A multi-resolution method for 3D multi-material topology optimization

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ScienceDirect Comput. Methods Appl. Mech. Engrg. 285 (2015) 571–586 www.elsevier.com/locate/cma

A multi-resolution method for 3D multi-material topology optimization Jaejong Park a , Alok Sutradhar a,b,∗ a Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, 43210, USA b Department of Plastic Surgery, The Ohio State University, Columbus, OH, 43210, USA

Received 3 June 2014; received in revised form 7 October 2014; accepted 10 October 2014 Available online 25 November 2014

Abstract This paper presents a multi-resolution implementation in 3D for multi-material topology optimization problem. An alternating active-phase algorithm where the problem at hand is divided into a series of the traditional material–void phase topology optimization is employed for the multi-material problem. Different levels of discretization are used for the displacement mesh, design variable mesh and density mesh which provides higher resolution designs for the solutions. A projection scheme is employed to compute the element densities from design variables and control the length scale of the material density. Simple block coordinate descent method similar to the Gauss–Seidel technique is used to solve the subproblems. Several 3D numerical examples are presented to demonstrate the ease and the effectiveness of the proposed implementation. Incorporating the multi-resolution method into the multi-material approach, robust designs with improved resolution can be achieved for real life problems with complex geometries. c 2014 Elsevier B.V. All rights reserved. ⃝

Keywords: Topology optimization; Multi-material; Multi-phase topology optimization; Multi-resolution; Multiple discretization; Alternating active phase algorithm

1. Introduction Topology optimization provides the best material distribution within a prescribed design domain by seeking where to put the material (solid) and where not to (void) in order to obtain the optimum structural performance. Since the pioneer paper by Bendsøe and Kikuchi [1], topology optimization has been used widely in designing mechanical components and other engineering applications e.g., thermoelasticity [2], fluid flow [3,4], acoustics [5], wave propagation [6], aerospace design [7], multi-functional material design [8–11], multi-physics systems [12,13], etc. New innovative materials and smart structures can also be designed, e.g., functionally graded piezo-composites have been developed using topology optimization and homogenization technique [14]. Recently, topology optimization has ∗ Correspondence to: Department of Plastic Surgery, Department of Mechanical and Aerospace Engineering, The Ohio State University, 915 Olentangy River Road, Suite 2109, Columbus, OH, 43212, USA. Tel.: +1 614 293 7655; fax: +1 614 293 9024. E-mail address: [email protected] (A. Sutradhar).

http://dx.doi.org/10.1016/j.cma.2014.10.011 c 2014 Elsevier B.V. All rights reserved. 0045-7825/⃝

