A method of substructuring large-scale computational micromechanical problems

Zohdi, Tarek I.; Wriggers, Peter; Huet, C.

doi:10.1016/s0045-7825(01)00189-x

Cited by 47 publications

(33 citation statements)

References 6 publications

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“…According to Zohdi et al [25], a specimen can be computationally tested to permit the solution to the elastic problem for determining the stress and strain micro fields. Solving the system of equations shown in Equation (8) determines the constants for the matrix of elasticity.…”

Section: The Plane Stress (Pt) and Plane Strain (Ps) Hypotheses And Tmentioning

confidence: 99%

Determination of the Effective Elastic Modulus for Nodular Cast Iron Using the Boundary Element Method

Betancur

Anflor

Pereira

et al. 2018

Metals

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Abstract:In this work, a multiscale homogenization procedure using the boundary element method (BEM) for modeling a two-dimensional (2D) and three-dimensional (3D) multiphase microstructure is presented. A numerical routine is specially written for modeling nodular cast iron (NCI) considering the graphite nodules as cylindrical and real geometries. The BEM is used as a numerical approach for solving the elastic problem of a representative volume element from a mean field model. Numerical models for NCI have generally been developed considering the graphite nodules as voids due to their soft feature. In this sense, three numerical models are developed, and the homogenization procedure is carried out considering the graphite nodules as non-voids. Experimental tensile, hardness, and microhardness tests are performed to determine the mechanical properties of the overall material, matrix, and inclusion nodules, respectively. The nodule sizes, distributions, and chemical compositions are determined by laser scanning microscopy, an X-ray computerized microtomography system (micro-CT), and energy-dispersive X-ray (EDX) spectroscopy, respectively. For the numerical model with real inclusions, the boundary mesh is obtained from micro-CT data. The effective properties obtained by considering the real and synthetic nodules' geometries are compared with those obtained from the experimental work and the existing literature. The final results considering both approaches demonstrate a good agreement.

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Section: The Plane Stress (Pt) and Plane Strain (Ps) Hypotheses And Tmentioning

confidence: 99%

Determination of the Effective Elastic Modulus for Nodular Cast Iron Using the Boundary Element Method

Betancur

Anflor

Pereira

et al. 2018

Metals

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“…16, 32 and 64 particle samples (the responses differed from one another, on average, by less than 1 %), we selected the 20-particle microstructures for further tests. 4 Following Zohdi et al [64], we then simulated 512 different samples, each time with a different random distribution of 20 nonintersecting particles occupying 22 % of the volume. Consistent with the previous test's mesh densities, we found that beyond approximately 2344 DOF per particle (24 × 24 × 24 mesh or 9 × 9 × 9 trilinear hexahedra or 46875 DOF per test sample), the numerical simulations were insensitive to further mesh refinements.…”

Section: Size Effects In Computational Materials Testingmentioning

confidence: 99%

Statistical ensemble error bounds for homogenized microheterogeneous solids

Zohdi

2005

Z. angew. Math. Phys.

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Typically, in order to characterize the homogenized effective macroscopic response of new materials possessing random heterogeneous microstructure, a relation between averages|Ω| Ω · dΩ, and where σ and are the stress and strain tensor fields within a statistically representative volume element (SRVE) of volume |Ω|. The quantity, I E * , is known as the effective property, and is the elasticity tensor used in usual macroscale analyses. In order to generate homogenized responses computationally, a series of detailed boundary value representations resolving the heterogeneous microstructure, posed over the SRVE's domain, must be solved. This requires an enormous numerical effort that can overwhelm most computational facilities. A natural way of generating an approximation to the SRVE's response is by first computing the response of smaller (subrepresentative) samples, each with a different random realization of the microstructural type under investigation, and then to ensemble average the results afterwards. Compared to a direct simulation of an SRVE, testing many small samples is a computationally inexpensive process since the number of floating point operations is greatly reduced, as well as the fact that the samples' responses can be computed trivially in parallel. However, there is an inherent error in this process. Clearly the population's ensemble average is not the SRVE's response. However, as shown in this work, the moments on the distribution of the population can be used to generate rigorous upper and lower error bounds on the quality of the ensemble-generated response. Two-sided bounds are given on the SRVE response in terms of the ensemble average, its standard deviation and its skewness.

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“…The work was an extension to their previous paper [31] in which the fine scale discretisation error was ignored. The natural error between the exact solution u and the coarsest scale solution u (0,h) is defined by [43,44]:…”

Section: Introductionmentioning

confidence: 99%

Scale selection in nonlinear fracture mechanics of heterogeneous materials

Akbari

Kerfriden

Bordas

2015

Philosophical Magazine

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A new adaptive multiscale method for the nonlinear fracture simulation of heterogeneous materials is proposed. The two major sources of error in the finite element simulation are discretisation and modelling errors. In the failure problems, the discretisation error increases due to the strain localisation which is also a source for the error in the homogenisation of the underlying micro-structure. In this paper, the discretisation error is controlled by an adaptive mesh refinement procedure following the Zienkiewicz-Zhu technique, and the modelling error, which is the resultant of homogenisation of micro-structure, is controlled by replacing the macroscopic model with the underlying heterogeneous micro-structure. The scale adaptation criterion which is based on an error indicator for homogenisation proposed by[1] is employed for our nonlinear fracture problem. The control of both discretisation and homogenisation errors is the main feature of the proposed multiscale method.

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A method of substructuring large-scale computational micromechanical problems

Cited by 47 publications

References 6 publications

Determination of the Effective Elastic Modulus for Nodular Cast Iron Using the Boundary Element Method

Determination of the Effective Elastic Modulus for Nodular Cast Iron Using the Boundary Element Method

Statistical ensemble error bounds for homogenized microheterogeneous solids

Scale selection in nonlinear fracture mechanics of heterogeneous materials

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