Boundary Element Method OUTLINE

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

Method of Weighted Residuals Green`s Theorem

Classification of Approximate Methods Original statement Weak statement Inverse statement

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

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

2D-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 2D-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 store elastic solution matrices are evaluated only once

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