Boundary Element Method

FINITE ELEMENT METHODS FOR ELECTRICAL ENGINEERING

IEM

BOUNDARY EL EMENT EMENT METHOD (BEM) The basic idea of the BEM is to discretize the integral equation using boundary elements. The well-known moment method is equivalent equivalent to BEM when using subsectional bases and the delta function as weighting functions. Thus, BEM can be regarded as a combination of the classical boundary integral equation method and the discretization concepts originated from FEM. Consider the case of the Laplace’s equation, i.e.

∇ 2u =0 in Ω u =u in Γ1

∂u =q in Γ2 ∂n Γ = Γ1 + Γ2 q=

By application of the weighted residual method we obtain:

∫ ∇ u ⋅W  ⋅ d Ω = 0 2

Ω

Using the following identities:

∇(∇uW ) = ∇ 2uW  + ∇u∇W  ∇(u∇W ) = u∇ 2W  + ∇u∇W   And applying the Gauss theorem, theorem, we can write:

∫ Ω

∫

W  ⋅ ∇ u ⋅ d Ω = W  ⋅ 2

Γ

∂u ∂W  ⋅ d Γ − ∫ u ⋅ ⋅ d Γ + ∫ u ⋅ ∇ 2W  ⋅ d Ω = 0 ∂n ∂n Γ Ω

The weighting function W is chosen to be the fundamental solution, determined earlier:

∇ 2W  = δ (r  − r ' ) Thus the domain integral in the above equation can be written as:

∫ u ⋅ ∇ W  ⋅ d Ω = −∫ u ⋅ δ (r − r ' ) ⋅ d Ω = −u 2

Ω

i

Ω

For any point inside the domain. Combining the above equations, the following integral relation is obtained:

∫

ui = W  ⋅ Γ

∂u ∂W  ⋅ d Γ − ∫ u ⋅ ⋅ d Γ ∂n ∂ n Γ

FINITE ELEMENT METHODS FOR ELECTRICAL ENGINEERING

IEM

Performing similar mathematical manipulations we can obtain the following integral relation for the Poisson equation (∇ 2u = −  f  ) :

∫

ui = W  ⋅ Γ

∂u ∂W  ⋅ d Γ − ∫ u ⋅ ⋅ d Γ − ∫  f  ⋅W  ⋅ d Ω ∂n ∂ n Γ Ω

Notes:

∂u ) are taken into account by the first integral ∂n

•

Neumann boundary conditions (

• •

Dirichlet boundary conditions ( u )  are taken into account by the second integral. Any source point ( f  ) inside the domain Ω   is taken into account by the domain integral in the above equation.

•

From the Poisson’s equation, the residual to be minimized can be written as:

∫ (∇ u +  f  )⋅W  ⋅ d Ω = 0 2

Ω

When the observation point “i” is located on the boundary Γ , the boundary integral becomes singular as R approaches zero. To extract the singularity on boundary, we rewrite the equation

∫

ui = W  ⋅ Γ

∂u ∂W  ⋅ d Γ − ∫ u ⋅ ⋅ d Γ ∂n ∂ n Γ

In the following form: ui =

∫ Γ − ∆Γ

W  ⋅

∂u ∂u ∂W  ∂W  ⋅ d Γ + ∫ W  ⋅ ⋅ d Γ − ∫ u ⋅ ⋅ d Γ − ∫ u ⋅ ⋅ d Γ ∂n ∂ ∂ ∂ n n n ∆Γ Γ − ∆Γ ∆Γ

We present in detail the two-dimensional case. In this case: W  = −

1 2π 

ln r 

And:

⎡θ  ⎤ 1 1 ∂u ∂u ∂u ln lim ln ε  ε  θ  ⋅ ⋅ Γ = − ⋅ ⋅ Γ = − ⋅ ⋅ ⋅ W  d  r  d  d  ⎢ ⎥=0 ∫∆Γ ∂n ∫ ∂n ∂n 2π  ∆Γ 2π  ε →0 ⎢⎣θ ∫ ⎥⎦ 2

1

⎡θ  1 ⎤ θ  − θ  ∂W  1 1 1 lim ε  θ  − ∫u ⋅ ⋅ d Γ = − ⋅ ⋅ Γ = − ⋅ ⋅ ⋅ = − 2 1 ⋅ ui u d  u d  ⎢ ⎥ ∫ ∫ 2π  ∆Γ r  2π  ε →0 ⎢θ  ε  2π  ∂n ⎥⎦ ∆Γ ⎣ 2

1

Substituting in the earlier equation we obtain:

the

FINITE ELEMENT METHODS FOR ELECTRICAL ENGINEERING

∫

ci ⋅ u i = W  ⋅ ∆Γ

IEM

∂u ∂W  ⋅ d Γ − ∫ u ⋅ ⋅ d Γ ∂n ∂ n ∆Γ

Where:

⎧ 1 ⎪ θ 2 − θ 1 ci = ⎨1 − 2π  ⎪ 0 ⎩

i ∈Ω i∈Γ i ∉Ω

If we consider Poisson’s equation:

∫  f  ⋅W  ⋅ d Ω = ∫  f  ⋅W  ⋅ d Ω + ∫  f  ⋅W  ⋅ d Ω

Ω

Ω − ∆Ω

∆Ω

⎡θ  ⎤ ⋅ ⋅ Ω = − ⋅ ⋅ Ω = − ⋅ ⋅ ⋅ ⋅ ln lim ln ε  ε  θ  ε   f  W  d   f  r  d   f  d  d  ⎢ ⎥=0 ∫∆Ω ∫ 2π  ∆Ω 2π  ε →0 ⎢θ ∫ ⎥⎦ ⎣ 1

1

2

1

The final equation is:

∫

ci ⋅ ui = W  ⋅ ∆Γ

In general either u or

∂u ∂W  ⋅ d Γ − ∫ u ⋅ ⋅ d Γ − ∫  f  ⋅ W  ⋅ d Ω ∂n ∂ n ∆Γ Ω

∂u  on the boundary must be known. Determining all values of ∂n

solution u and its normal derivatives on the boundary the solution at an arbitrary point of the domain can be calculated. The electrostatic and magnetostàtic problems are then defined by the following equations:

∫

 ρ  ∂Φ ∂W  ⋅ d Γ − ∫ Φ ⋅ ⋅ d Γ + ∫ ⋅ W  ⋅ d Ω ∂n ∂n ∆Γ Ω ε 

∫

∂ A ∂W  ⋅ d Γ − ∫  A ⋅ ⋅ d Γ + ∫ µ ⋅ J  ⋅ W  ⋅ d Ω ∂n ∂ n ∆Γ Ω

ci ⋅ Φ i = W  ⋅ ∆Γ

ci ⋅ Ai = W  ⋅ ∆Γ

Where Φ denotes the electrostatic potential and  A  is the magnetic potential vector.

Boundary element discretization The basic idea of the BEM is to discretize the boundary of the domain under consideration into a set of elements. The unknown solution over each element is approximated by an interpolation function, which is associated with the values of the functions at the element nodes, so that the integral equation can be converted into a system of algebraic equations. The boundary geometry can be discretized in a series of elements.