Shape sensitivity analysis for numerical solution of free boundary problems

Abstract

Kari Kärkkäinen tutki tehokkaita ja käyttökelpoisia ratkaisumenetelmiä vapaita pintoja sisältäville malleille. Hän tarkastelee väitöksessään numeerisessa simuloinnissa käytettävien vapaan pinnan ratkaisumenetelmien tehokkuutta ja niiden parantamista käyttäen matemaattista lähestymistapaa. Kärkkäinen ehdottaa ratkaisumenetelmää, jonka avulla vapaan pinnan tehtävien ratkaiseminen on helpompaa. Menetelmää voi myös soveltaa erilaisiin tilanteisiin.

This work is devoted to the development of efficient and robust solution algorithms for a class of free boundary problems. This consists of mathematical analysis of different model problems and the description of numerical implementation to generic free boundary problems along with the numerical results. The free boundary problems that are investigated in this work are elliptic stationary boundary value problems with overdetermined boundary conditions. Overdetermined boundary conditions are satisfied only in a special geometry which is a solution to the free boundary problem. The free boundary problems are nonlinear and can not be solved straightforwardly. Algorithms to solve this kind of free boundary problems are iterative, the solution is sought by iterating geometries. The suggested algorithm is based on the combination of continuous shape sensitivity analysis and automatic differentiation of discrete equations. The discrete linearized equations are derived tuning the finite element method to correspond to the continuous shape linearization of the problem. Efficiency of the presented algorithms are tested and illustrated through numerous numerical examples.