Principle of cluster minimum complementary energy of FEM-cluster-based reduced order method: fast updating the interaction matrix and predicting effective nonlinear properties of heterogeneous material

Abstract

The present paper studies the efficient prediction of effective mechanical properties of heterogeneous material by the FCA (FEM-Cluster based reduced order model Analysis) method proposed in Cheng et al. (Comput Methods Appl Mech Eng 348:157–168, 2019). The principle of minimum complementary energy and its cluster form for the RUC subjected to applied uniform eigenstrains and the PHBCs (Periodic Homogeneous Boundary Conditions) are developed. By using the known interaction matrix, an alternative form of the principle of cluster minimum complementary energy is constructed and proved very efficient for updating the interaction matrix and the effective elastic modulus when the material properties of clusters change. Moreover, the proposed principle of cluster minimum complementary energy is applied for the incremental nonlinear analysis of the cluster reduced order model, and thus greatly improves the prediction of nonlinear effective properties of the RUC in online stage computed in Cheng et al. (Comput Methods Appl Mech Eng 348:157–168, 2019). A number of numerical examples illustrate the effectiveness and efficiency of the FCA approach with the proposed principle of cluster minimum complementary energy.

Appendices

Appendix 1

Appendix 2

In the following, we prove that the solutions of the RUC problem calculated by the principle of cluster minimum complementary energy of two forms are the same.

The first problem is to directly apply the eigenstrain $\varvec{\varepsilon}^{*}$ to the RUC which is corresponding to the principle of cluster minimum complementary energy of the first form,

$$ \Pi _{ 1} (\varvec{\alpha ,\varepsilon}^{*}) = \frac{1}{2}\varvec{\alpha }^{\text{T}} {\mathbf{D}}^{\text{T}} {\mathbf{V}}{\mathbf{SD}}\varvec{\alpha }{ + }\varvec{\alpha }^{\text{T}} {\mathbf{D}}^{\text{T}} {\mathbf{V}}\varvec{\varepsilon}^{*} $$

(68)

which is Eq. (47). Its continuum formulation is as follows.

$$ \begin{array}{*{20}l} { ( {\text{a)}}\;\sigma_{ij,j}^{1} = 0,} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega ,} \hfill \\ { ( {\text{b)}}\;\varepsilon_{ij}^{1} = \frac{1}{2}\left( {u_{i,j}^{1} + u_{j,i}^{1} } \right),} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega ,} \hfill \\ {({\text{c}})\;\sigma_{ij}^{1} = C_{ijkl} \varepsilon_{kl}^{{{\prime }1}} ,\;\varepsilon_{kl}^{{{\prime }1}} = \varepsilon_{kl}^{1} - \varepsilon_{kl}^{1*} ,} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega ,} \hfill \\ {({\text{d}})\;u_{i}^{1} \in {\# ,}\;t_{i}^{1} \in - {\# } .} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega .} \hfill \\ \end{array} $$

(69)

where the superscript 1 denotes the first problem. ${\# , - \# }$ denote the periodic boundary conditions of displacement and traction, and their formulations can refer to Eq. (1)(d)(e). $\text{C}_{ijkl}^{{}} (x)$ denotes the elastic constitutive relation of the inhomogeneous medium. $\varepsilon_{kl}^{1} , \, \varepsilon_{kl}^{{{\prime }1}} , \, \varepsilon_{kl}^{1*}$ denote the total strain, mechanical strain and Eigen strain.

The second problem is corresponding to the principle of cluster minimum complementary energy of the alternative form

$$ \Pi _{{\text{c1}}} (\varvec{\alpha },\varvec{\varepsilon}^{*}) = \frac{1}{2}\varvec{\alpha }^{\text{T}} {\mathbf{D}}^{\text{T}} {\mathbf{V}}{\mathbf{SD}}\varvec{\alpha }{ + }\varvec{\alpha }^{\text{T}} {\mathbf{D}}^{\text{T}} {\mathbf{VSD}}\varvec{\varepsilon}^{*} $$

(70)

which is Eq. (50).

Comparing Eq. (68) with Eq. (70), we can observe that this problem is to apply the eigenstrain ${\mathbf{SD}}\varvec{\varepsilon }^{*}$ to the RUC. Note that the mechanical strain of the first problem in matrix form is $\varvec{\varepsilon^{\prime}}^{ 1} = - {\mathbf{SD}}\varvec{\varepsilon}^{*} .$ Therefore, the second problem is to apply the negative mechanical strain of the first problem to the RUC. Its continuum formulation can be described as

$$ \begin{array}{*{20}l} { ( {\text{a)}}\;\sigma_{ij,j}^{2} = 0 ,} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega ,} \hfill \\ { ( {\text{b)}}\;\varepsilon_{ij}^{2} = \frac{1}{2}\left( {u_{i,j}^{2} + u_{j,i}^{2} } \right),} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega ,} \hfill \\ {({\text{c}})\;\sigma_{ij}^{2} = C_{ijkl} \varepsilon_{kl}^{{{\prime }2}} ,\varepsilon_{kl}^{{{\prime }2}} = \varepsilon_{kl}^{2} - \varepsilon_{kl}^{2*} \, \varepsilon_{kl}^{2*} = - (\varepsilon_{kl}^{1} - \varepsilon_{kl}^{1*} )} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega ,} \hfill \\ { ( {\text{d)}}\;u_{i}^{2} \in {\# ,}t_{i}^{2} \in - {\# } .} \hfill & {\left( {x_{1} , \, x_{2} } \right) \in \varOmega } \hfill \\ \end{array} $$

(71)

where the superscript 2 refers to the second problem. Now it can be easily checked that the problem (71) has the following solutions

$$ \sigma_{ij}^{2} { = }\sigma_{ij}^{ 1} ,\varepsilon_{kl}^{{{\prime }2}} { = }\varepsilon_{kl}^{{{\prime }1}} , \, \varepsilon_{kl}^{2} { = }\varepsilon_{kl}^{{{\prime }2}} + \varepsilon_{kl}^{2 *} = \varepsilon_{kl}^{{{\prime }1}} - \varepsilon_{kl}^{1} + \varepsilon_{kl}^{1*} = 0,\quad {\text{u}}_{i}^{2} = 0 $$

(72)

The first two equalities of Eq. (72) shows that the stress and mechanical strain of the two problems are the same. In another word, we can solve the second problem instead of the first problem. The second problem can be solved by the principle of cluster minimum complementary energy of the alternative form (68).