Connected morphable components-based multiscale topology optimization

Abstract

The advances of manufacturing techniques, such as additive manufacturing, have provided unprecedented opportunities for producing multiscale structures with intricate latticed/cellular material microstructures to meet the increasing demands for parts with customized functionalities. However, there are still difficulties for the state-of-the-art multiscale topology optimization (TO) methods to achieve manufacturable multiscale designs with cellular materials, partially due to the disconnectivity issue when tiling material microstructures. This paper attempts to address the disconnectivity issue by extending component-based TO methodology to multiscale structural design. An effective linkage scheme to guarantee smooth transitions between neighboring material microstructures (unit cells) is devised and investigated. Associated with the advantages of components-based TO, the number of design variables is greatly reduced in multiscale TO design. Homogenization is employed to calculate the effective material properties of the porous materials and to correlate the macro/structural scale with the micro/material scale. Sensitivities of the objective function with respect to the geometrical parameters of each component in each material microstructure have been derived using the adjoint method. Numerical examples demonstrate that multiscale structures with well-connected material microstructures or graded/layered material microstructures are realized.

Key words： multiscale topology optimization; morphable component; material microstructure; homogenization

Introduction

Additive manufacturing, also known as 3D printing [¹,²], involves the progressive addition of materials layer by layer to fabricate components which can accommodate significantly more convoluted geometries than traditional manufacturing techniques. For example, additive manufacturing provides the ability to create parts with extremely sophisticated internal cellular/lattice structures that will not be possible using traditional manufacturing techniques [³–⁵]. Such latticed parts are considered as multiscale structures that comprise microstructure at a lower geometrical scale. The microstructure are usually cellular/lattice structures involving repetitive patterns of a particular cell shape or type, and possess specific properties such as additional weight reduction for raced cars and airplanes, increased surface area as a means of thermal insulation, and desired porosity/permeability for biomedical applications in bone scaffolds and implants [⁵–⁷]. It should be noted that lattice structures are not optimal for strictly stiffness optimization [⁸] but might be necessary for other design requirements like the diffusion of fluid media through the structural component or save printing time and material cost.

Due to the aforementioned advantages of cellular/lattice structures and the fabrication feasibility offered by additive manufacturing, there is an incentive to incorporate lattice structures into structural designs, which offers more design freedom. Advances in 3D printing and topology optimization (TO) techniques [⁹] allow engineers to come up with freeform shapes and complex lattice structures challenging conventional design limitations [⁴,⁷,¹⁰–¹³].

Incorporating lattice structures into structural TO design yields some research challenges associated with multiscale TO. The pioneering works for multiscale TO include homogenization-based TO [¹⁴,¹⁵] and free material optimization [¹⁶,¹⁷], both achieve optimal macrostructures by optimizing the material microstructure parameters. Recently, homogenization-based TO has been revisited by Ref. [¹⁸] to achieve high-resolution structures. Among the multiscale TO approaches, the hierarchical TO method [¹⁹] further optimizes the topologies of the material microstructures to obtain optimal macrostructure. Concurrent TO method [²⁰–²⁴] assumes that a multiscale structure is composed of homogeneous material, and then optimizes the macro and micro structures simultaneously. Other forms of multiscale TO methods exist including extensions to nonlinear problems [²⁵–²⁷].

In the aforementioned multiscale TO methods, mathematical homogenization-based on two-scale expansion is used to correlate the material microstructure scale with the macrostructure scale. Due to the scale separation assumption in mathematical homogenization, each unit cell is designed independently. This will result in the disconnectivity issue when the material microstructures are assembled for building up the macrostructures. In the concurrent TO method by Ref. [²³], the microstructure configuration of every point is assumed to be the same at the macro scale. This representation would simplify the analysis, optimization and manufacturing process. However, the assumption of only one type of porous homogeneous material limits the design space. The method of two-scale TO [²⁵] imposes constraints to ensure smooth transition of neighboring unit cells, however, the results of microstructure topologies look similar to each other.

To overcome the disconnectivity issue between neighboring cellular materials, researchers have designed manufacturable multiscale structures with one rectangular [¹⁰] or honeycomb [²⁸] cellular material throughout the component. These design schemes only allow one type of cellular material in the design domain, and the configuration/topology of the cellular material has to be predefined, which significantly restricts the design variables. Design methods to achieve multiscale structures with graded material microstructures have been proposed in Refs. [²⁹,³⁰]. In addition, there are methods to achieve multiscale structures with coating [³¹–³³], multiscale structures with graded material microstructures [²⁹,³⁰,³⁴,³⁵], and high-resolution multiscale structures [¹⁸].