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also been used in biomedical applications e.g., to design patient specific bone replacements [15], and implants [16]. However, most of the problems in these works consider only single material and void. For multi-physics and multifunctional materials, it is desired and interesting to look at designs with multi-materials i.e., multiple material phases which includes void. Commonly, multi-material topology optimization is defined when there are different material properties with specific volume fraction ratios. The most critical feature of a multi-material topology optimization method is representing the material properties based on the physical properties of each phase. Since its inception, mostly material interpolation schemes have been used to address multi-material problems, which is not a physicallybased technique. The choice of interpolation function is important when considering the distribution of multiple material phases in a given design domain. There is always a risk of having intermediate design elements with nonphysical materials [17]. For example, in the popular Solid Isotropic Material with Penalization (SIMP) model, the power-law interpolation for the single material is used. In this SIMP model, the elasticity tensor is expressed as p E (ρe ) = ρe E 0 , where it is assumed that material density ρ is the design variable and E 0 is the constant elasticity stiffness. This proportional elasticity tensor is penalized with p > 1 as the penalization factor. This interpolation becomes cumbersome with additional materials. There is no validated general rule for multi-material. In a two p material composite structure without a void, E (ρe ) = ρe E 1 + (1 − ρe ) p E 2 where E 1 and E 2 are the elasticity tensors of materials 1 and 2 respectively. This interpolation violates the Hashin–Shtrikman bounds for low values of ρe and large values of p. Also, for different combinations of Young’s moduli and Poisson’s ratio, the interpolation exhibits strange behavior [17]. To counter this problem, an interpolation scheme based on the weighted average of Hashin–Shtrikman lower and upper bounds has been suggested [17]. One way to penalize these structures is having one variable to control the topology using a point-wise material distribution while using another variable to control the mixture. For two material structures phase) an interpolation with two design variables  with a void(three material  1 2  p2 2  1 p 2 p 1 2 1 2 can be used, i.e. E ρe , ρe = (ρe ) (ρe ) E + 1 − ρe E . Material microstructures with extreme properties have been designed in [18,19] using an extension of the SIMP interpolation. A unified parameterized formulation for simultaneous topology and multi-material design as an extension to a direct generalization SIMP was proposed in [20]. Maximum stiffness designs of two-material structures have been reported in [21,22]. Also, a peak function approach [23] and a ‘color level set’ approach [24] have been proposed for multi-material problems. In the ‘color’ level set approach, different materials’ sub-domains are represented by different level sets which are composed of implicit functions. The evolution of the interface determines the optimal distribution of the sub-domains for each material. Using the phase field approach with intrinsic volume preserving property based on Cahn–Hilliard equation, a multi-phase topology optimization approach was presented in [25,26]. One of the main hurdles of the phase field approach to obtain multi-material design is slow convergence due to huge number of iterations (in the order of 104 ). For example, in [25], the number of iterations for a 3D three-material cantilever problem was 371,000. The dependence on different phenomenological constants in the phase field method is another obstacle. A multi-material topology optimization algorithm using alternating active phase algorithm based on the block coordinate descent approach has been presented in [27]. The method works similar to a Jacobi and Gauss–Seidel iterative scheme. The framework can be used to convert a single material (two material phase) topology optimization codes to multi-material problems with limited modification. The algorithm is straightforward, simple to implement and independent of the number of phases. A number of 2D examples were presented to show the efficacy of the method. More discussion on the advantages and disadvantages of different multi-material approach can be found in [27]. The most desired topology optimization framework is one which gives higher resolution designs with reduced computational cost for analysis and optimization. In general, a large number of finite elements are required in order to obtain higher resolution designs. Minimizing the computational cost of the finite element analysis can be done in a number of ways, by introducing parallel computing [28–31], by employing fast iterative solver [32,33], by performing FE analysis after certain intervals [34,35] or by reducing the number of degrees of freedom in the FE model [36–39]. Another notable scheme to increase efficiency is the adaptive mesh refinement approach. In this approach, the voids are represented by coarser mesh while the solids by relatively finer mesh during the process [37,40–42]. The concept of isogeometric analysis by Hughes et al. [43] has been used to minimize overall cost by reducing the dependence to the initial FE grid and the post-processing efforts of converting to NURBS based CAD models for problems with complex geometry [44–46]. An efficient approach named multi-resolution topology optimization (MTOP) has been developed where the models for FE analysis and the design optimization are decoupled with different levels of discretization [36]. Based on the foundations of a super-element approach, three different meshes are used, (i) finite element mesh

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for solving the governing equation, (ii) design variable mesh for the optimization and (iii) density element mesh for representing the material distribution. Evidently, in this technique, the design variable and density mesh are finer than the finite element mesh. Higher resolution designs are obtained due to the finer mesh during the optimization process while the use of coarser mesh during the FE analysis minimizes the computational cost. In [36], the same resolution is used for the design variable mesh and the density mesh. The method was further improved in [37], by employing three levels of mesh refinement, i.e., coarser FE mesh, moderately fine mesh for design variables and a finer mesh for the density variables. This method has been successfully implemented to obtain high resolution solutions for 3D complex geometries, e.g., craniofacial bone replacements [15,47]. Adaptive refinement of analysis mesh and density mesh can also provide higher resolution designs with better boundary description quality [48,49]. In the present work, a multi-material topology optimization method for three dimensional (3D) problems has been developed that utilizes the multi-resolution approach. The multi-material method uses the alternating active phase algorithm presented in [27]. To the best knowledge of the authors, this is the first work where multi-resolution technique is used for multi-material topology optimization problems, and alternating active phase algorithm is used for 3D problems. The remainder of this paper is structured as follows. Section 2 gives the general formulation of a multi-material optimization problem while the basic introduction of the multi-resolution approach is given in Section 3. Section 4 provides several numerical examples in 3D and finally Section 5 summarizes and concludes the paper. The numerical examples in this paper have been focused on compliance minimization. 2. Multi-material topology optimization 2.1. Problem formulation A common formulation for topology optimization problems is to minimize the compliance. In this formulation, the goal is to distribute a given amount of material to obtain a structure with maximum stiffness (or minimum compliance i.e., the energy of deformation). A multi-material problem with volume fraction constraints for each material phase is considered in this work. Consider a design domain Ω ∈ R2 or R3 . A multi-material topology optimization is to find the optimal material distribution of m number of different materials that minimizes the objective function (compliance) subjected to global per-material volume fraction constraints Vi . Following Bendsøe and Sigmund [50], classical material–void optimization will be referred as single material (two material phase) topology optimization from here on. Void is considered as a material phase. With the design domain Ω assumed to be discretized by finite elements, the design variables are the relative density of each element, ρ. These design variables can be combined into a vector, ρ. Since m different material phases are considered, densities at a given position are presented as an order of material phases ρ i (i = 1, . . . , m). The problem (objective function and constraints) can be mathematically expressed as follows,   minimize: C ρ i ,u = uT Ku ρi   subject to:  K ρi u = f (1) ρ i dΩ ≤ Vi , where (i = 1, . . . , m) , 

