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lectBEM - Chapter 16 Boundary Element Method We will...

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Chapter 16 Boundary Element Method We will proceed to learn about Object Oriented Programming and Design by look- ing at a specific example that revolves around the solution of the Laplace equation by the Boundary Element Method (BEM). The benefits of this approach is that it takes a 2D or 3D PDE and reduces it to an algorithm that involves 1D or 2D computations only, respectively. For simplicity we will focus on the 2D case only. 16.1 The equations and the boundary conditions The Laplace equation takes the form 2 φ = ∇ · ( φ ) = 0 (16.1) subject to boundary conditions of the type 1. Dirichlet where the unknown function is imposed on a portion of the bound- ary, i.e. φ ( x ) = a ( s ), s being a coordinate system along the boundary. 2. Neumann where the normal derivative of the function is known φ · n = q ( s ) where n is the outward unit normal. 3. Robin where a relationship of the form αφ + β φ · n = r ( s ). A well-posed problem requires the specification of only one type of boundary conditions on any portion of the boundary, that is either the function value is specified, or the normal derivative but not both. The data imposed on the boundary form are part of the problem’s specifi- cation and can be quite arbitrary except for some minor constraints. For example, if Neumann conditions are applied over the entire boundary, then the solution can be determined only up to an additive constant, that is if φ is a solution of 2 φ = 0 with φ · n = q , then so is φ + C where C is an 155
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156 CHAPTER 16. BOUNDARY ELEMENT METHOD arbitrary constant. In that case the flux have to satisfy a compatability con- dition that can be derived by applying the divergence theorem to equation 16.1: integraldisplay Ω 2 φ d A = integraldisplay Ω φ · n d s = integraldisplay Ω q d s = 0 (16.2) 16.2 From PDE to Integral Equation Consider a function G , solution to a Poisson equation of the form: 2 G = Q ( x x i ) (16.3) where Q ( x x i ) is a function that depends on the space variable x and on the source point x i where an impulse forcing is applied. I will keep these terms vague for the time being, and will delay specifying them until later. Multiplying equation 16.3 by φ and 16.1 by G and substracting the resulting expressions we get: φ 2 G G 2 φ = φG ( x x i ) (16.4) Integrating the above equation over the area of the domain Ω we get: integraldisplay Ω ( φ 2 G G 2 φ ) d A = integraldisplay Ω φG ( x x i ) d A (16.5) The chain rule of differentiation and the Gauss divergence theorem can be used to turn the left hand side into a boundary integral only. Indeed since for any differentiable functions a and b : ∇ · ( a b ) = a · ∇ b + a 2 b , we can then write: φ 2 G G 2 φ = ∇ · ( φ G ) − ∇ φ · ∇ G − ∇ · ( G φ ) + φ · ∇ G (16.6) = ∇ · ( φ G ) − ∇ · ( G φ ) = ∇ · ( φ G G φ ) (16.7) Furthermore the divergence form of the last equations permits the application of the Gauss divergence theorm to get: integraldisplay Ω ( φ 2 G G 2 φ ) d A = integraldisplay Ω
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