Multi-objective topology optimization of multi

Struct Multidisc Optim DOI 10.1007/s00158-014-1154-3

RESEARCH PAPER

Multi-objective topology optimization of multi-component continuum structures via a Kriging-interpolated level set approach David Guirguis & Karim Hamza & Mohamed Aly & Hesham Hegazi & Kazuhiro Saitou

Received: 4 October 2013 / Revised: 25 July 2014 / Accepted: 29 July 2014 # Springer-Verlag Berlin Heidelberg 2014

Abstract This paper explores a framework for topology optimization of multi-component sheet metal structures, such as those often used in the automotive industry. The primary reason for having multiple components in a structure is to reduce the manufacturing cost, which can become prohibitively expensive otherwise. Having a multi-component structure necessitates re-joining, which often comes at sacrifices in the assembly cost, weight and structural performance. The problem of designing a multi-component structure is thus posed in a multi-objective framework. Approaches to solve the problem may be classified into single and two stage approaches. Two-stage approaches start by focusing solely on structural performance in order to obtain optimal monolithic (single piece) designs, and then the decomposition into multiple components is considered without changing the base topology (identical to the monolithic design). Single-stage approaches simultaneously attempt to optimize both the base topology and its decomposition. Decomposition is an inherently discrete problem, and as such, non-gradient methods are needed for single-stage and second stage of two-stage approaches. This paper adopts an implicit formulation (level-sets) of the design variables, which significantly reduces the number of design variables needed in either single or two stage approaches. The number of design variables in the formulation is independent

An earlier version of this manuscript was presented in WCSMO-10 (May 2013), Orlando, FL, USA D. Guirguis : M. Aly Mechanical Engineering, American University in Cairo, Cairo, Egypt K. Hamza (*) : K. Saitou Mechanical Engineering, University of Michigan, Ann Arbor, USA e-mail: [email protected] H. Hegazi Mechanical Design and Production, Cairo University, Cairo, Egypt

from the meshing size, which enables application of nongradient methods to realistic designs. Test results of a short cantilever and an L-shaped bracket studies show reasonable success of both single and two stage approaches, with each approach having different merits. Keywords Multi-objective topology optimization . Multi-component structures . Level sets . Kriging . Genetic algorithms

1 Introduction Topology optimization of continuum structures aims at efficient allocation of structural material within a design domain to perform one or more desirable functions. Much work has been done since the pioneering work of Bendsøe and Kikuchi (1988) in terms of applications and methods. Example review articles may be found in (Rozvany 2001; 2009; Saitou et al. 2005; Sigmund and Maute 2013). Objectives in topology optimization have included compliance minimization (Bendsøe 1989; Sigmund 2001), Compliant Mechanisms (Sigmund 1997), stress (Allaire et al. 2004a), eigenfrequency (Diaz and Kikuchi 1992; Ma et al. 1995), crashworthiness (Mayer et al. 1996; Gea and Luo 2001) and reliability (Kharmanda et al. 2004). In most work, it is either assumed that the structure can be monolithic (manufactured as one piece), or that decomposition into multiple components is an independent problem from the topology optimization. Decomposition into multiple components is often necessary and/or desirable in large structures due to manufacturability considerations. Common assumptions of prior knowledge about joint locations (Jiang and Chirehdast 1997; Chickermane and Gea 1997; Li et al. 2001) or independence of the decomposition problem from base topology are debatable issues. A simple example (Hamza et al. 2013a) of an in-

D Guirguis et al.

plane loaded sheet metal structure with a coarse-mesh that allows full enumeration of all possible topologies is shown in Fig. 1. Out of more than two million topologies, about twenty thousand represent feasible multi-component designs, and out of which, 37 are Pareto-optimal in terms of compliance, weight, assembly and manufacturing costs. Notable observations in the example are: i)

ii)

Monolithic designs (such as solutions 1, 11, 20) are often Pareto-optimal for base topologies that have Pareto-optimal solutions. This is likely due to monolithic designs having zero assembly cost, and usually less weight and compliance (joints involve sheet folding, and are often less stiff than the base sheet metal) Monolithic designs are not always among the Paretooptimal solutions however. For example, solutions 16– 19 are Pareto-optimal solutions, but the monolithic design with the same base topology is not Pareto-optimal. This is because the monolithic design in question is dominated by solution 20. However, decompositions of that base topology into solutions 16–19 gives different merit compared to solution 20 and its decompositions.

The assumption of independence between the optimization problems of base topology and its decomposition can

significantly simplify the design task by adopting two-stage approaches. In two-stage approaches, the base topology is optimized first (via any of the established approaches) as a monolithic structure, and decomposition is considered in a second stage in order to explore options that have better manufacturability. However, two-stage approaches cannot generate all multi-component Pareto-optimal designs, as the ones whose base topology is sub-optimal (such as designs 16– 19 in Fig. 1) are not explored. More intuitively, if a design modification to the base topology results in (even small) compromise in structural performance but brings significant improvement in manufacturability, that modification is not explored in two-stage approaches. Single-stage optimization (simultaneously optimizing both the base topology and its decomposition) can eliminate this limitation (Yildiz and Saitou 2011; Zhou et al. 2011a; b) however, single-stage optimization faces a lot of difficulty in non-gradient problems with large number of design variables. Research in topology optimization may be classified from point of view of design variables into explicit and implicit representation. In explicit representation, one design variable is associated with each element in the finite element (FE) mesh, while implicit representations rely on indirect design realization via some domain classifier. Optimization methods that are used may be classified into gradient and non-gradient.