where C is the compliance, ρ i is the density vector for each different phases, f is the global load vector and u is the global displacement vector. K is the global stiffness matrix, Ω is the design domain, and Vi is the volume fraction constraint for phase i. The material density distribution is considered continuous in the design domain as opposed to discrete valued problem, as a result, ρ i = ρ i (x). Also,  ρ i (x) d ≤ Vi (i = 1, . . . , m) . (2) 

The problem is relaxed for density to have any value between 0 and 1 with small lower bound of ρ i min = 0.001 to avoid singularities during calculating for equilibrium. That is, 0 < ρ i min ≤ ρi ≤ 1

(i = 1, . . . , m) .

(3)

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Note that the summation of the densities at each point x ∈ , should be equal to unity, i.e., m 

ρi (x) = 1

(i = 1, . . . , m) .

(4)

i=1

Finally, the summation of per-material volume fraction constraints Vi should be equal to unity, hence, m 

Vi (x) = 1

(i = 1, . . . , m) .

(5)

i=1

2.2. Material interpolation The schemes for material interpolation allow intermediate material choices during the solution process. Typically, the local material properties are chosen as a local function of the densities of the contributing phases. As explained in Section 1, there are several ways to compute local material properties based on the properties of each phase. Some of the noteworthy methods are the SIMP power-law [17,50], the Hashin–Shtrikman bounds [17,19,51] and the homogenization method [52]. The SIMP approach for the single material topology optimization penalizes intermediate densities by expressing the material properties E of each element by E (ρ) = ρ p E 0 ,

(6)

E0

where represents the elastic modulus for the given isotropic material and p is the penalization factor. Following the concept of Vegard’s Law for Eigenstrain, Zhou and Wang [26] and recently Tavakoli and Mohseni [27] interpolated the elasticity stiffness tensor for multi-material as E (ρ) =

m 

p

ρi E i0 ,

(7)

i=1

where E i0 is the elastic modulus for phase i. The same interpolation scheme is used in this implementation. 2.3. Alternating active phase algorithm In the alternating active phase algorithm, the problem is decomposed into a number of binary phase topology optimization subproblems. In each stage, the updated solution from the previous subproblem is used as the input for the current binary phase subproblem similar to the Gauss–Seidel iterative method. Using this technique a standard single material topology optimization code can be recycled by converting into a multi-material code. Henceforth in this algorithm, the number of subproblems are the number of possible combinations of 2 materials from a set of m different materials which is m C2 or m(m −1)/2. In the solution process of each subproblem, only two phases are active at a time for the topology optimization and rest of the topologies for m − 2 phases are fixed. Let the two active phases be designated as ‘a’ and ‘b’. For each subproblem, the material properties for corresponding active phases need to be used accordingly. Compared to a single material topology optimization routine, only one additional iterative loop needs to be added in the algorithm. Using Eq. (4), the sum of the densities of two active phases ‘a’ and ‘b’ at each location x for each subproblem that will vary is ρa (x) + ρb (x) = 1 −

m 

ρi (x) .

(8)

i=1,i̸=a,b

The problem can be simplified in each subproblem by taking the density of one active phase ρ a as the only design variable. The density of the other phase can be calculated using Eq. (8). An alternate approach which is similar to the Jacobi method instead of the Gauss–Seidel can also be adopted [27]. It must be clarified that in each subproblem, the objective function should monotonically decrease to a convergent solution which is a feasible solution with respect to the governing equations and constraints. The flow chart of the algorithm is illustrated in Fig. 1.