Inspired by the moving morphable components-based TO framework [³⁶–³⁸], we propose in this work a new connected moving morphable component (CMC)-based multiscale TO design method with a linkage scheme to achieve well connected neighboring material microstructures. This linkage scheme has been employed to prevent design degeneration in flexible structural design [³⁹], and some preliminarily results have been obtained for multiscale TO [⁴⁰]. In the current paper, the effectiveness of the new linkage scheme in achieving connected neighboring material microstructures will be demonstrated using multiscale structure compliance minimization TO problems. In the proposed CMC-based multiscale design TO method, each unit cell is parameterized by several structural beam members and homogenization theory is used to compute the effective property of the unit cell. Sensitivities of the objective function with respect to the effective properties and sensitivity of the effective properties with respect to geometrical parameters of each structural beam member will be derived using the adjoint method. Another advantage of the proposed CMC-based multiscale TO design method is that it significantly reduces the number of design variables compared to an elemental density-based TO approach. This is because in CMC-based TO, the design variables are the geometrical parameters of each morphable structural member. By reducing the number of design variables, it transforms the topological design of multiscale structures into a manageable optimization problem.

Connected moving morphable component

The moving morphable components-based TO framework [³⁶–³⁸] has gained more popularity in recent years. For achieving well-connected pre-defined components in structural design, a linkage scheme working together with a particular moving morphable component, i.e., CMC, has been proposed in our earlier work [³⁹]. In this section, analytical expressions for the level set function and its derivatives of this particular CMC will be introduced. Additional details can also be found in Ref. [³⁷].

The schematic picture for the ith CMC in global coordinate \mathit{x}=(x,y) is shown in Fig. 1. The two ends of each CMC are circular, with a left end center point {\mathit{x}}_{i\mathrm{1}}=(x_{i\mathrm{1}},y_{i\mathrm{1}}) and a right end center point {\mathit{x}}_{i\mathrm{2}}=(x_{i\mathrm{2}},y_{i\mathrm{2}}). The thickness of the component is t_i. The geometrical parameters of the CMC, i.e., {\mathit{x}}_{i\mathrm{1}}, {\mathit{x}}_{i\mathrm{2}}, and t_i are treated as design variables.

Fig.1 A CMC in global coordinate

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The advantage of the CMC is that its level set function can be written explicitly as the signed distance function. For example, the level set function {\varphi }_i\left(\mathit{x}\right) of the ith CMC in Fig. 1 can be derived analytically as

(1)

with

(2)\{\begin{array}{l}\mathit{a}={\mathit{x}}_{i\mathrm{2}}-{\mathit{x}}_{i\mathrm{1}}\hfill \\ \mathit{b}=\mathit{x}-{\mathit{x}}_{i\mathrm{1}}\hfill \\ \mathit{e}=\mathit{x}-{\mathit{x}}_{i\mathrm{2}}\hfill \\ \mathit{g}=\mathit{b}-(\mathit{a}·\mathit{b})\mathit{a}/{\Vert \mathit{a}\Vert }^{\mathrm{2}}\hfill \end{array}.

The sensitivity of {\varphi }_i\left(\mathit{x}\right) with respect to design variables {\mathit{x}}_{i\mathrm{1}}, {\mathit{x}}_{i\mathrm{2}}, and t_i can be calculated as

(3)

and

(4)

and \partial {\varphi }_i\left(\mathit{x}\right)/\partial t_i=-\mathrm{0.5}.

The level set function of the CMC and its sensitivities with respect to the design variables will be used in the optimization formulation and sensitivity analysis of the proposed multiscale TO method.

CMC-based multiscale TO design method

In this section, a CMC-based multiscale structure TO method is defined. To illustrate the design scheme, a multiscale simply supported beam structure is shown in Fig. 2. For the macrostructure, three vertical loadcases are considered. The macrostructure is divided into M partitions (M=\mathrm{24}\times \mathrm{6} in Fig. 2), with each partition has a different porous material characterized by a unique microstructure/unit cell. Thereby, totally there are M porous materials. To parameterize the multiscale TO design problem, each unit cell is parameterized by N CMCs (N=\mathrm{8} in Fig. 2). The N CMCs in each unit cell can move, morph, and overlap, such that the final optimized configuration of each unit cell can be achieved. For illustrative purpose, two types of microstructures/unit cells (each with 8 CMCs) are shown in Fig. 2.

Fig.2 Design scheme for multiscale structure using CMC-based TO

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To correlate the micro and macro scales, the effective material properties of each porous material is calculated using the homogenization theory [⁴¹,⁴²]. Therefore, the optimizations of two scales are integrated into one fully coupled optimization. The unit cell I\in \{\mathrm{1},\mathrm{2},\mathrm{...},M\} composed of N CMCs in its domain Y, its homogenized effective modulus tensor E_{ijkl}^H is computed as

(5)E_{ijkl}^H={\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle {\int }_Y{E}_{ijpq}{ϵ}_{pq}({\mathit{v}}_{\mathrm{0}}^{kl}-{\mathit{\chi }}^{kl})H\left({\varphi }_I^s\right)\mathrm{d}Y},

where {\mathit{v}}_{\mathrm{0}}^{kl}=y_l{\delta }_{mk}{\mathit{e}}_m are the unit prestrains and {\chi }^{kl}\in V_Y is the solution of

