Application of Space-Time Finite Elements to Elastodynamics Problems

Abstract

Currently, commercial finite element codes for elastodynamics problems use a finite difference approximation in time and a finite element approximation in space. There are two disadvantages to existing methods: 1) extremely small integration time steps (e.g., nanoseconds) are necessary due to the Courant limit and 2) it is practically impossible to use different time steps for different spatial finite elements. This leads to the significant increase in computing time that could be crucial for time‐consuming calculations for real world elastodynamics problems. A new approach with space‐time finite elements for the solution of linear elastodynamics problems is suggested. Weak formulations for elastodynamics are based on continuous and discontinuous Galerkin methods. Using unstructured finite element meshes for space‐time domains allows an adaptive discretization simultaneously in space and time, i.e. a concentration of unknowns of a discrete model around small space‐time domains (where necessary) without a significant increase in a total number of unknowns for the whole problem. The so‐called ‘subcycling’ procedure, i.e. different time increments for different space domains, is automatically included. Also specific ‘inverse’ dynamics problems (when initial displacements and initial velocities are specified at different time) can be easily solved by the proposed technique. Our initial numerical results for one‐dimensional elastodynamics problems have demonstrated the feasibility of implementation of this new numerical method.