Abstract In this article, we propose a method to incorpo-rate fabrication cost in the topology optimization of lightand stiff truss structures and periodic lattices. The fabrica-tion cost of a design is estimated by assigning a unit costto each truss element, meant to approximate the cost of ele-ment placement and associated connections. A regularizedHeaviside step function is utilized to estimate the num-ber of elements existing in the design domain. This makesthe cost function smooth and differentiable, thus enablingthe application of gradient-based optimization schemes. Wedemonstrate the proposed method with classic examples instructural engineering and in the design of a material lattice,illustrating the effect of the fabrication unit cost on the opti-mal topologies. We also show that the proposed method canbe efficiently used to impose an upper bound on the allowednumber of elements in the optimal design of a truss system.Importantly, compared to traditional approaches in struc-tural topology optimization, the proposed algorithm reducesthe computational time and reduces the dependency on thethreshold used for element removal.
J. K. GuestDepartment of Civil Engineering, Johns Hopkins University,Baltimore, MD 21218, USAe-mail: [email protected]
1 Introduction
Structural topology optimization is a powerful method forthe identification of the optimized distributions of materialwithin a design domain. When applied to truss (or frame)structures or periodic lattices, topology optimization seeksthe best location and cross-section of each member withinthe design domain (the external shape of the structure orthe unit cell). This can be conveniently implemented withthe ground structure method (Dorn et al. 1964), in whicha very dense initial mesh is generated and inefficient ele-ments are subsequently removed from the design domainfollowing the optimization. The cross-sectional area of eachmember can be modeled as a continuous design variable,thus allowing the application of gradient-based schemes.
Although used extensively in literature, a drawbackof the ground structure approach is its tendency to pro-duce complex topologies for nontrivial problems optimizingstructural stiffness. Such topologies may result from con-vex problem formulations (Ben-Tal and Bendsøe 1993;Bendsøe et al. 1994), and thus can be proven to beglobally optimal for the considered ground structure. Apotential means of inhibiting topological complexity isto remove members with cross-sectional area below athreshold set by the designer. The problem can then besolved again using the previous final solution as the ini-tial guess, and material volume that is freed up throughthe thresholding can be re-assigned to members remainingin the topology. While a practical strategy, global opti-mality is lost in this process and the chosen thresholdmagnitude may have dramatic effect on the optimal topol-ogy. The threshold may also require multiple increasesbefore a structure with reasonable topological complexity isidentified.
Another strategy for preventing complexity is to penal-ize intermediate values of cross-sectional areas (i.e. cross-sectional areas between the minimum and maximumallowed values) using the continuum-based Solid IsotropicMaterial with Penalization (SIMP) method (Bendsøe 1989;Zhou and Rozvany 1991). In this approach, the stiffnessof an element is artificially modified so that it scalesnon-linearly with the cross-sectional area (thus remain-ing very low for all areas far from the maximum value).This method is extremely effective in continuum domainsin which the design variables are binary (i.e., local exis-tence or nonexistence of material), and has also been shownto reduce complexity of truss topologies with continu-ous design variables (Amir and Sigmund 2013). However,when small magnitudes of penalization are used, inter-mediate cross-sections may still appear in the final solu-tion. Although intermediate cross-sections are quite natu-ral for lattice structures, their stiffness is misrepresentedto the optimizer and thus the designed system will actu-ally exhibit suboptimal performance when analyzed underactual (non-penalized) conditions. When large magnitudesof penalization are used, the solution will tend towardsall members having the same (maximum) cross-section. Inthis case, one is essentially determining whether each ele-ment is to exist in the final topology or not (e.g., in thedesign of periodic lattices with beam elements (Sigmund1995)).
At the heart of the complexity property is that the groundstructure approach to maximum stiffness design is generallynon-convergent (Rozvany 2011), making solutions highlydependent on the choice of the ground structure. As newelements cannot be added during the optimization process,the ground structure must be complex enough to incor-porate a very large number of reasonable designs. Gener-ally speaking, the maximum achievable stiffness (minimumcompliance) improves as nodes and elements are added tothe ground structure and, typically, topological complexityfollows.
