Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method

Abstract

The continuous increase of computational capacity has encouraged the extensive use of multiscale techniques to simulate the material behaviour on several fields of knowledge. In solid mechanics, the multiscale approaches which consider the macro-scale deformation gradient to obtain the homogenized material behaviour from the micro-scale are called first-order computational homogenization. Following this idea, the second-order FE2 methods incorporate high-order gradients to improve the simulation accuracy. However, to capture the full advantages of these high-order framework the classical boundary value problem (BVP) at the macro-scale must be upgraded to high-order level, which complicates their numerical solution. With the purpose of obtaining the best of both methods i.e. first-order and second-order, in this work an enhanced-first-order computational homogenization is presented. The proposed approach preserves a classical BVP at the macro-scale level but taking into account the high-order gradient of the macro-scale in the micro-scale solution. The developed numerical examples show how the proposed method obtains the expected stress distribution at the micro-scale for states of structural bending loads. Nevertheless, the macro-scale results achieved are the same than the ones obtained with a first-order framework because both approaches share the same macro-scale BVP.

Appendix: Microscopic Kinematic Relationships for the EFOCH

In the following are described the kinematic relationships defined between master and slaves nodes of the RVE, required for the implementation of the EFOCH in a FEM software.

In the Fig. 17, it is possible identify easily master nodes (named with a letter) and slave nodes (named with a letter and number). This is when a structured FE mesh on the boundary of the RVE is used. In the vertices nodes, also it is possible to identify a master node (“1”) and seven slave nodes (“2” ,“3” , ...and “8”).

Fig. 17

Master and slaves nodes in a general hexagonal RVE

Using (91) is possible to write the displacement of the slave node “${\text {a}}_{1}$” as a function of the displacement of the master node “a” for EFOCH as

$$\begin{aligned} \bar{{\mathbf {u}}}_{a_{1}}=\bar{{\mathbf {u}}}_{a}+D_{2}\left( {\mathbf {F}}-{\mathbf {I}}\right) \cdot {\mathbf {N}}_{Y}^{+}+\frac{\left( D_{2}\right) ^{2}}{2} {\mathbf {N}}_{Y}^{+}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{Y}^{+}+D_{2}{\mathbf {X}}_{a}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{Y}^{+}. \end{aligned}$$

(94)

To simplify the final expressions is defined

$$\begin{aligned} {\mathbf {sm}}^{G}_{1}=\, & {} {\scriptstyle \frac{\left( D_{1}\right) ^{2}}{2} {\mathbf {N}}_{X}^{+}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{X}^{+}}, \quad {\mathbf {sm}}^{G}_{2}=\,{\scriptstyle \frac{\left( D_{2}\right) ^{2}}{2} {\mathbf {N}}_{Y}^{+}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{Y}^{+}},\\ {\mathbf {sm}}^{G}_{3}=\, & {} {\scriptstyle \frac{\left( D_{3}\right) ^{2}}{2} {\mathbf {N}}_{Z}^{+}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{Z}^{+}},\quad {\mathbf {sm}}^{G}_{12}=\, {\scriptstyle D_{1}D_{2} {\mathbf {N}}_{X}^{+}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{Y}^{+}},\\ {\mathbf {sm}}^{G}_{13}=\, & {} {\scriptstyle D_{1}D_{3} {\mathbf {N}}_{X}^{+}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{Z}^{+}}, \quad {\mathbf {sm}}^{G}_{23}=\, {\scriptstyle D_{2}D_{3} {\mathbf {N}}_{Y}^{+}\cdot {\mathbf {G}}\cdot {\mathbf {N}}_{Z}^{+}},\\ {\mathbf {SM}}^{G}_{1}=\, & {} {\scriptstyle D_{1}{\mathbf {N}}_{X}^{+}\cdot {\mathbf {G}}}, \quad {\mathbf {SM}}^{G}_{2}=\, {\scriptstyle D_{2}{\mathbf {N}}_{Y}^{+}\cdot {\mathbf {G}}}, \quad {\mathbf {SM}}^{G}_{3}=\, {\scriptstyle D_{3}{\mathbf {N}}_{Z}^{+}\cdot {\mathbf {G}}}. \end{aligned}$$

