Analytical Comparison of Two Multiscale Coupling Methods for Nonlinear Solid Mechanics

Abstract The aim of this work is to compare two existing multilevel computational approaches coming from two different families of multiscale methods in a nonlinear solid mechanics framework. A locally adaptive multigrid method and a numerical homogenization technique are considered. Both classes of methods aim to enrich a global model representing the structure’s behavior with more sophisticated local models depicting fine localized phenomena. It is clearly shown that even being developed with different vocations, such approaches reveal several common features. The main conceptual difference relying on the scale separation condition has finally a limited influence on the algorithmic aspects. Hence, this comparison enables to highlight a unified framework for multiscale coupling methods.

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Polyhedral finite elements for nonlinear solid mechanics using tetrahedral subdivisions and dual-cell aggregation

Computer Aided Geometric Design ◽

10.1016/j.cagd.2019.101812 ◽

2020 ◽

Vol 77 ◽

pp. 101812 ◽

Cited By ~ 2

Author(s):

Joseph E. Bishop ◽

N. Sukumar

Keyword(s):

Finite Elements ◽

Solid Mechanics ◽

Cell Aggregation ◽

Nonlinear Solid Mechanics

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Algebraic multigrid methods

Acta Numerica ◽

10.1017/s0962492917000083 ◽

2017 ◽

Vol 26 ◽

pp. 591-721 ◽

Cited By ~ 37

Author(s):

Jinchao Xu ◽

Ludmil Zikatanov

Keyword(s):

Minimization Problem ◽

Large Scale ◽

Multigrid Method ◽

Approximate Solutions ◽

Multigrid Methods ◽

Algebraic Multigrid ◽

Unified Framework ◽

Smoothed Aggregation ◽

Algebraic Multigrid Methods ◽

Coarse Space

This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother$R$for a matrix$A$, such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension$n_{c}$is the span of the eigenvectors corresponding to the first$n_{c}$eigenvectors$\bar{R}A$(with$\bar{R}=R+R^{T}-R^{T}AR$). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with$\bar{R}A$, and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.

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