Analysis of a Space-Time Discretization for Dynamic Elasticity Problems Based on Mass-Free Surface Elements

Hager, Corinna; Wohlmuth, Barbara

doi:10.1137/080715627

Cited by 21 publications

(23 citation statements)

References 16 publications

(26 reference statements)

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“…In contrast to standard mass lumping, the resulting mass matrix is singular, and Π 0 is a global operator. We note that recently in [12], the corresponding effect has been rigorously investigated for a linear elasticity problem with a singular mass matrix. As it turns out, the crucial ingredient for the analysis of the time-discrete problem is the quality of suitable associated elliptic stationary problems: Find y ∈ H div (Ω a ) with given normal components y ν on the boundary and y h ∈ RT 0 0 with Π 1 Γ y ν as the boundary condition such that…”

Section: A Priori Analysis Of the Discretization Error In Spacementioning

confidence: 99%

The Equivalence of Standard and Mixed Finite Element Methods in Applications to Elasto-Acoustic Interaction

Flemisch¹,

Triebenbacher²,

Wohlmuth³

2010

SIAM J. Sci. Comput.

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Two commonly used problem formulations for the description of acoustic wave propagation are investigated; one is based on fluid displacement, and the other one is based on the velocity potential as the primary variable. Their equivalence under general Neumann boundary conditions is shown on both the continuous and the discrete level. To obtain the equivalence in the discrete setting, a nonstandard mixed finite element formulation is introduced. Thus, the transfer of an already available analysis for coupled elasto-acoustic problems in the displacement formulation to the potential formulation can be achieved. For the applications, the potential formulation is of special interest because it allows the use of standard Lagrangian elements in both the stucture and the fluid subdomain. Moreover, the approach does not require any conformity of the subdomain meshes at the interface, which considerably simplifies the physics-adapted mesh generation. Several engineering examples demonstrate the applicability and efficiency of the resulting numerical scheme.

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Section: A Priori Analysis Of the Discretization Error In Spacementioning

confidence: 99%

The Equivalence of Standard and Mixed Finite Element Methods in Applications to Elasto-Acoustic Interaction

Flemisch¹,

Triebenbacher²,

Wohlmuth³

2010

SIAM J. Sci. Comput.

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“…The first approach can be found in and redistributes the mass globally by solving an L 2 minimization problem. The second approach is proposed and analyzed in and it is based on inexact quadrature formulas to assemble the entries of the mass matrix. However, both approaches cannot be applied to thin‐walled structures.…”

Section: Introductionmentioning

confidence: 99%

Hybrid‐mixed discretization of elasto‐dynamic contact problems using consistent singular mass matrices

Tkachuk

Wohlmuth

Bischoff

2013

Numerical Meth Engineering

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SUMMARYAn alternative spatial semi-discretization of dynamic contact based on a modified Hamilton's principle is proposed. The modified Hamilton's principle uses displacement, velocity and momentum as variables, which allows their independent spatial discretization. Along with a local static condensation for velocity and momentum, it leads to an approach with a hybrid-mixed consistent mass matrix. An attractive feature of such a formulation is the possibility to construct hybrid singular mass matrices with zero components at those nodes where contact is collocated. This improves numerical stability of the semi-discrete problem: the differential index of the underlying differential-algebraic system is reduced from 3 to 1, and spurious oscillations in the contact pressure, which are commonly reported for formulations with Lagrange multipliers, are significantly reduced. Results of numerical experiments for truss and Timoshenko beam elements are discussed. In addition, the properties of the novel discretization scheme for an unconstrained dynamic problem are assessed by a dispersion analysis.

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“…We apply no volume forces and enforce frictional contact with the coefficient F 0:5. The linearized contact conditions are assembled in an updated Lagrangian manner, and we employ a redistribution of the mass near the contact boundary c as described in [29]. We compute 10 time steps with a step size of Dt = 10 5 and prescribe a time dependent displacement of t 10Dt ð0; 0; 0:75 þ signðx 1 Þ dispÞ; disp 2 f0; 2g, on the top, whereas all other boundaries are free.…”

Section: Tire Applicationmentioning

confidence: 99%

“…Such a modification has already been successfully ap plied to dynamic contact problems (cf. Remark 12 and [30,28]), and the corresponding a priori error in space has been analyzed in [29]. The time subcycling algorithm with the standard mass ma trix is presented in [57].…”

Section: Local Time Subcyclingmentioning

confidence: 99%

Solving dynamic contact problems with local refinement in space and time

Hager

Hauret

Tallec

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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International audienceFrictional dynamic contact problems with complex geometries are a challenging task from the compu tational as well as from the analytical point of view since they generally involve space and time multi scale aspects. To be able to reduce the complexity of this kind of contact problem, we employ a non conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local struc ture and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme

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Analysis of a Space-Time Discretization for Dynamic Elasticity Problems Based on Mass-Free Surface Elements

Cited by 21 publications

References 16 publications

The Equivalence of Standard and Mixed Finite Element Methods in Applications to Elasto-Acoustic Interaction

The Equivalence of Standard and Mixed Finite Element Methods in Applications to Elasto-Acoustic Interaction

Hybrid‐mixed discretization of elasto‐dynamic contact problems using consistent singular mass matrices

Solving dynamic contact problems with local refinement in space and time

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