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Boundary Element Method (BEM)
Zoran Ilievski
Wednesday 28th June, 2006HG 6.96 (TU/e)
2
Talk Overview
• The idea of BEM and its advantages
• The 2D potential problem
• Numerical implementation
3
Idea of BEM
4
Idea of BEM
5
Advantages of BEM
1) Reduction of problem dimension by 1
• Less data preparation time.• Easier to change the applied mesh.• Useful for problems that require re-meshing.
6
Advantages of BEM
2) High Accuracy • Stresses are accurate as there are
no approximations imposed on the solution in interior domain points.
• Suitable for modeling problems of rapidly changing stresses.
7
Advantages of BEM
3) Less computer time and storage
• For the same level of accuracy as other methods BEM uses less number of nodes and elements.
8
Advantages of BEM
4) Filter out unwanted information.• Internal points of the domain are
optional.
• Focus on particularinternal region.
• Further reduces computer time.
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Advantages of BEM
1. Reduction of problem dimension by 1.2. High Accuracy.3. Less computer time and storage.4. Filter out unwanted information and so
focus on section of the domain you are interested in.
BEM is an attractive option.
10
The 2D potential problem
• Where can BEM be applied?• Two important functions.• Description of the domain.• Mapping of higher to lower dimensions.• Satisfaction of the Laplace equations
and how to deal with a singularity.• The boundary integral equation (BIE)
11
The 2D potential problem
Where can BEM be applied?
Where any potential problem is governed by a differential equationthat satisfies the Laplace equation.
(or any other behavior that has a related fundamental solution)
e.g. The following can be analyzed with the Laplace equation: fluid flow, torsion of bars,
diffusion and steady state heat conduction.
12
The 2D potential problem
02
2
2
22
yx
.2
The Laplace equation for 2D
yx,
Laplacian operator
Potential function
Cartesian coordinate axis
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The 2D potential problemTwo important functions.
)(2 PQ
The function describing the property under analysis. e.g. heat. (Unknown)
The fundamental solution of the Laplace equation. (These are well known)
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The 2D potential problem
),(
1ln
2
1),(
QprQp
),(
1ln
2
1),(
QprQp
Fundamental solution of the 2DLaplace equation for a concentrated source point at p is
22),( QpQp yYxXQpr
Where
Description of the domain
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The 2D potential problemMapping of higher to lower dimensions
dnn
dAA
22
• Boundary of any domain is of a dimension 1 less than of the domain.
• In BEM the problem is moved from within the domain to its boundary.
• This means you must, in this case, map Area to Line.
• The well known ‘Greens Second Identity’ is used to do this.
,
n
n
have continuous 1st and 2nd derivatives.unknown potential at any point.
known fundamental solution at any point.
unit outward normal.
derivative in the direction of normal.
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The 2D potential problemSatisfying the Laplace equation
The unknown
02 will satisfy
everywhere in the solution domain.
The known fundamental solution
satisfies
02 everywhere except the point p where it is singular.
),(
1ln
2
1),(
QprQp
22),( QpQp yYxXQpr
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The 2D potential problemHow to deal with the singularity
d
nndA
AA
22
• Surround p with a small circle of radius
ε , then
examine solution as ε 0
• New area is (A – Aε )• New boundary is (Γ + Γε )
Within area (A – Aε)
02 02 &
The left hand side of the equation is now 0 and the right is now …
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The 2D potential problemHow to deal with the singularity
d
nnd
nn
0
The second term must be evaluated and to do this let
dd
rn
r
rn
2
1.
And use the fact that
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The 2D potential problemHow to deal with the singularity
.1)2(2
1
1ln
1
2
1 2
0
dn
dnn
0
2/1
1
)(PC
Evaluated with p in the domain,on the boundary (Smooth surface),and outside the boundary.
2
)( PC For coarse surfaces
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The 2D potential problemThe boundary integral equation
)()(),()(),()()( 12 QdQQPKQdn
QPKPPC
),(),(
),(),(
2
1
QPQPKn
QPQPK
Where K1 and K2 are the known fundamental solutions and are equal to
2
)( PC
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The 2D potential problem• BEM can be applied where any potential problem is governed by
a differential equation that satisfies the Laplace equation. In this case the 2D form.
• A potential problem can be mapped from higher to lower
dimension using Green’s second identity.
• Shown how to deal with the case of the singularity point.
• Derived the boundary integral equation (BIE)
)()(),()(),()()( 12 QdQQPKQdn
QPKPPC
22
Numerical Implementation
• Dirichlet, Neumann and mixed case.• Discretisation• Reduction to a form Ax=B
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Numerical ImplementationDirichlet, Neumann and mixed case.
)()(),()(),()()( 12 QdQQPKQdn
QPKPPC
The unknowns of the above are values on the boundary and are n
,
Dirichlet Problem
Neumann Problem
n
is given every point Q on the boundary.
is given every point Q on the boundary.
Mixed case – Either are given at point Q
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Numerical ImplementationDiscretisation
N
jjjij
N
jjji
ji
jj
dQPKQdQPKn
QP
11
122
1 ),()(),()(
)(
Unknowns
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Numerical ImplementationDiscretisation
N
jijj
N
jij
ji KQK
n
QP
11
122
1 )()(
)(
jjiij dQPKKj
),(22 j
jjiij dQPKK ),(11
Unknowns
Let
26
Numerical Implementation
N
j
jijj
N
jijij n
QKQK
12
121
1
)()(
jiwhenQP ji )()(
BzAx
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Numerical Implementation
BzAx
cAx
Dirichlet Problem
Bzc
Neumann Problem
Mixed case
Matrix A and vector C are known
Matrix B and vector C are known
Unknowns and knowns can be separated in to same form as above
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Numerical Implementation
As each point p in the domain is expressed in terms ofthe boundary values, once all boundary values are known ANYpotential value within the domain can now be found.
)()(),()(),()()( 12 QdQQPKQdn
QPKPPC
29
Boundary Element Method (BEM)
THE END
Book: The Boundary Element Method in Engineering A.A.BECKER
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