1 / 21

Advanced Boundary Element Methods for Laplace's Equation: Weighted Residuals and Elastostatics

This comprehensive exploration covers essential techniques for solving Laplace's equation, emphasizing boundary conditions of the Essential Dirichlet and Natural Neumann types. It delves into various methods, including the Method of Weighted Residuals and Galerkin techniques, as well as boundary integral formulations. The application of fundamental solutions, discretization approaches, and elements such as linear, bilinear, and quadratic elements are discussed. Additionally, it addresses elastostatics, heat transfer, and the interplay between numerical methods and boundary conditions, providing a robust framework for tackling complex engineering problems.

reevesc
Télécharger la présentation

Advanced Boundary Element Methods for Laplace's Equation: Weighted Residuals and Elastostatics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related