Fig. 1 Example simple structure and Pareto-optimal multicomponent topologies

Load Always void

Always Material Design Domain

(a) Domain, Loading & Boundary Conditions

… (1)

(2)

(11)

(12)

(13)

(14)

(15)

… (16)

(17)

(18)

(19)

(20)

(b) Excerpt of Pareto-Optimal Designs

(21)

Multi-objective topology optim of multi-component structures

Gradient methods (Bendsøe and Kikuchi 1988; Bendsøe 1989; Diaz and Kikuchi 1992; Ma et al. 1995; Sigmund 2001; Allaire et al. 2004a; Liu et al. 2011) rely on computing/estimating the sensitivity of objective(s) to changes in the design variables from the results of FE solution, then adjusting the design variables in small increments towards perceived improvements. The main advantage of gradient methods is the capability to handle a very large number of design variables. However, some of the downsides to gradient optimization are possible entrapment in local optima if the objective is non-convex, and the inability to solve problems that have objectives with no known gradients. Non-gradient methods such as genetic algorithms (GA) (Deb and Gulati 2001; Madeira et al. 2005) on the other hand can solve for any defined objective(s) and are not easily trapped in local optima. The downsides of non-gradient methods include some inaccuracy and randomness in the obtained solutions (several runs may are often needed to obtain reliable results, which can require a lot more computational resources than gradient methods), and the inefficiency/inability to solve problems with a large number of design variables. Methods such as evolutionary structural optimization (Xie and Steven 1993) combine traits of both gradient and non-gradient methods to avoid downsides such as local optima, but end up inheriting some limitations of both types of methods such as reliance on analytical gradients and limitation (though not as severe) on the number of design variables that can be efficiently/reliably solved. Non-gradient methods using explicit representation received serious critique in (Sigmund 2011) with regards to applicability in continuum structures. The main issue is that continuum structures often require hundreds of elements in the FE mesh. Since the number of design variables is often equal to the number of elements (or comparatively large), nongradient methods run into serious difficulty even in simple problems. However, with the manufacturing and assembly objectives of interest in this paper not having known gradients, non-gradient methods are inevitable. While previous studies adopting explicit representation (Yildiz and Saitou 2011; Zhou et al. 2011a; b) achieved some success with a relatively coarse (20×10) FE mesh, the limitation of nongradient optimization is all but too clear even when adopting two-stage approaches. In order to overcome difficulties in non-gradient optimization, this paper adopts an implicit representation based on level-sets. Other meshing-independent approaches include projection maps (Guest et al. 2004; Almeida et al. 2009), adaptive design variable fields (Guest and Genut 2010), density filters (Bourdin 2001; Burns and Tortorelli 2001; Sigmund 2007), wavelets (Poulsen 2002), NURBS basis functions (Azegami et al. 2013), as well as others (for more detailed survey, readers are referred to Deaton and Grandhi 2014). Level-sets seemed a natural choice in this work since

they were originally introduced as a classification technique (Osher and Sethian 1988). Level sets were adopted in topology optimization in the early 2000’s (Wang et al. 2003; Allaire et al. 2004b; De Ruiter and Van Keulen 2004), in which a scalar level-set function is used to classify the design domain into “structural material” and “void” based on the value of the function relative to some threshold (illustration shown in Fig. 2, which was generated with a level set function that had 27 independent design variables). Typical level set approaches (Wang et al. 2003; Allaire et al. 2004b; Wang and Wang 2006; Wei and Wang 2009; Yamada et al. 2010; James and Martins 2012) involve a Hamilton-Jacobi system of partial differential equations, with a large number of degrees of freedom, which in turn requires a gradient-based solver. The approach adopted in this paper however, relies on a Kriginginterpolated level set (KLS) (Hamza et al. 2013b), which defines the design variables as the level-set function values at knot points that are interpolated in the rest of the design domain via a Kriging model (Simpson et al. 2004). The KLS formulation can significantly reduce the number of the design variables and makes it independent from the meshing of the FE model. The reduction in the number of design variables is perceived to improve the effectiveness of non-gradient optimization. Single-objective compliance minimization in three different problems (summarized in Fig. 3) was examined in (Hamza et al. 2013c) and was observed to be no worse than 6 % relative to gradient based topology optimization. Testing of KLS formulation for multi-component structures is the focus of this paper, and examines both single and two-stage approaches, along with a finer FE mesh than in (Yildiz and Saitou 2011; Zhou et al. 2011a; b). The paper started with a brief review of relevant work and the difficulties associated with the problem of interest.

Fig. 2 Example Kriging-interpolated level set function in a design domain and its realization into a structural design

KLS Topology (46 design variables)

0.45

4.1

0.6

1.6

Bridge Problem

Gradient-based Topology

0.45

1.1

0.6

0.0

Bookshelf Problem

Symmetric Cantilever Problem

Volume Fraction

Fig. 3 Summary of compliance minimization test problem results in (Hamza et al. 2013c)

KLS % Error

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0.45

6.1

0.6

0.0

Section 2 provides the details of the mathematical model of the objectives and design variables. Section 3 presents details of the adopted algorithms. Sections 4 and 5 conduct test runs and comparative analysis of two case studies of a short cantilever structure as well as an L-shaped bracket. The paper then concludes with a discussion of future work.