{\displaystyle {\int }_Y^{}{E}_{ijpm}{ϵ}_{pm}\left({\chi }^{kl}\right){ϵ}_{ij}\left(\mathit{v}\right)H\left({\varphi }_I^s\right)\mathrm{d}Y}

(6)={\displaystyle {\int }_Y^{}{E}_{ijkl}{ϵ}_{ij}\left(\mathit{v}\right)H\left({\varphi }_I^s\right)\mathrm{d}Y},\forall \mathit{v}\in V_Y,

or equivalently,

(7)E_{ijkl}^H={\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle {\int }_Y{E}_{pqrs}{ϵ}_{pq}({\mathit{v}}_{\mathrm{0}}^{kl}-{\chi }^{kl}){ϵ}_{rs}({\mathit{v}}_{\mathrm{0}}^{ij}-{\chi }^{ij})H\left({\varphi }_I^s\right)\mathrm{d}Y}.

In Eqs. (5)–(7), \left|Y\right| is the area of Y for the two dimensional case, \delta is the Kronecker delta, ϵ is the linear strain tensor, and V_Y=\{\mathit{v}:\mathit{v}\hspace{0.17em}\mathrm{is}\hspace{0.17em}Y\mathrm{-periodic}\}. H is the Heaviside function that will be numerically approximated and {\varphi }_I^s is the aggregate level set function of the unit cell I and described as

(8){\varphi }_I^s=\max ({\varphi }_{\mathrm{1}}^I,{\varphi }_{\mathrm{2}}^I,\mathrm{...},{\varphi }_N^I),

with {\varphi }_i^I,\hspace{0.17em}i=\mathrm{1},\mathrm{2},\mathrm{...},N denoting the level set function of the ith CMC, and N denotes the number of CMCs in the unit cell I.

Optimization formulation for CMC-based multiscale TO method

With types of unit cells) on the macro scale and N representing the number of CMCs in one type of unit cell then the optimization formulation for CMC-based multiscale TO method minimizing the structural compliance is stated as the following:

(9)\begin{array}{l}\mathrm{find}\mathit{d}=({\mathit{D}}^{\mathrm{1}},{\mathit{D}}^{\mathrm{2}},\mathrm{...},{\mathit{D}}^M)\hfill \\ \min \hspace{0.17em}\hspace{0.17em}J={\displaystyle \sum _{n=\mathrm{1}}^L{\displaystyle {\int }_{\mathrm{\Omega }}{ϵ}_{ij}\left({\mathit{u}}^n\right)E_{ijkl}^H{ϵ}_{kl}\left({\mathit{u}}^n\right)\mathrm{d}V}}\hfill \\ \hfill \\ \mathrm{s}\mathrm{.t}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{\mathrm{\Omega }}{ϵ}_{ij}\left({\mathit{u}}^n\right)E_{ijkl}^H{ϵ}_{kl}\left(\mathit{v}\right)\mathrm{d}V}={\displaystyle {\int }_{\mathrm{\Gamma }}{t}_i{u}_i^n}\mathrm{d}\mathrm{\Gamma },\hfill \\ \hfill \\ {\mathit{u}}^n\in U_{\mathrm{0}},\forall \mathit{v}\in U_{\mathrm{0}},n=\mathrm{1},\mathrm{2},\mathrm{...},L\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}g={\displaystyle {\int }_{\mathrm{\Omega }}{\displaystyle {\sum }_{I=\mathrm{1}}^M{\displaystyle {\int }_{Y}H\left({\varphi }_I^s\right)\mathrm{d}Y\mathrm{d}\mathrm{\Omega }}}}-\xi \left|\mathrm{\Omega }\right|\left|Y\right|\le \mathrm{0}\hfill \\ \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\underline{\mathit{d}}\le \mathit{d}\le \overline{\mathit{d}}\hfill \end{array}

where, \mathit{d} represents the vector of the design variables with {\mathit{D}}^I=({\mathit{d}}_{\mathrm{1}}^I,{\mathit{d}}_{\mathrm{2}}^I,\mathrm{...},{\mathit{d}}_{nc}^I),\hspace{0.17em}I=\mathrm{1},\mathrm{2},\mathrm{...},M and {\mathit{d}}_i^I=({\mathit{x}}_{i\mathrm{1}}^I,{\mathit{x}}_{i\mathrm{2}}^I,t_i^I),i=\mathrm{1},\mathrm{2},\mathrm{...},N representing the geometrical parameters of the ith CMC in unit cell I. \underline{\mathit{d}} and \overline{\mathit{d}} are the lower and upper bounds of \mathit{d}, respectively. J is the compliance of the macro structure, L is the number of load cases, \mathit{u} and \mathit{v} denote the trial and virtual displacement fields, respectively. {ϵ}_{ij}\left(\mathit{u}\right)=\left(u_{i,j}+u_{j,i}\right)/\mathrm{2} is the strain tensor. The effective Young’s modulus of the porous material E_{ijkl}^H is computed via mathematical homogenization in Eq. (7). \mathrm{\Omega } denotes the macro structural domain with volume measured as \left|\mathrm{\Omega }\right|, and \xi is the available material volume fraction.