From a practical standpoint, structures and lattices thatare topological complex may be prohibitively expensiveto manufacture. In the absence of a quantitative cost-benefit analysis that compares cost and performance ofnear-optimal designs of varying complexity, this optimiza-tion tool is of limited use to the engineer. To the authors’knowledge, the total cost of lattice structures has only beenmodeled as a function of the amount of material used, eitherthrough an explicit material volume (or mass) objectivefunction or constraint, or through a complexity parame-ter related to material use. Regarding the latter, Parkes(1975), for example, suggested to penalize the complexityof structures by adding a constant length to each elementat each joint (which was called the “joint radius”). Thepenalization of shorter elements is proportional to their
cross-sectional area, and hence is a linear function of mate-rial cost.
From a practical perspective, the cost of fabricatinga topologically complex lattice or a discrete structure isonly partly related to the cost of the material; i.e., twodesigns with the same mass (and hence amount of mate-rial) but different topological complexity will generallyhave very different fabrication costs. In this paper, wepropose a more comprehensive cost function, which com-bines both the cost of material and the fabrication costassociated with fabricating each structural member withinthe lattice. As the material cost scales linearly with themass of the structure, the proposed approach convenientlyenables a cost-benefit analysis between mass and designcomplexity. A regularized Heaviside step function is uti-lized to account for costs associated with each member,thus enabling the application of gradient-based approaches.We illustrate the advantages of the proposed cost func-tion with two classic examples of maximally stiff and lightstructures. Finally, to demonstrate the versatility of thesuggested function, a minimum deflection problem withan imposed upper bound on the number of elements isinvestigated.
It is worth pointing out that continuum topology opti-mization for maximum stiffness faces similar issues ofmesh dependency and the optimal results tending towardscomplex topologies. This has been studied extensively inliterature and is typically circumvented through restric-tion of the design space. Examples include the perime-ter constraint on topology boundaries (Ambrosio andButtazzo 1993; Haber et al. 1996), minimum memberlength scale constraints (Poulsen 2003), and minimumlength scale enforcing projection methods (Guest 2009;Guest et al. 2004) and nonlinear filters (Sigmund 2007).As these restrictions are tightened, design complexity andfabrication cost is reduced. In fact, the regularized Heav-iside function used herein is borrowed from the originalcontinuum Heaviside Projection algorithm for enforcing aminimum length scale on structural members (Guest et al.2004).
Although all the examples in this article refer to con-tinuous cross-section area variables and total cost opti-mization under stiffness constraints, the proposed methodis generally applicable to virtually any mechanical (andmultifunctional) constraints (e.g., bounds on yielding orbuckling strength), as well as problems where cross-sectionvariables must be selected from discrete set of availablesizes (Achtziger and Stolpe 2006, 2007a, b; Groenwoldet al. 1996; Stolpe 2004; Zhu et al. 2014). For furtherdetails on topology optimization of lattices, including othermechanical objectives and their challenges, readers arereferred to Bendsøe and Sigmund (2003) and Rozvany(1996).
