Workshop on

numerical methods for multiscale problems

 

     
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  Sören Bartels: Reliable and efficient approximation of polyconvex envelopes

Nonconvex variational problems occur in the mathematical modelling of phase transitions in crystalline solids. Since such problems typically lack classical solutions relaxations are considered to compute a generalised solution. This approach and its numerical analysis are now well understood for scalar problems for which the relaxed problem is (degenerately) convex. In vectorial problems the quasiconvex envelope of an energy density is needed to define the relaxed problem. Such envelopes are only known for a few special cases. The polyconvex envelope is a lower bound for the quasiconvex envelope and the talk is devoted to the reliable and efficient approximation of this lower bound. We present a quadratically convergent iterative algorithm with grid coarsening and local refinement to approximate polyconvex envelopes. The algorithm allows for the simultaneous (polyconvex) relaxation and approximation of certain vectorial variational problems and this approach results in a numerical scheme with two discrete scales.
Impressum
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