Multiscale finite element coarse spaces for the application to linear elasticity

Buck, Marco; Iliev, Oleg; Andrä, Heiko

doi:10.2478/s11533-012-0166-8

Cited by 25 publications

(25 citation statements)

References 35 publications

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“…The experimental results presented in the works of Buck et al (see also the work of Buck) justify expectations to obtain condition number bounds for the PDE system of linear elasticity similar to the existing ones for scalar elliptic PDEs in the work of Graham et al This issue is investigated in this paper for the case when the obstacles do not cross the elements boundary. Note that the framework provided in the work of Graham et al does not carry over to linear elasticity in a straightforward manner.…”

Section: Introductionsupporting

confidence: 74%

“…Assumption (C1) follows because

φ_{m}^{p, MsL}

coincides with a vector‐valued linear coarse basis function on ∂ T . As shown in the work of Buck et al, Assumption (C4) and (C5) holds because the PDE‐harmonic extension of vector‐valued linear boundary data to the interior of the coarse elements ensures that the coarse space preserves the rigid body modes globally in

\overset{Ω}{true}

.…”

Section: Coarsening Strategiesmentioning

confidence: 96%

“…The construction of boundary conditions for the multiscale finite element basis, which adapt to the heterogeneities in the PDE coefficients for the linear elasticity system, is addressed by Buck et al Multiscale basis functions with oscillatory boundary data are constructed. It is observed that the method extends the class of problems for which the coarse space in the work of Buck et al is robust to applications where inclusions touch or cross coarse element boundaries. Restrictions similar to those stated by Graham et al may apply.…”

Section: Introductionmentioning

confidence: 92%

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Domain decomposition preconditioners for multiscale problems in linear elasticity

Buck

Iliev

Andrä

2018

Numerical Linear Algebra App

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Summary We analyze two‐level overlapping Schwarz domain decomposition methods for vector‐valued piecewise linear finite element discretizations of the PDE system of linear elasticity. The focus of our study lies in the application to compressible, particle‐reinforced composites in 3D with large jumps in their material coefficients. We present coefficient‐explicit bounds for the condition number of the two‐level additive Schwarz preconditioned linear system. Thereby, we do not require that the coefficients are resolved by the coarse mesh. The bounds show a dependence of the condition number on the energy of the coarse basis functions, the coarse mesh, and the overlap parameters, as well as the coefficient variation. Similar estimates have been developed for scalar elliptic PDEs by Graham et al. The coarse spaces to which they apply here are assumed to contain the rigid body modes and can be considered as generalizations of the space of piecewise linear vector‐valued functions on a coarse triangulation. The developed estimates provide a concept for the construction of coarse spaces, which can lead to preconditioners that are robust with respect to high contrasts in Young's modulus and the Poisson ratio of the underlying composite. To confirm the sharpness of the theoretical findings, we present numerical results in 3D using vector‐valued linear, multiscale finite element and energy‐minimizing coarse spaces. The theory is not restricted to the isotropic (Lamé) case, extends to the full‐tensor case, and allows applications to multiphase materials with anisotropic constituents in two and three spatial dimensions. However, the bounds will depend on the ratio of largest to smallest eigenvalue of the elasticity tensor.

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Section: Introductionsupporting

confidence: 74%

“…Assumption (C1) follows because

φ_{m}^{p, MsL}

\overset{Ω}{true}

.…”

Section: Coarsening Strategiesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 92%

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Domain decomposition preconditioners for multiscale problems in linear elasticity

Buck

Iliev

Andrä

2018

Numerical Linear Algebra App

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“…Let us note that we are not considering robustness of FETI-DP for ACMS in this article. For overlapping domain decomposition methods and MsFEM, see, e.g., Aarnes and Hou [41] and Buck, Iliev, and Andrä [42,43].…”

Section: The Feti-dp Methodsmentioning

confidence: 99%

The approximate component mode synthesis special finite element method in two dimensions: Parallel implementation and numerical results

Heinlein

Hetmaniuk

Klawonn

et al. 2015

Journal of Computational and Applied Mathematics

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a b s t r a c tA special finite element method based on approximate component mode synthesis (ACMS) was introduced in Hetmaniuk and Lehoucq (2010). ACMS was developed for second order elliptic partial differential equations with rough or highly varying coefficients. Here, a parallel implementation of ACMS is presented and parallel scalability issues are discussed for representative examples. Additionally, a parallel domain decomposition preconditioner (FETI-DP) is applied to solve the ACMS finite element system. Weak parallel scalability results for ACMS are presented for up to 1024 cores. Our numerical results also suggest a quadratic-logarithmic condition number bound for the preconditioned FETI-DP method applied to ACMS discretizations.

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“…We expect that the preconditioning strategies developed herein can in principle be applied to the resulting discrete linear systems, however due to the presence of rigid body motions in mechanics, some additional developments will likely be needed (see, e.g., [9] for analogous extensions).…”

Section: Introductionmentioning

confidence: 99%

Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs

et al. 2017

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In simulation of fluid injection in fractured geothermal reservoirs, the characteristics of the physical processes are severely affected by the local occurence of connected fractures. To resolve these structurally dominated processes, there is a need to develop discretization strategies that also limit computational effort. In this paper, we present an upscaling methodology for geothermal heat transport with fractures represented explicitly in the computational grid. The heat transport is modeled by an advectionconduction equation for the temperature, and solved on a highly irregular coarse grid that preserves the fracture heterogeneity. The upscaling is based on different strategies for the advective term and the conductive term. The coarse scale advective term is constructed from sums of fine scale fluxes, whereas the coarse scale conductive term is constructed based on numerically computed basis functions. The method naturally incorporates the coupling between solution variables in the matrix and in the fractures, respectively, via the discretization. In this way, explicit transfer terms that couple fracture and matrix solution variables are avoided. Numerical results show that the upscaling methodology performs well, in particular for large upscaling ratios, and that it is applicable also to highly complex fracture networks.

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Multiscale finite element coarse spaces for the application to linear elasticity

Cited by 25 publications

References 35 publications

Domain decomposition preconditioners for multiscale problems in linear elasticity

Domain decomposition preconditioners for multiscale problems in linear elasticity

The approximate component mode synthesis special finite element method in two dimensions: Parallel implementation and numerical results

Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs

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