Exact energy and momentum conserving algorithms for general models in nonlinear elasticity

By Oscar Gonzalez. 2000

293 citations recorded by the source corpus.

Implicit time integration schemes that inherit the conservation laws of total energy, linear and angular momentum are considered for initial boundary-value problems in finite-deformation elastodynamics. Conserving schemes are constructed for general hyperelastic material models, both compressible and incompressible, and are formulated in a way that is independent of spatial discretization. Three numerical examples for Ogden-type material models, implemented using a finite element discretization in space, are given to illustrate the performance of the proposed schemes. These examples show that, relative to the standard implicit mid-point rule, the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy.

Subfields: Biomedical Engineering, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/0152ab6b2f86f8a18fc3643201ef975bc9da46a7

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In Biomedical Engineering

In Computational Mechanics

Citations in the corpus

An energy‐consistent material‐point method for dynamic finite deformation plasticity (2006, methodology): “Linear and angular momentum conservation are fairly straightforward and the fact that these quantities are conserved has been shown previously in the references [24], [30], [53].”
Transient three‐dimensional domain decomposition problems: Frame‐indifferent mortar constraints and conserving integration (2010, methodology): “Similar to the energy–momentum schemes for constrained mechanical systems developed by Gonzalez [22] and Betsch and Steinmann [24], we consider the following time-stepping scheme: qn+1−qn = tvn+1/2 M(vn+1−vn) = − t∇V (qn,qn+1)− t (D1g(qn+1/2,dn+1/2))Tkn+1 0 = D2(g(qn+1/2,dn+1/2))Tkn+1 0 = g(qn+1,dn+”
Energy-diminishing integration of gradient systems (2014, methodology): “An extension of discrete-gradient methods to problems with constraints has been proposed in Gonzalez (1999) within a Hamiltonian setting, and in Gonzalez (2000) for models in nonlinear elasticity.”
Development of a cell centred upwind finite volume algorithm for a new conservation law formulation in structural dynamics (2013, background): “Some modifications were made to allow for any hyperelastic materials [102,103] .”
Improved implicit integrators for transient impact problems—geometric admissibility within the conserving framework (2002, methodology): “In energy–momentum implementations of non-linear dynamics, the algorithmic approximation of the second Piola–Kirchho> stress Salgo may be calculated such that conservation properties are exactly preserved and so that the algorithm is second-order accurate in time; see Simo and Tarnow [3] and Gonzale”
Thermodynamically consistent algorithms for a finite‐deformation phase‐field approach to fracture (2014, background): “The underlying formulation provides enhanced numerical stability for applications in nonlinear structural dynamics, see [32–34] and the references therein, and has been applied to a wide range of applications, for example, for thermomechanical problem, see [31, 35–37].”