Hypoelastic Stabilization of Variational Algorithm for Construction of Moving Deforming Meshes

We suggest an algorithm for time-dependent mesh deformation based on the minimization of hyperelastic quasi-isometric functional. The source of deformation is time-dependent metric tensor in Eulerian coordinates. In order to attain the time continuity of deformation we suggest stress relaxation procedure similar to hypoelasticity where at each time step special choice of metric in Lagrangian coordinates eliminates internal stresses. Continuation procedure gradually introduces internal stresses back while forcing deformation to follow prescribed Eulerian metric tensor. At each step of continuation procedure functional is approximately minimized using few steps of preconditioned gradient search algorithm. Stress relaxation and continuation procedure are implemented as a special choice of factorized representation of Lagrangian metric tensor and nonlinear interpolation procedure for factors of this metric tensor. Thus we avoid solving time-dependent PDE for mesh deformation and under certain assumption guarantee that deformation from mesh on a one-time level to the next one converges to isometry when time step tending to zero.