Self-consistent Clustering Analysis-Based Moving Morphable Component (SMMC) Method for Multiscale Topology Optimization

Abstract

Current multiscale topology optimization restricts the solution space by enforcing the use of a few repetitive microstructures that are predetermined, and thus lack the ability for structural concerns like buckling strength, robustness, and multi-functionality. Therefore, in this paper, a new multiscale concurrent topology optimization design, referred to as the self-consistent analysis-based moving morphable component (SMMC) method, is proposed. Compared with the conventional moving morphable component method, the proposed method seeks to optimize both material and structure simultaneously by explicitly designing both macrostructure and representative volume element (RVE)-level microstructures. Numerical examples with transducer design requirements are provided to demonstrate the superiority of the SMMC method in comparison to traditional methods. The proposed method has broad impact in areas of integrated industrial manufacturing design: to solve for the optimized macro and microstructures under the objective function and constraints, to calculate the structural response efficiently using a reduced-order model: self-consistent analysis, and to link the SMMC method to manufacturing (industrial manufacturing or additive manufacturing) based on the design requirements and application areas.

Appendices

Appendix A: Self-consistent Clustering Analysis (SCA)

In order to predict the mechanical behaviors of microstructures, finite element and fast Fourier transform (FFT) methods are often used to analyze the mechanical response of representative volume elements (RVE). However, the huge computational cost associated with solving the microscale model at each integration point limits the application of these methods. To address this issue, Liu et al. proposed the self-consistent clustering analysis (SCA) based on a reduced-order RVE model [28]. Compared to the original DNS, SCA offers a great improvement in computational efficiency by significantly reducing the degrees of freedom through compressing the microstructural data into clusters in the offline stage. Furthermore, material nonlinear constitutive laws (such as plasticity and damage) can be obtained with the reduced-order model in the online stage. The basic idea behind SCA is as follows.

Offline Stage

In offline stage, the relationship between the elastic microscopic strain ${{\varvec{\varepsilon}}}^{\text{micro}}({\varvec{x}})$ and the homogeneous elastic macroscopic strain ${{\varvec{\varepsilon}}}^{\text{macro}}$ at each material point can be obtained using the strain concentration tensor A(x), that is

$${{\varvec{\varepsilon}}}^{\text{micro}}({\varvec{x}})={\varvec{A}}({\varvec{x}}):{{\varvec{\varepsilon}}}^{\text{macro}}$$

(A1)

where ${{\varvec{\varepsilon}}}^{\text{macro}}$ is the elastic macroscopic strain on the RVE with the boundary conditions, ${{\varvec{\varepsilon}}}^{\text{micro}}({\varvec{x}})$ is the elastic local strain at point x in the microscale domain Ω. Based on the strain concentration tensor A_m(x), the voxels are grouped into a few clusters using k-means clustering.

After clustering, the interaction tensors ${{\varvec{D}}}^{IJ}$, representing the influence of stress in the jth cluster on the strain in the ith cluster, can be computed. They can be written as an integral of Green’s function in an RVE domain Ω with periodic boundary conditions:

$${{\varvec{D}}}^{IJ}=\frac{1}{{c}^{I}|\Omega |}{\int }_{\Omega }{\int }_{\Omega }{\chi }^{I}({\varvec{x}}){\chi }^{J}\left({{\varvec{x}}}^{\boldsymbol{^{\prime}}}\right){\boldsymbol{\Phi }}^{0}\left({\varvec{x}},{{\varvec{x}}}^{\boldsymbol{^{\prime}}}\right)\text{d}{{\varvec{x}}}^{\boldsymbol{^{\prime}}}\text{d}{\varvec{x}}$$

(A2)

where ${\boldsymbol{\Phi }}^{0}\left({\varvec{x}},{{\varvec{x}}}{\prime}\right)$ is the fourth-order periodic Green’s function associated with an isotropic linear elastic reference material with stiffness tensor C⁰. $c^{{\varvec{I}}}$ is the volume fraction of the ith cluster, and $|\Omega |$ is the volume of the RVE domain. ${\chi }^{I}$=1 if x is in the ith cluster. Moreover, ${\boldsymbol{\Phi }}^{0}\left({\varvec{x}},{{\varvec{x}}}{\prime}\right)$ has a simple form in the Fourier space as

$${\widehat{{\varvec{\phi}}}}^{0}\left({\varvec{\xi}}\right)=\frac{1}{4{\mu }^{0}}{\widehat{{\varvec{\phi}}}}^{1}\left({\varvec{\xi}}\right)+\frac{{\lambda }^{0}+{\mu }^{0}}{{\mu }^{0}({\lambda }^{0}+2{\mu }^{0})}{\widehat{{\varvec{\phi}}}}^{2}\left({\varvec{\xi}}\right)$$