2 Problem formulation 2.1 Generation of multi-component finite element models Evaluating the structural objectives during optimization requires the generation of a multi-component FE model. A generic way of treating multi-component topology is to consider a “base topology” (also referred to as the “equivalent monolithic structure) which represents the union of all the structural components, and “joint topology”. The joint topology represents separation lines/curves that divide the base topology into multiple components. This work follows the meshing scheme in (Yildiz and Saitou 2011; Zhou et al. 2011a; b), where a continuum domain (Fig. 4a) is discretized

into (nElx × nEly) large square elements which are termed “main cells” (Fig. 4b), and represent candidate locations for “structural material”. Narrow vertical and horizontal rectangular strip elements, and small square diagonal elements represent candidate locations for joints (Fig. 4b). The discretized domain corresponds to a ground topology graph (Fig. 4c), where main cells are represented by the nodes of the graph, while vertical and horizontal elements are represented respectively by horizontal and vertical edges of the graph. Any instance of the design variables (discussed in detail in section 2.2) can be used to generate a product topology graph (Fig. 4d), which is a sub-graph of the ground topology graph (Fig. 4c). Graph search techniques are then used to assess the validity of the design, and if valid, a multi-component FE model (Fig. 4e) that corresponds to the product topology graph is generated, which can then be analyzed via FE analysis in order to evaluate structural performance objectives. In the current formulation, elements in the FE model are assigned “structural material”, “joint material” or “void material” (which has very low, but non-zero stiffness in order to avoid numerical singularities in the FE solver). Assigning e that is element material depends on an integer matrix X

Multi-objective topology optim of multi-component structures Fig. 4 FE modeling outline: a Continuum design domain, b Discretized design domain, c Ground topology graph, d a product topology graph, e a multicomponent structure

Design Domain (a)

(d) e has dimensions nElx×nEly obtained from the graph search. X and includes the IDs of structural components, with zero e from the product indicating void. Generation of the matrix X topology graph (Fig. 4d) is done via graph search techniques. The product topology graph is defined via three integer matrices X, Y, Z, which in turn are obtained from the KLS formulation (discussed in detail in section 2.2). Values in the matrices X, Y, Z are summarized as follows: X={Xij}, i=1 to nElx, j=1 to nEly, with Xij =1 implying the node (i, j) is part of the product topology graph. X, which controls the structural material distribution in the main cell elements, represents the base topology of the structural design Y={Yij}, i=1 to nElx—1, j=1 to nEly, with Yij =1 implying the horizontal edge between the nodes (i, j), (i+1, j) of the product topology graph is tentatively a structural joint Z={Zij}, i=1 to nElx, j=1 to nEly—1, with Zij =1 implying the vertical edge between the nodes (i, j), (i, j+1) of the product topology graph is tentatively a structural joint Given X, Y, Z, graph search techniques are applied in order to: i) ii) iii)

Check if structural material in main cells is “connected”. Example of a disconnected product topology graph (and corresponding structure) is shown in Fig. 5. If structure is disconnected, calculate a connectivity e ¼0 penalty function Pcon and set X e If structure is connected, form the matrix X

(c)

(b)

(e)

It should be noted that all the performed graph search tasks rely on the well-established one-to-all shortest path algorithms (Denardo 2003), and are carried out on temporary copies of the product topology graph. Using temporary copy ensures that adding/removing graph edges (discuss later in this section) does not lose information on the product topology graph. First, the connectivity check is done by setting edges of the temporary graph as “connected” (i.e. Y = 0, Z = 0), then performing the one-to-all shortest path search from the first node in the graph in an attempt to reach all the other nodes. If some nodes remain unreachable, this implies an invalid disconnected structure. If the structure is valid (passes the connectivity check), e starts by initializing it to zero, then forming the matrix X conducting a similar procedure to the connectivity check. However, in this check, the graph edges marked as joints in Y, Z are removed (i.e. a graph with more than one component becomes disjoint). All the reachable nodes from the first node in the graph are marked as the “first component” e , then the reachable nodes are removed (so that the in X remaining nodes represent the remaining components in the structure that are not part of the first component). The process is repeated for the remaining nodes in the tempoe as the rary graph, with the reachable nodes marked in X “next component”, until no more nodes remain in the temporary graph. It is noted that labeling components rather than joints has the advantage of automatically eliminating “internal joints” that do not separate structural components.

Fig. 5 Illustration of valid and invalid connectivity in product topology graph and structural designs

(a) Invalid

(b) Valid

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For an invalid structures (ones that don’t pass the connectivity check), a penalty function is proposed as: X Pcon ¼ Pcon1 ðnIslands −1Þ þ Pcon2 IslandDist ð1Þ Evaluation of the connectivity penalty function follows a e , but only to similar procedure to forming the matrix X calculate the number of disjoint material “islands” (nIslands) and the number of main cells in the shortest paths between them (IslandDist). The overall penalty function adds two penalty terms, one term increases with the number of disjointed island segments, and a second term that increases the further apart the disjointed segments are. The weighing of the penalty terms adopted in this work is by setting Pcon1 = 100, and Pcon2 =2 Pcon1 / nElx. It is noted that the connectivity penalty function only comes to play in single-stage approaches where the base topology X is simultaneously optimized alongside the joint topology. FE analysis is not conducted for disjoint designs, but the penalty function provides the optimization a way of preferring a disjoint design over another based on how close the disjoint designs are to being connected. 2.2 Design variables Explicit formulation of the matrices X, Y, Z (Yildiz and Saitou 2011; Zhou et al. 2011a; b) leads to a very large number of design variables even for a coarse FE mesh. In this work, level sets are used instead. For multi-component structures, there are two domain classifications needed; classification into structural material or void (defines the base topology), and classification of the structural material into multiple components. As such, two level set functions are adopted, with a separate subset of the design variables controlling each function. In the KLS formulation (Hamza et al. 2013b; c), the design variables are the values of the level set function at knot points within the design domain, with Kriging models interpolating the value of the functions fitting the knot points, as follows: ΦB ðxÞ ¼ wBo þ