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Fig. 1. Flow chart for multi-material topology optimization using the alternating active phase algorithm.

3. Multi-resolution topology optimization 3.1. Multiple discretization level The central idea of multi-resolution topology optimization technique is to obtain high-resolution optimal topology with relatively low computational cost by employing a coarser discretization for finite elements and a finer discretization for both density elements and design variables [36]. In the element-based approach, the uniform density of each displacement element is considered as a design variable. By contrast, the nodal-based approaches [53–55] consider the densities at the finite element nodes as the design variables. Three distinct levels of mesh resolutions for the topology optimization problem are employed in the multi-resolution scheme; (i) the displacement mesh to perform the FE analysis, (ii) the density mesh to represent material distribution and (iii) the design variable mesh to perform the optimization. In this scheme, the element densities (ρ) are computed from the design variables (d) by a projection function. The topology optimization problem definition in Eq. (1) is then rewritten accordingly   minimize: C ρ i ,u = uT Ku d   subject to: K ρ i u = f (9) ρ = f p (d)  

ρ i dΩ ≤ Vi ,

where (i = 1, . . . , m)

where d is the vector of design variables and f p (.) is the projection function. One way to obtain higher resolution design efficiently, usually a finer mesh for density distribution is prescribed than the displacement field. As a result, each displacement element consists of multiple density elements. The material density is assumed to be uniform within

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Fig. 2. The displacement mesh, density mesh and the design variable mesh for various multi-resolution elements are represented. An Q4/n16/d9 element which has a Q4 element as the displacement mesh, 16 elements as density mesh and 9 elements as design variable mesh is shown in (a). Similarly, Q4/n25/d25 element and B8/n125/d125 (also known as B8/n125) element are shown in (b) and (c) respectively. Possible density variations within single displacement element are shown for Q4/n16/d9 and Q4/n25/d25.

each density element. The discretization level for the displacement elements, the density elements, and the design variables can be chosen independently. For example, Fig. 2(a) shows a multi-resolution element (Q4/n16/d9) based on a Q4 displacement element which has 16 density elements and 9 design variables in it. Another multi-resolution element where one Q4 displacement element consists of 25 density elements and 25 design variables (Q4/n25/d25) is shown in Fig. 2(b). More multi-resolution elements in both 2D and 3D with varying discretization can be found in [37]. In this work, for 3D problems, 8-noded brick element is used for displacement mesh which is further discretized to have 125 uniform density elements and design variables (B8/n125) as shown in Fig. 2(c). 3.2. Stiffness and sensitivity For a multi-resolution element, the stiffness matrix will be the summation of the stiffness contributions from all the density elements using, i.e.,    Nn Nn   T BT D0 B dΩe/n , (10) Ke (ρe ) = B D(ρn )B dΩe/n = (ρn ) p · n=1

Ωe/n

n=1

Ωe/n

where Ke (ρe ) represents the stiffness matrix of the displacement element e, B is the shape function derivatives and D (ρn ) is the material property matrix of density element ρn , D0 is the material property matrix corresponding to ρn = 1, and Nn is the number of density elements within a displacement element. p is the penalization factor and Ωe/n refers to the domain of density element n inside displacement element e. In each density element, 8 Gauss points are utilized to compute stiffness. When multi-material topology optimization is blended with multi-resolution scheme, stiffness matrix of a displacement element can be computed using the material interpolation in Eq. (7) as follows,     Nn m   p 0 T Ke (ρe ) = B ρi Di B dΩe/n (11) n=1

Ωe/n

i=1

n

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where Di0 is the material property matrix corresponding to the phase i for ρi = 1. The sensitivity analysis is required for the gradient-based optimization problem. The sensitivity of the elemental stiffness matrix is required for the sensitivity analysis of the compliance. In the multi-resolution scheme, the sensitivity of the elemental stiffness and the volume fraction constraint with respect to the design variable is calculated as follows,  Nn Nn  ∂Ke ∂Ke ∂ρn  ∂ρn = = pρ p−1 · BT D0 B dΩe/n , ∂d N ∂ρn ∂d N n=1 ∂d N Ωe/n n=1 Nn  ∂ V ∂ρn ∂V = , ∂d N ∂ρn ∂d N n=1

