The Scaled Boundary Finite Element Method ...

– www.erdbebenschutz.ch The [email protected] Boundary Finite Element Method – Lecture A

Lecture

A1

D r. S a s s a n M o h a s s e b Vi s i t i n g P r o f e s s o r M . I . T. C a m b r i d g e December 10, 2013 ETH, IBK Zürich

The Scaled Boundary Finite Element Method – Lecture A

Overview  Finite Element Method: FEM

 Boundary Element Method: BEM  Scaled Boundary Finite Element Method: SBFEM

 Comparison of the three methods  Accuracy of Scaled Boundary Finite Element Method  Triangular wedge

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Spatial discretisation of

Finite Element Method

Figure 1.9

3 Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


 Spatial discretisation of domain  Shape functions for displacements, which are piecewise local  Stiffness matrices  Assembly of matrices / sparse / banded  Solving system of equations  Can handle large systems  Inhomogeneous, anisotropic materials 4 Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


Spatial discretisation of Boundary Element

Method

Figure 1.14



 Boundary discretisation only  Reducing the spatial dimension by one 3-D problems become 2-D, and 2-D become 1-D  Requires fundamental solution, is usually complicated, exhibits singularities  Shape functions for each boundary element for displacement and tractions  Resulting equations are fully populated and non-symmetric  Not well suited for inhomogeneous and isotropic material  The conditions at infinity are satisfied rigorously 6 Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


Scaled Boundary Finite Element Method. A new numerical method

Problem definition:

(a) bounded media (b) unbounded media

Figure 2.2

7

Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


Finite Elements Method  No fundamental solutions required  Symmetrical matrices, sparse , banded matrices  Convergence by increasing number of elements

Boundary Element Method  Reduction of the spatial dimension by one  Reduction of data preparation and computational efforts



Scaled Boundary Finite Element Method

 Combining advantages of FEM and BEM  Reduction of partial differential equations into ordinary differential equations  Analytical solution in radial direction



 Developed in the last years  Bounded and unbounded media  Static and dynamic problems  2D and 3D problems  Frequency and time domain solutions developed by Wolf and Song at EPFL  http://www.iitk.ac.in/nicee/wcee/article/11_70.PDF  Program SIMILAR downloaded from ftp://ftp.wiley.co.uk/pub/books/wolf/ and http://lchpc25.epfl.ch/ as well as http://www.civeng.unsw.edu.au/ staff/song.c/sbfem/SIMILAR/ and http://www.civil.uwa.edu.au/~deeks/sbfem/ 10 Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


Advantages of scaled boundary finite element method compared with those of finite element and boundary element methods Finite element method Reduction of the spatial dimension by one as only the boundary is discretised with surface finite elements, reducing the data preparation and computational efforts

Boundary element method

X

Analytical solution achieved inside domain No fundamental solution required, expanding the scope of application and avoiding singular integrals Radiation condition at infinity satisfied exactly when modelling unbounded (infinite or semi-infinite) media

Table 14.1

X X

X

X X

No discretisation of free and fixed boundaries and interfaces between different materials No approximation other than that of the surface finite elements on the boundary

Scaled boundary finite element method

X X

X

X 11


The Scaled Boundary Finite Element Method – Lecture A (ctd.): Advantages of scaled boundary finite element method compared with those of finite element and boundary element methods Finite element method

Boundary element method

Scaled boundary finite element method

Symmetric dynamic-stiffness and unit-impulse response matrices for unbounded media

X

(X)

X

Symmetric static-stiffness and mass matrices for bounded media (super element)

X

(X)

X

Body loads processed without additional domain discretisation and thus additional approximation

X

Straightforward calculation of stress concentrations and intensity factors based on their definition

X X

No fictitious eigenfrequencies for unbounded media

X

X

Straightforward coupling by standard assemblage of structure discretised with finite elements with unbounded medium

X

X

Table 14.1 (ctd.)

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Infinite plate with central circular hole subjected to uniaxial tensile stress

Bounded model representing infinite plate in uniaxial stress field

Reproduced by permission of John Wiley Sons LTD

Fig. 25.3

Fig. 25.4

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Raw (left) and recovered (right) stress as computed by scaled boundary finite element method for coarse mesh Plate 25.1

Raw (left) and recovered (right) stress as computed by finite element method for the intermediate mesh Plate 25.2 Reproduced by permission of John Wiley Sons LTD

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Plate 25.1

Table 25.1 15 Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


Cylindrical foundation embedded in half-space

Fig. 25.15



Vertical stress at 5% target error: scaled boundary finite element method (left) finite element method (right) (mesh a) Plate 25.7



Computational efficiency and accuracy of adaptive finite element analyses (meshes a to e) and adaptive scaled boundary finite element analysis (mesh f)

Table 25.4 18 Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


Model of the trapezoidal plate

19 Ref. Extension of the scaled boundary finite element method to plate bending problems, Rolf Dieringer et al., PAMM 11, 203-204 (2011)


Convergence study

20 Ref. Extension of the scaled boundary finite element method to plate bending problems, Rolf Dieringer et al., PAMM 11, 203-204 (2011)


read chapter 4

Problem Statement Out-of-plane motion of wedge

and truncated semi-infinite wedge of shear plate

Figure 4.1 21 Ref. The Scaled Boundary Finite Element Method, J.P. Wolf, Swiss Federal Institute of Technology, Lausanne, Switzerland, © 2003 J. Wiley & Sons Ltd., Chicester, West Sussex, England


Equilibrium equation

p: Body load per unit volume acting perpendicular to the planestress, strain relation

Reformulating equation 4.1



Substituting equation 4.2 into equation 4.1 we get

with the shear wave velocity

With the surface traction τn boundary conditions
