Topology optimization of micro-structure for composites applying a decoupling multi-scale analysis

Abstract

The present study proposes topology optimization of a micro-structure for composites considering the ma-cro-scopic structural response, applying a decoupling multi-scale analysis based on a homogenization approach. In this study, it is assumed that topology of macro-structure is unchanged and that topology of micro-structure is unique over the macro-structure. The stiffness of the macro-structure is maximized with a prescribed material volume of constituents under linear elastic regime. A gradient-based optimization strategy is applied and an analytical sensitivity approach based on numerical material tests is introduced. It was verified from a series of numerical examples that the proposed method has great potential for advanced material design.

Appendix:Finite element analysis for a unit cell with external material points

For preparation of the finite element analysis (FEA) for the micro-scale BVP, the spatial domain of the unit cell is discretized to generate its FE mesh. The principle of virtual work for micro-structure is formulated considering the first equation in (10) and anti-periodicity condition (8) with some mathematical rearrangement as follows:

$$ \displaystyle{ \int\limits_{Y} \delta \boldsymbol{\varepsilon}^{*} :\boldsymbol{\sigma} \, \mathrm{d}y \,=\, \int\limits_{Y} \nabla_{y}^{\mathrm sym} \delta \boldsymbol{u}^{*} :\boldsymbol{\sigma} \, \mathrm{d}y \,=\, {0}}, $$

(33)

where δ u∗ and δ ε∗ denote the virtual fluctuation displacement field and its strain field satisfying the periodic condition, respectively. The virtual work expression (33) is discretized in the finite element sense assuming the following approximation:

$$ \boldsymbol{w} = {\displaystyle\sum\limits_{\alpha =1}^{n_{\text{node}}}} \, {N}_{\alpha} \hat{w}_{\alpha}^{\mathrm{e}} \quad \text{or} \quad \boldsymbol{w} \,=\, \boldsymbol{N} \hat{\boldsymbol{w}}^{\mathrm{e}}, $$

(34)

$$\delta\boldsymbol{w} = {\displaystyle\sum\limits_{\alpha =1}^{n_{\text{node}}}} \, {N}_{\alpha} \delta \hat{w}_{\alpha}^{\mathrm e} \quad \text{or} \quad \delta\boldsymbol{w} \,=\, \boldsymbol{N} \delta \hat{\boldsymbol{w}}^{\mathrm e}, $$

(35)

$$ \boldsymbol{\varepsilon} = {\displaystyle\sum\limits_{\alpha =1}^{n_{\text{node}}}} \, {B}_{\alpha} \hat{w}_{\alpha}^{\mathrm e} \quad \text{or} \quad \boldsymbol{\varepsilon} \,=\, \boldsymbol{B} \hat{\boldsymbol{w}}^{\mathrm e},$$

(36)

$$ \delta\boldsymbol{\varepsilon} = {\displaystyle\sum\limits_{\alpha =1}^{n_{\text{node}}}} \, {B}_{\alpha} \delta \hat{w}_{\alpha}^{\mathrm e} \quad \text{or} \quad \delta\boldsymbol{\varepsilon} \,=\,\boldsymbol{B} \hat{\boldsymbol{w}}^{\mathrm e}, $$

(37)

where N is the general shape function and B the B-operator, respectively. $\hat {\boldsymbol {w}}^{\mathrm e}$ indicates the nodal micro-displacement vector of an element in a unit cell. Furthermore, we similarly discretize δ u∗ and δ ε∗ as $\delta \boldsymbol {u}^{*} =\boldsymbol {N} \left ( \delta \hat {\boldsymbol {d}}^{*}{}^{\mathrm e} \right )$ and $\delta \boldsymbol {\varepsilon }^{*} =\boldsymbol {B} \left ( \delta \hat {\boldsymbol {d}}^{*}{}^{\mathrm e} \right )$. The discretized formulation of (33) can be written by inserting these equations as follows:

$$ \displaystyle{ \sum\limits_{\mathrm e =1}^{n_{\mathrm{ele}}} \Biggl\{ \left( \delta \hat{\boldsymbol{d}}^{*}{}^{\mathrm e} \right)^{\mathrm{T}} \int\limits_{\mathrm{Y_{e}}} \boldsymbol{B}^{\mathrm{T}} \mathbb{C} \boldsymbol{B} \mathrm{d}y \left( \hat{\boldsymbol{w}}^{\mathrm e} \right)} \Biggr\} = 0. $$

(38)

As the virtual fluctuation displacement $\delta \hat {\boldsymbol {d}}^{*}{}^{\mathrm e}$ is arbitrary, the discretized formulation of Eq. (33) is expressed by assembling Eq. (38) over the unit cell as:

$$\begin{array}{@{}rcl@{}} \boldsymbol{K}^{\mathrm m} \hat{\boldsymbol{w}} = \boldsymbol{0} \qquad \text{with} \qquad \boldsymbol{K}^{\mathrm m} = \sum\limits_{\mathrm e =1}^{n_{\mathrm{ele}}} \int\limits_{\mathrm{Y_{e}}} \boldsymbol{B}^{\mathrm{T}} \mathbb{C} \boldsymbol{B} \mathrm{d}y, \qquad \end{array} $$

(39)

where K ^m is the global stiffness matrix of a unit cell and $\hat {\boldsymbol {w}}$ is the global nodal micro-displacement vector.

