A selection of benchmark problems in solid mechanics and applied mathematics
In this contribution we provide benchmark problems in the field of computational solid mechanics. In detail, we address classical fields as elasticity, incompressibility, material interfaces, thin structures and plasticity at finite deformations. For this we describe explicit setups of the benchmarks and introduce the numerical schemes. For the computations the various participating groups use different (mixed) Galerkin finite element and isogeometric analysis formulations. Some programming codes are available open-source. The output is measured in terms of carefully designed quantities of interest that allow for a comparison of other models, discretizations, and implementations. Furthermore, computational robustness is shown in terms of mesh refinement studies. This paper presents benchmarks, which were developed within the Priority Programme of the German Research Foundation ‘SPP 1748 Reliable Simulation Techniques in Solid Mechanics—Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis’.
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The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non- standard discretization methods, mechanical and mathematical analysis” under the projects ‘Coordination Funds’—project number 255431921 (SCHR 570/23-1&2), ‘Novel finite elements—Mixed, Hybrid and Virtual Element formulations at finite strains for 3D applications’—project number 255431921 (SCHR 570/23-1&2, WR 19/50-1&2), ‘Approximation and Reconstruction of Stresses in the Deformed Configuration for Hyperelastic Material Models’—project number 392587488 (SCHR 570/34-1), ‘First-order system least squares finite elements for finite elasto-plasticity’—project number 255798245 (SCHW 1355/2-1, SCHR 570/24-1), ‘Hybrid discretizations in solid mechanics for non-linear and non-smooth problems’—project number 643861 (RE 1057/30-1&2), ‘High-order immersed-boundary methods in solid mechanics for structures generated by additive processes’—project number 255496529 (DU 405/8-1&2, RA 624/27-1&2, SCHR 1244/4-1&2), ‘Adaptive isogeometric modeling of discontinuities in complex-shaped heterogeneous solids’—project number 255853920 (KA 3309/3-1&2), ‘Advanced Finite Element Modelling of 3D Crack Propagation by a Phase Field Approach’ - project number 255846293 (MU1370/11-1&2 and KU 3221/1-1&2), ‘Structure Preserving Adaptive Enriched Galerkin Methods for Pressure-Driven 3D Fracture Phase-Field Models’—project number 392587580 (WA 4200/1-1, WI 4367/2-1, and WO 1936/5-1), Timo Heister was partially supported by the National Science Foundation (NSF) Award DMS-2028346, OAC-2015848, EAR-1925575, by the Computational Infrastructure in Geodynamics initiative (CIG), through the NSF under Award EAR-0949446 and EAR-1550901 and The University of California—Davis, and by Technical Data Analysis, Inc. through US Navy STTR Contract N68335-18-C-0011.