Therefore, it can be shown that the slaves nodes are

$$\begin{aligned} \bar{{\mathbf {u}}}_{a_{1}}=\, & {} \bar{{\mathbf {u}}}_{a}+{\mathbf {sm}}_{2}+{\mathbf {sm}}^{G}_{2}+{\mathbf {SM}}^{G}_{2}\cdot {\mathbf {X}}_{a},\\ \bar{{\mathbf {u}}}_{a_{2}}=\, & {} \bar{{\mathbf {u}}}_{a}+{\mathbf {sm}}_{2}+{\mathbf {sm}}_{3}+{\mathbf {sm}}^{G}_{2}+{\mathbf {sm}}^{G}_{3} + {\mathbf {sm}}^{G}_{23} \\&+\,({\mathbf {SM}}^{G}_{2}+{\mathbf {SM}}^{G}_{3}) \cdot {\mathbf {X}}_{a},\\ \bar{{\mathbf {u}}}_{a_{3}}=\, & {} \bar{{\mathbf {u}}}_{a}+{\mathbf {sm}}_{3}+{\mathbf {sm}}^{G}_{3}+{\mathbf {SM}}^{G}_{3}\cdot {\mathbf {X}}_{a},\\ \bar{{\mathbf {u}}}_{b_{1}}=\, & {} \bar{{\mathbf {u}}}_{b}+{\mathbf {sm}}_{1}+{\mathbf {sm}}^{G}_{1}+{\mathbf {SM}}^{G}_{1}\cdot {\mathbf {X}}_{b},\\ \bar{{\mathbf {u}}}_{b_{2}}=\, & {} \bar{{\mathbf {u}}}_{b}+{\mathbf {sm}}_{1}+{\mathbf {sm}}_{3}+{\mathbf {sm}}^{G}_{1}+{\mathbf {sm}}^{G}_{3} + {\mathbf {sm}}^{G}_{13}\\&+\,({\mathbf {SM}}^{G}_{1}+{\mathbf {SM}}^{G}_{3}) \cdot {\mathbf {X}}_{b},\\ \bar{{\mathbf {u}}}_{b_{3}}=\, & {} \bar{{\mathbf {u}}}_{b}+{\mathbf {sm}}_{3}+{\mathbf {sm}}^{G}_{3}+{\mathbf {SM}}^{G}_{3}\cdot {\mathbf {X}}_{b},\\ \bar{{\mathbf {u}}}_{c_{1}}=\, & {} \bar{{\mathbf {u}}}_{c}+{\mathbf {sm}}_{1}+{\mathbf {sm}}^{G}_{1}+{\mathbf {SM}}^{G}_{1}\cdot {\mathbf {X}}_{c},\\ \bar{{\mathbf {u}}}_{c_{2}}=\, & {} \bar{{\mathbf {u}}}_{c}+{\mathbf {sm}}_{1}+{\mathbf {sm}}_{2} + {\mathbf {sm}}^{G}_{1}+{\mathbf {sm}}^{G}_{2} + {\mathbf {sm}}^{G}_{12}\\&+ ({\mathbf {SM}}^{G}_{1}+{\mathbf {SM}}^{G}_{2}) \cdot {\mathbf {X}}_{c},\\ \bar{{\mathbf {u}}}_{c_{3}}=\, & {} \bar{{\mathbf {u}}}_{c}+{\mathbf {sm}}_{2}+{\mathbf {sm}}^{G}_{2}+{\mathbf {SM}}^{G}_{2}\cdot {\mathbf {X}}_{c},\\ \bar{{\mathbf {u}}}_{d_{1}}=\, & {} \bar{{\mathbf {u}}}_{d}+{\mathbf {sm}}_{3}+{\mathbf {sm}}^{G}_{3}+{\mathbf {SM}}^{G}_{3}\cdot {\mathbf {X}}_{d},\\ \bar{{\mathbf {u}}}_{e_{1}}=\, & {} \bar{{\mathbf {u}}}_{e}+{\mathbf {sm}}_{1}+{\mathbf {sm}}^{G}_{1}+{\mathbf {SM}}^{G}_{1}\cdot {\mathbf {X}}_{e},\\ \bar{{\mathbf {u}}}_{f_{1}}=\, & {} \bar{{\mathbf {u}}}_{f}+{\mathbf {sm}}_{2}+{\mathbf {sm}}^{G}_{2}+{\mathbf {SM}}^{G}_{2}\cdot {\mathbf {X}}_{f}. \end{aligned}$$

