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Boundary Element Method

Boundary Element Method. OUTLINE. 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.

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Boundary Element Method

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  1. Boundary Element Method OUTLINE

  2. Motivation Laplace`s equation with boundary conditions EssentialDirichlet type NaturalNeumann type

  3. Method of Weighted Residuals Green`s 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 General weak weighted residual formulations Finite element Galerkin techniques Inverse statement Boundary integral Trefftz method

  6. BEM formulation whereu* 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: matrixAis 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

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