Incorporating fabrication cost into topology optimization of discrete structures and lattices 387
2 Minimum cost in structural topology optimization
We consider the common problem in topology optimiza-tion of trusses (or frames) of minimizing weight (or materialcost) under allowable deflection constraints. As discussed inthe introduction, nearly every topology optimization algo-rithm for discrete domains, such as frame structures, startswith a dense ground structure mesh; as the optimization pro-cedure progresses, inefficient elements are removed fromthe design domain. Without loss of generality we only con-sider truss elements, characterized by a single design vari-able (the cross-sectional area). Extension of the proposedalgorithm to beam elements is straightforward. The FiniteElement method is employed for calculation of displace-ments. The weight (or material cost) optimization problemfor a discretized domain (�) can then be expressed as:
minρ
W (ρ) = minρ
∑
∀e∈�
αeW γ eρeve (1)
s.t. K (ρ) d = f (2)
Ci (ρ) ≤ C∗i f or i = 1..N (3)
0 ≤ ρe ≤ 1 ∀e ∈ � (4)
Here ρ is the vector of design variables, assembling thenon-dimensionalized cross-sectional areas of each element,ρe (ρe = Ae/Amax , with Amax the maximum allowedcross-sectional area); W is the weight of the lattice in thedesign domain; γ e is the weight density of the material usedfor element e; ve is a quantity giving the volume of the ele-ment e if multiplied by ρe (i.e., ve = AmaxLe); C∗
i is themaximum allowable deflection constraint i, N is the num-ber of deflection constraints, αe
W is the material cost per unitweight of element e, and Ci is the deflection constraint i,computed as follows:
Ci (ρ) = LTi d (ρ) (5)
where d is the vector of nodal displacements, obtained bysolving the discretized equilibrium equation, Kd = f withK and f the global stiffness matrix and applied load vector,respectively; finally, Li is a vector that extracts the desireddisplacement (or combines multiple displacements) fromd. When constraining the deflection at a single degree offreedom i, the vector Li has only one non-zero componentcorresponding to degree of freedom i. The element stiffnessmatrix defined in the parent domain is given as
ke(ρe
) = ((1 − ρmin) ρe + ρmin
) EAmax
Le
[1 −1
−1 1
](6)
where E is the Young’s modulus of the constituent mate-rial, Le is the element length of element e, and ρmin is a
small positive number to maintain positive definiteness ofthe global stiffness matrix during the optimization process.Once the optimization algorithm solving (1)–(4) converges,an element e remains in the structure only if its designvariable ρe is greater than a threshold value.
The fabrication cost of the entire domain can be esti-mated by accounting for the presence of all the remainingelements. The idea is that any element existing in thestructure incurs a placement cost and cost for making twoconnections, one at either end. This model is particularlysuitable to the construction of structures, such as bridges,in which installation costs for each element can be readilyquantified. Importantly, the same model is also meaningfulfor Additive Manufacturing (AM) processes. Although theplacement cost of an element is not well defined in AM pro-cesses, increasing the number of elements in a structure (orlattice unit cell) of a given size will generally reduce theirdimension and hence increase the structural hierarchy of thedesign. This obviously increases the fabrication time, andhence the manufacturing cost. As the number of elementsis not a differentiable function, a regularized Heaviside stepfunction is used to ensure that the fabrication cost functionbe smooth and consequently differentiable, thus maintain-ing the advantage of using gradient-based optimizers. Onecan then extend the minimum cost problem stated in (1)–(4)to minimum material and fabrication cost minimization asfollows:
minρ
W (ρ) + F (ρ) = minρ
∑
∀e∈�
αeWγ eρeve
+∑
∀e∈�
αeF H
(ρe
)(7)
s.t. K (ρ) d = f (8)
Ci (ρ) ≤ C∗i f or i = 1..N (9)
0 ≤ ρe ≤ 1 ∀e ∈ � (10)
where αeF is the fabrication cost associated with element e,
F (ρ) is total fabrication cost, and H is a regularization ofthe Heaviside step function, defined as follows (Guest et al.2004):
H(ρe
) = 1 − exp(−βρe
) + ρe exp (−β) (11)
where β is a shaping parameter for the regularization. Asβ → ∞, the above function approaches the Heavisidefunction and any element with ρe > 0 will count towardsthe fabrication cost (see Guest et al. (2011) for detaileddiscussion).
Equation (7) represents a simple smooth function thatcombines fabrication cost and material cost. In this equa-tion, the value of αe
F is dependent on the fabrication pro-
388 A. Asadpoure et al.
cedure for element e. In the construction of structures, forexample, this parameter for each structural element mayinclude the cost of crane time and labor necessary to posi-tion the element, and the labor cost of making two connec-tions. More elaborate functions combining the two costs canbe implemented, but this simple function captures the nec-essary features and has a strong influence on the optimizedtopology as will be demonstrated in the numerical examplesbelow. It is worth mentioning that the problem formulationin (7)–(10) considers only deflection constraints under lin-ear elasticity, and ignores possible strength constraints (forbuckling or yielding) or other design requirements. Thesecan be added as needed, without changing the structure ofthe model or conclusions. The scope of this work is to inves-tigate how the fabrication cost function influences topology.The reader is referred to Bendsøe and Sigmund (2003) for areview of other such constraints and solution strategies.
