

270

360

T (degrees)

(a) (b) Figure 2. Boundary solutions of (a) displacements and (b) electric potentials along R1 = 0.5 m on x3 = 0. -5 20 x10

Total electric displacement D0 (C/m2)

2.5

von-Mises stress (MPa)

2.0 1.5 1.0 0.5 ANSYS BEM

0.0

ANSYS BEM 15

10

5

0 0

90

180

270

360

T (degrees)

0

90

180

270

360

T (degrees)

(a) (b) Figure 3. Internal solutions along R2 = 0.75 m on x3 = 0: (a) von Mises stress, (b) total electric displacement. Conclusions For studying the piezoelectric effect in 3D anisotropic solids, this article employs the methodology of dimension expansion to combine all related matrices such that the conventional BIE for dealing with elastostatics may also apply with expanded dimensions. For solving the generalized BIE, the present analysis adopts the fundamental solutions of the 3D anisotropic elasticity formulated by real variable Radon-Stroh formalism [9]. Our successful implementation of the methodology is illustrated by a numerical example, showing satisfactory agreements with the ANSYS analysis. Undoubtedly, our successful implementation has demonstrated its promising extension to more complicated 3D problems. Acknowledgement The authors would like to thank the Ministry of Science and Technology, Taiwan, Republic of China, for the support through Grant MOST 104-2221-E-006-138-MY3. References [1] [2]

D. Barnett Physica status solidi, 49, 741-748 (1972). K.C. Wu Journal of elasticity, 51, 213-225 (1998).


[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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