Multilevel Wavelet-based Numerical Method of Local

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 111 (2015) 569 – 574

XXIV R-S-P seminar, Theoretical Foundation of Civil Engineering (24RSP) (TFoCE 2015)

Multilevel wavelet-based numerical method of local structural analysis for three-dimensional problem Marina L. Mozgalevaa, Pavel A. Akimova,b,c * a Moscow State University of Civil Engineering; 26, Yaroslavskoe Shosse, Moscow, 129337, Russia Russian Academy of Architecture and Construction Sciences; 24-1, Ulitsa Bolshaya Dmitrovka, Moscow, 107031, Russia c Scientific Research Centre “StaDyO”, 18, 3-ya Ulitsa Yamskogo Polya, Moscow, 125040, Russia

b

Abstract High-accuracy solution of three-dimensional problems of structural analysis is normally required in some pre-known domains (regions of structure with the risk of significant stresses that could potentially lead to the destruction, regions subjected to specific operational requirements). This paper is devoted to wavelet-based numerical method of local analysis of three-dimensional structures. Initial discrete operational and variational formulation of the considering problem and corresponding formulations with the use of discrete Haar basis are presented. Due to special algorithms of averaging within multigrid approach, correct reduction of the problem is presented. © 2015 The Authors. Published by Elsevier B.V. © 2015 The Authors. by Elsevier Ltd. This is ancommittee open access of article theR-S-P CC BY-NC-ND Peer-review underPublished responsibility of organizing the under XXIV seminar,license Theoretical Foundation of Civil (http://creativecommons.org/licenses/by-nc-nd/4.0/). Engineering (24RSP) under responsibility of organizing committee of the XXIV R-S-P seminar, Theoretical Foundation of Civil Engineering (24RSP) Peer-review Keywords: Multilevel method; Wavelet-based method; Numerical method; Discrete method; Local structural analysis; Three-dimensional problems; Haar basis; Averaging; reduction

1. Formulation of the Problem Effective qualitative multilevel analysis of local and global stress-strain states of the three-dimensional structure is normally required in various technical problems [1]. Modern stage of development in structural mechanics is connected with the widespread use of numerical methods. Considering problems of analysis, normally leads to rather complex three-dimensional systems of equations, and only numerical solutions of these systems can be obtained.

* Corresponding author. Tel.: +7-499-183-5994; fax: +7-499-183-5994. E-mail address: [email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the XXIV R-S-P seminar, Theoretical Foundation of Civil Engineering (24RSP)

doi:10.1016/j.proeng.2015.07.044

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Marina L. Mozgaleva and Pavel A. Akimov / Procedia Engineering 111 (2015) 569 – 574

Corresponding number of degrees of freedom can exceed several millions (for instance for the coupled problems of fluid-structure interaction, foundation-structure interaction, analysis of high-rise and unique buildings, long-span bridges etc.).Static structural analysis with the use of finite element method (FEM) [2,3,4,5] requires solution of systems of linear algebraic equations with immense number of unknowns [6,7]. Generally this is the most timeconsuming stage of the computing [8,9], especially if we take into account the limitation in power of the contemporary software and in performance of personal computers or even advanced supercomputers and necessity to obtain correct and accurate solution in a reasonable time. However, practically in many cases it is impossible or unreasonable to obtain such solution for the entire structure and due to structural or loading conditions the location and approximate dimensions of critical and most vital for designers regions of the structure can be determined. The stress-strain state in these regions is of paramount importance from the standpoint of analysis and design, and may lead to structural failure or cause impairment in structural performance [10]. The distinctive paper is devoted to correct multilevel wavelet-based numerical method of three-dimensional local structural analysis, which can be applied for all types of structures. This method allows reducing the size of the problem and obtaining accurate results in selected regions simultaneously. It is rather efficient approach for evaluation of local phenomenon such as stress concentration or concentrated force or even stress in special member, in unique or complex buildings and structures. Furthermore, the proposed method allows qualitative and quantitative assessments of the degree of localization of various kinds of design factors and evaluation of the effect of each degree of freedom on behavior of the structure. It should be noted that wavelet analysis has become one of the most effective and rather popular contemporary sources for this kind of research in the recent years. Generally it is a powerful computational-analytical tool for the ecomposition and multilevel mathematical modeling of functions, which has extraordinary characteristics and combines advantages of functional analysis, Fourier transform, spline analysis, harmonic analysis and numerical analysis as well [11-14] and can be successfully employed for the goal considering in this paper. In accordance with the method of extended domain, proposed by Prof. Alexander B. Zolotov [15,16,17,18], the given domain : , occupied by considering structure, is embordered by extended one Z of arbitrary shape, particularly elementary (for instance, parallelepiped, cylinder and others). Operational formulation of the problem in the extended domain Z normally has the form

