Title: Boundary Element Method
1Boundary Element Method
OUTLINE
2Motivation
Laplaces equation with boundary
conditions Essential Dirichlet
type Natural Neumann type
3Method of Weighted Residuals
Greens Theorem
4Classification of Approximate Methods
- Original statement
- Weak statement
- Inverse statement
5Original 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
6BEM formulation
where u is the fundamental solution
Note
7Dirac delta function
8Boundary integral equation
Fundamental solution for Laplaces equation
9Discretization
Nodes
Element
10Matrix form
Note matrix A is nonsymmetric
112D-Interpolation Functions
- Linear element
- Bilinear element
- Quadratic element
- Cubic element
12Elastostatics
Bettis theorem
Field equations
Boundary conditions
Lames equation
13Fundamental solution
Lames equation
2D-Kelvins solution
displacement
traction
stress
14Somiglians formulation
On boundary
For internal points
displacement
stress
15Internal cell
16Numerical Example
17Discretization
FEM BEM
18Results
19Results
20BEM elastoplasticity-initial strain problem
Governing equations
Equation used in iterative procedure
where
Note vectors store elastic solution
matrices are evaluated only once
21Other problems
2D, 3D, axisymmetric
Plate bending
Diffusion
- Time discretization time independent
fundamental solution
time dependent fundamental solution
Heat transfer
Coupled heat and vapor transfer
Consolidation