简介：Inthispaper,asimplebutinherentrelationbetweentheL-integralandtheBueeknerworkconjugateintegralisprovedforcrackproblemsinisotropic,anisotropic,anddissimilarmateri-als,respectively.Itisfoundthat,intheabove-mentionedthreecases,theL-integral,fromthemath-ematicalpointofviewaswellasinprinciple,arisesfromBetti’sreciprocaltheorem.ThismeansthattheBuecknerworkconjugateintegralisamoregeneralpath-independentintegralthantheotherssinceanyotherpath-independentintegralscouldbederivedbyusin8theBuecknerintegralwhilechoosingadifferentsubsidiarystress-displacementfield.

简介：

简介：InthispaperwestudythesolutionofSATproblemsformulatedasdiscretedecisionanddiscreteconstrainedoptimizationproblems.Constrainedformulationsarebetterthantraditionalunconstrainedformulationsbecauseviolatedconstraintsmayprovideadditionalforcestoleadasearchtowardsasatisfiableassignment.Wesummarizethetheoryofextendedsaddlepointsinpenaltyformulationsforsolvingdiscreteconstrainedoptimizationproblemsandtheassociateddiscretepenaltymethod(DPM).Wethenexaminevariousformulationsoftheobjectivefunction,choicesofneighborhoodinDPM,strategiesforupdatingpenalties,andheuristicsforavoidingtraps.ExperimentalevaluationsonhardbenchmarkinstancespinpointthattrapscontributesignificantlytotheinefficiencyofDPMandforceatrajectorytorepeatedlyvisitthesamesetofornearbypointsintheoriginalvariablespace.Toaddressthisissue,weproposeandstudytwotrap-avoidancestrategies.Thefirststrategyaddsextrapenaltiesonunsatisfiedclausesinsideatrap,leadingtoverylargepenaltiesforunsatisfiedclausesthataretrappedmoreoftenandmakingtheseclausesmorelikelytobesatisfiedinthefuture.Thesecondstrategystoresinformationonpointsvisitedbefore,whetherinsidetrapsornot,andavoidsvisitingpointsthatareclosetopointsvisitedbefore.Itcanbeimplementedbymodifyingthepenaltyfunctioninsuchawaythat,ifatrajectorygetsclosetopointsvisitedbefore,anextrapenaltywilltakeeffectandforcethetrajectorytoanewregion.Itspecializestothefirststrategybecausetrapsarespecialcasesofpointsvisitedbefore.Finally,weshowexperimentalresultsonevaluatingbenchmarksintheDIMACSandSATLIBarchivesandcompareourresultswithexistingresultsonGSAT,WalkSAT,LSDL,andGrasp.TheresultsdemonstratethatDPMwithtrapavoidanceisrobustaswellaseffectiveforsolvinghardSATproblems.

简介：