Lu

F,

(1)

where L is the operator of boundary problem, which takes into account the boundary conditions; u is the unknown function; F is the given right-side function. Corresponding variational formulation has the form (solution of (1) is the critical point of functional)

) ( u)

0.5 ( Lu, u) ( F, u) ,

(2)

where symbol ( f , g ) denotes dot product of function f and function g . Discrete formulation of the problem has the form:

Au f ,

(3)

where A {ai , j }i , j 1, 2, ..., ngl is the difference approximation of operator L ; u function; f

[ u1 u2 ... ungl ]T is the unknown mesh

[ f1 f 2 ... f n ]T is the given right-side mesh function; n gl is dimension of problem. gl

Various methods can be used to form the matrix of the discrete operator. We recommend method of basis (local) variations, proposed by Prof. Alexander B. Zolotov [5]. Its major peculiarities include universality and computer orientation. We can use the following formulas for linear problems:

ai , j

Ɏ( e ( i ) e ( j ) ) Ɏ( e ( i ) ) Ɏ( e ( j ) ) Ɏ( 0 )] ;

fi

0.5 [Ɏ( e ( i ) ) Ɏ( e ( i ) )] ,

(4)

571


e (i )

[ e1( i ) e2( i ) ... en( i ) ]T , i 1, 2, ..., ngl ; e (ji ) gl

G i , j , j 1, 2, ... n gl ;

(5)

where e ( i ) , i 1, 2, ..., ngl are basis mesh vectors; 0 is the null function; G i, j is the Kronecker delta. 2. Direct and Inverse Discrete Haar Transforms Let us consider the three-dimensional rectangular region Z { ( x1 , x 2 , x 3 ) : 0 d x1 d l1 , 0 d x 2 d l 2 , 0 d x 3 d l 3 } , where x1 , x 2 , x 3 are coordinates; l1 , l 2 , l 3 are dimensions along x1 , x 2 , x 3 . Let us divide Z into (n 1) equal parts along x1 , into (n 1) equal parts along x 2 and into (n 1) equal parts along x 3 , where n 2 M , M is the number of levels in the Haar basis. Therefore we have the following formulas for coordinates of mesh nodes: 1, 2, ..., n ; x 2 ,i

x1,i

(i1 1)h1 , i1

h1

l1 /(n 1) ; h2

(i2 1)h2 , i2

1, 2, ..., n ; x 3,i

(i 3 1)h2 , i 3

1, 2, ..., n ,

(6)

where

l 2 /(n 1) ; h3

l 3 /(n 1) .

Haar mesh functions \ sp3 ,s2 ,s1 , j3 , j2 , j1 (i3 , i2 , i1 ) , s1 , s 2 , s 3

s1

s2

3

0,1 , j1 , j2 , j3

1,2,..., N p , i1 , i2 , i3

1, 2 , ... , N (except

0 ) can be defined by formulas:

s3

\ sp , s

(7)

2 , s1 , j3 , j 2 , j1

(i3 , i 2 , i1 )

