Title

An algorithm for optimizing CVBEM and BEM nodal point locations
Authors

 T.P. Kendall
 T. V. Hromadka II
 D.D. Phillips
Journal / Anthology

 Engineering Analysis with Boundary Elements
 Year: 2012
 Volume: 36
 Issue: 6
 Page range: 979-984
Description

The Complex Variable Boundary Element Method or CVBEM is a numerical technique for approximating particular partial differential equations such as the Laplace or Poisson equations(which frequently occur in physics and engineering problems,among many other fields of study).The advantage in using the CVBEM over traditional domain methods such as finite difference or finite element based methods includes the properties that the resulting CVBEM approximation is a function:(i)defined throughout the entire plane,(ii)that is analytic throughout the problem domain and almost everywhere on the problem boundary and exterior of the problem domain union boundary;(iii)is composed of conjugate two-dimensional real variable functions that are both solutions to the Laplace equation and are orthogonal suchastoprovidethe‘‘flownet’’ofpotentialandstreamfunctions,amongmanyother features.Inthispaper,aprocedureisadvancedthatlocatesCVBEMnodalpointlocationsonand exterior oftheproblemboundarysuchthaterrorinmatchingproblemboundaryconditionsisreduced. That is,locatingthenodalpointsispartofmodelingoptimizationprocess,wherenodesarenot restrictedtobelocatedontheproblemboundary(asisthetypicalcase)butinsteadlocationsare optimizedthroughouttheexterioroftheproblemdomainaspartofthemodelingprocedure.The presentedprocedureresultsinnodallocationsthatachieveconsiderableerrorreductionovertheusual methodsofplacingnodesontheproblemboundarysuchasatequallyspacedlocationsorothersuch procedures.Becauseofthesignificanterrorreductionobserved,thenumberofnodesneededinthe modelissignificantlyreduced.Itisnotedthatsimilarresultsoccurwiththerealvariableboundary elementmethod(orBEM). The CVBEMandrelevantnodallocationoptimizationalgorithmisprogrammedtorunonprogram Mathematica,whichprovidesextensiveinternalmodelingandoutputgraphingcapabilities,and considerablelevelsofcomputationalaccuracy.TheMathematicasourcecodeisprovided.
Subject

 Unclassified