Numerical model reduction for FEanalysis of the viscoplasticity problem

Abstract

Some numerical problems require, despite modern computational power, a lot of time to solve. The nonlinear viscoplasticity model is one such problem, where the plasticity in a material depends on the rate at which the load is applied. A method for reducing the system and decreasing the simulation time would therefore be an advantage. This thesis aims to adopt one such method to reduce the computational cost for the viscoplasticity problem and evaluate it for some test cases. A mixed weak form together with the Finite Element Method (FEM) on monolithic form is established. Thereby, displacements and viscoplastic strains are solved for simultaneously rather than in the standard nested fashion. Proper Orthogonal Decomposition (POD) is performed on snapshots of the viscoplastic strains from a set of finite element training simulations carried out in an offline phase. The Nonuniform Transformation Field Analysis (NTFA) approach expresses the displacements in a corresponding reduced basis. The numerical computations have been implemented in Julia and tested in 2D for varying load combinations. It was shown that it was possible to reduce the solve time and still obtain good approximations of the solution. However, there is a crucial dependency on the training, with higher accuracy for targeted simulations similar to the training. Still, the robustness of the procedure was illustrated by near monotonic error convergence. Although more research would be needed, the results show promise for the development of highly efficient approximations of the viscoplasticity problem.