Numerical results for a time-discontinuous Galerkin space–time finite element formulation for second-order hyperbolic partial differential equations are presented. Discontinuities are allowed at finite, but not fixed, time increments. A method for h-adaptive refinement of the space–time mesh is proposed and demonstrated. Numerical results are presented for linear elastic problems in one space dimension. Numerical verification of unconditional stability, as proven in , is rendered. Comparison is made with analytic solutions when available. It is shown that the accuracy of the numerical solution can be increased without a major penalty on computational cost by using an adaptively refined mesh. Results are presented for a type of solid–solid dynamic phase transition problem where the trajectory of a moving surface of discontinuity is tracked.
Discontinuous Galerkin FEM Space–time finite element method Dynamic fracture
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Johnson C (1993) Discontinuous Galerkin finite element methods for second order hyperbolic problems. Comput Methods Appl Mech Eng 107: 117–129zbMATHCrossRefGoogle Scholar
Huang H, Costanzo F (2002) On the use of space–time finite elements in the solution of elasto-dynamic problems with strain-discontinuities. Comput Methods Appl Mech Eng 191(46): 1649–1679CrossRefMathSciNetGoogle Scholar
Costanzo F, Huang H (2005) Proof of unconditional stability for a single-field discontinuous galerkin finite element formulation for linear elasto-dynamics. Comput Methods Appl Mech Eng 194(18–20): 2059–2076zbMATHCrossRefMathSciNetGoogle Scholar
Li XD, Wiberg N-E (1998) Implementation and adaptivity of a space–time finite element method for structural dynamics. Comput Methods Appl Mech Eng 156: 211–229zbMATHCrossRefMathSciNetGoogle Scholar
Wiberg N-E, Li XD (1999) Adaptive finite element procedures for linear and non-linear dynamics. Int J Numer Methods Eng 46: 1781–1802zbMATHCrossRefGoogle Scholar
Thompson LL, He D (2005) Adaptive space–time finite element methods for the wave equation on unbounded domains. Comput Methods Appl Mech Eng 194: 1947–2000zbMATHCrossRefMathSciNetGoogle Scholar
Idesman AV (2005) Solution of linear elastodynamics problems with space–time finite elements on structured and unstructured meshes. Comput Methods Appl Mech Eng 196: 1787–1815CrossRefGoogle Scholar
Abedi R, Petracovici B, Haber RB (2006) A space–time discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance. Comput Methods Appl Mech Eng 195: 3247–3473zbMATHCrossRefMathSciNetGoogle Scholar