A novel Lagrange-multiplier based method for consistent mesh tying

Parks, Michael L.; Romero, Louis A.; Bochev, Pavel B.

doi:10.1016/j.cma.2007.03.013

Cited by 12 publications

(12 citation statements)

References 13 publications

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“…Mixed Galerkin methods for mesh tying that are consistent when applied to geometries with curved interfaces may significantly increase the complexity of the overall solution [17], and lead to indefinite linear systems [1][2][3]15,19]. We formulated and analyzed an LSFEM for mesh tying that is optimally accurate, patch test consistent for arbitrary order discretizations, and gives rise to sparse symmetric positive definite matrices.…”

Section: Discussionmentioning

confidence: 99%

“…A disadvantage of these methods is that in order to maintain accuracy, typically six levels of uniform 3D mesh refinement are required near the boundary to pass the patch test approximately. The generalized Lagrange multiplier method of [19] avoids the mesh refinement but requires an interface balancing procedure to cancel out the signed areas of the gaps and overlaps.…”

Section: Specifics Of Mesh Tyingmentioning

confidence: 99%

“…One example is certification of aerospace structures where creating a monolithic mesh is hugely impractical and time consuming. In such cases, for practical and efficiency reasons, grid generation on is replaced by independent meshing of its subdomains [3,4,8,10,16,19,20]. Other examples that lead to mesh-tying settings include transmission, contact, and domain-bridging problems [1,6,15,17].…”

Section: Introductionmentioning

confidence: 99%

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Analysis and computation of a least-squares method for consistent mesh tying

Day

Bochev

2008

Journal of Computational and Applied Mathematics

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In the finite element method, a standard approach to mesh tying is to apply Lagrange multipliers. If the interface is curved, however, discretization generally leads to adjoining surfaces that do not coincide spatially. Straightforward Lagrange multiplier methods lead to discrete formulations failing a first-order patch test [T.A. Laursen, M.W. Heinstein, Consistent mesh-tying methods for topologically distinct discretized surfaces in non-linear solid mechanics, Internat. J. Numer. Methods Eng. 57 (2003) 1197-1242].This paper presents a theoretical and computational study of a least-squares method for mesh tying [P. Bochev, D.M. Day, A leastsquares method for consistent mesh tying, Internat. J. Numer. Anal. Modeling 4 (2007) 342-352], applied to the partial differential equation −∇ 2 + = f . We prove optimal convergence rates for domains represented as overlapping subdomains and show that the least-squares method passes a patch test of the order of the finite element space by construction. To apply the method to subdomain configurations with gaps and overlaps we use interface perturbations to eliminate the gaps. Theoretical error estimates are illustrated by numerical experiments.

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Section: Discussionmentioning

confidence: 99%

Section: Specifics Of Mesh Tyingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis and computation of a least-squares method for consistent mesh tying

Day

Bochev

2008

Journal of Computational and Applied Mathematics

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“…Тогда в глобальной системе уравнений появятся дополнительные степени свободы, связанные с аппроксимацией этого множителя. Варианты использования такого подхода можно найти в работах [3,14]. С другой стороны, два последних равенства из (2.10) определяют множитель b через уже введенные переменные и можно априори положить .…”

Section: вариационный принципunclassified

Hybrid Finite Element for the Analysis on Non-Matched Meshes

Semenov¹,

Techsoft²,

Moscow³

2011

PSP

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Предложен вариационный принцип для «склеивания» несогласованных интерфейсов. Множители Лагранжа, введенные на уровне общего функционала, явно определены через базовые независимые переменные, и таким образом глобальная система уравнений включает в себя только обычные узловые степени свободы. Формулировка проходит обычный и контактный patchтесты.

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“…The reader is referred to, for example, and references therein for a comprehensive overview in the context of fluid–structure interaction (FSI). However, only mortar finite element methods (FEMs) as addressed, for example, in and other closely related Lagrange multiplier methods as discussed, for example, in assure mathematical optimality of the interface coupling in the sense that the global discretization error is bounded by the sum of the individual subdomain errors.…”

Section: Introductionmentioning

confidence: 99%

A dual mortar approach for mesh tying within a variational multiscale method for incompressible flow

Ehrl

Popp

Gravemeier

et al. 2014

Numerical Methods in Fluids

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SUMMARY In the present work, the coupling of computational subdomains with non‐conforming discretizations is addressed in the context of residual‐based variational multiscale finite element methods for incompressible fluid flow. A mortar method using dual Lagrange multipliers is introduced for handling the coupling conditions at arbitrary fluid–fluid interfaces. Recently, mortar methods have been successfully applied in the field of nonlinear solid mechanics, for example, to weakly impose interface constraints for finite deformation contact. The focus of this study is on both the integration of the dual mortar approach into an existing variational multiscale finite element framework and the investigation of the resulting interplay between variational multiscale and coupling terms. We analyze the effects of either constraining only velocity or both velocity and pressure degrees of freedom at the internal interfaces using dual Lagrange multipliers. In analogy to other problem classes, we exploit the fact that the dual mortar approach allows for an efficient condensation of the additional Lagrange multiplier degrees of freedom from the global system of equations. As a result, the typical but often undesirable saddle‐point structure of this system is completely removed. The proposed method is validated numerically for various three‐dimensional examples, including a complex patient‐specific aneurysm, and its accuracy and efficiency in comparison with standard conforming discretizations is demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.

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A novel Lagrange-multiplier based method for consistent mesh tying

Cited by 12 publications

References 13 publications

Analysis and computation of a least-squares method for consistent mesh tying

Analysis and computation of a least-squares method for consistent mesh tying

Hybrid Finite Element for the Analysis on Non-Matched Meshes

A dual mortar approach for mesh tying within a variational multiscale method for incompressible flow

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