Traditionally a length scale is required for optimization formulation in Eq. (9) being well-posed. For TO this is often enforced by a regularization in the form of a filter [⁴³]. However, regularization is not needed for Eq. (9) as a componential thickness is already defined in Section 2 through the parameterization.

Sensitivity analysis

To compute the sensitivities, both the sensitivities of the objective function J with respect to the effective property E_{ijkl}^H and the sensitivities of E_{ijkl}^H with respect to the geometrical parameters of each CMC should be derived.

The sensitivities of the objective function J with respect to the effective property E_{ijkl}^H can be derived easily using the adjoint method [⁹] as

(10)J^{\prime }=-{\displaystyle \sum _{n=\mathrm{1}}^L{\displaystyle {\int }_{\mathrm{\Omega }}{ϵ}_{ij}\left({\mathit{u}}^n\right)\left(E_{ijkl}^H{)}^{\prime }{ϵ}_{kl}\right({\mathit{u}}^n)\mathrm{d}V}}.

To calculate (E_{ijkl}^H{)}^{\prime } in Eq. (10), e.g., the sensitivities of E_{ijkl}^H with respect to the geometrical parameters ({\mathit{x}}_{i\mathrm{1}}^I,{\mathit{x}}_{i\mathrm{2}}^I,t_i^I) of the ith CMC in unit cell I, taking the first order variation of E_{ijkl}^H in Eq. (7) yields,

E_{ijkl}^H{)}^{\prime }={\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle {\int }_Y{E}_{pqrs}{ϵ}_{pq}({\mathit{v}}_{\mathrm{0}}^{kl}-{\chi }^{kl}){ϵ}_{rs}({\mathit{v}}_{\mathrm{0}}^{ij}-{\chi }^{ij})\delta \left({\varphi }_I^s\right)({\varphi }_I^s{)}^{\prime }\mathrm{d}Y}

(11)\begin{array}{l} -{\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle {\int }_Y{E}_{pqrs}{ϵ}_{pq}\left(\right({\chi }^{kl}{)}^{\prime }\left){ϵ}_{rs}\right({\mathit{v}}_{\mathrm{0}}^{ij}-{\chi }^{ij}\left)H\right({\varphi }_I^s)\mathrm{d}Y}\hfill \\ -{\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle {\int }_Y{E}_{pqrs}{ϵ}_{pq}({\mathit{v}}_{\mathrm{0}}^{kl}-{\chi }^{kl}){ϵ}_{rs}\left(\right({\chi }^{ij}{)}^{\prime }\left)H\right({\varphi }_I^s)\mathrm{d}Y}.\hfill \end{array}

In Eq. (6), let \mathit{v}=({\chi }^{rs}{)}^{\prime } (as they belong to the same functional space) yields,

{\displaystyle {\int }_Y{E}_{ijpq}{ϵ}_{pq}\left({\chi }^{kl}\right){ϵ}_{ij}\left(\right({\chi }^{rs}{)}^{\prime }\left)H\right({\varphi }_I^s)\mathrm{d}Y}

(12)

Switching index i with r, and j with s in Eq. (12), we can find that the last two terms in Eq. (11) will vanish. Thus, for the unit cell I we will have

(13)\begin{array}{l}={\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle \sum _{i=\mathrm{1}}^N{\displaystyle {\int }_Y{E}_{pqrs}{ϵ}_{pq}({\mathit{v}}_{\mathrm{0}}^{kl}-{\chi }^{kl}){ϵ}_{rs}({\mathit{v}}_{\mathrm{0}}^{ij}-{\chi }^{ij})\delta \left({\varphi }_I^s\right)({\varphi }_i^I{)}^{\prime }\mathrm{d}V}}.\hfill \end{array}

where {\varphi }_I^s is defined in Eq. (8), {\varphi }_i^I and ({\varphi }_i^I{)}^{\prime } can be found in Section 2.

Linkage schemes for manufacturable multiscale structures

The CMC-based TO method itself cannot guarantee obtaining well-connected material microstructures at the micro scale. A linkage scheme needs to be devised to remove the microstructure disconnectivity. Thus, a linkage scheme is proposed in this work as illustrated in Fig. 3. The principle is to ensure well-connected microstructures fixing the end points located on the boundary of the unit cell. As shown in Fig. 3, the red dots represent fixed end points which are nondesignable and cannot move during the optimization process. The green dots denote the movable end points. By fixing the red end points for the neighboring unit cells then one assures that they connect with each other. In the meantime, by allowing the green end points to move, different CMCs can move and overlap and thus the topology of the unit cell can be varied. For example, Fig. 4 shows the optimized topology of the unit cell after the optimization process, from which it can be seen that the optimized unit cells will stay well connected through the fixed red end points.

