Boundary Element Method OUTLINE Motivation Laplaces equation with

Boundary Element Method OUTLINE

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

Method of Weighted Residuals Green`s Theorem

Classification of Approximate Methods • Original statement • Weak statement • Inverse statement

Original statement Basis functions for u and w are different for u and w are the same Finite differences Original Galerkin Method of moments Weak formulation General weighted residual Finite element General weak weighted Galerkin techniques residual formulations Inverse statement Trefftz method Boundary integral

BEM formulation where u* is the fundamental solution Note:

Dirac delta function

Boundary integral equation Fundamental solution for Laplace`s equation

Discretization Nodes Element

Matrix form Note: matrix A is nonsymmetric

2 D-Interpolation Functions • Linear element • Bilinear element • Quadratic element • Cubic element

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

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

Somiglian`s formulation On boundary For internal points displacement stress

Internal cell

Numerical Example

Discretization FEM BEM

Results

Results

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

Other problems 2 D, 3 D, axisymmetric Plate bending Diffusion • Linear • Nonlinear - Time discretization – time independent fundamental solution – time dependent fundamental solution Heat transfer Coupled heat and vapor transfer Consolidation