Discrete exterior calculus and exact controllability for time-harmonic acoustic wave simulation
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2023Copyright
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Diskreetti ulkoinen laskenta (engl, discrete exterior calculus, DEC) on differentiaaliyhtälöiden ratkaisemiseen soveltuva diskretointimenetelmä,
joka säilyttää tiettyjä fysikaalisten mallien geometrisia ominaisuuksia ja tuottaa laskennallisesti tehokkaita algoritmeja. Tutkielmassa tarkastellaan keinoja tehostaa DEC:n suorituskykyä aikaharmonisten ratkaisujen selvittämisessä akustisten aaltojen sirontaa kuvaaville tehtäville. Tähän käytetään harmonisille malleille suunniteltuja operaattoreita ja eksaktiin kontrolloitavuuteen perustuvaa optimointimenetelmää. Toteutetuissa kokeiluissa harmoniset operaattorit lisäävät menetelmän tarkkuutta huomattavasti, kun taas kontrollimenetelmä osoittautuu hyväksi valinnaksi vain tehtävissä, joiden geometria on huomattavan epäkonveksia. Laskentaverkon laadulla on suuri merkitys tutkittujen menetelmien tarkkuudessa ja stabiilisuudessa. Discrete exterior calculus (DEC) is a discretization method for differential equations which preserves important geometric properties of physical models and produces computationally efficient algorithms. We investigate ways to improve the performance of DEC in the context of time-periodic solutions to acoustic wave scattering problems by using operators purpose-built for harmonic problems and an optimization approach based on exact controllability. The harmonic operators are found to improve accuracy significantly, while the controllability method is found to be a good choice only in problems with highly nonconvex geometry. Computation mesh quality is identified as a key issue in the accuracy and stability of these methods.
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