And, taking into account that the position vector of the master vertex node “1” is: ${\mathbf {X}}_{1}=\,-\frac{D_{1}}{2}{\mathbf {N}}_{X}^{+}-\frac{D_{2}}{2}{\mathbf {N}}_{Y}^{+}-\frac{D_{3}}{2}{\mathbf {N}}_{Z}^{+}$, the slaves vertices nodes are

$$\begin{aligned} \bar{{\mathbf {u}}}_{2}=\, & {} \bar{{\mathbf {u}}}_{1}+{\mathbf {sm}}_{1}-\frac{{\mathbf {sm}}^{G}_{12}}{2}-\frac{{\mathbf {sm}}^{G}_{13}}{2},\\ \bar{{\mathbf {u}}}_{3}=\, & {} \bar{{\mathbf {u}}}_{1}+{\mathbf {sm}}_{1}+{\mathbf {sm}}_{2}-\frac{{\mathbf {sm}}^{G}_{13}}{2}-\frac{{\mathbf {sm}}^{G}_{23}}{2},\\ \bar{{\mathbf {u}}}_{4}=\, & {} \bar{{\mathbf {u}}}_{1}+{\mathbf {sm}}_{2}-\frac{{\mathbf {sm}}^{G}_{12}}{2}-\frac{{\mathbf {sm}}^{G}_{23}}{2},\\ \bar{{\mathbf {u}}}_{5}=\, & {} \bar{{\mathbf {u}}}_{1}+{\mathbf {sm}}_{3}-\frac{{\mathbf {sm}}^{G}_{13}}{2}-\frac{{\mathbf {sm}}^{G}_{23}}{2},\\ \bar{{\mathbf {u}}}_{6}=\, & {} \bar{{\mathbf {u}}}_{1}+{\mathbf {sm}}_{1}+{\mathbf {sm}}_{3}-\frac{{\mathbf {sm}}^{G}_{12}}{2}-\frac{{\mathbf {sm}}^{G}_{23}}{2},\\ \bar{{\mathbf {u}}}_{7}=\, & {} \bar{{\mathbf {u}}}_{1}+{\mathbf {sm}}_{1}+{\mathbf {sm}}_{2}+{\mathbf {sm}}_{3},\\ \bar{{\mathbf {u}}}_{8}=\, & {} \bar{{\mathbf {u}}}_{1}+{\mathbf {sm}}_{2}+{\mathbf {sm}}^{G}_{3}-\frac{{\mathbf {sm}}^{G}_{12}}{2}-\frac{{\mathbf {sm}}^{G}_{13}}{2}. \end{aligned}$$

Fig. 18

Master and slaves nodes on the negative faces of the RVE

The extra boundary restrictions must be also satisfied. These boundary conditions are integral boundary constraints on each negative face of the RVE. Equation (92) can be rewritten as

$$\begin{aligned} \begin{array}{l} {\mathbf {A}}_{yz}\cdot \bar{{\mathbf {u}}}_{yz}={\mathbf {H}}_{yz},~in~\partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{X}^{-}},\\ {\mathbf {A}}_{xz}\cdot \bar{{\mathbf {u}}}_{xz}={\mathbf {H}}_{xz},~in~\partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{Y}^{-}},\\ {\mathbf {A}}_{xy}\cdot \bar{{\mathbf {u}}}_{xy}={\mathbf {H}}_{yz},~in~\partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{Z}^{-}} \end{array} \end{aligned}$$

(95)