i 1 i 1 § i1 1 · ( j1 1), 2 p 1 ( j 2 1), 3 p 1 ( j3 1) ¸ ; p 1 2 2 ©2 ¹

D p\ s s s ¨ 3 2 1

§ i1 1 i 2 1 i3 1 · , M , M ¸; M 2 2 ¹ © 2

\ 0M, 0, 0, j , j , j (i3 , i2 , i1 ) D M\ 0, 0, 0 ¨ 3

2

1

1 , 0 d x1 , x 2 , x 3 1 / 2 °( 1) s , 1 / 2 d x 1 0 d x 1 / 2 0 d x 1 / 2 1 2 3 ° s x x x ( 1 ) , 0 1 / 2 1 / 2 1 0 d d d 1 2 3 1/ 2 ° s s , 1 / 2 d x1 1 1 / 2 d x 2 1 0 d x 3 1 / 2 °( 1) ° s ®( 1) , 0 d x1 , x 2 1 / 2 1 / 2 d x 3 1 °( 1) s s , 1 / 2 d x1 1 0 d x 2 1 / 2 1 / 2 d x 3 1 °( 1) s s , 0 d x 1 / 2 1 / 2 d x 1 1 / 2 d x 1 1 2 3 ° s s s , 1 / 2 d x1 1 1 / 2 d x 2 1 1 / 2 d x 3 1 °( 1) °¯ 0 , x1 , x 2 , x 3 0 x1 , x 2 , x 3 t 1;

(8)

(9)

1

2

1

\ s ,s 3

2 , s1

(i3 , i2 , i1 )

2

3

1

(10)

3

2

3

1

2

3

where N p u N p u N p is the number of Haar functions [19,20] at level p ,

Np

n / 2 p 1 2 M ( p 1) , 0 d p M ® 1, p M ; ¯

Dp

° 1/2 p 1 / 2 p 1 , 0 d p M ® °¯ 1/n / n , p M .

Let f (i3 , i2 , i1 ) be an arbitrary mesh function. Consequently we have

(11)

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Marina L. Mozgaleva and Pavel A. Akimov / Procedia Engineering 111 (2015) 569 – 574 M 1

v1M,1,1\ 1p, 0, 0, 0 (i3 , i2 , i1 ) ¦

f (i3 , i2 , i1 )

p 0

Np

Np

Np

¦¦¦ ¦ v

p k , j3 , j2 , j1

\ kp, j , j , j (i3 , i2 , i1 ) ; 3

2 1

(12)

j3 1 j2 1 j1 1 1d k d7

where

\ sp , s 3

2 , s1 , j3 , j 2 , j1

(i3 , i 2 , i1 ) \ kp, j3 , j2 , j1 (i3 , i 2 , i1 ) ; v sp3 , s2 , s1 , j3 , j2 , j1 v kp, j3 , j2 , j1 ;

1, s 3 °2 , s 3 °3 , s 3 ° ®4 , s 3 °5 , s 3 °6 , s 3 ° ¯7, s 3

4 s 3 2 s 2 s1

k

N

vsp ,s ,s , j , j , j 3 2 1 3

2 1

N

s 2 0 , s1 1 0, s 2 1, s1 0, s 2 1, s1 1, s 2 0 , s1 1, s 2 0 , s1 1, s 2 1, s1 1, s 2 1, s1

0 1 0 1 0 1;

(13)

(14)

N

¦¦¦ f (i , i , i )\ 3

2

1

p k , j3 , j2 , j1

(i3 , i2 , i1 )

(15)

i3 1 i2 1 i1 1

v sp3 , s2 , s1 , j3 , j2 , j1 are Haar expansion coefficients.

Let’s use the following notation: f is the vector with values of mesh functions f (i3 , i2 , i1 ) at nodes (i3 , i2 , i1 ) , i3 1, 2, ..., n , i 2 1, 2, ..., n , i1 1, 2, ..., n ; Q is the matrix of Haar functions written in columns; v is the corresponding vector of Haar expansion coefficients of function f . Thus we have ( E is identity matrix) f

Qv ; v

QQT

QTQ

QTf ;

E ; Q 1

(16)

QT .

(17)

3. Haar-based Formulation of the Problem Let us consider Haar-based formulation of the problem. After change of basis we will get

) ( u ) 0.5 ( Au, u ) (f , u ) 0.5 ( AQ v, Qv ) (f , Qv ) 0.5 (Q* AQ v, v ) (Q*f , v )

(18)

and finally ~ )( v )

0.5 (Q*LQ v, v ) (Q*f , v ) ,

(19)

where v is vector of Haar expansion coefficients of the vector f . Corresponding operational formulation of the problem has the form

~ Av

~ ~ f , A

Q* AQ ;

~ f

Q* f .

