Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

Chabassier, Juliette; Imperiale, Sébastien

doi:10.1051/m2an/2021030

Cited by 10 publications

(13 citation statements)

References 38 publications

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“…The mortar element method [1][2][3][4][5] is originally a domain decomposition approach enabling coupling of non-conforming space discretizations. Recently Reference 11 it has been used to incorporate ETC between two solids and we extend this approach to fluid-solid coupling.…”

Section: Mortar Elements and Coupled Weak Formulationmentioning

confidence: 99%

“…Let us start with the mortar unknowns. Using the simple relation a 2 + b 2 ≤ (a + b) 2 for any real values a and b with positive product, we obtain 𝜂m 8…”

Section: Lemma 6 Definingmentioning

confidence: 99%

“…Various tools and techniques can be found in the literature to overcome some of these computational bottlenecks. For instance, authors in References 1 and 2 use an implicit scheme dedicated to the thin layer. Their approach alleviates the dependency of the timestep w.r.t .…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by this seminal body of work, our approach combine the use of ETC—as in Reference 9—with an adequate form of the mortar element method—as in References 1 and 2. It is in essence the extension of Reference 11 to configurations with fluid and solid propagators.…”

Section: Introductionmentioning

confidence: 99%

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An energy preserving time scheme based on the mortar element method for effective transmission conditions between fluid and solid domains in transient wave propagation problems

Impériale

2023

Numerical Meth Engineering

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We are interested in providing an efficient numerical scheme to represent wave propagation in time domain for configurations where a fluid domain and a solid domain are separated by a thin coating layer. Various computational challenges and bottlenecks occur in this context: incorporating fluid–solid coupling, enabling non‐conform space discretization when wavelengths greatly differ from the fluid and the solid domain, and rendering robust time discretization w.r.t. the thin layer thickness. In order to address these issues, the proposed approach combines the mortar element method with so‐called effective transmission conditions, where the effect of the coating layer is incorporated through spring—mass coefficients at the interface between the fluid and solid domain. Using discrete energy arguments, we are able to prove that the associated fully discrete scheme is stable upon a stability condition independent of the thin coating layer. Moreover, we provide an efficient time marching algorithm leading to significantly improved performances, illustrated in 2D and 3D configurations linked to ultrasonic testing experiments modeling.

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Section: Mortar Elements and Coupled Weak Formulationmentioning

confidence: 99%

“…Let us start with the mortar unknowns. Using the simple relation a 2 + b 2 ≤ (a + b) 2 for any real values a and b with positive product, we obtain 𝜂m 8…”

Section: Lemma 6 Definingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An energy preserving time scheme based on the mortar element method for effective transmission conditions between fluid and solid domains in transient wave propagation problems

Impériale

2023

Numerical Meth Engineering

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“…Hochbruck and Sturm proved optimal error estimates for the CNLF scheme from [41] when combined with a centered or an upwind discontinuous Galerkin (DG) FE discretization of Maxwell's equations [29,28]. In [6], Chabassier and Imperiale proposed fourth-order energy-preserving IMEX schemes for the wave equation. Here, the computational domain is divided by a fixed artificial boundary into a coarse and fine region with a Lagrange multiplier along the interface.…”

Section: Introductionmentioning

confidence: 99%

Stabilized leapfrog based local time-stepping method for the wave equation

Grote¹,

Michel²,

Sauter³

2020

Preprint

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Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. Recently, a leapfrog based explicit local time-stepping (LF-LTS) method was proposed for the time integration of second-order wave equations. Optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step, ∆t, depends on the smallest elements in the mesh. In general one cannot improve upon that stability constraint, as the LF-LTS method may become (linearly) unstable at certain discrete values of ∆t. To remove those critical values of ∆t, we propose a slight modification of the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, secondorder accurate, satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition where ∆t no longer depends on the mesh size inside the locally refined region.

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An implicit–explicit time discretization for elastic wave propagation problems in plates

Methenni,

Imperiale,

Imperiale

2023

Numerical Meth Engineering

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We propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest.

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Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

Cited by 10 publications

References 38 publications

An energy preserving time scheme based on the mortar element method for effective transmission conditions between fluid and solid domains in transient wave propagation problems

An energy preserving time scheme based on the mortar element method for effective transmission conditions between fluid and solid domains in transient wave propagation problems

Stabilized leapfrog based local time-stepping method for the wave equation

An implicit–explicit time discretization for elastic wave propagation problems in plates

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