Improving algebraic multigrid interpolation operators for linear elasticity problems

Baker, Allison H.; Kolev, Tzanio V.; Yang, Ulrike Meier

doi:10.1002/nla.688

Cited by 43 publications

(47 citation statements)

References 23 publications

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“…Consider the Sobolev space V = [H 1 (Ω)] 3 of vector-valued functions whose components are square-integrable with weak first-order partial derivatives in the Lebesgue space L 2 (Ω). We define…”

Section: Weak Formulationmentioning

confidence: 99%

“…A multigrid approach based on a finite difference discretization of the elasticity system has been proposed in [43]. Important works concerning classical AMG methods for linear elasticity are given in [1,5].…”

Section: Introductionmentioning

confidence: 99%

“…We shall assume that Γ = ∂Ω admits the decomposition into two disjoint subsets Γ D and Γ N , Γ = Γ D ∪ Γ N and meas Γ D > 0 for = 1 2 3. The system given in equation (1) follows the boundary conditions…”

mentioning

confidence: 99%

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Multiscale finite element coarse spaces for the application to linear elasticity

2013

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Abstract:We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169-189] to the PDE system of linear elasticity. The application, motivated by the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction of robust coarse spaces in the context of two-level overlapping domain decomposition preconditioners. We motivate and explain the construction and show that the constructed multiscale coarse space contains all the rigid body modes. Under the assumption that the material jumps are isolated, that is they occur only in the interior of the coarse grid elements, our numerical experiments show uniform convergence rates independent of the contrast in Young's modulus within the heterogeneous material. Elsewise, if no restrictions on the position of the high coefficient inclusions are imposed, robustness cannot be guaranteed any more. These results justify expectations to obtain coefficient-explicit condition number bounds for the PDE system of linear elasticity similar to existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner and Scheichl [Graham I.G., Lechner P.O., Scheichl R., Domain decomposition for multiscale PDEs, Numer. Math., 2007, 106(4), 589-626]. Furthermore, we numerically observe the properties of the MsFEM coarse space for linear elasticity in an upscaling framework. Therefore, we present experimental results showing the approximation errors of the multiscale coarse space w.r.t. the fine-scale solution. MSC:65N55, 35R05, 74B05, 74S05

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Section: Weak Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multiscale finite element coarse spaces for the application to linear elasticity

2013

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“…This approach was first proposed for linear elasticity in [39] and has been extended in [24,31]. An extension framework to improve AMG for elasticity problems was proposed in [8], which uses a hybrid approach with nodal coarsening, but interpolation based on the unknown-based approach.…”

Section: Algebraic Multigrid Methodsmentioning

confidence: 99%

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

et al. 2011

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Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms. In this paper, we consider the simulation of asphalt concrete, which is a mixture of components with large differences in material stiffness. The discontinuities in material stiffness give rise to many small eigenvalues that negatively affect the convergence of iterative solution algorithms such as the preconditioned conjugate gradient (PCG) method. This paper considers the deflated preconditioned conjugate gradient (DPCG) method in which the rigid body modes of sets of elements with homogeneous material properties are used as deflation vectors. As preconditioner we consider several variants of the algebraic multigrid smoothed aggregation method. We evaluate the performance of the DPCG method on a parallel computer using up to 64 processors. Our test problems are derived from real asphalt core samples, obtained using CT scans. We show that the DPCG method is an efficient and robust technique for solving these challenging linear systems. T. B. Jönsthövel (B)·

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“…Other works have shown AMG to be an efficient method for solving linear elasticity (Baker et al, 2009), but the performance is closely coupled to the selection of proper parameters. The Hypre -AMG preconditioner (BoomerAMG) provides a convenient interface for selecting the appropriate parameters.…”

Section: Iterative Solversmentioning

confidence: 99%

Parallel Performance of Linear Solvers and Preconditioners

Crone¹,

Munday²

2014

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Approved for public release; distribution is unlimited.ii REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. REPORT DATE (DD-MM-YYYY)January 2014 ARL-TR-6778 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution is unlimited. SUPPLEMENTARY NOTES ABSTRACTIn this report we examine the performance of parallel linear solvers and preconditioners available in the Hypre, PETSc, and MUMPS libraries to identify the combination with the shortest wall clock time for large-scale linear systems. The linear system of equations in this work is produced by a finite element code solving a linear elastic boundary value problem (BVP). The boundary conditions for the linear elastic BVP are produced by a discrete dislocation dynamics (DDD) simulation and change with each timestep of the DDD simulation as the dislocation structure evolves. However, the coefficient-or stiffness matrixremains constant during the DDD simulation and some expensive matrix factorizations only occur once during initialization. Our results show that for system sizes of less than three million degrees of freedom (DOF), the MUMPS direct solver is 20× faster than the best iterative solvers per timestep, but has a large upfront cost for the LU decomposition. Systems larger than three million DOFs require iterative solvers. The Hypre algebraic multigrid (AMG) preconditioner packaged was the best performing iterative solver, but was found to be sensitive to the AMG parameters. The PETSc Block Jacobi preconditioner showed good performance with the default preconditioning setting. SUBJECT TERMS

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Improving algebraic multigrid interpolation operators for linear elasticity problems

Cited by 43 publications

References 23 publications

Multiscale finite element coarse spaces for the application to linear elasticity

Multiscale finite element coarse spaces for the application to linear elasticity

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

Parallel Performance of Linear Solvers and Preconditioners

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