Fig.3 A linkage scheme to ensure well-connected microstructures

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Fig.4 Topology of the unit cell after optimization

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For the unit cell in Fig. 3, it seems that the linkage is only valid for structures that can be meshed into structured rectangular finite element mesh (or blocked finite element mesh for 3D problems), as the shape of the unit cell is rectangular. However, it should be mentioned that the linkage scheme proposed for the material unit cell design is independent on the macro finite element mesh. The linkage is valid and effective as long as the macrostructure can be divided into different partitions sharing vertical or horizontal boundaries. As shown in Fig. 5, a macrostructure with arbitrary shape can be divided into different material partitions. Each partition will have a cellular material with a specific material microstructure. When the material microstructures in neighboring partitions are compatible then they can be manufactured.

Fig.5 A macrostructure divided into different microstructure material partitions

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To make the linkage scheme in Fig. 3 effective, another key factor is linking the thickness design variables of different CMCs in the same unit cell to have the same value. If different CMCs have different thickness values, when neighboring unit cells are assembled together, it may happen that one sharing red end point is attached to one unit cell having a CMC while the other unit cell has no CMC, which will also result in a disconnection.

Flowchart and numerical implementation

The process for the proposed CMC-based multiscale TO design is implemented according to the flowchart in Fig. 6. After setting the initial values for the geometrical parameters for all the CMCs in all types of unit cells, the homogenized effective material properties defined by homogenization equations in Eqs. (6) and (7) are calculated. With the effective material properties obtained, the macro governing equations can be solved to get structural response and evaluate the objective function. Then the sensitivities of the effective properties with respect to the geometrical parameters are calculated via Eq. (13), and the sensitivities of the objective function with respect to the geometrical parameters of the CMCs can be computed via Eq. (10). With the sensitivities obtained, the designs are updated using mathematical programming in the form of the method of moving asymptotes (MMA) [⁴⁴].

Fig.6 Flowchart for CMC-based multiscale TO

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For the numerical implementation of the sensitivity calculation in Eq. (13), the volume integral is transformed into a boundary integral as proposed by Ref. [³⁹]. Using the boundary integration, Eq. (13) is written as

(14)E_{ijkl}^H{)}^{\prime }={\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle \sum _{i=\mathrm{1}}^N{\displaystyle {\int }_{{\mathrm{\Gamma }}_i^I}{E}_{pqrs}{ϵ}_{pq}({\mathit{v}}_{\mathrm{0}}^{kl}-{\chi }^{kl}){ϵ}_{rs}({\mathit{v}}_{\mathrm{0}}^{ij}-{\chi }^{ij})({\varphi }_i^I{)}^{\prime }\mathrm{d}S}},

where {\mathrm{\Gamma }}_i^I represents the portion of the boundary of CMC i which is also shared by the microstructure’s boundary in unit cell I. Equation (14) is evaluated using Gaussian quadrature.

Numerical examples

In this section, a simply supported beam as shown in Fig. 7 is initially considered. Both designs with/without linkage scheme are investigated for demonstrating the effectiveness of the linkage scheme in ensuring well-connected microstructures. Next, the example is modified to demonstrate how to extend CMC-based multiscale TO to obtain multiscale structures with graded cellular materials or layered materials. Finally, a femur is optimized for three design scenarios.

Fig.7 A simply supported beam example with 3 load cases

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Simply supported beam example

Initially, the simply supported beam example in Fig. 7 is used to show the result without using the linkage scheme proposed in Section 3.3. The 4 by 1 macrostructure is meshed into 24 by 6 bilinear quadrilateral macro finite elements. Each macro element has a different material microstructure. The macrostructure has three loadcases with unit loads, and only the right half macrostructure is considered due to symmetry. Therefore, there are 72 types of unit cells. Each unit cell is meshed into 100 by 100 linear quadrilateral finite elements, and in each type of unit cell there consists of 8 CMCs. The Young’s modulus and Poisson’s ratio of the base material is 1 and 0.3, respectively. The objective is to minimize the compliance of the structure having 40% available material volume fraction.

The optimized structural topology and corresponding microstructure topologies are shown in the left part and right part of structure in Fig. 8, respectively. Figure 8 shows that the proposed CMC-based multiscale TO method clearly determines the material distribution in different porous materials by optimizing the geometrical parameters of the CMCs. Reasonable and easily interpretational macro and micro designs are obtained, demonstrating the proposed CMC-based multiscale TO method for obtaining multiscale TO designs.