An important beneficial byproduct of using this objec-tive function is that the optimizer avoids designing memberswith very small cross-sections when a sufficiently large fab-rication cost is introduced. This makes identifying elementsfor removal extremely objective circumventing the usualreliance on the arbitrary thresholding parameter. This willbe demonstrated quantitatively in the numerical example(Section 3).
Derivatives of (7) with respect to design variable can becomputed by:
d
dρe(W (ρ) + F (ρ)) = αe
Wγ eve + αeF
(β exp
(−βρe)
+ exp (−β)) (12)
where d/dρe denotes the (full) derivative with respect to ρe.Derivatives of (9) can be computed using the adjoint methodas:
dCi
dρe= −λT
i
dKdρe
d = −λeTi
dKe
dρede (13)
where
Kλi = Li (14)
and the superscript e for each vector or matrix denotes theelemental level of that vector or matrix for element e.
A conceptually different approach to accounting for fab-rication cost is to constrain the maximum allowable numberof elements appearing in the optimized design. Practically,this optimization represents a situation where the maximumallowable fabrication cost is well defined or, for example,there is a maximum number of elements that can be installeddue to transportation considerations, time to construction, oravailable connection components. The functions describedpreviously in this section can be equivalently applied to
this formulation, to handle a discrete variable (the numberof elements) with a gradient-based approach. If the objec-tive is stiffness maximization (i.e., deflection or complianceminimization), this alternative optimization problem can bestated as follows:
minρ
C (ρ) = minρ
LT d (ρ) (15)
s.t. K (ρ) d = f (16)
∑
∀e∈�
H(ρe
x
) ≤ n∗el (17)
0 ≤ ρe ≤ 1 ∀e ∈ � (18)
where n∗el is the maximum allowable number of elements,
and all the other variables are defined as before. Derivativesof (15) and (17) can be obtained using (13) and the sec-ond term in the parentheses of the right hand side of (12),respectively.
3 Numerical examples
We apply the proposed minimum cost design algorithm (7)–(10) to two classic 2D truss problems and to the design ofa periodic lattice. We also provide an example for the opti-mization problem stated by (15)–(18). For all examples, theMethod of Moving Asymptotes (MMA) (Svanberg 1987;1995) is used to solve the optimization problem. For prob-lems in which the fabrication cost is significant, i.e. morethan 5 % of the total cost, we use a modification of MMAproposed by Guest et al. (2011). Using this modificationallows avoiding a continuation step on the parameter β in(11), which would otherwise be required to ensure conver-gence. After convergence, elements with design variable ρe
below the threshold are removed and the problem is solvedagain using the converged solution as the initial guess. Thisis repeated until no element with design variable lower thanthe threshold is found.
In practice, unit material costs αeW and unit fabrication
costs αeF are dictated by local markets and methods where
the structure or lattices are to be fabricated. To clearly illus-trate the algorithm, however, all examples herein use a fixedmagnitude of αe
W = αW and only αeF is varied with αe
F =αF , i.e. uniform weight cost and fabrication cost functionfor all elements, unless otherwise stated. Without loss ofgenerality, symmetry about a vertical axis passing throughthe load application point is imposed to reduce the num-ber of design variables. Nonetheless, the whole symmetricstructure is modeled, allowing the development of elementscrossing the line of symmetry in optimized solutions (e.g.,horizontal elements connecting pairs of symmetrical nodes
Incorporating fabrication cost into topology optimization of discrete structures and lattices 389
Fig. 1 A simply supported domain with a point load on the top boundary (a) geometry and boundary conditions; (b) initial fully connected meshwith 17 × 5 nodes and 3,570 truss elements
on two sides of the line of symmetry). In the followingexamples, initial meshes used for optimization consist ofoverlapping elements and no specific algorithm, other thanthe proposed, is used to avoid development of overlappingelements in the final structure.
3.1 A simply supported truss structure with a point loadon the top boundary
Figure 1a illustrates the geometry and boundary conditionsof a simply supported domain with a point load appliedat the middle of the top boundary. The ground structurefeatures a fully connected lattice with 17 × 5 nodes, con-sisting of 3570 truss elements connecting all the possiblepairs of nodes as shown in Fig. 1b. The removal thresholdfor cross-sectional area of elements is set at ρe
th = 10−5 anda single maximum allowable deflection constraint is appliedat midspan with C∗ = 1600P/EL, where P is the magni-tude of the applied load and E the Young’s modulus of thematerial.