(12)

where the term ∂ρn /∂d N can be obtained from the projection function. For multi-material, phase ‘a’ is the only design variable (d N ) in each subproblem among active phases ‘a’ and ‘b’. Using information provided in Eqs. (8) and (12), the sensitivities for multi-material formulation with respect to design variable can be obtained as follows,  Nn Nn  ∂Ke ∂Ke ∂ρ n a  ∂ρ n a = = , p (ρa ) p−1 · BT (Da0 −D0b )B dΩe/n ∂d N ∂ρ n a ∂d N n=1 ∂d N Ωe/n n=1 Nn  ∂V ∂ V ∂ρ n a = , ∂d N ∂ρ n a ∂d N n=1

(13)

where ρ n a is the density of phase ‘a’ inside density element n. The sensitivities of the objective function for the multi-material multi-resolution topology optimization can then be simply calculated from the adjoint equation as follows ∂C ∂Ke = −uTe ue , ∂d N ∂d N

(14)

where ue is the displacement vector of displacement element e. 3.3. Minimum length scale In order to achieve mesh independency, projection functions can be used [56–60]. Projection function also eliminates numerical instabilities and checkerboard effects [56]. In this work, we use the density filter to obtain minimum length scale and mesh independency. The element density ρn is obtained from the design variables d N as follows, ρn = f p (d N ) ,

(15)

where f p (.) is the projection function. Here, d N denotes the design variable associated with the design variable mesh, while ρn represents the density of element n associated with the density element mesh. A projection function centered at a design variable N with a radius of rmin for a 2D domain with multi-resolution elements (Q4/n25/d25) is shown in Fig. 3. For a linear projection function, the uniform density of a density element n is computed as the weighted average of the design variables in the neighborhood as follows,  d N w(rn N ) ρn =

N ∈Sn



w(rn N )

,

(16)

N ∈Sn

where Sn is the sub-domain corresponding to the density element n and rn N is the distance from the point associated with design variable N to the center of density element n. The corresponding weight function w is defined as   rmin − rn N if rn N ≤ rmin , w(rn N ) = (17) r 0 min otherwise.

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Fig. 3. Schematic of projection function from a design variable to the density elements for Q4/n25/d25 multi-resolution element.

Notice the physical radius rmin (see Fig. 3) is independent of the mesh. The sensitivities of element density with respect to design variables are derived as ∂ρn w(rn N ) =  . ∂d N w(rn N )

(18)

n∈Sn

Employing this projection function scheme with minimum length scale, the solution becomes mesh independent and checkerboard pattern is avoided. 4. Numerical examples In this section, the numerical implementation is demonstrated with five numerical examples. The examples include classical benchmark problems in topology optimization, i.e. cantilever beam, MBB beam, L shaped beam and a bridge problem. The first example is a cantilever beam with concentrated vertical load at the center of the free end. The second problem considers a MBB beam with a distributed load on the center. An L-shaped beam with edge load is solved in the third example. In the fourth example, a bridge with a prescribed deck is optimized. For the last example, optimization of the design of a stool in a cubic domain is examined. For each of the examples, the single material (binary phase: material–void) topology optimization result is first provided followed by multi-material results. In all the examples, the Optimality Criteria (OC) has been used as the optimizer with a move limit of 0.2 and a numerical damping coefficient of 0.5. All the quantities are assumed to be dimensionless. In the single material topology optimization simulations, the Young’s modulus (E) and the Poisson’s ratio for the material is chosen as 1 and 0.3 respectively for all examples. In the multi-material problems, the elasticity modulus is chosen as E = 3 for the stiff material and E = 1 for the soft material with the Poisson’s ratio of 0.3 for both the materials. The void is modeled with E = 1e − 9. As a common practice in 3D topology optimization problems, solutions are represented by constructing an iso-surface from the resulting density field. The density value for each iso-surface is chosen as one quarter of the maximum value (e.g. if maximum density value is 1 the iso-surface value is chosen to be 0.25) unless stated otherwise. The termination of the algorithm is controlled with either the maximum iteration number (iter max) or the minimum change in density field (tol). The maximum iteration number is set to be 100 and the tol is chosen to be 0.1% for all numerical examples. The evolution of the solution along with the convergence history of the objective function is provided for the cantilever problem. We also examine the effect of volume fraction constraint with the cantilever problem. Finally, strain contours from the FE analyses are compared with multi-material topology optimization solution for the bridge problem in order to validate the result. 4.1. Cantilever beam problem Cantilever beam problem, a classical benchmark problem is solved in 3D space. A three-dimensional cantilever beam is considered with a domain of 3L × L × L as shown in Fig. 4(a). The left end is constrained in all the degrees of freedom and a vertically downward unit force is applied at the center of the free end. The domain is discretized

