A Selection of Benchmark Problems in Solid Mechanics and Applied Mathematics

Schröder, Jörg; Wick, Thomas; Reese, Stefanie; Wriggers, Peter; Müller, Ralf; Kollmannsberger, Stefan; Kästner, Markus; Schwarz, Alexander; Igelbüscher, Maximilian; Viebahn, Nils; Bayat, Hamid Reza; Wulfinghoff, Stephan; Mang, Katrin; Rank, Ernst; Bog, Tino; D’Angella, Davide; Elhaddad, M.; Hennig, Paul; Düster, Alexander; Garhuom, Wadhah; Hubrich, Simeon; Walloth, Mirjam; Wollner, Winnifried; Kuhn, Charlotte; Heister, Timo

doi:10.1007/s11831-020-09477-3

Cited by 52 publications

(30 citation statements)

References 51 publications

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“…The 1D application case is related with a Riemann problem at finite strains [9,11]. On the other hand, the 2D benchmarks are related with the plane strain Cook's membrane [37] under a dynamic load, a plane strain solid block with low stiffness under the action of gravity and a large deformation vibration of a cantilever beam under its own weight [34]. The 3D benchmark is related with a thick-walled cylinder pinched by two opposite dynamic line loads.…”

Section: Validation Benchmarks and Efficiency Analysismentioning

confidence: 99%

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A component-free Lagrangian finite element formulation for large strain elastodynamics

et al. 2021

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Standard finite element formulation and implementation in solid dynamics at large strains usually relies upon and indicial-tensor Voigt notation to factorized the weighting functions and take advantage of the symmetric structure of the algebraic objects involved. In the present work, a novel component-free approach, where no reference to a basis, axes or components is made, implied or required, is adopted for the finite element formulation. Under this approach, the factorisation of the weighting function and also of the increment of the displacement field, can be performed by means of component-free operations avoiding both the use of any index notation and the subsequent reorganisation in matrix Voigt form. This new approach leads to a straightforward implementation of the formulation where only vectors and second order tensors in $${\mathbb {R}}^3$$ R 3 are required. The proposed formulation is as accurate as the standard Voigt based finite element method however is more efficient, concise, transparent and easy to implement.

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Section: Validation Benchmarks and Efficiency Analysismentioning

confidence: 99%

“…The values considered for the Young modulus E, density ρ 0 , and maximum of the applied load f max are obtained from [37]. The time step considered for the simulation is Δt = 0.02s.…”

Section: Cook's Membrane Under a Dynamic Loadingmentioning

confidence: 99%

A component-free Lagrangian finite element formulation for large strain elastodynamics

et al. 2021

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“…These results lead to the assumption that |J(u FM ) − J(u h )| = O( ) as presented in [146,218,226,237]. In addition, two phase-field fracture configurations were proposed as prototype models for comparison in the recent benchmark collection [265].…”

Section: Numerical Analysis For Pfmentioning

confidence: 99%

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Diehl¹,

Lipton²,

Wick³

et al. 2021

Preprint

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Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, The Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities with their relative advantages and challenges are summarized.

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“…To facilitate the communication of different research communities, serve as an entry to a specific problem type, and allow for a validation of different modeling strategies, benchmark problems can be of great benefit, as they are often designed to reproduce characteristic mechanisms by means of a rather simple setup [ 19 , 20 , 21 ]. For multi-physics problems involving strong coupling phenomena, they allow us to analyze and discuss relevant aspects regarding the modeling as well as numerical simulations: How does a specific modeling framework perform compared to other, well-established approaches?…”

Section: Introductionmentioning

confidence: 99%

Benchmark for the Coupled Magneto-Mechanical Boundary Value Problem in Magneto-Active Elastomers

et al. 2021

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Within this contribution, a novel benchmark problem for the coupled magneto-mechanical boundary value problem in magneto-active elastomers is presented. Being derived from an experimental analysis of magnetically induced interactions in these materials, the problem under investigation allows us to validate different modeling strategies by means of a simple setup with only a few influencing factors. Here, results of a sharp-interface Lagrangian finite element framework and a diffuse-interface Eulerian approach based on the application of a spectral solver on a fixed grid are compared for the simplified two-dimensional as well as the general three-dimensional case. After influences of different boundary conditions and the sample size are analyzed, the results of both strategies are examined: for the material models under consideration, a good agreement of them is found, while all discrepancies can be ascribed to well-known effects described in the literature. Thus, the benchmark problem can be seen as a basis for future comparisons with both other modeling strategies and more elaborate material models.

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A Selection of Benchmark Problems in Solid Mechanics and Applied Mathematics

Cited by 52 publications

References 51 publications

A component-free Lagrangian finite element formulation for large strain elastodynamics

A component-free Lagrangian finite element formulation for large strain elastodynamics

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Benchmark for the Coupled Magneto-Mechanical Boundary Value Problem in Magneto-Active Elastomers

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