The optimization is performed for αW = (10/L)3
and αF = {0, 10−1, 100, 101}, resulting in a total
of 4 independent optimization runs. The optimized solu-tions for different values of αF are shown in Fig. 2.As expected, the number of elements in the optimal designis reduced as the fabrication cost of elements increases.This happens at the cost of a heavier structure, i.e. morematerial is needed to meet the same stiffness constraint(Fig. 3). Both material and fabrication costs are normal-ized with the corresponding values for the αF = 0 opti-mized structure in Fig. 2a. Notice that for αF ≤ 10−1,the optimal topology (and hence the material and fabrica-tion cost) is constant for the considered ground structure.If αF is increased to 100, the number of elements in theoptimal design drops by 70 %, with only a 5 % increasein the material cost (and hence the weight of the structure).A further increase to αF = 101 reduces the num-ber of elements to the minimum possible for the prob-lem (3), but this requires a 35 % increase in weight.For any larger αF , the solution will stay the same.Notice that even for situations where the fabrication costis not well defined, this study would reveal the optimumcompromise between mechanical efficiency and ease offabrication (in this case, the design in Fig. 2b).
Fig. 2 Optimized truss structures for the simply supported domain shown in Fig. 1 for different values of fabrication unit cost αF
390 A. Asadpoure et al.
Fig. 3 Material cost and the number of elements in optimized struc-tures for the domain shown in Fig. 1 for different fabrication unit costsof αF ; All values are normalized with the corresponding value for theoptimized structure shown in Fig. 2a with zero fabrication cost, i.e.αF = 0
An interesting byproduct of the proposed approach toincorporating fabrication cost is that elements with verysmall but nonzero cross-sectional areas are naturally drivento zero area by the optimizer, to eliminate their contributionto the fabrication cost. This creates a separation between“structural” and “non-structural” elements, thereby reduc-ing the influence of the arbitrarily selected threshold used
in conventional approaches to identify elements that areto be removed from the ground structure. This approachalso reduces the number of optimization iterations to con-vergence, thereby substantially reducing the computationalcost. As an example, consider the optimized designs shownin Fig. 2a for αF = 0 and 10−1. The optimized struc-ture for these two values of αF is identical and thereforeboth designs have the same amount of material, but the con-vergence rate for αF = 10−1 is about 20 % higher thanfor αF = 0. This behavior is quantitatively illustrated inFig. 4, which shows the distribution of design variablesafter the first optimization convergence, i.e. before the firstround of element removal. Clearly, increasing αF results inmore elements dropping significantly below the thresholdand fewer elements around the threshold, resulting in fasterconvergence.
3.2 A simply supported truss structure with a point loadon the bottom boundary (Wheel-like problem)
Geometry and boundary conditions for a simply supporteddomain with a point load applied at the middle of the bot-tom boundary are illustrated in Fig. 5a. This design domainis similar to the structural part of the well-known wheelproblem (Jog et al. 1994), with the difference that only
Fig. 4 Distribution of design variables for the simply supported truss domain shown in Fig. 1 after the first optimization convergence, i.e. beforefirst element removal, for different values of αF . Dashed lines in each figure indicate the threshold for removing elements at the end of eachoptimization iteration
Incorporating fabrication cost into topology optimization of discrete structures and lattices 391
Fig. 5 (a) Geometry and boundary conditions for a simply supported truss domain with a point load on the bottom boundary; (b) initial fullyconnected mesh (ground structure) with 9 × 7 nodes and 1,953 truss elements
the domain between two supports is modeled. A 9 × 7-node mesh with 1,953 truss elements, shown in Fig. 5b,is used as the ground structure. As in the previous exam-ple, every pair of nodes in the mesh is connected by a trusselement. The removal threshold for the elements is set toρe
th = 10−5 , and the maximum allowable deflection at themidspan of the bottom boundary is C∗ = 240P/EL.
