Boundary Element Method

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Boundary Element Method
OUTLINE
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Motivation
Laplaces equation with boundary
conditions Essential Dirichlet
type Natural Neumann type
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Method of Weighted Residuals
Greens Theorem
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Classification of Approximate Methods

• Original statement
• Weak statement
• Inverse statement

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Original statement
Basis functions for u and w are different
Basis functions for u and w are the same
Finite differences Method of moments General
weighted residual
Original Galerkin
Weak formulation
Finite element Galerkin techniques
General weak weighted residual formulations
Inverse statement
Trefftz method
Boundary integral
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BEM formulation
where u is the fundamental solution
Note
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Dirac delta function
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Boundary integral equation
Fundamental solution for Laplaces equation
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Discretization
Nodes
Element
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Matrix form
Note matrix A is nonsymmetric
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2D-Interpolation Functions

• Linear element
• Bilinear element
• Quadratic element
• Cubic element

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Elastostatics
Bettis theorem
Field equations
Boundary conditions
Lames equation
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Fundamental solution
Lames equation
2D-Kelvins solution
displacement
traction
stress
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Somiglians formulation
On boundary
For internal points
displacement
stress
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Internal cell
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Numerical Example
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Discretization
FEM BEM
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Results
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Results
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BEM elastoplasticity-initial strain problem
Governing equations
Equation used in iterative procedure
where
Note vectors store elastic solution
matrices are evaluated only once
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Other problems
2D, 3D, axisymmetric
Plate bending
Diffusion

• Linear
• Nonlinear

- Time discretization time independent
fundamental solution
time dependent fundamental solution
Heat transfer
Coupled heat and vapor transfer
Consolidation