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Fig. 4. Cantilever beam problem. (a) Geometry and boundary conditions, (b) single material (material–void) and (c) multi-material (2 materials and void) topology optimization results. Green and tan colors correspond to materials with E = 3 and E = 1 respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Compliance convergence history for the cantilever beam problem.

with 3000 B8/n125 elements into a 30 × 10 × 10 grid. The volume fraction constraint (Vs ) is chosen to be 5% of the whole domain for single material topology optimization problem. The minimum length scale (filter size, rmin ) of 1 and the penalization factor ( p) of 3 are employed. Single material topology optimization is depicted in Fig. 4(b). The result is similar to the classical 2D topology optimization result. The same boundary conditions are adopted in the multi-material topology optimization. Instead of having a single material, two different materials with Young’s moduli of E = 3 and E = 1 for stiff and soft materials respectively are employed. Volume fraction constraints are selected to be 2.5% for both the stiff and the soft material resulting in 95% allowed for the void. The same value for rmin and p as in single material topology optimization are selected in the multi-material study. The optimized solution using the multi-material approach is shown in Fig. 4(c). The result of the multi-material topology optimization is different than that from single material case. The stiff materials spans out from the support to the load, while the soft material that is similar to scissor tongs in shape is located in between the stiff material structures. The history of compliance convergence for the cantilever problem is plotted in Fig. 5. The trajectory shows that compliance converges rapidly within 10 iterations and becomes plateau afterwards. In Fig. 6, selected topologies during the solution process for cantilever problem are shown. One can observe the significant topological change between iteration 24 and 100 (even though their compliance values are similar). It is interesting to see the evolution in the merging and dissolution of solids and voids.

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Fig. 6. Topology in selected iterations to show the solution evolution for cantilever beam problem.

Fig. 7. Modified cantilever beam boundary conditions and multi-material topology optimization result with volume fraction constraints of 30%, 15% and 7.5% respectively. The color purple corresponds to stiff material with E = 3 whereas the tan is for soft material with E = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4.1.1. Effect of volume fraction constraint A cantilever problem with a domain of 2L × L × L is analyzed to study the effect in changing volume fraction ratio. The domain is discretized with 2000 B8/n125 elements into a 20 × 10 × 10 grid. Two materials are considered  with the Young’s modulus ratio E stiff /E soft of 3. This problem is optimized with the volume fraction ratio of 30%, 15% and 7.5%. It is assumed that the volume fraction ratio is shared equally between the stiff and the soft materials. rmin = 1 and p = 3 are selected. The geometry, prescribed domain boundary condition and three respective results are shown in Fig. 7. It is worth to note that as the volume fraction constraint varies the shape of the topology changes significantly. 4.2. Messerschmitt–Bolkow–Blohm (MBB) beam problem The classical MBB beam problem is designed in three dimensions. Due to the symmetry, half of the design domain is considered for the topology optimization. The geometry and appropriate symmetry conditions on the face of symmetry are shown in Fig. 8(a). A distributed load at the center of the beam along the direction of the width is applied. The domain is discretized with 3000 B8/n125 elements into a 30×10×10 grid. Vs is 10%, rmin = 1 and p = 3 are employed. Fig. 8(b) shows the result of the single material topology optimization with an iso-surface drawn with a density value of 0.1. This solution is similar to the well-known 2D MBB beam solution [61]. For the multi-material  topology optimization, two different materials are considered with Young’s moduli ratio E stiff /E soft of 3 with volume fraction constraints of 7% and 3% for the stiff and the soft material respectively. rmin and p remains the same

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Fig. 8. MBB Beam problem. (a) Geometry and boundary condition, (b) single material topology optimization result, result on the Y Z plane matches the well-known MBB beam solution in 2D. (c) Result using multi-material (2 materials and void) topology optimization in various planes. Green and tan colors correspond to materials with E = 3 and E = 1 respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. The MBB beam results using the single and the multi-material topology optimization.