(A3)

with

$${\widehat{\phi }}_{ijkl}^{1}\left({\varvec{\xi}}\right)=\frac{1}{{\left|{\varvec{\xi}}\right|}^{2}}({\delta }_{ik}{\xi }_{j}{\xi }_{l}+{\delta }_{il}{\xi }_{j}{\xi }_{k}+{\delta }_{jl}{\xi }_{i}{\xi }_{k}+{\delta }_{jk}{\xi }_{i}{\xi }_{l})$$

(A4)

$${\widehat{\phi }}_{ijkl}^{2}\left({\varvec{\xi}}\right)=-\frac{{\xi }_{i}{\xi }_{j}{\xi }_{k}{\xi }_{l}}{|\varvec{\xi} {|}^{4}}$$

(A5)

where ${\lambda }^{0}$ and ${\mu }^{0}$ are Lamé constants of the reference material, ${\varvec{\xi}}$ is the Fourier space corresponding to x in real space, and ${\delta }_{ij}$ is the Kronecker delta function.

Online Stage

The equilibrium condition in the RVE can be rewritten as a continuous Lippmann–Schwinger integral equation by introducing a homogeneous isotropic linear elastic reference material [28] as follows

$$\begin{aligned}\begin{aligned}\Delta {\varvec{\varepsilon}}_{m}({\varvec{x}})+&\int {\boldsymbol{\Phi}}^{0}\left({\varvec{x}},{\varvec{x}}{\prime}\right):\left[\Delta \sigma_{m}\left({\varvec{x}}{\prime}\right)-{\varvec{C}}^{0}\right.\\&\left.:\Delta {\varvec{\varepsilon}}_{m}\left({\varvec{x}}{\prime}\right)\right]{\textrm{d}}{\varvec{x}}{\prime}-\Delta {\varvec{\varepsilon}}^{0}=\varvec{0}\end{aligned}\end{aligned}$$

(A6)

where $\Delta {{\varvec{\varepsilon}}}^{0}$ is the far-field strain increment controlling the evolution of the local strain. The far-field strain is uniform in the RVE. The stiffness tensor ${{\varvec{C}}}^{0}$ of the isotropic linear elastic reference material can be described using Lamé parameters ${\lambda }^{0}$ and ${\mu }^{0}$. By averaging the incremental integral equation and considering the effect of macro-strain constraints or macro-stress constraints, Eq. (A6) can be rewritten. For more details, please refer to [28]. Following the clustering idea in the offline stage, the discretized integral equation of the ith cluster can be derived as

$$\begin{aligned}&\Delta \varvec{\varepsilon }_{m}^{I}+\sum_{I=1}^{k}{{\varvec{D}}}^{IJ}:\left[\Delta \varvec{\sigma }_{m}^{J}-{{\varvec{C}}}^{0}:\Delta \varvec{\varepsilon }_{m}^{J}\right]\\ & \quad -\Delta \varvec{\varepsilon }^{0}=\mathbf{0} \end{aligned}$$

(A7)

where $\Delta {{\varvec{\varepsilon}}}_{m}^{I}$ and $\Delta {{\varvec{\sigma}}}_{m}^{J}$ are the microscopic strain and stress increments in the jth cluster, respectively. ${{\varvec{D}}}^{IJ}$ is the interaction tensor computed from the offline stage. After discretization, the far-field strain increment, which is equal to the average strain in the RVE, can be expressed as

$$ \Delta \varvec{\varepsilon} ^{0} = \sum\limits_{{I = 1}}^{k} {c^{I} \Delta \varvec{\varepsilon} _{m}^{I} } $$

(A8)

For the macro-strain constraint, the discrete form can be written as

$$ \mathop \sum \limits_{{I = 1}}^{k} c^{I} \Delta {\varvec{\varepsilon }}_{m}^{I} = \Delta {\varvec{\varepsilon }}_{M} \,{\text{or}}\,\Delta {\varvec{\varepsilon} }^{0} = \Delta {\varvec{\varepsilon }}_{M} $$

(A9)

where $\Delta {{\varvec{\varepsilon}}}_{M}$ is the macroscopic strain increment. Similarly, the discrete form of macro-stress constraint is

$$ \sum\limits_{{I = 1}}^{k} {c^{I} \Delta \varvec{\sigma} _{m}^{I} } = \Delta \varvec{\sigma} _{M} $$

(A10)

where $\Delta {{\varvec{\sigma}}}_{M}$ is the macroscopic strain increment. For the continuous Lippmann–Schwinger equation in Eq. (A6), the stiffness tensor of the reference material ${{\varvec{C}}}^{0}$ does not influence the solution. However, once the equation is transformed into a discrete form, the equilibrium condition is not strictly satisfied at each point in the RVE, and the solution depends on ${{\varvec{C}}}^{0}$. Therefore, the effect of ${{\varvec{C}}}^{0}$ on the reduced-order system needs to be considered.

