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Caustics, pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination
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| OpticsandLasersinEngineering |
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The methodofcausticsisapowerfulexperimentalmethodinelasticityandparticularlyinfracture mechanics forcrackproblems.Therelatedmethodofpseudocausticsisalsoofinterest.Hereweapply the computationalmethodofquantifier eliminationimplementedinthecomputeralgebrasystem Mathematica in ordertodetermine(i)thenon-parametricequationandtwopropertiesofthecausticata cracktipandespecially(ii)theilluminatedandthedarkregionsrelatedtocausticsandpseudocausticsin plane elasticityandplateproblems.Thepresentcomputationsconcern:(i)Thederivationofthenon- parametric equationoftheclassicalcausticaboutacracktipthroughtheeliminationoftheparameter involved(herethepolarangle)aswellastwogeometricalpropertiesofthiscaustic.(ii)Thederivationof the inequalitiesdefining theilluminatedregiononthescreenintheproblemofanelastichalf-plane loaded normallybyaconcentratedloadwiththeboundaryofthisilluminatedregionrelatedtosome extenttothecausticformed.(iii)Similarlyfortheproblemofaclampedcircularplateunderauniform loading withrespecttothecausticandthepseudocausticformed.(iv)Analogouslyfortheproblemofan equilateraltriangularplateloadedbyuniformlydistributedmomentsalongitswholeboundary,which defines therelatedpseudocaustic.(v)Thedeterminationofquantitiesofinterestinmechanicsfromthe obtained causticsorpseudocaustics.Thekindofcomputationsintheapplications(ii)to(iv),i.e.the derivationofinequalitiesdefining theilluminatedregiononthescreen,seemstobecompletelynew independently oftheusehereofthemethodofquantifier elimination.Additionalapplicationsarealso possible, butsomeofthemrequiretheexpansionofthepresentsomewhatlimitedpowerofthe quantifier eliminationalgorithmsin Mathematica. Thisisexpectedtotakeplaceinthefuture.
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