At this stage the boundary conditions (12) and (13) have not been included in (39) yet. In order to establish the extended micro-scale BVP aforementioned, the relative displacement q is embedded to (39) as constraints by replacing pairs of DOFs in $\hat {\boldsymbol {w}}$. For this purpose, each external material point is also ‘discretized’ to an element with a single node which has two DOFs and no mass. Since the external material points enable us to control the components of the macro-scale stress and deformation, as explained above, the node corresponding to an external material point is referred to as a control node in this study. Thus, we obtain an extended system of FE-discretized equations involving four additional DOFs of two control nodes. In the following, we introduce some specific usages of the two control nodes to solve the extended system.

First, the macro-scale strain is assumed to be known; that is, all the components of the macro-strain E are given as data. Using (13), we obtain all the components of the nodal relative displacement vector q ^[k] at the two control nodes. Then, given all the components $q_{i}^{[k]}$, we solve the extended system of FE equations with the appropriate number of ‘two-point’ constraints realized by (12).

This procedure starts from applying a transformation matrix Φ such as,

$$ \tilde{\boldsymbol{w}} = \boldsymbol{\Phi} \hat{\boldsymbol{w}}, $$

(40)

where Φ is an operator which transforms $\hat {\boldsymbol {w}}$ to $\tilde {\boldsymbol {w}}$ and $\tilde {\boldsymbol {w}}$ is the nodal displacement vector in which some components are replaced by the components of q ^[k] . For example, concentrating only on q ^[1] , we impose the corresponding two points on the boundaries ∂ Y ^{[− 1]} and ∂ Y ^[1] whose DOFs are defined as (w _a , w _b ) and (w _c , w _d ), respectively, to be periodically constrained. Then, we can establish $\tilde {\boldsymbol {w}}$ from $\hat {\boldsymbol {w}}$ as shown in Fig. 3, namely

$$ \hat{\boldsymbol{w}}=\{w_{1} \,\,\,\, w_{2},\ldots, w_{a}\,\,\,\, w_{b},\ldots, w_{c}\,\,\,\, w_{d},\ldots, w_{N}\} $$

(41)

and

$$ \tilde{\boldsymbol{w}}=\{w_{1} \,\,\,\, w_{2},\ldots, w_{a}\,\,\,\, w_{b},\ldots, q^{[1]}_{1}\,\,\,\, q^{[1]}_{2},\ldots, w_{N}\} $$

(42)

$$ \text{with} \qquad w_{c} -w_{a}=q^{[1]}_{1} \quad\text{and}\quad w_{d} -w_{b}=q^{[1]}_{2}. $$

(43)

where N is the total number of DOFs in the unit cell without the number of DOFs of the external material points. Of course, as the above example is simply for one set of the constrained points on the ∂ Y ^[−1] and ∂ Y ^[1] , all other constrained points have to be considered in $\tilde {\boldsymbol {w}}$ at the same time. Note that one set of q ^[k] (k = 1or 2) can control all constrained points staying on ∂ Y ^[−k] and ∂ Y ^[k] , see again Fig. 3.

Then, inserting (40) into (39) and pre-multiplying the operator Φ by both sides of (39) yields,

$$\tilde{\boldsymbol{K}}^{\mathrm m} \tilde{\boldsymbol{w}} = \boldsymbol{0} \quad \text{with} \quad \tilde{\boldsymbol{K}}^{\mathrm m} = \boldsymbol{\Phi} \boldsymbol{K}^{\mathrm m} \boldsymbol{\Phi}^{\mathrm T}. $$

(44)

This linear equations can be condensed depending on the distribution of components of q ^[k] in $\tilde {\boldsymbol {w}}$. After solving this linear equation, unknown components in $\tilde {\boldsymbol {w}}$ are determined. The results of the FEA contain not only the micro-scale displacement $\tilde {\boldsymbol {w}}$, strain ε and stress σ, but also the reaction force R ^[k] as aforementioned. This means that R ^[k] can be obtained by (15) since the traction force vector t ^[k] on ∂ Y ^[k] is obtained by the Cauchy law t ^[k] = σ e _k . Therefore, the macro-stress Σ _jk can be computed from (16), without performing a numerical integration on (1). Finally, as shown in (19), the macro-material stiffness $\mathbb {C}^{\mathrm {H}}$ can be obtained by computing the macro-stress Σ _jk separately three times according to the corresponding relative displacements.