Fig. 10. The MBB beam results using different volume fraction constraints between materials (a) Vstiff = 3% and Vsoft = 7%, (b) Vstiff = 4% and Vsoft = 6%, (c) Vstiff = 5% and Vsoft = 5%, (d) Vstiff = 6% and Vsoft = 4%.

above. The result is shown in Fig. 8(c). Again the density value of 0.1 is chosen for the iso-surface. In order to study the results of the full single and multi-material designs, the complete beam is redrawn in Fig. 9 by mirroring results shown in Fig. 8(b) and (c) about the predefined plane of symmetry. In order to explore the change in the solutions, different combinations of the volume fractions constraints for the stiff and the soft material are used. The sum of the volume fraction constrain of materials is preserved as 10%. The volume fraction constraint of the stiff material is varied to 3%, 4%, 5%, and 6%. Iso-surface representations with 0.1 density value of respective solutions are provided in Fig. 10. Modifying the volume fraction constraints of the stiff and the soft materials results in significant change in the final topology. 4.3. L-shaped beam problem An L-shaped beam with a distributed edge load is examined. Due to sharp corner located in the middle of the domain, this problem causes stress concentration which makes it an interesting benchmark problem that is often used in the literature [62]. Although the problem formulation does not consider stress constraints, the effect of introducing another material phase stimulates curiosity. The domain is an extruded square discretized with 2000 B8/n125 elements

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Fig. 11. L-shaped beam problem. (a) Geometry of design domain and boundary condition (b) single material topology optimization result (c) multimaterial topology optimization result. Three holes in single material result are merged to have two holes in multi-material topology optimization. Purple and tan colors correspond to materials with E = 3 and E = 1 respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

into a 20 × 20 × 5 grid. The volume fraction constraint is chosen as 12%. rmin = 1 and p = 3 are selected. The geometry and the boundary conditions are illustrated in Fig. 11(a). Additional material is added into the algorithm to the problem above. Ratio of the Young’s modulus (E stiff /E soft ) of 3 is employed. Volume fraction constraint of 12% used in single material topology optimization is shared equally by the stiff and the soft material while rest of them is assigned as void. The single and multi-material topology optimization results are given in Fig. 11(b) and (c). 4.4. Bridge problem In this 3D bridge problem, a simply supported deck is subjected to a uniformly distributed load. Here, in the z direction, there are 10 displacement elements and 50 (10 × 5) density elements. The deck is designed with a nondesignable layer having a thickness of 1 density element. The domain is discretized with 4000 B8/n125 elements into a 40 × 10 × 10 grid as shown in Fig. 12(a). rmin = 1 and p = 3 are employed. Vs is chosen to be 9.5%. Three materials phases including the void are considered in multi-material analysis. Young’s moduli (E) for stiff, soft, and void phases are selected to be 3, 1, and 1e-9 respectively. The E of non-designable layer (deck) is assumed to be 3. The volume fraction of the stiff material is chosen as 5.75%, while 3.75% is chosen for the soft material. The remaining 90.5% is allowed for the void. The single and multi-material topology optimization results are given in Fig. 12(b) and (c) with an iso-surface value of 0.1. 4.4.1. Correlation with FEA with homogeneous material Typically for linear elastic design, the practice is to put relatively stiffer material at locations with larger displacement or stress. A FE analysis with the same geometry and the boundary conditions as in Fig. 12(a) is performed using homogeneous elastic material to explore the critical regions. The algorithm in this work minimizes the strain energy by increasing the densities in the regions with higher sensitivities. As a result, the areas with higher sensitivities can be comparable to the locations with higher strain values in the FE analysis. The strain contour from FEA of the domain and the boundary conditions of bridge problem is shown in Fig. 13(a). The contour plot as

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Fig. 12. Bridge problem. (a) Geometry of design domain and boundary condition, (b) single material topology optimization result and Y Z plane view, (c) multi-material topology optimization result and its Y Z and X Y views. Orange and tan colors correspond to materials with E = 3 and E = 1 respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. Strain contour from FEA of the bridge problem is shown. Topology optimization result to the bridge problem with multi-materials. The places where multi-material assigned stiff material coincides with the high strain locations in the FEA result. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

