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[email protected] www.erdbebenschutz.ch

L e c t u r e A 1

December 10, 2013

ETH, IBK

Zrich

The Scaled Boundary Finite Element Method Lecture A

D r. Sa s sa n M oh a ss e b

V i 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

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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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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

Figure 1.9

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Spatial discretisation of

Finite Element Method


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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


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Spatial discretisation of

Boundary Element

Method

Figure 1.14

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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

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7/23Ref. 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

Figure 2.2

Scaled Boundary Finite Element

Method. A new numerical method

Problem definition:

(a) bounded media

(b) unbounded media

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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

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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

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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/

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Finite

element

method

Boundary

element

method

Scaled

boundary 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

X X

Analytical solution achieved inside domain X

No fundamental solution required, expanding the scope of

application and avoiding singular integralsX X

Radiation condition at infinity satisfied exactly when

modelling unbounded (infinite or semi-infinite) media X X

No discretisation of free and fixed boundaries and interfaces

between different materialsX

No approximation other than that of the surface finite

elements on the boundaryX X

Advantages of scaled boundary finite element method compared with those of finite element and

boundary element methods


Table 14.1 11


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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 approximationX X

Straightforward calculation of stress concentrations and

intensity factors based on their definition 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

(ctd.): Advantages of scaled boundary finite element method compared with those of finite element and

boundary element methods


Table 14.1 (ctd.) 12


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Infinite plate with central circular hole

subjected to uniaxial tensile stress



Fig. 25.3

Bounded model representing infiniteplate in uniaxial stress field Fig. 25.4

Reproduced by permission of John Wiley Sons LTD

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Plate 25.1

Raw (left) and recovered

(right) stress as computed by

scaled boundary finite element

method for coarse mesh

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

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Table 25.1

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Cylindrical foundation

embedded in half-space


Fig. 25.15

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Plate 25.7

Vertical stress at 5%

target error:

scaled boundary finiteelement method (left)

finite element method

(right) (mesh a)

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Computational efficiency and accuracy of adaptive

finite element analyses (meshes a to e) and adaptive

scaled boundary finite element analysis (mesh f)

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Table 25.4

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Model of the trapezoidal plate

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

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

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Convergence study

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Problem Statement

Out-of-plane motion of wedge

and truncated semi-infinite

wedge of shear plate

Figure 4.1

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read

chapter 4


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Equilibrium equation

Reformulating equation 4.1

p: Body load per unit volume acting perpendicular

to the planestress, strain relation

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Substituting equation 4.2 into equation 4.1 we get

with the shear wave velocity

With the surface traction nboundary conditions

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