Multigrid methods for Prandtl-Reuss plasticity (Q2760365)

This paper develops accurate and efficient numerical procedures for the solution of Prandtl-Reuss plasticity problems. The numerical solution uses incremental finite element technique with separate discretization in time and space. The discretization in time is performed by diagonally implicit Runge-Kutta methods, for which a B-stability result is shown. The computation of increments of displacements, stresses and internal parameters within load steps is based on global equilibrium condition and local evaluation of material response. The nonlinearity in the equilibrium equation is solved by Newton method with numerically computed tangent operator. The displacements and other variables are approximated by finite elements with a discussion of use of standard and stabilized finite elements, and use of adaptive local refinement techniques with a new technique for interpolation of material history. The solution of plasticity problems then uses a finite element software for linear elasticity problems (in this case the UG software including adaptivity and multigrid solvers), plus a plasticity interface which involves material response functions for computation of stresses and internal variables, interpolation of these variables from coarse to refined grids, and the computation of tangent operator for Newton method. NEWLINENEWLINENEWLINEThe paper contains unique numerical experiments showing convergence of Newton methods and multigrid solvers and efficiency of parallelization up to 256 processors. These experiments also give comparisons of influence of various finite elements and load increment sizes on the accuracy of computed results. The numerical experiments use extremely fine discretization: some examples use more than \(10^3\) load steps and nearly \(10^7\) spatial degrees of freedom.