Multiscale seamless‐domain method for linear elastic analysis of heterogeneous materials

Mechanical Engineering Journal

Todoroki

Mizutani

2016

Self Cite

In this paper, we applied a multiscale numerical scheme called the seamless-domain method (SDM) to nonlinear elliptic boundary value problems. Although the SDM is meshfree, it can obtain a high-resolution solution whose dependent-variable gradient(s) is sufficiently smooth and continuous. The SDM models with only coarse-grained points can produce accurate solutions for both linear heat conduction problems and linear elastic problems. This manuscript presents a simple nonlinear solver for the SDM analysis of heterogeneous materials. Although the solver can easily approximate the solutions to nonlinear multiscale problems, it does not require an iterative multiscale analysis at every convergence calculation. In other words, the proposed scheme does not completely interactively couple the multiple scales. We present numerical examples of nonlinear stationary heat conduction analyses of heterogeneous fields and compare the SDM model, the direct finite-element model, and the homogenized model based on the homogenization theory. For a real heterogeneous structure (graphite fiber composite) that did not have strong material nonlinearities, the SDM model using only 925 points gave a solution with similar precisions as an ordinary finite element solution using hundreds of thousands of nodes. To investigate the limitations of the method, we also applied the SDM to imaginary materials with various strengths of thermal property nonlinearities.

Section: Discussionmentioning

confidence: 99%

“…In previous work (Suzuki and Soga, 2016, Suzuki, 2016a, 2016b, a multiscale analysis called the seamless-domain method (SDM) was developed and applied to linear analyses. When conducting a macroscopic analysis of an entire field, the SDM constructs a meshfree model that is represented by only a small number of coarse-grained points (CPs).…”

Section: Introductionmentioning

confidence: 99%

Multiscale seamless-domain method for solving nonlinear heat conduction problems without iterative multiscale calculations

Mechanical Engineering Journal

Todoroki

Mizutani

2016

Self Cite

“…By giving enforced-displacement boundary conditions ( 1 u and 4 u ) and solving the below simultaneous equations, we can obtain unknown displacements ( 2 u and 3 u ). (Suzuki and Soga, 2016;Suzuki, 2016a;Suzuki, 2016b). Suzuki, Mechanical Engineering Journal, Vol.4, No.4 (2017) [DOI: 10.1299/mej.…”

Section: L-shaped Beammentioning

confidence: 99%

“…The detailed formulations are presented in subsection 3.3.2. The different features of the SDM in itself compared to other numerical solvers were illustrated in (Suzuki and Soga, 2016;Suzuki, 2016a).…”

Section: Differences Compared With the Standard Super-elementsmentioning

confidence: 99%

“…Fig. 1 Schematic of the original two-scale seamless-domain method for a heterogeneous analytical domain having a periodic microstructure: (a) heterogeneous entire domain; (b) two examples of local domains extracted from the entire domain; and (c) calculated "seamless" dependent-variable distribution (Suzuki and Soga, 2016;Suzuki, 2016a;. Fig.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A domain decomposition technique based on the multiscale seamless-domain method

Mechanical Engineering Journal

2017

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We present a domain decomposition technique combining the finite-element analysis and a multiscale simulation scheme that is termed the seamless-domain method. This technique is applicable to any type of linear problems and has the below different features from conventional substructuring techniques used for finite-element methods such as the super-elements. Adjacent substructures do not need to have the same finite-element mesh or the same node layout at their interface. Therefore, the proposed technique allows us to connect a substructure having a coarse mesh and another substructure having a fine mesh. Additionally, the neighboring substructures are partially overlapped each other around their interfacial region. The overlapped region attempts to generate continuous dependent-variable distribution(s) at the interfacial region. We present four types of numerical experiments related to linear elasticity to investigate practical feasibility of the proposed technique and evaluate an effect of different finite-element meshes in region shared by two substructures on the computational accuracy.

Multiscale seamless‐domain method for solving nonlinear problems using statistical estimation methodology

Numerical Meth Engineering

2017

Self Cite

This article presents a nonlinear solver combining regression analysis and a multiscale simulation scheme. First, the proposed method repeats microscopic analysis of a local simulation domain, which is extracted from the entire global domain, to statistically estimate the relation(s) between the value of a dependent variable at a point and values at surrounding points. The relation is called regression function. Subsequent global analysis reveals the behavior of the global domain with only coarse-grained points using the regression function quickly at low computational cost, which can be accomplished using a multiscale numerical solver, called the seamless-domain method. The objective of the study is to solve a nonlinear problem accurately and at low cost by combining the 2 techniques. We present an example problem of a nonlinear steadystate heat conduction analysis of a heterogeneous material. The proposed model using fewer than 1000 points generates a solution with precision similar to that of a standard finite-element solution using hundreds of thousands of nodes. To investigate the relationship between the accuracy and computational time, we apply the seamless-domain method under varying conditions such as the number of iterations of the prior analysis for statistical data learning.KEYWORDS elliptic, multiscale, nonlinear solvers, partial differential equations, regression analysis | INTRODUCTIONPrevious work 1-4 presented a multiscale numerical solver called the seamless-domain method (SDM). The SDM model is meshless and represented by only a small number of coarse-grained points (CPs). The method constructs a structure as a "seamless" analytical domain whose distributions of a dependent variable and its gradient are almost continuous Nomenclature: Symbol, Explanation; # −1 , inverse of #; # T , transpose of #; Ω G ⊂ R d , domain for global analysis; Ω L ⊂ R d , domain for local analysis; Ω R ⊂ Ω L , region of influence; Γ G , boundary of the global domain; Γ L , boundary of the local domain; Γ R , boundary of the region of influence; a ∈ R 1 × m , weighting coefficient matrix; d ∈ {1, … , 3}, dimension of domain; f i , i-th response-surface function that determines the dependent-variable value at a coarse-grained point (CP) from the values at surrounding CPs (u R m ð Þ ); f L i , i-th response-surface function that determines u L m ð Þ from u R m ð Þ ; m, number of CPs in a region of influence; m l , number of iterations of the prior local analysis for constructing each response surface; m r , number of response surfaces; n, number of CPs in a global domain; N ∈ R 1 × m , interpolating function matrix for u R m ð Þ at temperature u ref ; R, set of all real numbers; u(x) ∈ R, dependent variable at point x; u G n ð Þ ∈R n , dependent variable of CPs in a global domain (Ω G ); u L m ð Þ ∈R m , dependent variable for all CPs near a local domain 0 s boundary (Γ L ); u R m ð Þ ∈R m , dependent variable for all CPs near a boundary of a region of influence (Γ R ); u ref i , reference temperature for CP i; x ∈ R d ,...