The influence of fabrication cost on an optimized struc-ture is demonstrated for αW equal to (15/L)3 and αF equalto {0, 100, 101, 102, 103}, for a total of 5 indepen-dent optimization runs. Figure 6 displays optimized topolo-gies for different values of αF . For this discretization, the
solution uses a relatively small number of elements whenfabrication cost is not considered (αF = 0) (Fig. 6a). Forsmall values of the fabrication cost (αF = 101), the opti-mal structure remains unchanged. When the unit fabricationcost is increased to αF = 101, the number of elementsis reduced by 15 %, with further reductions observed forαF = 102 and αF = 103. As the αF = 103 structure isthe simplest kinematically determinate topology, no furtherreduction in the number of elements is possible. The evo-lution of topology as a function of the unit fabrication costis quantitatively displayed in Fig. 7. Values of fabricationcost and material cost (or weight) are normalized with the
Fig. 6 Optimized trussstructures for the simplysupported domain shown inFig. 5 for different values of αF
392 A. Asadpoure et al.
Fig. 7 Material cost and the number of elements in optimized struc-tures for the domain shown in Fig. 5 for different values of αF ; Allvalues are normalized with the corresponding value for the optimizedstructure shown in Fig. 6a with zero fabrication cost, i.e. αF = 0
corresponding value for the optimized αF = 0 struc-ture (Fig. 6a). As expected, design complexity decreasesand material cost increases with increasing fabrication costparameter αF in Fig. 7.
3.3 A periodic lattice subjected to effective axial and shearstiffness constraints
In this example we study a two-dimensional periodic lattice,defined by a unit cell, subjected to effective axial and shearstiffness constraints. This example is motivated by optimaldesign of sandwich panel cores, which demand a relativelyhigh shear stiffness and a lower axial stiffness. We use thegeneral optimization formulation stated in (7)–(10) with twototal stiffness constraints: the effective Young’s modulus inthe horizontal direction and the effective shear modulus areconstrained to 1 % and 10 % of the constituent materialmoduli, respectively. The effective properties of the unit cellare estimated by numerical homogenization (e.g., Guedesand Kikuchi (1990)), leading to an optimization problemthat is often referred to as inverse homogenization (see
e.g., Sigmund (1995) for additional discussion and detailedequations).
It is well known that unit cell topologies offering maxi-mal stiffness may be non-unique, and this example displaysmultiple global minima with the same mass (material cost).As these minima are associated with different topologies, itmakes the problem well suited for examining different fab-rication cost scenarios, two of which are considered herein.The first scenario is uniform unit fabrication cost, where allelements have αe
F = αF = 1. In the second scenario, weexpress the element unit cost as a function of the elementlength. Specifically, elements whose length is half the lengthof the unit cell size incur the least fabrication cost, whilethe unit cost of longer and shorter elements increase by asmuch as a factor of twenty. The unit cost function is shownin Fig. 8a and given as αe
F = 1+19|Le/L−0.5|/(√2−0.5)
where L is the length of the unit cell, Le is the length ofelement e and | • | denotes the absolute value of •. This sce-nario represents the situation where a specific length, hereL/2, has minimum fabrication cost and any deviation fromit results in extra fabrication costs. In both scenarios, it isassumed that αe
W = αW = 1.Figure 8b depicts the ground structure of a fully con-
nected lattice with 5 × 5 nodes, consisting of 300 trusselements connecting all the possible pairs of nodes in theunit cell. Periodic boundary conditions are applied usingthe same equation numbers for degrees of freedom oppos-ing boundaries (e.g., Sigmund (1995)). Figure 9 illustratesthe optimized unit cell for each of the fabrication cost sce-narios, with each unit cell having the same material cost.When uniform fabrication cost is used, i.e. αe
F = αF =1, a unit cell featuring a few long elements emerges. Onthe other hand, when unit fabrication cost is expressed asa non-uniform function of element length, as parameter-ized in Fig. 8a, with maximum unit fabrication cost of11, the optimization algorithm designs a topology usingseveral elements whose lengths are closer to L/2, thelength corresponding to the cheapest unit cost. Despiterequiring more elements, the total cost of this design is
Fig. 8 (a) Unit cost functionused for different fabricationcost scenarios as a function ofelement length; (b) initial fullyconnected mesh (groundstructure) with 5 × 5 nodes and300 truss elements within theunit cell used for optimal designof a minimum-cost periodiclattice under effective axial andshear stiffness constraints. Inthis mesh, the length of elementsvaries from L/4 to
√2L
Incorporating fabrication cost into topology optimization of discrete structures and lattices 393
Fig. 9 Optimized periodic unit cells for the ground structure shownin Fig. 8b subjected to axial and shear stiffness constraints for twodifferent fabrication cost scenarios plotted in Fig. 8a. First row showsthe optimized solution for uniform unit cost of αe
F = 1 and the
second row shows the optimized solution for nonuniform cost of αeF =
1 + 19|Le/L − 0.5|/(√2 − 0.5). Both optimized unit cells have thesame minimal material cost
10 % lower than for the topology with fewer, longerelements, when evaluated using this nonuniform costmodel. It should nevertheless be mentioned that each ofthese unit cells has minimum material cost, and thuscould have been found by simply minimizing materialcost using different initial guesses for the design vari-ables. Considering fabrication cost, however, may lead toa unique minimum of the total cost problem, as illus-trated here.