NB X wBi ϕi ðxÞ

ð2Þ

i¼1

Φ J ðxÞ ¼ wJo þ

NJ X wJi ϕi ðxÞ

ð3Þ

i¼1

where x is the spatial position within the design domain. ΦB, ΦJ are respectively the Kriging interpolations of the base topology and joint topology level set functions. wBi for i=1 to NB, and wJi for i=1 to NJ are the weights of the Kriging models, with NB and NJ being the number of knot points used

for the base topology and joint topology level set functions respectively. A Gaussian covariance function ϕi(x) is calculated for each knot point as: φi ðxÞ ¼ e−c

2

kx−V i k2

ð4Þ

where Vi is the spatial position of knot point i, ||.|| is the Euclidean distance and c is a tunable constant that controls the degree overlap between the fitting of knot points, and by extent, the smoothness of level set functions. A parametric study on the tuning of the constant c for single-objective monolithic structures in [32] suggested that the obtainable results via ELS are fairly indifferent to setting of a dimensionless parameter ec within the range of 0.35 and 0.7, with the value of c computed according to the formula: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 c¼ ln 1=ec ð5Þ er where er is the minimum Euclidean distance between two knot points. Since the knot points used in the base topology level set function may be different than the ones used for joint topology, it follows that two different values for the constant c are computed for the Gaussian covariance functions in (Equations 2, 3). Calculation of the Kriging model weights is done by solving the linear system(s) of equations:      CB 1 wB yB ð6Þ ¼ 1T 0 wBo 1



CJ 1T

1 0



wJ wJo



 ¼

yJ 1

 ð7Þ

where y={yBT, yJT}T is the vector of design variables of the optimization, yB ={yBi}T for i=1 to NB and yJ ={yJi}T for i=1 to NJ are the values of the level set function at the base topology and joint topology knot points respectively. The matrices CB and CJ are matrices of cross-covariance values between knot points, i.e. C={Cij}, with Cij =ϕi(Vj). All design variables in the adopted formulation have the ranges −1≤yi ≤ 1, i=1 to NB +NJ. Classification threshold for the base topology level set function ΦT is calculated via a line search procedure (Hamza et al. 2013b). For an instance of the design variables yB dictating the profile of ΦB(x), increasing the threshold ΦT means fewer main cell elements will contain structural material, and decreasing it will increase the number of main cell elements containing structural material. Thus, for any instance of the design variables, ΦT can be adjusted to obtain a desired structural material fraction volume V in the main cells. The classification threshold for the joint

Multi-objective topology optim of multi-component structures

topology level set function is simply set to zero, which implies that regions of the design domain that have different sign for ΦJ will belong to different components, with ΦJ ≈0 being the separation lines. Formation of the matrices X, Y, Z from the level set functions is done as:   X ij ¼ X¯ij þ X ij −X¯ij X ij X ij ð8Þ 

1 0

X¯ij ¼ (

1 0

Y ij ¼

 Z ij ¼

1 0



if ΦB x1 ≥ ΦT otherwise

if Φ J

ð9Þ

 

x1 ⋅Φ J x2 < 0 otherwise

ð10Þ



if Φ J x1 ⋅Φ J x3 < 0 otherwise

ð11Þ

   ði − 1Þa þ b ia þ b ; x2 ¼ ; x3 ð j − 1Þa þ b ð j − 1Þa þ b   ði − 1Þa þ b ¼ ja þ b

x1 ¼



2.3 Optimization problem ð12Þ

where X ij and X ij are binary masking matrices for structural material and void respectively (optional input by user to facilitate treatment of a more complex design domain than a simple rectangle). It should be noted that the multiplication in (Equation 8) is an element by element multiplication (not matrix product). As such, setting an element X ij ¼ 0 forces the corresponding main cell to be void irrespective of the value in Xij . Likewise, setting X ij ¼ 1 forces the main cell to include structural material (unless overridden by setting X ij ¼ 0 ). x1,x2,x3 are respectively the position vectors (Fig. 6) for the center of the main cell ij, the cell to its right and the cell above it. a is the unit cell distance between the centers of main cell elements while b is half the length of one main cell element. Xij in (Equation 9) sets the base material marker (which may

a x3 1

x

be overridden by the masks X ij and X ij in (Equation 8) based on the value of the base topology level set function at the center of a main cell element. (Equations 9, 10) on the other hand set the joint markers based on a change of sign in the joints level set function between the center of the cell ij and at the centers of the adjacent cells to the right and top. When considering single-stage approaches, the number of design variables is equal to NB +NJ, and given an instance of the design variables y, (Equations 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) are used for forming the matrices X, Y, Z, which are then analyzed via graph search (as discussed in section 2.1) and employed to generate a multi-component FE model in order to estimate the values of objectives, as discussed in section 2.3. When considering two-stage approaches, the matrix X is first determined via some other topology optimization approach while only considering structural objectives and monolithic structures. In the second stage of two stage approach, the matrix X is constant and the number of design variable is equal to NJ. Only (Equations 3, 4, 5, 7, 10, 11, and 12) are needed in order to form the matrices Y, Z.

b

The multi-component topology optimization is posed as multi-objective problem as: Minimize f ðyÞ

ð13Þ

Subject to g ðyÞ≤ 0

ð14Þ

where g(y) is a vector of constraints, which may contain several constrains on structural performance as a designer sees fit. g(y) however contains at least one constraint for the structural connectivity: g1 ðyÞ ¼ Pcon ðyÞ≤ 0

f (y) is a vector of objectives, which may contain one or more structural performance objectives { f1(y), …, fnStrObj (y)} pertaining to one or more load cases and/or types of FE analysis. These objectives are calculated from the results of FE analysis. Other than structural performance objectives, the following are considered: fnStrObj+1

