SECOND-ORDER CONCURRENT COMPUTATIONAL HOMOGENIZATION METHOD AND MULTISCALE HYDROMECHANICAL MODELING FOR SATURATED GRANULAR MATERIALS

2020 ◽  
Vol 18 (2) ◽  
pp. 199-240 ◽  
Author(s):  
Xikui Li ◽  
Songge Zhang ◽  
Qinglin Duan
Keyword(s):  
Granular Materials ◽  
Homogenization Method ◽  
Computational Homogenization ◽  
Second Order

A novel scheme for imposing periodic boundary conditions on RVE in second-order computational homogenization for granular material

2019 ◽  
Vol 36 (8) ◽  
pp. 2835-2858 ◽  
Author(s):  
Xikui Li ◽  
Songge Zhang ◽  
Qinglin Duan
Keyword(s):  
Boundary Conditions ◽  
Granular Materials ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Cosserat Continuum ◽  
Computational Homogenization ◽  
Second Order ◽  
Macroscopic Strain ◽  
Particle Assembly ◽  
Content Type

Purpose This paper aims to present a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the representative volume element (RVE) of discrete particle assembly in the frame of the second-order computational homogenization methods for granular materials. Design/methodology/approach The proposed scheme is based on the generalized Hill’s lemma of gradient Cosserat continuum and the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE of discrete particle assembly. The generalized Hill’s lemma conducts to downscale the macroscopic strain or stress measures and to impose the periodic boundary conditions on the RVE boundary so that the Hill-Mandel energy equivalence condition is ensured. Because of the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE, the periodic boundary displacement and traction constraints together with the downscaled macroscopic strains and strain gradients, micro-rotations and curvatures are imposed in the point-wise sense without the need of introducing the Lagrange multipliers for enforcing the periodic boundary displacement and traction constraints in a weak sense. Findings Numerical results demonstrate that the applicability and effectiveness of the proposed scheme in imposing the periodic boundary conditions on the RVE. The results of the RVE subjected to the periodic boundary conditions together with the displacement boundary conditions in the second-order computational homogenization for granular materials provide the desired estimations, which lie between the upper and the lower bounds provided by the displacement and the traction boundary conditions imposed on the RVE respectively. Research limitations/implications Each grain in the particulate system under consideration is assumed to be rigid and circular. Practical implications The proposed scheme for imposing periodic boundary conditions on the RVE can be adopted solely for estimating the effective mechanical properties of granular materials and/or integrated into the frame of the second-order computational homogenization method with a nested finite element method-discrete element method solution procedure for granular materials. It will tend to provide, at least theoretically, more reasonable results for effective material properties and solutions of a macroscopic boundary value problem simulated by the computational homogenization method. Originality/value This paper presents a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the RVE of discrete particle assembly for granular materials without need of introducing Lagrange multipliers for enforcing periodic boundary conditions in a weak (integration) sense.

A data-driven computational homogenization method based on neural networks for the nonlinear anisotropic electrical response of graphene/polymer nanocomposites

2018 ◽  
Vol 64 (2) ◽  
pp. 307-321 ◽  
Author(s):  
Xiaoxin Lu ◽  
Dimitris G. Giovanis ◽  
Julien Yvonnet ◽  
Vissarion Papadopoulos ◽  
Fabrice Detrez ◽  
...  
Keyword(s):  
Neural Networks ◽  
Polymer Nanocomposites ◽  
Electrical Response ◽  
Homogenization Method ◽  
Computational Homogenization ◽  
Data Driven

Studies of microstructural size effect and higher-order deformation in second-order computational homogenization

2010 ◽  
Vol 88 (23-24) ◽  
pp. 1383-1390 ◽  
Author(s):  
Łukasz Kaczmarczyk ◽  
Chris J. Pearce ◽  
Nenad Bićanić
Keyword(s):  
Size Effect ◽  
Higher Order ◽  
Computational Homogenization ◽  
Second Order ◽  
Order Deformation

scholarly journals Fully-coupled micro–macro finite element simulations of the Nakajima test using parallel computational homogenization

2021 ◽  
Vol 68 (5) ◽  
pp. 1153-1178
Author(s):  
Axel Klawonn ◽  
Martin Lanser ◽  
Oliver Rheinbach ◽  
Matthias Uran
Keyword(s):  
Computing Time ◽  
Homogenization Method ◽  
Computational Homogenization ◽  
Forming Limit ◽  
Contact Conditions ◽  
Blank Holder ◽  
Macroscopic Level ◽  
Contact Formulation ◽  
Balancing Domain Decomposition ◽  
Nakajima Test

AbstractThe Nakajima test is a well-known material test from the steel and metal industry to determine the forming limit of sheet metal. It is demonstrated how FE2TI, our highly parallel scalable implementation of the computational homogenization method FE$$^2$$ 2 , can be used for the simulation of the Nakajima test. In this test, a sample sheet geometry is clamped between a blank holder and a die. Then, a hemispherical punch is driven into the specimen until material failure occurs. For the simulation of the Nakajima test, our software package FE2TI has been enhanced with a frictionless contact formulation on the macroscopic level using the penalty method. The appropriate choice of suitable boundary conditions as well as the influence of symmetry assumptions regarding the symmetric test setup are discussed. In order to be able to solve larger macroscopic problems more efficiently, the balancing domain decomposition by constraints (BDDC) approach has been implemented on the macroscopic level as an alternative to a sparse direct solver. To improve the computational efficiency of FE2TI even further, additionally, an adaptive load step approach has been implemented and different extrapolation strategies are compared. Both strategies yield a significant reduction of the overall computing time. Furthermore, a strategy to dynamically increase the penalty parameter is presented which allows to resolve the contact conditions more accurately without increasing the overall computing time too much. Numerically computed forming limit diagrams based on virtual Nakajima tests are presented.

scholarly journals A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Xiong Liu ◽  
Wenming He
Keyword(s):  
Elliptic Problem ◽  
Homogenization Method ◽  
Second Order ◽  
Homogenization Theory ◽  
High Demand ◽  
Periodic Coefficients ◽  
Order Elliptic Problem ◽  
Multiscale Homogenization

In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form ∂ / ∂ x i a i j x / ε , x ∂ u ε x / ∂ x j = f x . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution u 0 , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution u 0 satisfies than the classical homogenization theory needs.

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