Numerical Modeling.Numerical modelingin nondestructive testing is an outgrowthof the failure of analytical models reliablyto predict the necessary field interactionswith any degree of generality. A numericalmodel uses a digital computer to solve thegoverning equations directly, with fewsimplifying assumptions. This in itself isenough to explain the value of suchmodels. Numerical modeling allows thesolution of very complex problems and, atthe same time, does not require the userto know the intricacies of electromagnetictheory or differential calculus. All that theuser is required to do is input the problemvariables and, if necessary, verify theresults experimentally.General Overview of Analyticaland Numerical ModelingThe existence of models for eddy currenttesting phenomena depends entirely onthe ability to solve Maxwell’s equationswith or without approximations. Thevalue of such models in solving theinversion problem satisfactorily is beyonddispute. Solution of this importantproblem is possible only with thedevelopment of good theoretical models,capable of predicting the complexinteractions of a multitude of factors inthe test object. A good, reliable theoreticalmodel for nondestructive testing shouldbe able to satisfy the following conditions.1. The model should describe the physicsof interaction between the appliedalternating current field, inducedcurrents and discontinuities in the testobject.2. The model should serve as atheoretical test bed for situationsdifficult or impossible to replicateexperimentally.3. The model should generate eddycurrent output signals for a widevariety of discontinuities andspecimen shapes, avoiding costlysample preparation and helping todetermine discontinuitycharacterization parameters.4. The model should provide trainingdata for automated discontinuitycharacterization systems andequipment.5. The model should aid in the design ofeddy current probes for specificapplications.For the purpose of deriving such amodel, two main avenues are available:the analytical approach and the numericalapproach.Analytical ModelingAnalytical models are derived from basicfield and circuit theory considerations. Ineffect, an attempt is made to solveMaxwell’s equations directly. Theseequations are generally three-dimensional,nonlinear, partial differential equationswithin complex boundaries anddiscontinuity shapes. In addition, formoving probe problems, the solution isboth time and position dependent. It istherefore not surprising to find that suchsolutions are only possible for the mostelementary of test geometries, withsimplifying assumptions in terms ofgeometry, dimensionality, discontinuityshapes and sources. Thisoversimplification accounts for the factthat analytical models are limited inscope, applicable only to selectedproblems and not easily extended to othergeometries. On the other hand, thesolution to problems for which ananalytical model applies is relatively
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