To build a ${{\varvec{C}}}^{0}$-based iteration, a self-consistent scheme is used in the online stage [28]. The stiffness tensor of the reference material ${{\varvec{C}}}^{0}$ is approximately the same as the macroscale tangent stiffness tensor $\bar{{\varvec{C}}}$.

Appendix B: Moving Morphable Component (MMC) Method

The framework of SMMC is based on the idea of moving morphable component (MMC) method proposed by Zhang et al. [23] This section gives a short review of the MMC method.

In the MMC method, the morphable structural components, which can move, deform, and overlap freely, are used as the primary building blocks for topology optimization. The geometries/topologies of structural components are described using a topology description function (TDF) with a set of explicit parameters. In the MMC framework, topology optimization is achieved by optimizing the shapes, lengths, widths, orientations, and layout of the components. For the ith component in the structure, the TDF can be expressed as [22]

$$\chi^{i} \left( \varvec{x} \right) = 1 - \left[ {\left( {\frac{{\left( {x - x_{{i0}} } \right)\cos \theta _{i} + \left( {y - y_{{i0}} } \right)\sin \theta _{i} }}{{L_{i} }}} \right)^{m} + \left( {\frac{{ - \left( {x - x_{{i0}} } \right)\sin \theta _{i} + \left( {y - y_{{i0}} } \right)\cos \theta _{i} }}{{t_{i} }}} \right)^{m} } \right] $$

(B1)

where $\left({x}_{i0},{y}_{i0}\right)$ represents the coordinates of the center of the component, ${L}_{i}$, ${\theta }_{i}$, and ${t}_{i}$ denote the half-length, inclined angle, and variable thickness of the component, respectively, as shown in Fig. 2. $m$ is a large even integer number (we take p = 6). With the use of ${\chi }^{i}({\varvec{x}})$, the region ${\Omega }^{i}$ occupied by the ith component can be expressed as

$$\left\{\begin{array}{l}{\chi }^{i}\left(x\right)>0,\mathrm{ if} x\in {\Omega }^{i}\\ {\chi }^{i}\left(x\right)=0, \mathrm{if} x\in \partial {\Omega }^{i}\\ {\chi }^{i}\left(x\right)<0, \mathrm{if} x\in D\backslash \left({\Omega }^{i}\cup \partial {\Omega }^{i}\right)\end{array}\right.$$

(B2)

Then, the corresponding structural topology can be described as

$$\left\{\begin{array}{l}{\chi }^{s}\left(x\right)>0, \mathrm{if} x\in {\Omega }^{i}\\ {\chi }^{s}\left(x\right)=0, \mathrm{if} x\in \partial {\Omega }^{i}\\ {\chi }^{s}\left(x\right)<0, \mathrm{if} x\in D\backslash \left({\Omega }^{i}\cup \partial {\Omega }^{i}\right)\end{array}\right.$$

(B3)

where ${\Omega }^{s}={\cup }_{i=1}^{n}{\Omega }^{i}$ is the region occupied by the solid structural components, and ${\chi }^{s}$ is the TDF of the whole structure, which can be formulated as [8]

$${\chi }^{s}({\varvec{x}})={\max}({\chi }^{1},{\chi }^{2},...,{\chi }^{n})$$

(B4)

where n is the total number of components in the structure.