2

x

fnStrObj+2 fnStrObj+3

Fig. 6 Center point positions of main cell ij and adjacent cells

ð15Þ

is the structural weight, calculated from the FE model is an estimate of assembly cost of all the components in the structure is an estimate of manufacturing cost of all the components in the structure

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The assembly cost objective is estimated as a weighed sum of the number of components in a structure (pertains to the number of joining operations and assembly stations) and the total length of welding to be performed. This is summarized as: f nStrObjþ2 ðyÞ ¼ a1 ðnComp −1Þ þ a2 Lwelds

ð16Þ

The manufacturing cost objective is calculated as the sum of manufacturing cost of all components, which in turn depends on the size and shape complexity of the parts. The model adopted in (Zhou et al. 2011a), (b) assumes a polynomial as: f nStrObjþ3 ðyÞ ¼

nComp X

X

i¼1

j;k

mjk Aij ψki

ð17Þ

where a1 and a2 are assembly cost coefficients associated with the number of joining operations and length of welds respectively. nComp is the number of components in the structure. mjk are the polynomial coefficients for estimating the manufacturing cost, while Ai and ψi for i=1 to nComp, are respectively the convex hull area in which a component fits, and the shape complexity factor. The shape complexity factor is calculated based on the model in (Boothroyd et al. 1994) as the ratio of square of the perimeter to the convex hull area: ψi ¼

p2i Ai

ð18Þ

When only considering up to the first order terms in the polynomial, the cost estimate reduces to: f nStrObjþ3 ðyÞ ¼

nComp X

ðm00 þ m01 Ai þ m10 ψi Þ

ð19Þ

i¼1

It should be noted that values for cost estimation used in the numerical studies in section 4 of this work are only demonstrative and do not represent any particular real system. For a real system, the cost coefficients in (Equations 16, 17 (or 19)) may be estimated via linear/polynomial regression of estimated or actual costs of sample products (requires at least as many sample products as the number of coefficients).

3 Optimization algorithms 3.1 Overview Algorithms for optimization of the multi-objective problem formulated in section 4 adopted in this paper rely on NSGA-II

(Deb et al. 2002) along with the crossover and mutation parameters recommended by Michalewiz and Fogel (2000), and a univariate local search (Hamza et al. 2013b) which is conducted after the GA runs. In previous studies (Yildiz and Saitou 2011; Zhou et al. 2011a; b) bit was observed that the number of quasi-Pareto solutions to the optimization problem in (Equations 13, 14) can be very large, which can pose a lot of difficulty for single-stage approaches in producing meaningful results. A main perceived reason for this difficulty is that ultra-lightweight designs that have very poor structural performance and/or designs that have decent structural performance but are overly heavyweight (with “dangling” masses) are both easy to generate, and hard to eliminate from the set of undominated solutions in GA, which can clutter the set of undominated solutions and hinders its evolution towards the proper Pareto frontier. To eliminate this difficulty, single-stage approaches in this paper adopt a search intensification strategy, where separate GA runs are conducted for preset/fixed values of V={V1, … , VnVolFrac} for the material fraction volume of main cell elements. In the second stage of twostage approaches, the obtainable solutions are bound to a fixed base topology from the first stage, and values from V are used (material fraction volume is a required input in most topology optimization approaches) in the first stage to generate the base topology. Summary of the inputs to optimization approaches adopted in this paper are: 123456-

Domain length, nElx, nEly Material/void masks: X ij and X ij Material properties of the sheet metal Stiffness and mass properties of joints Loading cases and boundary conditions Cost estimation coefficients for assembly and manufacturing 7- Material fraction volume of main cells to explore V={V1, … , VnVolFrac} 8- Tuning parameters for the search Tuning parameters for the search include the number and layout of knot points Vi for i=1 to NB +NJ, GA population size nPop and number of generations nGen, number of runs nSubRuns, number of Pareto solutions nSubPareto to retain for each value of the material fraction volume, as well as the overall number of Pareto solutions nPareto to report. A parametric study is conducted in (Hamza et al. 2013b) on the choice of knots points, population size and number of generations of the GA for the KLS formulation. Increasing the number of knot points allows more freedom for the level set function, which in turn allows it to properly represent more complex topologies thereby improving the quality of “obtainable” solutions. However, excessively increasing the number of knot points has the downside of increasing the number of design variables, which in turn requires more computational

Multi-objective topology optim of multi-component structures

resources for the GA to reliably produce good results. Choices of the tuning parameters in this paper are based off the recommendations in (Hamza et al. 2013b) despite that study being based on single objective problems. It may be possible to attain better performance by further tuning of the search parameters, but such a task is more difficult in multi-objective problems and is beyond the scope of this work.

3.2 Single-stage approach Main steps of the adopted algorithm for single-stage optimization are as presented follows: 1. For i=1 to nVolFrac %% Single Stage: Multi-component structure 2. Si ←{} 3. For j=1 to nSubRuns 4. Conduct a run of NSGA-II (with a new random number seed) with population size and number of generations nPop , nGen respectively with consideration of all objectives f(y) and constraints g(y), and setting Vi as the material fraction volume for calculating the threshold ΦT 5. Attempt to add all solutions in final population to Si 6. Limit the number of solutions retained in Si to only nSubPareto solutions 7. Next j 8. Next i 9. 9. e S ← { S1, … , SnVolFrac} 10. Limit the number of solutions retained in e S to only nPareto solutions 11. S←{} 12. For i=1 to nPareto 13. Run univariate search from starting point solution i in e S 14. Attempt to add result of the univariate search to S 15. Next i 16. Return S The univariate search (line 13), adopted from (Hamza et al. 2013b) attempts to adjust each design variable at a time. If adjustment of a variables results in improving the solution, the new solution replaces current one and the process is repeated. Reducing the number of solutions in a set of undominated solutions (lines 6, 10) is based on retaining the solutions having largest “crowding distance” (Deb et al. 2002), which keeps the retain ed so lutions well spread-out on the undominated frontier. The “attempt to add solution” (lines 5, 14) involves two checks: i)

