1. Boundary Element Method
OUTLINE

2. with boundary conditions
Motivation
Laplace`s equation
with boundary conditions
Essential Dirichlet type
Natural Neumann type

3. Method of Weighted Residuals
Green`s Theorem

4. Classification of Approximate Methods
Original statement
Weak statement
Inverse statement

5. Original statement Finite differences Weak formulation Finite element
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

6. BEM formulation
where u* is the fundamental solution
Note:

7. Dirac delta function

8. Boundary integral equation
Fundamental solution for Laplace`s equation

9. Discretization
Nodes
Element

10. Matrix form
Note: matrix A is nonsymmetric

11. 2D-Interpolation Functions
Linear element
Bilinear element
Quadratic element
Cubic element

12. Elastostatics Betti`s theorem Field equations Boundary conditions
Lame`s equation

13. Fundamental solution Lame`s equation 2D-Kelvin`s solution displacement
traction
stress

14. Somiglian`s formulation
On boundary
For internal points
displacement
stress

15. Internal cell

16. Numerical Example

17. Discretization
FEM BEM

18. Results

19. Results

20. BEM elastoplasticity-initial strain problem
Governing equations
Equation used in iterative procedure
where
Note: vectors store elastic solution
matrices are evaluated only once

21. 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