Fig.8 Optimized microstructure topologies on the right and macro-scale structural topology on the left without linkage scheme

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However, one issue with the microstructures (unit cells) in Fig. 8 is that some neighboring microstructures are not well connected when assembled on the macro scale. Secondly, an issue is that some microstructures are only constructed of isolated beam members. Materials constructed by these kinds of microstructures are unstable and hard to be manufactured.

To address the disconnectivity issue, the linkage scheme proposed in Section 3.3 is employed to the same example as in Fig. 7. The optimized structural topology and corresponding microstructure topologies are shown in the left part and right part of Fig. 9, respectively. From the assembly of different unit cells in the right part of Fig. 9, it can be concluded that the neighboring microstructures are now well-connected through the fixed end points. Additionally, unstable beam members are removed and now all the microstructures are stable.

Fig.9 Optimized microstructure topologies on the right and structural topology on the left with linkage scheme

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The optimization iteration histories from using the two approaches, without vs. with linkage schemes, are compared in Fig. 10. It is noted that the objective function decreases monotonically in both designs indirectly validating the proposed CMC-based multiscale TO method. The final objective function value of the design using linkage scheme is 0.66 which is higher than that without linkage scheme 0.54, as the linkage scheme will further restrict the design space for the design variables. This study demonstrates the effectiveness of the proposed multiscale TO framework based on CMC method with linkage scheme.

Fig.10 Objective iteration curves for designs with/without linkage scheme

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Simply supported beam with graded materials

The example in Fig. 7 is revisited, but now the unit cell topology of each cellular material is assumed to be changing gradually. This is done by linking the design variables in all types of unit cells, with the exception that there is an independent thickness design variable for the CMCs in each type of unit cells.

Both designs using CMC-based TO method without/with the linkage scheme are obtained. The final optimized macro and micro topology results for the two designs for simply supported beam with graded materials are shown in Figs. 11(a) and 11(b). The objective iteration curves for the designs with/without linkage scheme for simply supported beam with graded materials are shown in Fig. 12. From Fig. 11, it is observed that now every microstructure has a similar topological configuration, i.e., a multiscale structure with graded cellular materials has been successfully obtained. From Fig. 12, we note that the objective function decreases monotonically in both designs, which validates the proposed CMC-based multiscale TO method in achieving multiscale structures with graded materials.

Fig.11 Optimized microstructure topologies on the right and structural topology on the left. (a) Without linkage scheme; (b) with linkage scheme

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By comparing the objective function values in Fig. 12, we find that the final objective function value of the design with linkage scheme 0.86 is slightly higher than that without linkage scheme 0.82, even though the two values are very close. The macro and micro topologies look similar for the designs with/without the linkage scheme in Fig. 11. This may be due to the reason that the design spaces for the two designs are similar. For example, in terms of the number of design variable, for the design with the linkage scheme is 80 (72 for thickness design variables and 8 for coordinate design variables), and the design without linkage scheme is slightly larger being 104 (72 for thickness design variables and 32 for coordinate design variables).

Fig.12 Objective iteration curves for designs with/without linkage scheme

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Simply supported beam with layered materials

In this section, the proposed CMC-based multiscale TO method is used to design multiscale structure with layered materials, which means their mechanical properties layer by layer by varying the material microstructures in order to adapt to loading and boundary conditions. This is realized by dividing the macrostructure into several layers, and each layer has a unique type of material microstructure (unit cell), which will greatly reduce the number of the design variables. The example in Fig. 7 is considered again, and the macrostructure is divided into 6 layers with equal height in the vertical direction.

Fig.13 Optimized microstructure topologies on the right and structural topology on the left. (a) Without linkage scheme. (b) with linkage scheme

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Figures 13(a) and 13(b) show the final optimized macro and micro topology results for the design without/with linkage scheme for simply supported beam with layered materials. From Fig. 13, it is noted that both designs produce layered multiscale structure, which validates the CMC-based multiscale TO method. In Fig. 13(a), we note the microstructures in the first layer from the bottom are not well connected to the microstructures in the second layer from the bottom. On the contrary, in Fig. 13(b), the microstructures in each layer are well connected to its neighboring microstructures. This further validates the proposed linkage scheme. The objective convergence behavior is similar as those in the previous two examples. The final objective function values of the design with and without linkage scheme are 1.1 and 0.87, respectively.

Femur example

A simplified femur example is investigated in the present section. The femur macrostructure with different partitions and the associated finite element mesh are shown in Fig. 14. The femur is fixed at the bottom, and there are 42 unit vertical loads applied on the top shown in the left Fig. 14. The maximum width and height of the femur is set at 245 and 265, respectively. The femur is divided into 39 partitions with each partition having a different microstructure. Therefore, there are 39 types of unit cells. Each unit cell is meshed into 100 by 100 linear quadrilateral finite elements, and in each type of unit cell there reside 8 CMCs. Each partition is meshed into finite elements with 735 macro finite elements in total. The Young’s modulus and Poisson’s ratio of the base material is 10000 and 0.3, respectively. The objective is to minimize the compliance of the structure with 60% available material volume fraction.