If candidate solution is Pareto-dominated by any solution in the solution set, it is not added

ii)

If candidate solution dominates any solution in the solution set, the dominated solutions are removed

3.3 Two-stage approach Main steps of the adopted algorithm for two-stage optimization are as presented follows: 1. For i=1 to nVolFrac %% First Stage: Base topology monolithic structure 2. Run topology optimization considering only the structural objectives and monolithic structure to obtain a base topology Xi for material fraction volume Vi 3. Next i 4. For i=1 to nVolFrac %% Second Stage: Multi-component structure 5. Si ←{} 6. For j=1 to nSubRuns 7. Conduct a run of NSGA-II (with a new random number seed) with population size and number of generations nPop , nGen respectively with consideration of all objectives f(yJ | Xi) 8. Attempt to add all solutions in final population to Si 9. Limit the number of solutions retained in Si to only nSubPareto solutions 10. Next j 11. Next i 12. e S ← { S1, … , SnVolFrac} 13. Limit the number of solutions retained in e S to only nPareto solutions 14. S←{} 15. For i=1 to nPareto 16. Run univariate search from starting point solution i in e S 17. Attempt to add result of the univariate search to S 18. Next i 19. Return S

4 Example studies 4.1 Short cantilever A study of the performance of single and two stage approaches is conducted for a short cantilever structure whose boundary conditions and loading are shown in Fig. 7. Only one loading case is considered and one structural performance objective, which is the minimization of compliance at the loading point. Material properties (assuming mild steel) and cost estimation coefficients are summarized in Table 1. It is noted that the joints model assumes lap joints, hence the 2.0

D Guirguis et al. Table 2 Tuning parameters for optimization of short cantilever

Design Domain Fig. 7 Design domain and boundary conditions for short cantilever problem

multiplier on the mass of joint elements due to overlapping sheets and the isoperimetric in-plane stiffness behavior, with somewhat less stiffness than the base sheet metal (Xu and Deng 2004). Tuning parameters for the optimization algorithms are listed for single and two-stage approaches in Table 2. A total of ten complete runs were conducted for both single and two stage approaches following the steps discussed in sections 3.2 and 3.3 respectively. Normalized Pareto-plots (objectives scaled with respect to the maximum and minimum among the undominated set of solutions) from all runs are shown in Fig. 8, and select structural designs are shown along with spider-web diagrams of the objective values in Fig. 9. It should be noted that while the objectives are all to be minimized (smaller values are better), axis of the spider-web diagrams have been reverse scaled (for better visibility of results), so “better performance” is towards the outer circumference. The base topology in two-stage approaches was generated via power-law method using publicly available code (Sigmund 2001) for the respective values of material fraction volume. Monolithic designs corresponding to the base topology are observed among the undominated set of solutions (designs B1, B4, B7, B10 in Fig. 9). It is observed in Fig. 8

Table 1 Material properties and cost estimation coefficients

Parameter

Single-stage

Material fraction volume in main cells Base topology knot points Joint topology knot points nPop nGen nSubRuns nSubPareto nPareto

{0.45, 0.5, 0.55, 0.6} {0.45, 0.5, 0.55, 0.6} 41 points, zigzag grid 15 points, square grid 150 400 5 30 50

Two-stage

– 15 points, square grid 80 120 5 30 50

that the Pareto frontier may be including a very large number of solutions, and in earlier tests, single stage approaches had most of the solutions drifting towards low manufacturing cost 1.0

g2

f2: structural weight

Always Material

Single-stage

g2

Two-stage

g2

g2 0.0 0.0

f1: structural compliance

1.0

1.0

Value

Unit

Young’s modulus Poisson’s ratio Density Design domain length nElx nEly Sheet metal thickness (t) Length to width ratio of rectangular elements Joint elements mass multiplier Joint elements stiffness multiplier Assembly cost coefficient (a1) Assembly cost coefficient (a2) Manufacturing cost coefficient (m00) Manufacturing cost coefficient (m10) Manufacturing cost coefficient (m01)

207 0.3 7,800 2.0 40 20 1.2 10:1 2.0 0.65 1.00 5.00 0.00 1.0 4.0×10−4

GPa

Single-stage

3

kg/m m

mm

$ $/m $ $/m2 $

Two-stage

f4: manufacturing cost

Parameter

0.0 0.0

f3: assembly cost

1.0

Fig. 8 Normalized Pareto-plots for single and two stage approaches of short cantilever problem

Multi-objective topology optim of multi-component structures Fig. 9 Details of select designs from single and two stage runs undominated solutions in short cantilever problem

f1 Design A1

Design B1

f4 f2 Design A2

Design B2

f3 Design A3

Design B3

(a) V1 = 0.45 f1

Design A4

Design B4

f2

f4 Design A5

Design B5

f3 Design A6

Design B6

(b) V2 = 0.50 f1 Design A7

Design B7

f2

f4 Design A8

Design B8

f3 Design A9

Design B9

(c) V3 = 0.55 f1 Design A10

Design A11

Design B10

Design B11

f2

f4

f3 Design A12

Design B12

(d) V4 = 0.60

D Guirguis et al.

