Asynchronous variational integration using continuous assumed gradient elements

Wolff, Sebastian; Bucher, Christian

doi:10.1016/j.cma.2012.11.004

Cited by 4 publications

(6 citation statements)

References 26 publications

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“…For variational integrators in combination with spatial discretization we refer to Ref. [18,51,77] and the references therein. Besides these semi-discrete approaches, there exists a covariant space-time discretization method by Marsden et al [58].…”

Section: Overview Of Existing Methodsmentioning

confidence: 99%

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

2016

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We develop a class of C1‐continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1‐continuity for the displacement field and C0‐continuity for the velocity field. From the discrete virtual action we finally construct a class of one‐step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one‐dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate — denoted as p2‐scheme — converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long‐term energy behavior.

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Section: Overview Of Existing Methodsmentioning

confidence: 99%

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

2016

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“…The velocity vector shown in Figure is computed from the discrete momenta by v k = M − 1 j k . Hence they can be interpreted as – losely spoken – purely ‘numerical’ quantities required to enforce continuity of the piecewise linear trajectory of q ( t ), see . A ‘better’ approximation of the tip velocity would be to take an average increment of the displacements over the interval length of a contact time step, that is, v k = ( q ( θ k + h c ) − q ( θ k )) ∕ h c , which reduces the oscillations significantly.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The asynchronous time integrator is then given through the time stepping scheme

\begin{matrix} leftalign \\ rightalign-odd j_{k + 1} & align-even = j_{k} - \sum_{i \in I MathClass-open(k MathClass-close)} (t_{i}^{A MathClass-open(k, i MathClass-close) + 1} - t_{i}^{A MathClass-open(k, i MathClass-close)}) \nabla V_{i} (q_{k}^{+}) & rightalign-label (2) \end{matrix}

\begin{matrix} leftalign \\ rightalign-odd q_{k + 1}^{+} & align-even = q_{k}^{+} + (θ_{k + 1} - θ_{k}) M^{- 1} j_{k + 1} & rightalign-label (3) \end{matrix}

see for a detailed illustration of the formulation and notation. In words, at time θ k one determines all potentials that are part of the set

scriptI (k)

, that is, which are active at this time.…”

Section: Asynchronous Variational Integrationmentioning

confidence: 99%

“…see [12,30] for a detailed illustration of the formulation and notation. In words, at time Â k one determines all potentials that are part of the set I.k/, that is, which are active at this time.…”

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confidence: 99%

“…Another type are external forces, for example, time-dependent loads that may be Figure 2. Illustration of a space-time front in one spatial dimension [30]: The colored area is the spacetime domain to be integrated (material space X time t). The highlighted area is the domain that is already integrated.…”

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Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

Wolff

Bucher

2013

Numerical Meth Engineering

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This article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints. Asynchronous discretizations allow individual time step sizes for each spatial region, improving the efficiency of explicit time stepping for finite element meshes with heterogeneous element sizes. The article first introduces asynchronous variational integration being expressed by drift and kick operators. Linear nodal restraint conditions are solved by a simple projection of the forces that is shown to be equivalent to RATTLE. Unilateral contact is solved by an asynchronous variant of decomposition contact response. Therein, velocities are modified avoiding penetrations. Although decomposition contact response is solving a large system of linear equations (being critical for the numerical efficiency of explicit time stepping schemes) and is needing special treatment regarding overconstraint and linear dependency of the contact constraints (for example from double-sided node-to-surface contact or self-contact), the asynchronous strategy handles these situations efficiently and robust. Only a single constraint involving a very small number of degrees of freedom is considered at once leading to a very efficient solution. The treatment of friction is exemplified for the Coulomb model. Special care needs the contact of nodes that are subject to restraints. Together with the aforementioned projection for restraints, a novel efficient solution scheme can be presented. The collision integrator does not influence the critical time step. Hence, the time step can be chosen independently from the underlying time-stepping scheme. The time step may be fixed or time-adaptive. New demands on global collision detection are discussed exemplified by position codes and node-to-segment integration. Numerical examples illustrate convergence and efficiency of the new contact algorithm. Copyright © 2013 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.

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Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration

Leitz

Ober‐Blöbaum

Leyendecker

2014

Computational Methods in Applied Sciences

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Asynchronous variational integration using continuous assumed gradient elements

Cited by 4 publications

References 26 publications

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

C¹‐continuous space‐time discretization based on Hamilton's law of varying action

Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration

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Asynchronous variational integration using continuous assumed gradient elements

Cited by 4 publications

References 26 publications

C1‐continuous space‐time discretization based on Hamilton's law of varying action

C1‐continuous space‐time discretization based on Hamilton's law of varying action

Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration

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Product

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C¹‐continuous space‐time discretization based on Hamilton's law of varying action

C¹‐continuous space‐time discretization based on Hamilton's law of varying action