where,

$$\begin{aligned} {\mathbf {H}}_{yz}=\, & {} {\displaystyle - \frac{1}{2}D_{1}D_{2}D_{3}({\mathbf {F}}-{\mathbf {I}})\cdot {\mathbf {N}}_{X}^{+} + \frac{1}{8}(D_{1})^{2}D_{2}D_{3}{\mathbf {G}}:{\mathbf {N}}_{X}^{+}\otimes {\mathbf {N}}_{X}^{+}}\\&+ {\displaystyle \frac{1}{24}(D_{2})^{3}D_{3}{\mathbf {G}}:{\mathbf {N}}_{Y}^{+}\otimes {\mathbf {N}}_{Y}^{+} + \frac{1}{24}D_{2}(D_{3})^{3}{\mathbf {G}}:{\mathbf {N}}_{Z}^{+}\otimes {\mathbf {N}}_{Z}^{+}},\\ {\mathbf {H}}_{xz}=\, & {} {\displaystyle - \frac{1}{2}D_{1}D_{2}D_{3}({\mathbf {F}}-{\mathbf {I}})\cdot {\mathbf {N}}_{Y}^{+} + \frac{1}{8}D_{1}(D_{2})^{2}D_{3}{\mathbf {G}}:{\mathbf {N}}_{Y}^{+}\otimes {\mathbf {N}}_{Y}^{+}}\\&+ {\displaystyle \frac{1}{24}(D_{1})^{3}D_{3}{\mathbf {G}}:{\mathbf {N}}_{X}^{+}\otimes {\mathbf {N}}_{X}^{+} + \frac{1}{24}D_{1}(D_{3})^{3}{\mathbf {G}}:{\mathbf {N}}_{Z}^{+}\otimes {\mathbf {N}}_{Z}^{+}},\\ {\mathbf {H}}_{xy}=\, & {} {\displaystyle - \frac{1}{2}D_{1}D_{2}D_{3}({\mathbf {F}}-{\mathbf {I}})\cdot {\mathbf {N}}_{Z}^{+} + \frac{1}{8}D_{1}D_{2}(D_{3})^{2}{\mathbf {G}}:{\mathbf {N}}_{Z}^{+}\otimes {\mathbf {N}}_{Z}^{+}}\\&+ {\displaystyle \frac{1}{24}(D_{1})^{3}D_{2}{\mathbf {G}}:{\mathbf {N}}_{X}^{+}\otimes {\mathbf {N}}_{X}^{+} + \frac{1}{24}D_{1}(D_{2})^{3}{\mathbf {G}}:{\mathbf {N}}_{Y}^{+}\otimes {\mathbf {N}}_{Y}^{+}} \end{aligned}$$

and,

$$\begin{aligned} {\mathbf {A}}_{yz}= & {} {\displaystyle \int _{{\mathbf {N}}_{X}^{-}}{\mathbf {N}}_{yz}\,dA_{yz}}, \quad {\mathbf {A}}_{xz} = {\displaystyle \int _{{\mathbf {N}}_{Y}^{-}}{\mathbf {N}}_{xz}\,dA_{xz}},\\ {\mathbf {A}}_{xy}= & {} {\displaystyle \int _{{\mathbf {N}}_{Z}^{-}}{\mathbf {N}}_{xy}\,dA_{xy}}. \end{aligned}$$

Here, ${\mathbf {N}}_{yz}$, ${\mathbf {N}}_{xz}$ and ${\mathbf {N}}_{xy}$ are the shape functions on the negative face YZ, XZ and XY of the RVE, respectively. And, from Fig. 18 it is possible to write the displacement vectors of the nodes on the different negative faces as

$$\begin{aligned} \bar{{\mathbf {u}}}_{yz}=\, & {} \{\bar{{\mathbf {u}}}_{1}|\bar{{\mathbf {u}}}_{4}|\bar{{\mathbf {u}}}_{5}|\bar{{\mathbf {u}}}_{8}|\bar{{\mathbf {u}}}_{b}|\bar{{\mathbf {u}}}_{b_{3}}|\bar{{\mathbf {u}}}_{c}|\bar{{\mathbf {u}}}_{c_{3}}|\bar{{\mathbf {u}}}_{e}\},\\ \bar{{\mathbf {u}}}_{xz}=\, & {} \{\bar{{\mathbf {u}}}_{1}|\bar{{\mathbf {u}}}_{2}|\bar{{\mathbf {u}}}_{5}|\bar{{\mathbf {u}}}_{6}|\bar{{\mathbf {u}}}_{a}|\bar{{\mathbf {u}}}_{a_{3}}|\bar{{\mathbf {u}}}_{c}|\bar{{\mathbf {u}}}_{c_{1}}|\bar{{\mathbf {u}}}_{f}\},\\ \bar{{\mathbf {u}}}_{xy}=\, & {} \{\bar{{\mathbf {u}}}_{1}|\bar{{\mathbf {u}}}_{2}|\bar{{\mathbf {u}}}_{3}|\bar{{\mathbf {u}}}_{4}|\bar{{\mathbf {u}}}_{a}|\bar{{\mathbf {u}}}_{a_{1}}|\bar{{\mathbf {u}}}_{b}|\bar{{\mathbf {u}}}_{b_{1}}|\bar{{\mathbf {u}}}_{d}\}. \end{aligned}$$

