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History and recent developments of energy-momentum schemes
KIT Karlsruhe, Germany
Energy-Momentum (EM) schemes belong to the class of structure-preserving numerical discretization techniques. This specific discretization approach aims at the preservation of main structural properties of the underlying continuous system. Originally EM schemes were developed in the context of nonlinear elastodynamics to provide enhanced numerical stability and robustness in the iterative solution procedure. In particular, the seminal paper by Simo and Tarnow (J.C. Simo, N. Tarnow, The Discrete Energy-Momentum Method. Conserving Algorithms for Nonlinear Elastodynamics, ZAMP, 43, 757-792, 1992) sparked numerous works on the design of EM schemes for nonlinear structural dynamics and elastodynamics. EM schemes are capable to exactly reproduce major balance laws in the discrete setting, independent of the mesh size (in space and time). In nonlinear elastodynamics the focus is on the balance laws for energy and angular momentum. Due to their success in nonlinear applications EM schemes have been further developed to cover more elaborate problems such as large deformation contact and flexible multibody dynamics. In recent years EM schemes have been extended to coupled field problems such as large deformation thermo-mechanics and electro-mechanics. This extension gives rise to the notion of Energy-Momentum-Entropy (EME) schemes that typically provide enhanced numerical stability and physical reliability of the numerical results. The talk will address the development of EM schemes from their origins in nonlinear elastodynamics to their extension to EME schemes for coupled nonlinear field problems.