(20)

4. Reduction of the Problem. Formulas of Averaging As we have already mentioned above, is not necessary to obtain global solution in the domain in some cases. Local solution for several prescribed subdomains is normally required. But if we do not need to find a complete

573


solution we can reduce the number of unknowns without significant loss of accuracy or with a small error in local solutions. Algorithm of averaging in three-dimensional case is described below. Let us assume that it is necessary to make averaging at some level number q . For all p 0, 1, 2, ..., q ,

1, 2, ..., N p , s 3 , s 2 , s1

j3 , j 2 , j1

0, 1 (except s 3

s2

0 ) we suppose

s1

( D1u p ) 2 j3 k3 1, 2 j2 k2 1, 2 j1k11 | ( D1u~ p ) 2 j3 1, 2 j2 1, 2 j11 , k3

0, 1, k 2


0, 1, k 2

0, 1, k1

0, 1 ;

(22)


0, 1, k 2

0, 1, k1

0, 1 ;

(23)

0, 1 ;

0, 1, k1

(21)

( D2 ,1u p ) 2 j3 k3 1, 2 j2 k2 1, 2 j1k11 | ( D2 ,1u~ p ) 2 j3 1, 2 j2 1, 2 j11 , k3

0, 1, k 2

0, 1, k1

0, 1 ;

(24)

( D3,1u p ) 2 j3 k3 1, 2 j2 k2 1, 2 j1k11 | ( D3,1u~ p ) 2 j3 1, 2 j2 1, 2 j11 , k3

0, 1, k 2

0, 1, k1

0, 1 ;

(25)

( D3, 2u p ) 2 j3 k3 1, 2 j2 k2 1, 2 j1k11 | ( D3, 2u~ p ) 2 j3 1, 2 j2 1, 2 j11 , k3

0, 1, k 2

0, 1, k1

0, 1 ;

(26)

( D3 D2 D1 u p ) 2 j3 k3 1, 2 j2 k2 1, 2 j1k11 | ( D3 D2 D1 u~ p ) 2 j3 1, 2 j2 1, 2 j11 , k3

0, 1, k 2

0, 1, k1

vsp3 ,s2 ,s1 , 2 j3 k3 1, 2 j2 k2 1, 2 j1k11

vsp3 ,s2 ,s1 , 2 j3 1, 2 j2 1, 2 j11 , k3

0, 1, k 2

0, 1, k1

0, 1 ;

0, 1 ;

(27) (28)

where

1 1 1 1 p ¦¦¦ u 2 j 1k ,2 j 1k , 2 j 1k ; 8k 0k 0k 0

u~2pj 1, 2 j 1, 2 j 1 3

2

1

3

3

2

1 T3 T2 D1 ; D2 4

D1

2

2

1

(29)

1

1 T2 T1 D3 ; 4

1 T3 T1 D2 ; D3 4

1 T3 D2 D1 ; D3,1 2

D2 ,1

3

1

1 T2 D3 D2 ; D3, 2 2

( D1 u p ) j3 , j2 , j1

(u jp3 , j2 , j11 u jp3 , j2 , j1 ) / h ; ( D2 u p ) j3 , j2 , j1

(T1 u p ) j3 j2 j1

u jp3 , j2 , j11 u jp3 , j2 , j1 ; (T2 u p ) j3 j2 j1

(30)

1 T1 D3 D 2 ; 2

(31)

(u jp3 , j2 1, j1 u jp3 , j2 , j1 ) / h ; ( D3 u p ) j3 , j2 , j1 u jp3 , j2 1, j1 u jp3 , j2 , j1 ; (T3 u p ) j3 j2 j1

(u jp3 1, j2 , j1 u jp3 , j2 , j1 ) / h ;

u jp3 1, j2 , j1 u jp3 , j2 , j1 ;

(32) (33)

Then it can be checked that final formulas of averaging have the form

vkp, 2 j 1k , 2 j 1k , 2 j 1k 3

E1

3

E2

2

2

E4

1

1

1 4 2

E k vkp,j1, j , j , k1 , k 2 , k 3 3

; E3

0,1, j1 , j 2 , j3

2 1

E5

E6

1 8 2

; E7

1 16 2

1, 2, ..., N p1 ;

(34)

.