Fig.14 Femur macrostructure with different partitions (left) and finite element mesh (right)

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Three designs are obtained using three different test schemes. When each partition has a different material microstructure and the linkage scheme is used, the obtained design is defined as Design I. When all partitions have the same material microstructure and the linkage scheme is used, the design is defined as Design II. When all partitions have the same material microstructure and the linkage scheme is not used, the design is defined as Design III. The final microstructure optimized topologies for all three designs are illustrated in Fig. 15. It is noted from the left Design I in Fig. 15 that the microstructures are stiffer having more base material usage in the areas with a high energy density. The microstructures are well connected through vertical/horizontal edges when applying the linkage scheme. For Designs II and III, there is only one optimized microstructure with the same member thicknesses due to the use of the linkage scheme, while in Design I different microstructures can have beam members of different thickness values.

Fig.15 Final designs

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The objective function and volume constraint iteration curves for Designs I, II, and III are shown in Fig. 16. It is observed that all cases show stable convergence. The final objective function values for Designs I, II, and III are 11.17, 19.42, and 17.61, respectively. This is reasonable because Design I has the largest design space for the design variables compared with Designs II and III.

Fig.16 Objective and volume constraint iteration curves for different designs

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Conclusions and future work

A new CMC-based multiscale TO method has been proposed to extend the component-based TO methodology to multiscale structural design. The optimized multiscale structure is obtained by designing the geometrical parameters of the CMCs in each type of unit cell. With the CMC representation, the number of design variables is significantly reduced, compared to traditional density-based multiscale structural design. Furthermore, an effective linkage scheme has been devised and integrated for multiscale TO ensuring the smooth transitions/connectivity of the neighboring microstructures. The proposed method in achieving well-connected material microstructures has been validated through multiple example problems. Multiscale structures with graded or layered material microstructures are also obtained using the proposed CMC-based multiscale TO design methodology.

Sensitivities, both those of the objective function with respect to the effective properties of the cellular materials and those of the effective properties with respect to the geometrical parameters of each CMC have been derived using the adjoint method. The optimization convergence shows a stable iteration history indirectly demonstrating the accuracy of the sensitivities.

Future research direction includes devising other linkage schemes, investigating full-scale component-based multiscale TO without scale separation assumptions, and applying component-based multiscale TO to engineering problems. It will also be interesting to fully exploiting the benefits of the proposed methodology to achieve manufacturable multifunctional multiscale structures with other functionalities such as sufficient permeability to transfer cells and nutrients in biomedical applications, enhanced thermal management, improved structural stability or energy absorption performance due to the utilization of cellular materials, etc.

Acknowledgements

The authors would also like to thank the Digital Manufacturing and Design Innovation Institute (DMDII) at Northwestern University for their support through award number 15-07-07.

References

1 Kruth J P, Leu M C, Nakagawa T. Progress in additive manufacturing and rapid prototyping. CIRP Annals-Manufacturing Technology, 1998, 47(2): 525–540

2 Gibson I, Rosen D W, Stucker B. Additive Manufacturing Technologies. New York: Springer, 2010

3 Rosen D W. Computer-aided design for additive manufacturing of cellular structures. Computer-Aided Design and Applications, 2007, 4(5): 585–594

4 Chu C, Graf G, Rosen D W. Design for additive manufacturing of cellular structures. Computer-Aided Design and Applications, 2008, 5(5): 686–696

5 Chu J, Engelbrecht S, Graf G, A comparison of synthesis methods for cellular structures with application to additive manufacturing. Rapid Prototyping Journal, 2010, 16(4): 275–283

6 Murr L, Gaytan S, Medina F, Next-generation biomedical implants using additive manufacturing of complex, cellular and functional mesh arrays. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1917, 2010(368): 1999–2032

7 Wang X, Xu S, Zhou S, Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: A review. Biomaterials, 2016, 83: 127–141

8 Sigmund O, Aage N, Andreassen E. On the (non-) optimality of Michell structures. Structural and Multidisciplinary Optimization, 2016, 54(2): 361–373

9 Bendsøe M P, Sigmund O. Topology Optimization: Theory, Methods, and Applications. 2nd ed. Berlin: Springer, 2004

10 Brackett D, Ashcroft I, Hague R. Topology optimization for additive manufacturing. In: Proceedings of the Solid Freeform Fabrication Symposium. Austin, 2011

11 Gaynor A T, Meisel N A, Williams C B, Multiple-material topology optimization of compliant mechanisms created via PolyJet three-dimensional printing. Journal of Manufacturing Science and Engineering, 2014, 136(6): 061015

12 Almeida H A, Oliveira E S. Sustainability based on biomimetic design models. In: Muthu S, Savalani M, eds. Handbook of Sustainability in Additive Manufacturing. Singapore: Springer, 2016, 65–84