Based on the MMC topology optimization method, the topology optimization problem can be formulated as [22, 23]:

$$ \begin{aligned} & {\text{Find}}\,\varvec{d} = \left( {\left( {\varvec{d}^{1} } \right)^{{ \top }} , \ldots ,\left( {\varvec{d}^{\varvec{i}} } \right)^{{ \top }} , \ldots ,\left( {\varvec{d}^{\varvec{n}} } \right)^{{ \top }} } \right)^{{ \top }} ,\varvec{b} = \left( {\left( {\varvec{b}^{1} } \right)^{{ \top }} , \ldots ,\left( {\varvec{b}^{\varvec{i}} } \right)^{{ \top }} , \ldots ,\left( {\varvec{b}^{\varvec{n}} } \right)^{{ \top }} } \right)^{{ \top }} ,\varvec{~u}\left( {\varvec{x}} \right) \\ & {\text{Minimize}}\,C = \mathop \sum \limits_{{i = 1}}^{n} \mathop \int \limits_{{\Omega ^{i} \backslash \;\left( { \cup _{{1\; \le j < i}} \;\left( {\;\Omega ^{i} \cap \Omega ^{j} } \right)} \right)\;}} \varvec{f}^{\varvec{i}} \cdot \varvec{u}{\text{d}}V + \mathop \int \limits_{{\Gamma _{{\mathfrak{t}}} }} \varvec{t} \cdot \varvec{u}{\text{d}}S \\ & {\text{s}}{\text{.t}}{\text{.}} \\ & g_{j} \left( \varvec{d} \right) \le 0,j = 1,...,m \\ & \varvec{d} \subset {\mathcal{U}}_{d} \\ \end{aligned}$$

(B5)

where ${\varvec{d}}_{i}$ = (${x}_{i0},{y}_{i0}$,${L}_{i}$, ${t}_{1i}$, ${t}_{2i}$, ${t}_{3i}$,${\theta }_{i}$) ($i=1,\dots ,n$) denotes the vector of design variables for the ith component. ${g}_{j}$($j=1,\dots ,m$) is the constraint function/functional, and ${\mathcal{U}}_{d}$ represents the admissible set of d.

For the compliance optimization problem, Eq. (B5) can be rewritten as:

$$ \begin{gathered} {\text{Find}}\,\varvec{d} = \left( {\left( {\varvec{d}^{1} } \right)^{ \top } , \ldots ,\left( {\varvec{d}^{i} } \right)^{ \top } , \ldots ,\left( {\varvec{d}^{n} } \right)^{ \top } } \right)^{ \top } ,\varvec{u}\left( {\varvec{x}} \right) \hfill \\ {\text{Minimize}}\,C = \mathop \sum \limits_{{i = 1}}^{n} \mathop \int \limits_{{\Omega ^{i} \backslash \;\left( { \cup _{{1\; \le j < i}} \;\left( {\;\Omega ^{i} \cap \Omega ^{j} } \right)} \right)\;}} \varvec{f}^{i} \cdot \varvec{u}{\text{d}}V + \mathop \int \limits_{{\Gamma _{{\mathfrak{t}}} }} \varvec{t} \cdot \varvec{u}{\text{d}}S \hfill \\ {\text{s}}.{\text{t}}. \hfill \\ \mathop \sum \limits_{{i = 1}}^{n} \mathop \int \limits_{{\Omega ^{i} \backslash \;\left( { \cup _{{1\; \le j < i}} \;\left( {\;\Omega ^{i} \cap \Omega ^{j} } \right)} \right)}} \varvec{E}^{\varvec{i}} :\varvec{\varepsilon }\left( \varvec{u} \right):\varvec{\varepsilon }\left( \varvec{v} \right){\text{d}}V \hfill \\ \quad = \mathop \sum \limits_{{\varvec{i} = 1}}^{\varvec{n}} \mathop \int \limits_{{\Omega ^{i} \backslash \;\left( { \cup _{{1\; \le j < i}} \;\left( {\;\Omega ^{i} \cap \Omega ^{j} } \right)} \right)}} \varvec{f} \cdot \varvec{v}{\text{d}}V+\mathop \int \limits_{{\Gamma _{t} }} \varvec{t} \cdot \varvec{v}{\text{d}}S,\varvec{~}\forall \varvec{v} \in {\mathbf{\mathcal{U}}}_{{{\text{ad}}}} \hfill \\ \quad V\left( \varvec{d} \right) \le \bar{V} \hfill \\ \quad \varvec{d} \subset {\mathcal{U}}_{d} \hfill \\ \quad \varvec{u} = {\bar{\varvec{u}}},\,{\text{on}}\,\Gamma _{u} \hfill \\ \end{gathered} $$

(B6)