A second study of the performance of single and two stage approaches is conducted for an L-shaped bracket (Le et al. 2010; Wang and Li 2013) whose boundary conditions and loading are shown in Fig. 10. Material properties and cost parameters are the same as in Table 1 (with the exception of domain length being 1.0 m and width ratio of rectangular elements 5:1), and the GA and search parameters are summarized in Table 3. It is perhaps worth noting that knot points and material fraction volume are set for the design domain (hatched area in Fig. 10) only. A total of ten complete runs were conducted for both single and two stage approaches. Normalized Pareto-plots from all runs are shown in Fig. 11, and select structural designs are shown along with spider-web diagrams of the objective values in Fig. 12. For better readability, axis of the spider-web diagrams have been reverse scaled (i.e. “better performance” in minimization objectives appears towards the outer circumference). The base topology in two-stage approaches was generated via power-law method using the publicly available code in (Andreassen et al. 2011) for the considered material fraction volume. Similar to the study in section 4.1, a constraint g2(y) was imposed to eliminate the uninteresting regions of the Pareto set that have poor structural performance.

Always Material Load 1.0

Design Domain

Parameter

Single-stage

Two-stage

Material fraction volume in main cells Base topology knot points Joint topology knot points nPop nGen nSubRuns nSubPareto nPareto

{0.44, 0.58}

{0.44, 0.58}

41 points, zigzag grid 16 points, square grid 150 400 5 30 50

– 16 points, square grid 80 120 5 30 50

1.0

g2

f2: structural weight

4.2 L-shaped bracket

Table 3 Tuning parameters for L-shaped bracket

Single-stage Two-stage

g2 0.0 0.0

f1: structural compliance

1.0

1.0 Single-stage Two-stage

f4: manufacturing cost

with poor structural properties. To remedy this, an additional constraint g2(y) was imposed to eliminate those uninteresting regions of the Pareto set. g2(y) required the compliance to be no more than three times the minimum for the respective material fraction volume (from designs B1, B4, B7, B10). Limits of g2(y) are sketched in Fig. 8. Figure 9 is organized in increasing order of material fraction volume and type of approach used: Designs A1 to A12 are from single-stage runs, while designs B1 through B12 are from two-stage runs.

0.4

0.4 1.0

Fig. 10 Design domain and boundary conditions for L-shaped bracket problem

0.0 0.0

f3: assembly cost

1.0

Fig. 11 Normalized Pareto-plots for single and two stage approaches of L-shaped bracket

Multi-objective topology optim of multi-component structures Fig. 12 Details of select designs from single and two stage runs undominated solutions in Lshaped bracket problem

f1

Design A1

Design B1

f4 f2

f3 Design A2

Design B2

(a) V1 = 0.44 f1

Design A3

Design B3

f2

f4

f3 Design A4

Design B4

(b) V2 = 0.58 5 Discussion It is understandably evident (Figs. 8, 11) that material fraction volume in the main cells is the major contributing factor in the structural weight objective f2, with decomposition into multiple components having some, but minor effect on it. It is also observed that fixing the base topology all but locks the value of the structural compliance objective f1. With weight and stiffness not changing much via decomposition of a base topology, two-stage results appear lumped into narrow spots in the f1—f2 plot in Figs. 8, 11. While designs obtained via the two-stage approach appear to dominate those obtained from single-stage approach in f1—f2 plot, the opposite appears to be true in f3 – f4 plot. This appears to imply that single and two stage approaches are converging to different regions of the Pareto frontier, which may be a natural outcome from the way the approaches are set. In the two-stage approach, the base topology is restricted to that of the best structural-only

monolithic structure. It therefore lacks the capability to make compromises (improving assembly and manufacturing) that involve a change of the base topology. On the other hand, observing the resulting topologies via KLS formulation of base topology (Fig. 3) shows that KLS may lack some freedom in representing complex shapes, which plays in favor of compromises that improve manufacturing cost. Further examination of the design freedom and capability to represent complex shapes in base/joints topology is an ongoing work (Guirguis et al. 2014) whose preliminary results indicate that near-perfect representation of the base topologies of designs B1-B12 in Fig. 9 is possible with approximately double the density of knot points used in this work. One notable limitation of current KLS formulation of joints topology is that T-joints (three components meeting) cannot be exactly represented, as illustrated in Fig. 13. Possible future remedies to this may be use of multiple threshold values for joints topology (instead of the current one threshold at zero value of the level set function).

D Guirguis et al. Table 5 C-metric analysis for L-shaped bracket problem

Component 1

Component 1

Run ID Joint Comp. 3

Comp. 2

Comp. 3

Comp. 2

Gap

Can be represented

Cannot be represented

Fig. 13 Illustration of a limitation of current joints topology formulation that relies on change of sign in level set function

A quantitative comparison is conducted between the solutions obtained from the various runs via the “C-metric” (Zitler and Thiele 1999), which is a means for comparing sets of solutions for multi-objective problems. Given two sets of solutions S1 and S2, the C-metric C12 is the fraction of solutions in S1 that are dominated by at least one solution in S2. Large values (closer to 1.0) of C12 indicate poor quality in the set of solutions S1 compared to S2, however, small values (closer to zero) do not necessarily indicate superiority of S1 since the solutions may be skewed towards different zones of the Pareto frontier. Since pairwise comparisons (set of solutions from one GA run verses another run) may not be very indicative, a modified C-metric is defined between the results of one set of solutions and the undominated frontier from groups of runs as: CiS

CiT

CiA

is the percentile of solutions in solution set i that are dominated by at least one solution in all the results of single-stage runs is the percentile of solutions in solution set i that are dominated by at least one solution in all the results of two-stage runs is the percentile of solutions in solution set i that are dominated by at least one solution in all runs from both single and two-stage approaches

