Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation

Nguyễn, Vân Dung; Béchet, Éric; Geuzaine, Christophe; Noels, Ludovic

doi:10.1016/j.commatsci.2011.10.017

Cited by 204 publications

(119 citation statements)

References 19 publications

Supporting

Mentioning

119

Contrasting

Order By: Relevance

“…While computational algorithms to implement DBC and PBC are well established and discussed by many authors [241,246,462,464,465], special care should be taken to deal with the stiffness matrix singularity due to prescribing a pure Neumann boundary condition on the RVE to implement TBC. Several authors have treated this problem using either mass-type diagonal perturbation to regularize the stiffness matrix [462], construction of a free-flexibility matrix to preserve the rigid body modes [466], adding very soft materials to the microstructure, or in the most extreme case, completely fixing enough degrees-of-freedom to make the problem well defined.…”

Section: Microdeformation Implementationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

176

138

View full text Add to dashboard Cite

The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

show abstract

Section: Microdeformation Implementationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

176

138

View full text Add to dashboard Cite

show abstract

“…Furthermore, the applicability of the method is restricted on periodic microelement meshes. To alleviate such problems, a procedure has been established for the generalization of the periodic boundary condition assumption allowing its application to nonstructured, non-periodic meshes [42]. Also, refined boundary condition assumptions such as the oversampling technique [19] and the generalized periodic boundary condition method (combining periodic boundary conditions with oversampling) have been effectively used in [63] for non-periodic media.…”

Section: Appendixmentioning

confidence: 99%

A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials

Triantafyllou

Chatzi

2014

Comput Mech

View full text Add to dashboard Cite

A new multiscale finite element formulation is presented for nonlinear dynamic analysis of heterogeneous structures. The proposed multiscale approach utilizes the hysteretic finite element method to model the microstructure. Using the proposed computational scheme, the micro-basis functions, that are used to map the microdisplacement components to the coarse mesh, are only evaluated once and remain constant throughout the analysis procedure. This is accomplished by treating inelasticity at the micro-elemental level through properly defined hysteretic evolution equations. Two types of imposed boundary conditions are considered for the derivation of the multiscale basis functions, namely the linear and periodic boundary conditions. The validity of the proposed formulation as well as its computational efficiency are verified through illustrative numerical experiments.

show abstract

“…In a more general setting, the conformity of mesh distributions on opposite boundaries of the representative volume element cannot always be guaranteed, leading to a non-periodic mesh. In that case, the periodic boundary conditions can be enforced using the polynomial interpolation method [37]. More details on the periodic boundary conditions enforcement are given in Section 4.…”

Section: Problem At the Microscopic Scalementioning

confidence: 99%

“…In [4,7,9], the periodic boundary condition is only applied in case of conforming meshes -two opposite sides of the RVE have the same mesh distribution. On arbitrary meshes another method, as the polynomial interpolation method [37], must be used to constrain the boundary conditions. In that case the matrix C has an arbitrary form.…”

Section: Solution At the Microscopic Scalementioning

confidence: 99%

Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation

Nguyễn

Becker

Noels

2013

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

When considering problems of dimensions close to the characteristic length of the material, the size effects can not be neglected and the classical (so-called first-order) multiscale computational homogenization scheme (FMCH) looses accuracy, motivating the use of a second-order multiscale computational homogenization (SMCH) scheme. This second-order scheme uses the classical continuum at the micro-scale while considering second-order continuum at the macro-scale. Although the theoretical background of the second-order continuum is increasing, the implementation into a finite element code is not straightforward because of the lack of high-order continuity of the shape functions. In this work, we propose a SMCH scheme relying on the discontinuous Galerkin (DG) method at the macro-scale, which simplifies the implementation of the method. Indeed, the DG method is a generalization of weak formulations allowing for inter-element discontinuities either at the C 0 level or at the C 1 level, and it can thus be used to constrain weakly the C 1 continuity at the macro-scale. The C 0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C 0 displacement-based finite elements. Therefore, two formulations can be used at the macro-scale: (i) the full-discontinuous Galerkin formulation (FDG) with weak C 0 and C 1 continuity enforcements, and (ii) the enriched discontinuous Galerkin formulation (EDG) with high-order term enrichment into the conventional C 0 finite element framework. The micro-problem is formulated in terms of standard equilibrium and periodic boundary conditions. A parallel implementation in three dimensions for non-linear finite deformation problems is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way.

show abstract

Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation

Cited by 204 publications

References 19 publications

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials

Multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation

Contact Info

Product

Resources

About