smteam@gmx.ch – www.erdbebenschutz.ch

Course 2.092 / 2.093

Fall Semester 2015

Massachusetts Institute of Technology

M.I.T., Cambridge USA

The Scaled Boundary Finite Element Method – Introduction

I n t r o d u c t i o n

Dr. Sa s s a n Mohas seb

2

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

The Scaled Boundary Finite Element Method – Introduction

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

3

Spatial discretisation of

Finite Element Method

The Scaled Boundary Finite Element Method – Introduction

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

4

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

The Scaled Boundary Finite Element Method – Introduction

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

5

The Scaled Boundary Finite Element Method – Introduction

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

6

The Scaled Boundary Finite Element Method – Introduction

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

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 2.2

Scaled Boundary Finite Element

Method. A new numerical method

Problem definition:

(a) bounded media

(b) unbounded media

7

The Scaled Boundary Finite Element Method – Introduction

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

8

The Scaled Boundary Finite Element Method – Introduction

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

Combining advantages of FEM and BEM

Reduction of partial differential equations into ordinary differential

equations

Analytical solution in radial direction

9

The Scaled Boundary Finite Element Method – Introduction

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

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

The Scaled Boundary Finite Element Method – Introduction

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 materials X

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

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

Table 14.111

The Scaled Boundary Finite Element Method – Introduction

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 definitionX

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

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

Table 14.1 (ctd.) 12

The Scaled Boundary Finite Element Method – Introduction

13

Infinite plate with central circular hole

subjected to uniaxial tensile stress

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

The Scaled Boundary Finite Element Method – Introduction

Fig. 25.3

Bounded model representing infinite

plate in uniaxial stress field Fig. 25.4

Reproduced by permission of John Wiley Sons LTD

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

14

The Scaled Boundary Finite Element Method – Introduction

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

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

Plate 25.1

15

The Scaled Boundary Finite Element Method – Introduction

Table 25.1

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

16

Cylindrical foundation

embedded in half-space

The Scaled Boundary Finite Element Method – Introduction

Fig. 25.15

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

Plate 25.7

Vertical stress at 5%

target error:

scaled boundary finite

element method (left)

finite element method

(right) (mesh a)

17

The Scaled Boundary Finite Element Method – Introduction

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

Computational efficiency and accuracy of adaptive

finite element analyses (meshes a to e) and adaptive

scaled boundary finite element analysis (mesh f)

18

The Scaled Boundary Finite Element Method – Introduction

Table 25.4

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

Problem Statement

Out-of-plane motion of wedge

and truncated semi-infinite

wedge of shear plate

Figure 4.1

19

read

chapter 4

The Scaled Boundary Finite Element Method – Introduction

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

Reformulating equation 4.1

p: Body load per unit volume acting perpendicular

to the planestress, strain relation

20

The Scaled Boundary Finite Element Method – Introduction

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

Substituting equation 4.2 into equation 4.1 we get

with the shear wave velocity

With the surface traction τn boundary conditions

21

The Scaled Boundary Finite Element Method – Introduction