Interestingly, the topology significantly changes if weadjust the parameterization of the nonuniform fabricationcost scenario. For example, we further increase the expenseof members whose length is different than L/2 by chang-ing the unit cost structure to αe
F = 100 + 10000|Le/L −0.5|/(√2 − 0.5). This makes the fabrication cost forelements of length of L/2 equal to 100 units, and increasesthe maximum cost occurring for long diagonal elements to10100 units. The resulting optimized solution is depicted in
394 A. Asadpoure et al.
Fig. 10 Optimized unit cell for the ground structure shown in Fig. 8b for the fabrication cost with αeF = 100 + 10000|Le/L − 0.5|/(√2 − 0.5).
This optimized unit cell consists of elements that are either L/2 or√
1.25L/2 long
Fig. 10 and it is clearly seen that the algorithm has designeda topology that uses many more elements having length L/2or
√1.25L/2. This design change is done at the cost of
material efficiency, as this design uses approximately 67 %more material to satisfy the stiffness constraints. The overalltotal cost, however, is significantly lower than the total costof the previous unit cell designs when evaluated using thisnew fabrication cost model. Clearly the optimized latticesolution is dependent on the fabrication cost model, and thesolution for the minimum material cost may not be optimalfor the minimum total cost problem. On the other hand, forthe uniform fabrication cost scenario, the optimized solu-tion does not change by increasing the fabrication cost asit simultaneously possesses the least number of elementsand the least material cost as opposed to previous structuralexamples.
3.4 Restricting the number of elements in a simplysupported truss structure with a point load on the topboundary
As a final example, we look to design maximum stiffness(minimum deflection) structures using a limited number ofmembers, as specified by the designer. This is achievedby solving the optimization problem stated by (15)–(18),with L = F, for the initial structure shown in Fig. 1.External geometry and boundary conditions are the same asin Section 3.1. The results found when varying the maxi-mum allowable number of members between 3 and 100 arepresented in Fig. 11. The maximum deflection and the mass
of the structure, both normalized against the solution forn∗
el = 3, are plotted against the maximum allowable numberof elements, n∗
el . Obviously, as the allowable number of ele-ments is increased, the maximum deflection of the structuredecreases (and hence the stiffness increases), while the massof the structure increases. Note that weight is not minimizedin this optimization problem, leading the optimizer to assignall members appearing in the final solution to have the max-imum allowable cross-sectional area, Amax . Interestingly,stiffness and mass both increase nearly linearly with n∗
el ;the implication is that all the designs in Fig. 11 have nearlyidentical structural efficiency (i.e., stiffness/mass). In this
Fig. 11 Amount of material and maximum deflection of optimizedstructures for minimizing maximum deflection for the initial struc-ture shown in Fig. 1 for different maximum number of elements, i.e.n∗
el . All values are normalized with the corresponding value for theoptimized structure with three elements, i.e. n∗
el = 3
Incorporating fabrication cost into topology optimization of discrete structures and lattices 395
case, the conclusion is that increasing the complexity ofthe topology presents insignificant benefit in terms of spe-cific stiffness (albeit, there might be advantages from theperspective of strength and/or robustness).