In the previous displacement vectors of the nodes on the negative faces of the RVE it is possible identify masters and slaves nodes. Therefore, the boundary constraints (95) obtained above can be written in terms of master nodes as

$$\begin{aligned} {\mathbf {A}}^{m}_{yz}\cdot \bar{{\mathbf {u}}}^{m}_{yz}=\, & {} {\mathbf {H}}^{m}_{yz},\,\, in \,\, \partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{X}^{-}}, \\ {\mathbf {A}}^{m}_{xz}\cdot \bar{{\mathbf {u}}}^{m}_{xz}=\, & {} {\mathbf {H}}^{m}_{xz},\,\, in \,\, \partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{Y}^{-}}, \\ {\mathbf {A}}^{m}_{xy}\cdot \bar{{\mathbf {u}}}^{m}_{xy}=\, & {} {\mathbf {H}}^{m}_{yz},\,\, in \,\, \partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{Z}^{-}}. \end{aligned}$$

(96)

where,

$$\begin{aligned} \bar{{\mathbf {u}}}^{m}_{yz}= & {} \{\bar{{\mathbf {u}}}_{1}|\bar{{\mathbf {u}}}_{b}|\bar{{\mathbf {u}}}_{c}|\bar{{\mathbf {u}}}_{e}\},\\ \quad \bar{{\mathbf {u}}}^{m}_{xz}= & {} \{\bar{{\mathbf {u}}}_{1}|\bar{{\mathbf {u}}}_{a}|\bar{{\mathbf {u}}}_{c}|\bar{{\mathbf {u}}}_{f}\},\\ \quad \bar{{\mathbf {u}}}^{m}_{xy}= & {} \{\bar{{\mathbf {u}}}_{1}|\bar{{\mathbf {u}}}_{a}|\bar{{\mathbf {u}}}_{b}|\bar{{\mathbf {u}}}_{d}\}. \end{aligned}$$

and, as an example, the term of the matrix ${\mathbf {A}}^{m}_{yz}$ for the $\bar{{\mathbf {u}}}_{b}$ of the master nodes on the negative face YZ is

$$\begin{aligned} {\mathbf {A}}^{m}_{yz|b} = {\mathbf {A}}_{yz|b} +{\mathbf {A}}^{m}_{yz|b_{3}}, \end{aligned}$$

and the contribution to ${\mathbf {H}}^{m}_{yz}$ for the $\bar{{\mathbf {u}}}_{b_{3}}$ of the slave nodes on the negative face YZ is

$$\begin{aligned} {\mathbf {H}}^{m}_{yz|b}={\mathbf {A}}^{m}_{yz|b_{3}}\cdot ({\mathbf {sm}}_{3}+{\mathbf {sm}}^{G}_{3}+{\mathbf {SM}}^{G}_{3}\cdot {\mathbf {X}}_{b}). \end{aligned}$$

The master nodes on the different negative faces of the RVE must verify (96). Therefore, with the aim to find redundant unknowns, it is possible to identify another slave extra node by each negative face which can be obtained as a function of the other master nodes. Then,

$$\begin{aligned} \bar{{\mathbf {u}}}^{s_{1}}_{yz}= & {} - [{\mathbf {A}}^{s_{1}}_{yz}]^{-1} \cdot {\mathbf {A}}^{(m-1)}_{yz}\cdot \bar{{\mathbf {u}}}^{(m-1)}_{yz}+{\mathbf {H}}^{m}_{yz},\,\, in \,\, \partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{X}^{-}}, \\ \bar{{\mathbf {u}}}^{s_{2}}_{xz}= & {} - [{\mathbf {A}}^{s_{2}}_{xz}]^{-1} \cdot {\mathbf {A}}^{(m-1)}_{xz}\cdot \bar{{\mathbf {u}}}^{(m-1)}_{xz}+{\mathbf {H}}^{m}_{xz},\,\, in \,\, \partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{Y}^{-}}, \\ \bar{{\mathbf {u}}}^{s_{3}}_{xy}= & {} - [{\mathbf {A}}^{s_{3}}_{xy}]^{-1} \cdot {\mathbf {A}}^{(m-1)}_{xy}\cdot \bar{{\mathbf {u}}}^{(m-1)}_{xy}+{\mathbf {H}}^{m}_{yz},\,\, in \,\, \partial \varOmega ^{h}_{\mu |{\mathbf {N}}_{Z}^{-}}. \end{aligned}$$

(97)