A Comparative Study of a Backward Euler and Exponential Update for a Viscoelastic Maxwell Model Within the Material Point Method
Marvin Koßler, Ralf Müller, Jörg Schröder
Source abstract
ABSTRACT The material point method (MPM) offers a distinct simulation method compared to conventional techniques such as the finite element method (FEM). In the MPM, physical bodies are represented by a collection of material points while the governing equations are solved on an independent computational mesh. Each simulation step follows a three‐stage process. First, the material points transfer their kinematic properties to the nodes of the background grid. Next, the necessary calculations are performed on the grid, and the quantities are mapped back to the material points, updating their location, velocity, internal stress, and deformation states. Since the grid does not retain information between steps, it is regenerated anew at each time step. This reinitialization prevents issues like mesh distortion, which frequently occur in FEM simulations under large deformations. In this contribution, a comparative study of two integration schemes in the framework of a finite viscoelastic Maxwell model is performed. The first scheme involves an evolution equation regarding the internal viscous variables that are solved using a backward Euler scheme. In the second scheme, an exponential map is used for the integration of the viscous variables over time. Both integrators are compared using a typical numerical example within the MPM in terms of the stress field and volume preservation regarding the viscous flow.
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