13 Zegard T, Paulino G H. Bridging topology optimization and additive manufacturing. Structural and Multidisciplinary Optimization, 2016, 53(1): 175–192

14 Bendsøe M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202

15 Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224

16 Bendsoe M P, Guedes J M, Haber R B, An analytical model to predict optimal material properties in the context of optimal structural design. Journal of Applied Mechanics, 1994, 61(4): 930–937

17 Ringertz U T. On finding the optimal distribution of material properties. Structural Optimization, 1993, 5(4): 265–267

18 Groen J P, Sigmund O. Homogenization-based topology optimization for high-resolution manufacturable microstructures. International Journal for Numerical Methods in Engineering, 2018, 113(8): 1148–1163

19 Rodrigues H, Guedes J M, Bendsoe M P. Hierarchical optimization of material and structure. Structural and Multidisciplinary Optimization, 2002, 24(1): 1–10

20 Deng J, Chen W. Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty. Structural and Multidisciplinary Optimization, 2017, 56(1): 1–19

21 Deng J, Yan J, Cheng G. Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Structural and Multidisciplinary Optimization, 2013, 47(4): 583–597

22 Guo X, Zhao X, Zhang W, Multi-scale robust design and optimization considering load uncertainties. Computer Methods in Applied Mechanics and Engineering, 2015, 283: 994–1009

23 Liu L, Yan J, Cheng G. Optimum structure with homogeneous optimum truss-like material. Computers & Structures, 2008, 86(13‒14): 1417–1425

24 Niu B, Yan J, Cheng G. Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Structural and Multidisciplinary Optimization, 2009, 39(2): 115–132

25 Schury F, Stingl M, Wein F. Efficient two-scale optimization of manufacturable graded structures. SIAM Journal on Scientific Computing, 2012, 34(6): B711–B733

26 Nakshatrala P B, Tortorelli D, Nakshatrala K. Nonlinear structural design using multiscale topology optimization. Part I: Static formulation. Computer Methods in Applied Mechanics and Engineering, 2013, 261‒262: 167–176

27 Xia L, Breitkopf P. Concurrent topology optimization design of material and structure within FE² nonlinear multiscale analysis framework. Computer Methods in Applied Mechanics and Engineering, 2014, 278: 524–542

28 Zhang P, Toman J, Yu Y, Efficient design-optimization of variable-density hexagonal cellular structure by additive manufacturing: Theory and validation. Journal of Manufacturing Science and Engineering, 2015, 137(2): 021004

29 Wang Y, Chen F, Wang M Y. Concurrent design with connectable graded microstructures. Computer Methods in Applied Mechanics and Engineering, 2017, 317: 84–101

30 Wang Y, Zhang L, Daynes S, Design of graded lattice structure with optimized mesostructures for additive manufacturing. Materials & Design, 2018, 142: 114–123

31 Clausen A, Aage N, Sigmund O. Exploiting additive manufacturing infill in topology optimization for improved buckling load. Engineering, 2016, 2(2): 250–257

32 Wu J, Clausen A, Sigmund O. Minimum compliance topology optimization of shell-infill composites for additive manufacturing. Computer Methods in Applied Mechanics and Engineering, 2017, 326: 358–375

33 Wang Y, Kang Z. A level set method for shape and topology optimization of coated structures. Computer Methods in Applied Mechanics and Engineering, 2018, 329: 553–574

34 Li H, Luo Z, Gao L, Topology optimization for functionally graded cellular composites with metamaterials by level sets. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 340–364

35 Vogiatzis P, Ma M, Chen S, Computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 477–497

36 Guo X, Zhang W, Zhong W. Doing topology optimization explicitly and geometrically—A new moving morphable components based framework. Journal of Applied Mechanics, 2014, 81(8): 081009

37 Norato J, Bell B, Tortorelli D. A geometry projection method for continuum-based topology optimization with discrete elements. Computer Methods in Applied Mechanics and Engineering, 2015, 293: 306–327

38 Zhang W, Yuan J, Zhang J, A new topology optimization approach based on moving morphable components (MMC) and the ersatz material model. Structural and Multidisciplinary Optimization, 2016, 53(6): 1243–1260

39 Deng J, Chen W. Design for structural flexibility using connected morphable components based topology optimization. Science China. Technological Sciences, 2016, 59(6): 839–851

40 Deng J. Topology optimization of emerging complex structures. Dissertation for the Doctoral Degree. Evanston: Northwestern University, 2016

41 Bensoussan A, Lions J L, Papanicolaou G. Asymptotic Analysis for Periodic Structures. Amsterdam: AMS Chelsea Publishing, 1978

42 Terada K, Kikuchi N. A class of general algorithms for multi-scale analyses of heterogeneous media. Computer Methods in Applied Mechanics and Engineering, 2001, 190(40‒41): 5427–5464

43 Sigmund O, Petersson J. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization, 1998, 16(1): 68–75

44 Svanberg K. The method of moving asymptotes—A new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373