Another interesting conclusion that emerges from theresults plotted in Fig. 11 is that intervals of n∗
el existwithin which the solution does not change (for example,n∗
el ∈ [3 12], n∗el ∈ [15 20], and n∗
el ∈ [25 30]). Thishappens because adding very few extra elements to theseoptimized structures is not sufficient to develop new loadcarrying paths, and thus the optimizer has no incentive toadd elements in these cases. Notice that in some instances,adding members actually leads to a decrease in materialcost (mass), even though these members achieve Amax . Thisis because elements have different lengths and there is noconstraint imposed on the total material cost in the structure.
Finally, it is worth emphasizing the importance of usingthe nonlinear (regularized) Heaviside step function in (17)to approximate the number of elements used in the design.The output of this function for a small value, e.g. ρe = 0.01,and a large value, e.g. ρe = 1, would be approximately onein both cases as long as a sufficiently large value for param-eter β is employed (e.g., β = 1000 is used herein). Thisis critical if members with these cross-sectional areas areto each count as one member used in the design; the samefeature could not be achieved with (for example) a simplesummation on all ρe.
4 Conclusions
This article proposes a method to incorporate fabricationcost in topology optimization of discrete structures, i.e.,trusses and/or periodic lattices. The fabrication cost is mod-eled by assigning each element in the mesh a unit cost. Theidea is that any designed element requires installation andtwo connections, and that these are treated as discrete, fixedper element costs. A regularized Heaviside step function isadopted to render the objective function smooth and differ-entiable. The proposed algorithm is demonstrated on classicexamples from structural engineering and on the design ofa unit cell of a periodic material; in both cases, the evo-lutions of optimal topology, mass (or material cost) andfabrication cost of the structure are tracked as a functionof the fabrication unit cost (and hence the relative ‘cost ofcomplexity’). As expected, increasing the fabrication unitcost results in optimized structures with fewer elements andsimpler designs. Typically, this comes at the expense ofstructural efficiency, and thus induces an increase in mass(or material cost) in order to satisfy structural constraints.
Interestingly, if sufficiently large fabrication unit costsare chosen, the proposed objective function significantlyaccelerates convergence and reduces computational time
for mass minimization problems under stiffness constraints.This is due to the fact that the optimizer pushes low mag-nitude design variables (i.e., small cross-sectional areas)to zero as such elements are inefficient in terms of fab-rication cost. This also makes the process of identifyingelements to be removed very objective, thereby reducingsolution dependence on the arbitrary threshold variable typ-ically used to identify such elements. It may therefore serveas a useful and physically meaningful penalization functionto promote elimination of inefficient elements in generaltruss and frame topology optimization.
It is worth noting that the methodology requires quantifi-cation of parameters αe
W and αeF , the unit material cost and
unit fabrication cost, respectively. Ideally, these unit costswould be input by the designer based on local market ratesfor materials and labor (or automated fabrication processes),and the algorithm would output the lowest cost structure.While materials costs may be straightforward to estimate,fabrication costs are significantly more complicated, typi-cally requiring input from local construction experts. Wehave also limited our discussion of fabrication costs tomember placement and connections. These functions cansurely be made more complicated, such as making place-ment and connection costs a function of member mass andcross-section dimensions.
Regardless, even for situations where the fabrication unitcost (or the allowable number of elements) is unknowna priori, the presented approach can provide the designerwith a quick assessment of the complexity/cost benefitsin topology optimization. Finally, it is anticipated thatthis algorithm would work for more complex problems,where more elaborate mechanical objectives and constraintsare formulated. For example, it would also be interestingto combine the approach with recent work focusing onimproving the robustness of truss structures under fabri-cation flaws. Robustness may often be improved throughdiversification of the load path, resulting in increased com-plexity of the optimized topology (Asadpoure et al. 2011;Jalalpour et al. 2011). Introducing a cost to this complex-ity, as done here, could lead to more realistic optimizedstructures.
Acknowledgments This work was financially supported by theOffice of Naval Research under Grant No. N00014-11-1-0884 (pro-gram manager: D. Shifler). This support is gratefully acknowledged.The authors are also thankful to Krister Svanberg for providing theMMA optimizer subroutine.
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