边界元简介BEM

1、1Boundary Element Method (BEM)Zoran IlievskiWednesday 28th June, 2006HG 6.96 (TU/e)2Talk Overview The idea of BEM and its advantages The 2D potential problem Numerical implementation3Idea of BEM4Idea of BEM5Advantages of BEM1) Reduction of problem dimension by 1Less data preparation time.Easier to c

2、hange the applied mesh. Useful for problems that require re-meshing.6Advantages of BEM2) High Accuracy Stresses are accurate as there are no approximations imposed on the solution in interior domain points. Suitable for modeling problems of rapidly changing stresses.7Advantages of BEM3) Less compute

3、r time and storage For the same level of accuracy as other methods BEM uses less number of nodes and elements.8Advantages of BEM4) Filter out unwanted information. Internal points of the domain are optional. Focus on particularinternal region. Further reduces computer time.9Advantages of BEMReductio

4、n of problem dimension by 1.High Accuracy.Less computer time and storage.Filter out unwanted information and so focus on section of the domain you are interested in.BEM is an attractive option.10The 2D potential problem Where can BEM be applied? Two important functions. Description of the domain. Ma

5、pping of higher to lower dimensions. Satisfaction of the Laplace equations and how to deal with a singularity. The boundary integral equation (BIE)11The 2D potential problemWhere can BEM be applied?Where any potential problem is governed by a differential equationthat satisfies the Laplace equation.

6、 (or any other behavior that has a related fundamental solution)e.g. The following can be analyzed with the Laplace equation: fluid flow, torsion of bars, diffusion and steady state heat conduction.12The 2D potential problem022222yx.2The Laplace equation for 2Dyx,Laplacian operatorPotential function

7、Cartesian coordinate axis13The 2D potential problemTwo important functions.)(2PQ The function describing the property under analysis. e.g. heat. (Unknown)The fundamental solution of the Laplace equation. (These are well known)14The 2D potential problem),(1ln21),(QprQp),(1ln21),(QprQp Fundamental sol

8、ution of the 2DLaplace equation for a concentrated source point at p is22),(QpQpyYxXQprWhereDescription of the domain15The 2D potential problemMapping of higher to lower dimensionsdnndAA22Boundary of any domain is of a dimension 1 less than of the domain.In BEM the problem is moved from within the d

9、omain to its boundary.This means you must, in this case, map Area to Line.The well known Greens Second Identity is used to do this.,nnhave continuous 1st and 2nd derivatives.unknown potential at any point.known fundamental solution at any point.unit outward normal.derivative in the direction of norm

10、al.16The 2D potential problemSatisfying the Laplace equationThe unknown 02will satisfy everywhere in the solution domain.The known fundamental solutionsatisfies02everywhere except the point p where it is singular.),(1ln21),(QprQp22),(QpQpyYxXQpr17The 2D potential problemHow to deal with the singular

11、itydnndAAA22Surround p with a small circle of radius , thenexamine solution as 0New area is (A A )New boundary is ( + ) Within area (A A)0202&The left hand side of the equation is now 0 and the right is now 18The 2D potential problemHow to deal with the singularitydnndnn0The second term must be eval

12、uated and to do this letddrnrrn21.And use the fact that19The 2D potential problemHow to deal with the singularity. 1)2(211ln12120dndnn02/11)(PCEvaluated with p in the domain,on the boundary (Smooth surface),and outside the boundary.2)(PCFor coarse surfaces20The 2D potential problemThe boundary integ

13、ral equation)()(),()(),()()(12QdQQPKQdnQPKPPC),(),(),(),(21QPQPKnQPQPKWhere K1 and K2 are the known fundamental solutions and are equal to2)(PC21The 2D potential problemBEM can be applied where any potential problem is governed by a differential equation that satisfies the Laplace equation. In this

14、case the 2D form. A potential problem can be mapped from higher to lower dimension using Greens second identity.Shown how to deal with the case of the singularity point.Derived the boundary integral equation (BIE)()(),()(),()()(12QdQQPKQdnQPKPPC22Numerical Implementation Dirichlet, Neumann and mixed

15、 case. Discretisation Reduction to a form Ax=B23Numerical ImplementationDirichlet, Neumann and mixed case.)()(),()(),()()(12QdQQPKQdnQPKPPCThe unknowns of the above are values on the boundary and aren,Dirichlet ProblemNeumann Problemnis given every point Q on the boundary.is given every point Q on t

16、he boundary.Mixed case Either are given at point Q24Numerical ImplementationDiscretisationNjjjijNjjjijijjdQPKQdQPKnQP111221),()(),()()(Unknowns25Numerical ImplementationDiscretisationNjijjNjijjiKQKnQP111221)()()(jjiijdQPKKj),(22jjjiijdQPKK),(11UnknownsLet26Numerical ImplementationNjjijjNjijijnQKQK121211)()(jiwhenQPji)()(BzAx 27Numerical ImplementationBzAx cAx Dirichlet ProblemBzc Neumann ProblemMixed caseMatrix A and vector C are knownMatrix B and vector C are known Unknowns and knowns can be separated in to same form as above28Numerical Implementation As each point p in the doma