(35)

574


5. Conclusions Results of verification [16,17] show an efficiency of the proposing method for localization and reducing the size of the problem. By comparing between conventional FEM and multilevel wavelet-based numerical method of local structural analysis, it is clear that the localization of the problem provides good results for selected regions even in high level of reduction in wavelet coefficients. This localization can be imposed to any desired point (region) in the structure and, by choosing an optimum reduction matrix, high accuracy solution of the problem with an acceptable reduced size can be obtained. However results of such local analysis may be unacceptable in the other (unselected) regions. The effect of reduction on results, out of the selected (desired) area, depends on behaviour of fundamental function of corresponding differential operator of boundary problem. Acknowledgements This work was financially supported by the Grants of the Russian Academy of Architecture and Construction Sciences (7.1.7, 7.1.8). References [1] P.A. Akimov, A. M. Belostosky, V. N. Sidorov, M. L. Mozgaleva, O. A. Negrozov, Application of discrete-continual finite element method for global and local analysis of multilevel systems. // Applied Mechanics and Materials; AIP Conference Proceedings 1623, 3 (2014). [2] K. J. Bathe, Finite Element Procedures, Prentice Hall, 2007, 1050 pages. [3] J. Fish, T. Belytschko, A First Course in Finite Elements, John Wiley & Sons Ltd, 2007, 344 pages. [4] Smith I.M. Programming the Finite Element Method, John Wiley & Sons Ltd, fourth edition, 2004, 650 pages. [5] A. B. Zolotov, P. A. Akimov, V. N. Sidorov, M. L. Mozgaleva, Numerical and Analytical Methods of Structural Analysis. Moscow, ASV Publishing House, 2009, 336 pages (in Russian). [6] H. A. Van der Vorst, Iterative Krylov Methods for Large Linear Systems, volume 13. Cambridge University Press, 2003. [7] T. A. Davis, Direct Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics Philadelphia (SIAM), 2006, 230 pages. [8] A. V. Terekhov, A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method, Volume 39, Issues 6-7, June – July 2013, pp. 245-258. [9] S. Yu. Fialko, Iterative Methods for Solving Large-Scale Problems of Structural Mechanics Using Multi-Core Computers. // Archives of Civil and Mechanical Engineering, Volume 14, Issue 1, January 2014, pp. 190-203. [10] J. A. Collins, Failure of Materials in Mechanical Design: Analysis, Prediction, Prevention. Second edition, John Wiley & Sons, 1993, 654 pages. [11] C. S. Burrus, R. A. Copinath, H. Guo, Introduction to Wavelets and Wavelet Transforms. New Jersey, Prentice-Hall Inc., 1998, 268 pages. [12] C. K. Chui, An introduction to wavelets. Academic press, 1992, 366 pages. [13] M. W. Frazier, An introduction to wavelets through linear algebra. Springer, 1999, 520 pages. [14] I. Daubechies, Ten Lectures on Wavelets. New Jersey, SIAM, 1992, 357 pages. [15] P. A. Akimov, M. L. Mozgaleva, Method of Extended Domain and General Principles of Mesh Approximation for Boundary Problems of Structural Analysis. Applied Mechanics and Materials, Vols. 580-583, 2014, pp. 2898-2902. [16] P. A. Akimov, M. L. Mozgaleva, Mojtaba Aslami, O. A. Negrozov, Modified Wavelet-based Multilevel Discrete-Continual Finite Element. Part 1: Continual and Discrete-Continual Formulations of the Problems Method for Local Structural Analysis. Applied Mechanics and Materials, Vols. 670-671, 2014, pp. 720-723. [17] P. A. Akimov, M. L. Mozgaleva, Mojtaba Aslami, O. A. Negrozov, Modified Wavelet-based Multilevel Discrete-Continual Finite Element. Part 2: Reduced Formulations of the Problems in Haar Basis Method for Local Structural Analysis. Applied Mechanics and Materials, Vols. 670-671, 2014, pp. 724-727. [18] P. A. Akimov, M. L. Mozgaleva, V. N. Sidorov, Fundamentals of Wavelet-based Multilevel Numerical Methods of Local Structural Analysis. International Journal for Computational Civil and Structural Engineering. Volume 7, Issue 3, 2011, pp. 20-42 (in Russian). [19] P. A. Akimov, Mojtaba Aslami, Theoretical Foundations of Correct Wavelet-Based Approach to Local Static Analysis of Bernoulli Beam. // Applied Mechanics and Materials, Vols. 580-583, 2014, pp. 2924-2927. [20] D. N. Alexeev Numerical Methods of Local Structural Analysis. PhD Thesis, MSUCE, 2002 (in Russian).