A Locking-Free MHM Method for Elasticity

Pereira, Weslley; Valentin, Frédéric

doi:10.5540/03.2017.005.01.0336

Cited by 3 publications

(8 citation statements)

References 6 publications

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“…Assume the FE spaces E γ = S γ × U γin × Q γin verify the stability constraints (21)- (22). Then, the MHM-WS(E γ ) scheme defined by the downscaling local solvers (28)- (31) and (32)- (35), and by the global upscaling system (36)- (37), has a unique solution.…”

Section: Mhm-ws(e γ ) As a Mfem-ws(e γ ) Formulationmentioning

confidence: 99%

“…(i) By construction, the strong enforcement of the Neumann boundary conditions (31) and (35) is the reason to assume, from start, thatσ =T σ (λ) +T σ (f ) ∈ S γ , i. e., that the stress is globally H(div)conforming. This is an important property of the MHM-WS(E γ ) solutions that, for instance, distinguish them from those of the multiscale mortar domain decomposition method [30].…”

Section: Remarksmentioning

confidence: 99%

“…(iv) The local contributionT (f ) of the numerical solution, defined in (32)- (35), is one of the important properties of the proposed multiscale method. Notably, such a perspective is paramount when f changes rapidly or embeds multiple scales.…”

Section: Remarksmentioning

confidence: 99%

“…Our method is based on a divide-and-conquer strategy combined with bubble enrichment techniques and static condensation, which are general-purpose tools widely adopted in multiscale simulations. It shall be denoted by the acronym MHM-WS, for its design is in the spirit of Multiscale Hybrid Mixed (MHM) methods (already applied for Darcy problems [20,27], for displacement-based elasticity formulations [26,35,36], and other contexts therein cited). In summary, this means that the MHM-WS scheme shares with these MHM methods the following characteristics:…”

Section: Introductionmentioning

confidence: 99%

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NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

et al. 2021

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This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforced via the rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in the H (div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in the L 2 -norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.

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Section: Mhm-ws(e γ ) As a Mfem-ws(e γ ) Formulationmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

et al. 2021

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“…Particularly adapted to handle multiscale and/or high-contrast coefficients on coarse meshes, the MHM method permits face-crossing interfaces endowed in local boundary conditions. Originally proposed for the Laplace problem in [31], the MHM method has been analyzed in [4] and further extended to other elliptic problems in [30,32] and mixed problems in [5,43]. Also, its robustness with respect to (small) physical parameters has been established in [42], and an abstract general setting to develop and analyze MHM methods has been proposed in [33].…”

mentioning

confidence: 99%

The Multiscale Hybrid-Mixed method for the Maxwell Equations in Heterogeneous Media

Lanteri¹,

Paredes²,

Scheid³

et al. 2018

Multiscale Model. Simul.

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In this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the multiscale hybrid-mixed (MHM) finite element method for the two-and three-dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse-mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a "divide-and-conquer" framework. Several two-dimensional numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy to solve heterogeneous media problems on coarse meshes.

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The MHM Method for Linear Elasticity on Polytopal Meshes

Gomes

Pereira

Valentin

2022

IMA Journal of Numerical Analysis

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The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.

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A Locking-Free MHM Method for Elasticity

Cited by 3 publications

References 6 publications

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

The Multiscale Hybrid-Mixed method for the Maxwell Equations in Heterogeneous Media

The MHM Method for Linear Elasticity on Polytopal Meshes

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