Table 4 C-metric analysis for short cantilever problem Run ID

1 2 3 4 5 6 7 8 9 10

Single stage runs

Two stage runs

Joint

Single stage runs

Two stage runs

CiS

CiT

CiA

CiS

CiT

CiA

0.426 0.391 0.362 0.455 0.298 0.400 0.596 0.489 0.386 0.189

0.000 0.022 0.106 0.159 0.021 0.067 0.128 0.064 0.136 0.027

0.426 0.391 0.426 0.545 0.298 0.422 0.681 0.489 0.432 0.216

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.479 0.298 0.333 0.260 0.327 0.367 0.245 0.520 0.292 0.347

0.479 0.298 0.333 0.260 0.327 0.367 0.245 0.520 0.292 0.347

1 2 3 4 5 6 7 8 9 10

CiS

CiT

CiA

CiS

CiT

CiA

0.857 0.841 0.451 0.463 0.944 0.265 0.640 0.717 0.600 0.393

0.167 0.455 0.176 0.037 0.306 0.061 0.240 0.208 0.180 0.107

0.857 0.841 0.569 0.481 0.972 0.306 0.680 0.736 0.600 0.500

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.025 0.073 0.049 0.073 0.049 0.049 0.024 0.024 0.049 0.073

0.025 0.073 0.049 0.073 0.049 0.049 0.024 0.024 0.049 0.073

Tables 4 and 5 summarize the (CiS, CiT, CiA) values for the results of single and two-stage runs for the short cantilever and L-shaped bracket problems, respectively. A notable observation in Table 4 is that none of the twostage runs contain solutions that are dominated by any of the solutions generated by single-stage runs, which is quite understandable since the f1 value in all solutions from two-stage runs (Fig. 8) is less than its best value in single-stage runs at each material fraction volume. Most (but not all) single-stage runs do include solutions dominated by two-stage runs. However, the maximum dominance percentile appears to be 16 % at most. On the other hand, dominance among runs of the same approach appear to reach much higher percentiles (up to 60 %). This is indicative that the random variations in GA results are more pronounced than the dominance between single and two stage approaches, which supports the assessment that single and two stage approaches are generating solutions at different regions of the Pareto frontier. Overall dominance from among the best of all runs (CiA) shows a fairly large percentile of dominated solutions among all runs (between 29 and 68 %), which may hint that further tuning is needed for (nPop, nGen, nSubRuns), or that multiple runs per problem are recommended. Gradient-based generation of base topology NSGA-II Local Search

Two Stage

Single Stage 0

5

10

15

20

25

Run time (hr) Fig. 14 Bench-marked run times for short cantilever problem

30

Multi-objective topology optim of multi-component structures

Results of the L-shaped bracket problem affirm some of the previous observations in the short cantilever problem, with the increased shape complexity in the problem highlighting some issues more than others. In particular, single-stage runs include higher percentiles of dominated solutions (CiA), with the tendency for most of the dominance to come from other single-stage runs. Similar to the short cantilever problem, no dominance of two-stage runs via single-stage runs was observed to occur, and the dominance of single-stage runs via two-stage runs remains at fairly low percentiles. This further enforces the hypothesis that single and two-stage runs drift towards different zones of the Pareto frontier. What appears different in the results of the L-shaped bracket problem is that percentile dominance among two-stage runs (CiA) is much lower compared to the short cantilever problem. Examining the topologies of the obtained solutions show only simple multi-component decompositions (Fig. 12), which may imply that the complex base topology of the L-shaped bracket might require more knot points (NJ) for the joints level-set function. The apparent high consistency in the solutions of the twostage runs is likely an indication that the solution space lacks some freedom in exploration capability. Overall, from a designer’s perspective, when structural performance is of outmost importance, it is likely best to opt for a two-stage approach, since the solutions returned include only minimal compromises from best attainable structural performance. Two-stage approach also involves fewer design variables, which means less requirement for computational resources. Run-time benchmarking for the short cantilever problem on an Intel i5 processor produced approximately 4.2 and 30 h for two and single stage approaches respectively (breakdown of the runtime is provided in Fig. 14). On the other hand, when cost is important and some compromise in performance is acceptable, the single-stage approach can be useful in exploring alternative base topologies that two-stage approach does not consider.

6 Conclusion This paper adopted the KLS formulation to multi-objective topology optimization of multi-component structures. Since assembly and manufacturing objectives having no known analytical gradients, non-gradient optimization approaches such as GA are needed. The KLS formulation can reduce the number of design variables and makes them independent from the meshing size, which is beneficial to both single and two stage optimization approaches. This enables use of mesh sizes that appropriately model realistic sheet metal structures. Testing of single and two stage approaches in two study problems showed them to have different merits. Two-stage approaches can take advantage of well-established topology

optimization methods to produce designs that have the best structural objectives and/or minimal compromise from it. Single stage approach may require additional constraints on minimum acceptable structural performance in order to eliminate uninteresting solutions and require almost an order of magnitude more computational resources than the two stage approach. The advantage of single-stage approach however, is that it can search for alternative base topologies and hence opportune for improvement in manufacturing and assembly costs despite some compromise in structural performance objectives. While adoption of KLS formulation could be an appreciable step in multi-component topology optimization, future work may flow in one or more directions: i) more general FE models such as shell elements, ii) mapped regions instead of a rectangular domains, iii) multiple thicknesses and material options for the components, iv) multiple joint types, v) more refined models for assembly and manufacturing such as inclusion of higher order polynomial terms, and vi) algorithmic refinements such as GA population seeding and tuning of the search parameters.

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