In Eq. (B6), $\bar{{{\varvec{u}}}}$ is the prescribed displacement on the Dirichlet boundary ${\Gamma }_{u}$. The symbol ${\varvec{\varepsilon}}$ is the second-order linear strain tensor. ${\varvec{f}}^{i}$ is the body force density in ${\Omega }^{i}$, while t is the surface traction on the Neumann boundary ${\Gamma }_{t}$. ${\varvec{u}}$ and ${\varvec{v}}$ are the displacement field and the corresponding test function defined on $\Omega ={\cup }_{i=1}^{n}{\Omega }^{i}$. The fourth-order isotropic elasticity tensor of the ith component’s material is expressed as ${{\varvec{E}}}^{{\varvec{i}}}={E}^{i}/(1+{\nu }_{i})\left[{\varvec{I}}+{\nu }^{i}/\left(1-2{\nu }^{i}\right){\varvec{\delta}}\otimes{\varvec{\delta}}\right]$ (${\varvec{I}}$ and ${\varvec{\delta}}$ are the fourth- and second-order identity tensors, respectively). The upper bound of the available volume of solid material is defined as$\bar{V}$. ${{\varvec{E}}}^{i}$ is Young’s modulus, and ${\nu }^{i}$ is Poisson’s ratio. When ${\varvec{x}}\in {\Omega }^{i}\backslash \hspace{0.33em}({\cup }_{1\hspace{0.33em}\le j<i}\hspace{0.33em}(\hspace{0.33em}{\Omega }^{i}\cap {\Omega }^{j}))$, the assumption is made that ${\varvec{E}}({\varvec{x}})={{\varvec{E}}}^{{\varvec{i}}}$ and${\varvec{f}}={{\varvec{f}}}^{{\varvec{i}}}$. Besides, it is assumed that ${{\varvec{E}}}^{{\varvec{i}}}=\cdots ={{\varvec{E}}}^{{\varvec{n}}}={\varvec{E}}$ and ${\nu }^{i}=\cdots ={\nu }^{n}=\nu$. To simplify the problem, $\bar{{{\varvec{u}}}}=0$ is set in the following discussion.

If only single-phase material is taken into account, Eq. (B6) can be transferred into the following equations.

$$ \begin{aligned} & {\text{Find}}\,{\varvec{d}} = \left( {\left( {{\varvec{d}}^{1} } \right)^{{ \top }} , \ldots ,\left( {{\varvec{d}}^{{\varvec{i}}} } \right)^{{ \top }} , \ldots ,\left( {{\varvec{d}}^{{\varvec{n}}} } \right)^{{ \top }} } \right)^{{ \top }} ,\\ & {\varvec{b}} = \left( {\left( {{\varvec{b}}^{1} } \right)^{{ \top }} , \ldots ,\left( {{\varvec{b}}^{{\varvec{i}}} } \right)^{{ \top }} , \ldots ,\left( {{\varvec{b}}^{{\varvec{n}}} } \right)^{{ \top }} } \right)^{{ \top }} ,\varvec{ u}\left( {{\varvec{x}}} \right) \\ & {\text{Minimize}}\,C = \mathop \int \limits_{{\Omega }} H\left( {\phi^{s} \left( {{\varvec{x}};{\varvec{d}}} \right)} \right){\varvec{f}} \cdot {\varvec{u}}{\text{d}V} + \mathop \int \limits_{{\Gamma_{{{t}}} }} {\varvec{t}} \cdot {\varvec{u}}{\text{d}S} \\ & {\text{s}}.{\text{t}}. \\ & \mathop \int \limits_{\Omega } \left( {H\left( {\phi^{s} \left( {{\varvec{x}};{\varvec{d}}} \right)} \right)} \right)^{q} {\varvec{E}}:{\varvec{\varepsilon}}\left( {\varvec{u}} \right):{\varvec{\varepsilon}}\left( {\varvec{v}} \right){\text{d}V} \\ & \quad = \mathop \int \limits_{\Omega } H\left( {\phi^{s} \left( {{\varvec{x}};{\varvec{d}}} \right)} \right){\varvec{f}} \cdot {\varvec{v}}{\text{d}{V + }}\mathop \int \limits_{{\Gamma_{t} }} {\varvec{t}} \cdot {\varvec{v}}{\text{d}S},\varvec{ }\forall {\varvec{v}} \in {{\mathcal{\bf U}}}_{{{\text{ad}}}} \\ & \varvec{ }\mathop \int \limits_{\Omega } H\left( {\phi^{s} \left( {{\varvec{x}};{\varvec{d}}} \right)} \right){\text{d}V} \le \bar{V} \\ & {\varvec{d}} \subset {\mathcal{U}}_{d} \\ & {\varvec{u}} = \bar{\varvec{u}},\,{\text{on}}\,\Gamma_{u} \\ \end{aligned} $$

(B7)

where $H=H(x)$ is the Heaviside function, and we take q as an integer lager than 1.