Convergence of a Finite Element-Based Space-Time Discretization in Elastodynamics

Prohl, Andreas

doi:10.1137/070685166

Cited by 11 publications

(2 citation statements)

References 19 publications

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“…Dolzmann and Friesecke used an implicit time scheme and wondered if a Galerkin type method could be used instead. Later, Feireisl and Petzeltová showed that this can be accomplished [12]; see also the following recent papers [10,18,20] which involved similar existence problems and results.…”

Section: Elastodynamicsmentioning

confidence: 87%

Newton's second law with a semiconvex potential

Hynd¹

2018

Preprint

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We make the elementary observation that the differential equation associated with Newton's second law mγ(t) = −DV (γ(t)) always has a solution for given initial conditions provided that the potential energy V is semiconvex. That is, if −DV satisfies a one-sided Lipschitz condition. We will then build upon this idea to verify the existence of solutions for the Jeans-Vlasov equation, the pressureless Euler equations in one spatial dimension and the equations of elastodynamics under appropriate semiconvexity assumptions.

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Section: Elastodynamicsmentioning

confidence: 87%

Newton's second law with a semiconvex potential

Hynd¹

2018

Preprint

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“…Existence of Young-measure-valued solutions to the peridynamic problem is shown by applying a Galerkin approximation and investigating the limit of approximate solutions. Young measures in (visco)elastodynamics have been studied, e.g., in [9,32,33]. However, talking about classical nonlinear elasticity theory, mainly gradient Young measures have been considered.…”

Section: Introductionmentioning

confidence: 99%

Measure-valued and weak solutions to the nonlinear peridynamic model in nonlocal elastodynamics

Emmrich

Puhst

2014

Nonlinearity

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The peridynamic equation of motion is considered for a large class of nonlinear pairwise force functions modeling isotropic microelastic material. Existence of Youngmeasure-valued solutions to the peridynamic equation of motion is proven if the underlying basic function space is L p. Moreover, if the underlying basic function space is W σ ,p (0 < σ < 1), existence of weak solutions is shown. The choice of the function space depends on the singularity of the pairwise force function. The results do not rely upon any convexity assumption whatsoever.

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Convergence of a full discretization for a second-order nonlinear elastodynamic equation in isotropic and anisotropic Orlicz spaces

Ruf

2017

Z. Angew. Math. Phys.

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In this paper, we study a second-order, nonlinear evolution equation with damping arising in elastodynamics. The nonlinear term is monotone and possesses a convex potential but exhibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation. Moreover, we show uniqueness in a class of sufficiently smooth solutions and provide an a priori error estimate for the temporal semidiscretization.Mathematics Subject Classification (2010). 35L20; 47J35; 47H05; 65M12; 65M06; 35M60.for ε > 0, as well as for ε = 0 and presents numerical experiments [28]. The limit case ε = 0 constitutes the elastodynamic equationfor which one cannot expect smooth solutions even for smooth initial data (see [2,26]). For an excellent survey of the literature concerning equation (1.3) see [8]. The remainder of this paper is structured as follows: In Section 2, we introduce the necessary notation, give a brief introduction to Orlicz spaces and compare the growth condition (1.2) with the restrictive ∆ 2 -condition. The description of the numerical method we employ, the construction of the Galerkin scheme, the proof of existence and uniqueness of the numerical solution, and the derivation of a priori estimates for the fully discrete solution and the discrete time derivative follow in Section 3. Finally, in Section 4 we show convergence towards and, thus, existence of an exact solution, as well as its uniqueness (under additional regularity assumptions) and an error estimate for the temporal semidiscretization. The appendix contains an elementary lemma concerning the separability of the space for wich we want to construct the Galerkin scheme.

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Convergence of a Finite Element-Based Space-Time Discretization in Elastodynamics

Cited by 11 publications

References 19 publications

Newton's second law with a semiconvex potential

Newton's second law with a semiconvex potential

Measure-valued and weak solutions to the nonlinear peridynamic model in nonlocal elastodynamics

Convergence of a full discretization for a second-order nonlinear elastodynamic equation in isotropic and anisotropic Orlicz spaces

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