expected indicates that higher strain is expected near the support and the top center. The multi-material topology optimization solution is illustrated which shows that the stiff materials (orange) are assigned in the areas with higher strains. Comparing Fig. 13(a) and (b), it is evident that the optimized multi-material topology gives reasonable result. 4.5. 4-legged stool problem In this example, a 3D cube which is simply supported at the 4 corners of the base and subjected to a unit vertical loading at the center of the bottom surface is considered. This usually generates a solution that resembles a 4-legged stool. The four corner nodes of the bottom surface of the cube are fixed in all degrees of freedom. This cubic domain is discretized into a 12 × 12 × 12 grid resulting in a total of 1728 B8/n125 elements. The geometry and the prescribed boundary condition are shown in Fig. 14(a). Volume fraction constraint is 10%, rmin = 1, and p = 3 are chosen. For the multi-material topology optimization, 5% volume fraction constraints are assigned for both the stiff and the soft material. The ratio of the Young’s modulus (E stiff /E soft ) of 3 is selected. Filter size and penalization factor remains the same. The single and multi-material topology optimization results are given in Fig. 14(b) and (c). As shown in Fig. 14, overall configurations of the results are similar but they also exhibit distinct features at the center region. 5. Conclusion This paper presents a new topology optimization procedure in 3D that blends a multi-resolution approach with a multi-material topology optimization technique. It has successfully incorporated the multi-resolution topology optimization method into an alternating active phase algorithm based on the concept similar to the Gauss–Seidel iterative scheme. The method generates higher resolution designs in an efficient manner. The iso-surfaces obtained from the 3D numerical examples using the proposed approach demonstrate crisp topologies. It is found that introducing multi-material concept into topology optimization brings limited complexity but also opens up new opportunities for designers and engineers as it suggests potential design alternatives that are not intuitive. The objective

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Fig. 14. 4-legged stool problem. (a) Geometry and boundary condition, (b) single material topology optimization result with views from Y Z and X Y plane, (c) result using multi-material (2 material) topology optimization, also views from Y Z plane and Z X planes are shown for clarity. Green and tan colors correspond to materials with E = 3 and E = 1 respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

function considered in this study can be easily modified to accommodate multi-scale and/or multi-objective problems. For example, the algorithm can be used to create feasible multi-material topologies for smart structures which can efficiently withstand mechanical, thermal and/or hydrostatic loadings. Another promising area that can be benefited with this procedure is the bone tissue engineering and implant technology which needs to be optimized for mechanical, chemical and biological considerations [16,63,64]. Innovative scaffold for bone growth may be engineered with multimaterials by taking advantage of composite materials with recent developments in multi-material 3D printers. Acknowledgments The work was partially funded by National Science Foundation through the CMMI Award No. 1032884. We would like to acknowledge Dr. Tam H. Nguyen for useful discussions regarding the multi-resolution topology optimization method. We also thank Professor Glaucio H. Paulino (University of Illinois at Urbana–Champaign) for inspiring us to work in the topology optimization area. We would like to thank Dr. Michael J. Miller (Chair, Department of Plastic Surgery, The Ohio State University) for his unconditional support to continue this research work. References [1] M.P. Bendsoe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Math. 71 (1988) 197–224. [2] P. Pedersen, N.L. Pedersen, Interpolation/penalization applied for strength design of 3D thermoelastic structures, Struct. Multidiscip. Optim. 45 (2012) 773–786. [3] T. Borrvall, J. Petersson, Topology optimization of fluids in Stokes flow, Internat. J. Numer. Methods Fluids 41 (2003) 77–107. [4] N. Aage, T.H. Poulsen, A. Gersborg-Hansen, O. Sigmund, Topology optimization of large scale stokes flow problems, Struct. Multidiscip. Optim. 35 (2008) 175–180. [5] E. Wadbro, M. Berggren, Topology optimization of an acoustic horn, Comput. Methods Appl. Math. 196 (2006) 420–436. [6] O. Sigmund, J.S. Jensen, Systematic design of phononic band-gap materials and structures by topology optimization, Philos. Trans. R. Soc. A 361 (2003) 1001–1019. [7] B. Stanford, P. Beran, Conceptual design of compliant mechanisms for flapping wings with topology optimization, Aiaa J. 49 (2011) 855–867. [8] J.K. Guest, J.H. Prevost, Design of maximum permeability material structures, Comput. Methods Appl. Math. 196 (2007) 1006–1017. [9] Y.H. Chen, S.W. Zhou, Q. Li, Multiobjective topology optimization for finite periodic structures, Comput. Struct. 88 (2010) 806–811. [10] Y.H. Chen, S.W. Zhou, Q. Li, Computational design for multifunctional microstructural composites, Internat. J. Modern. Phys. B 23 (2009) 1345–1351.

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