Multi-time-step explicit-implicit method for non-linear structural dynamics

Gravouil, Anthony; Combescure, Alain

doi:10.1002/1097-0207(20010110)50:1<199::aid-nme132>3.0.co;2-a

Cited by 239 publications

(215 citation statements)

References 31 publications

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“…12 We now show that the Lagrange multipliers λ (n+γ) are bounded ∀n. There are several ways in proving this result.…”

Section: Stability Analysismentioning

confidence: 91%

“…The FETI method is a popular way of implementing the dual Schur domain decomposition method which has been shown to possess good convergence and scability properties [9,6,8]. An interesting approach (in the context of structural dynamics) has been developed in [12,21]. Any of the aforementioned implementation methodologies can be employed for the methods presented in this paper.…”

Section: Dual Schur Domain Decomposition Methodsmentioning

confidence: 99%

“…dual Schur domain decomposition method; coupling algorithms; differential-algebraic equations; generalized trapezoidal family. 1 problems [9,10,19], designing preconditioners [24,15], parallel implementation [9,6], efficient computer implementation [9,12,21], solvers [25], etc. On the other hand, domain decomposition techniques for transient problems are not as mature as those associated with static problems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On dual Schur domain decomposition method for linear first-order transient problems

Nakshatrala¹,

Prakash²,

Hjelmstad³

2009

Journal of Computational Physics

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Abstract. This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. The spatially discrete system of equations resulting from a dual Schur domain decomposition method can be expressed as a system of differential algebraic equations (DAEs). In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Unlike the continuous formulation, the discretized formulation of the transient problem is unable to enforce simultaneously the continuity of both the primary variable and its rate along the subdomain interface (except for the backward Euler method).Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the "energy" method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs). Finally, we present numerical examples to corroborate our theoretical predictions.

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“…12 We now show that the Lagrange multipliers λ (n+γ) are bounded ∀n. There are several ways in proving this result.…”

Section: Stability Analysismentioning

confidence: 91%

Section: Dual Schur Domain Decomposition Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On dual Schur domain decomposition method for linear first-order transient problems

Nakshatrala¹,

Prakash²,

Hjelmstad³

2009

Journal of Computational Physics

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“…A suboptimal choice can lead to numerical instability or to artificial dissipation at the interface, even if the time integrators in the subdomains are stable and energy conserving [31 33]. In [34,35], a time sub stepping scheme with linear interpolation of the velocity and the Lagrange multipliers has been proposed which is stable but dissi pative at the interface. An improved energy conserving scheme with linear interpolation of the multipliers has been analyzed in [36,37].…”

Section: Introductionmentioning

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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“…This can be accurate in an elastic case, but can sacrifice accuracy to achieve stability, if the material at the timestep interface does not deform elastically. Recently, Gravouil [12,13] has presented a relatively complicated method of subcycling the implicit or explicit Newmark algorithm using Lagrange multipliers to enforce an appropriate constraint on an element interface. GravouilÕs timestep interface can be proved to be stable, but dissipative.…”

Section: Introductionmentioning

confidence: 99%

A partial velocity approach to subcycling structural dynamics

Daniel

2003

Computer Methods in Applied Mechanics and Engineering

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Subcycling, or the use of different timesteps at different nodes, can be an effective way of improving the computational efficiency of explicit transient dynamic structural solutions. The method that has been most widely adopted uses a nodal partition, extending the central difference method, in which small timestep updates are performed interpolating on the displacement at neighbouring large timestep nodes. This approach leads to narrow bands of unstable timesteps or ''statistical stability''. It also can be in error due to lack of momentum conservation on the timestep interface. The author has previously proposed energy conserving algorithms that avoid the first problem of statistical stability. However, these sacrifice accuracy to achieve stability. An approach to conserve momentum on an element interface by adding partial velocities is considered here. Applied to extend the central difference method, this approach is simple, and has accuracy advantages. The method can be programmed by summing impulses of internal forces, evaluated using local element timesteps, in order to predict a velocity change at a node. However, it is still only statistically stable, so an adaptive timestep size is needed to monitor accuracy and to be adjusted if necessary. By replacing the central difference method with the explicit generalized alpha method, it is possible to gain stability by dissipating the high frequency response that leads to stability problems. However, coding the algorithm is less elegant, as the response depends on previous partial accelerations. Extension to implicit integration, is shown to be impractical due to the neglect of remote effects of internal forces acting across a timestep interface.

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Multi-time-step explicit-implicit method for non-linear structural dynamics

Cited by 239 publications

References 31 publications

On dual Schur domain decomposition method for linear first-order transient problems

On dual Schur domain decomposition method for linear first-order transient problems

Solving dynamic contact problems with local refinement in space and time

A partial velocity approach to subcycling structural dynamics

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