Boundary Element Method - PowerPoint PPT Presentation

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Boundary Element Method

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Title: Boundary Element Method


1
Boundary Element Method
OUTLINE
2
Motivation
Laplaces equation with boundary
conditions Essential Dirichlet
type Natural Neumann type
3
Method of Weighted Residuals
Greens Theorem
4
Classification of Approximate Methods
  • Original statement
  • Weak statement
  • Inverse statement

5
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
6
BEM formulation
where u is the fundamental solution
Note
7
Dirac delta function
8
Boundary integral equation
Fundamental solution for Laplaces 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
Bettis theorem
Field equations
Boundary conditions
Lames equation
13
Fundamental solution
Lames equation
2D-Kelvins solution
displacement
traction
stress
14
Somiglians 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
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