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Mathematical foundations of FEM-cluster based reduced order analysis method and a spectral analysis algorithm for improving the accuracy

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Abstract

As a reduced order homogenization approach, the FEM-Cluster based reduced order Analysis method (FCA) proposed by Cheng et al. provides an efficient approach to predict the nonlinear effective properties of heterogeneous materials. In its improved version, the clustered Minimum Complementary Energy (MCE) approach is adopted for the computations of incremental strain–stress relation. This work mainly focuses on the mathematical foundations of FCA with clustered MCE. The completeness of the interaction matrix as the clustered self-equilibrium stress space is proved, and the prediction error of FCA solution is analyzed. The deductions also reveal that some bases of the clustered self-equilibrium stress space denoted by the interaction matrix may contradict the basic hypothesis of the cluster-based reduce order methods, and they are mainly responsible for the error of FCA. According to this observation, a spectral analysis algorithm is developed to refine the interaction matrix and improve the accuracy of FCA.

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Abbreviations

\({0}_{m}\in {\mathbb{R}}^{m}\) :

\(m\)-order vector with all elements equaling zero

\({I}_{m}\in {\mathbb{R}}^{m\times m}\) :

\(m\)-order identity matrix

\(rank\left(a\right)\) :

Rank of the matrix a

\(dim\) :

Dimension of the problem

\(d\) :

Number of degree of freedom in strain or stress

\(N\) :

Number of clusters

\(n\) :

Number of elements in the finite element model

\({e}_{i}\in {\mathbb{R}}^{d}\) :

Unit orthogonal vector

\(\Omega \subset {\mathbb{R}}^{dim}\) :

Domain of the representative unit cell

\({\Omega }_{I}\subset\Omega \) :

Domain of the Ith cluster

\(\left|\Omega \right|\) :

Volume of Ω

\(\left|{\Omega }_{I}\right|\) :

Volume of Ωl

\({\chi }_{I}\left(x\right)\) :

Characteristic functions of Ωl, see (5)

\(\varepsilon \left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Strain

\({\varepsilon }^{mech}\left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Mechanical strain

\({\varepsilon }^{0}\in {\mathbb{R}}^{d}\) :

Uniform loaded eigenstrain

\(\sigma \left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Stress

\({\tilde{\sigma }}_{I}\in {\mathbb{R}}^{d}\) :

Average of σ in Ωl, see (7)

\(\overline{Q }\) :

Cluster-average operator, see (6)

\(\widehat{Q}\) :

Cluster-approximation operator, see (18)

\(\overline{A }\) :

Cluster-average of A whose columns are stress, see (15)

\(\widehat{A}\) :

Cluster-approximation of \(A\) whose columns are stress, see (12)

\(\langle \cdot \rangle \) :

Integration over Ω that \( \left\langle {a\left( x \right)} \right\rangle = \mathop \smallint \limits_{{x \in \Omega }}^{{}} a\left( x \right)dx \)

\(span\left(A\right)\) :

Space spanned by the columns of A

\({\mathbb{E}}\) :

Complete self-equilibrium stress set

\(\Sigma \) :

Generalized orthogonal complete self-equilibrium stress set

\({\Sigma }_{i}\left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Stress bases (columns) of Σ, see (21)

\({\lambda }_{i}\) :

Eigenvalues in spectral analysis algorithm, see (21)

\(S\left(x\right):\Omega \to Sym(d)\) :

Inverse of algorithmic tangent moduli

\({S}^{E}\in Sym(d)\) :

Material compliance in linear elastic stage

\({\mathbb{S}}\left(x\right):\Omega \to Sym(d)\) :

Unique inverse of algorithmic tangent moduli, see (36)

\(\overline{S }\in Sym(Nd)\) :

Block diagonal matrix of inverse of algorithmic tangent moduli in all clusters, see (12)

\(\overline{V }\in Sym(Nd)\) :

Block diagonal matrix of clusters’ volume, see (12)

\(\mathcal{A}\) :

Strain-concentration factors

\(D\) :

Interaction matrix, see (13)

\({D}^{*}\) :

Interaction matrix obtained with spectral analysis algorithm, see (54)

\({\mathbb{E}}^{D}\) :

Self-equilibrium stress set for creating D

\({\mathbb{E}}_{I,i}^{D}\left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Columns of \({\mathbb{E}}^{D}{\mathbb{E}}^{D}\)

\({\varepsilon }_{I,i}^{0}\left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Eigenstrain for obtaining \({\mathbb{E}}_{{I,i}}^{D}{\mathbb{E}}_{{I,i}}^{D}\)

\({\Vert \cdot \Vert }_{S}\) :

Distance metric for stress, see (24)

\(\Delta {\sigma }_{DNS}\left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Stress solution of DNS approach

\(\Delta {\widehat{\sigma }}_{FCA}\left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Stress solution of FCA approach with D

\(\Delta {\widehat{\sigma }}_{FCA}^{*}\left(x\right):\Omega \to {\mathbb{R}}^{d}\) :

Stress solution of FCA approach with D

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Acknowledgements

The present work was supported by the National Natural Science Foundation of China (Grant Nos. 11602050, 11702053 and 11821202), the 111 Project (Grant No. B14013), and the Fundamental Research Funds for the Central Universities (Grant No. DUT2019TD37).

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Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, International Research Center for Computational Mechanics, Dalian University of Technology, Dalian, 116024, China

    Zheng Li, Yinghao Nie & Gengdong Cheng

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  1. Zheng Li
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  2. Yinghao Nie
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  3. Gengdong Cheng
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Correspondence to Gengdong Cheng.

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Li, Z., Nie, Y. & Cheng, G. Mathematical foundations of FEM-cluster based reduced order analysis method and a spectral analysis algorithm for improving the accuracy. Comput Mech 69, 1347–1363 (2022). https://doi.org/10.1007/s00466-022-02144-3

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