A Full-space Barrier Method for Stress-constrainedDiscrete Material Design Optimization

Graeme J. Kennedy ∗

Abstract

In this paper, we present a full-space barrier method designed for stress-constrained massminimization problems with discrete material options. The advantages of the full-space barriermethod are twofold. First, in the full-space the stress constraints are provably concave, whichfacilitates the construction of convex subproblems within the optimization algorithm. Second,by using the full-space, it is no longer necessary to employ stress constraint aggregation tech-niques to reduce adjoint-gradient evaluation costs. The proposed optimization algorithm usesa Newton method where an approximate linearization of the KKT conditions is solved inex-actly at each iteration using a preconditioned Krylov subspace method. Sparse constraints thatarise in the discrete material parametrization are treated using a null-space method. Resultsof the proposed algorithm are demonstrated on a series of three topology and multimaterialoptimization problems with selection between isotropic and orthotropic materials, as well asdiscrete ply-angle selection.

1 IntroductionThe objective of this paper is to develop efficient optimization methods for stress-constrained massminimization of structures with discrete material options. Discrete material optimization (DMO)methods seek the optimal distribution of a discrete set of materials within a structure. DMO meth-ods, which were first proposed by Stegmann and Lund [51], are an extension of topology opti-mization methods and are a generalization of multiphase composite design methods developedby Sigmund and Torquato [48] and Gibiansky and Sigmund [18]. Since then, DMO techniqueshave been applied to both multimaterial design [48, 18, 51, 25] and laminate design, where thelamination angles are restricted to a discrete set of manufacturable angles [51, 36, 26, 31, 49, 50].Various authors have developed extensions and enhancements of the original DMO method. Theseextensions include alternative penalization schemes [26, 31], interpolation schemes that requirepartition of unity constraints [25], and methods that combine multimaterial and thickness opti-mization [49, 50]. Other authors have developed alternative discrete material methods. Bruyneel[8] and Bruyneel et al. [10] proposed the Shape Function Parametrization (SFP) techniques whichuse finite-element shape functions combined with SIMP- or DMO-type material interpolation thatreduces the number of design variables compared with conventional DMO.∗Assistant Professor, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, email:

graeme.kennedy@ae.gatech.edu

1

Most discrete material design applications have focused on compliance minimization [51, 49,26], buckling design [36, 31], or compliant mechanism design [19]. Relatively few authors haveaddressed the extension of multimaterial design to stress-constrained mass minimization problems.Ramani [44] developed a heuristic design method for multimaterial problems with strength require-ments. Guo et al. [21], used a color level set representation and a global stress measure to obtainmultiphase stress-constrained designs. Kennedy [29] developed a discrete thickness method whichincorporated stress-constrained design. In this paper, we develop a full-space design optimiza-tion approach that is tailored specifically for stress-constrained multimaterial mass-minimizationproblems.

Stress-constrained design problems are more challenging to solve than compliance-based prob-lems, due to the local nature of the stress constraints which must be imposed at many points withinthe structure. Stress constraints formulated specifically for topology optimization have been de-veloped by several authors [11, 14, 35]. More recently, stress-constrained mass minimizationproblems have been solved using level-set methods [20, 55, 57]. Two issues must be addressedwhen performing stress-constrained topology optimization: (1) formulating the stress constraintsto avoid the so-called stress-singularity, and (2) handling the large number of local design- andstate-dependent stress constraints.

The stress singularity arises when the stress constraint associated with a material is activewhile the corresponding material vanishes. Without proper reformulation, the design space isdegenerate and the true optimum may be an isolated point that is inaccessible [11, 9]. Commonapproaches to resolve this stress-singularity issue include ε-relaxation and relaxed-stress methods,which regularize the design space in the neighborhood of the singular points; see, for example Leet al. [35].

Many authors have used constraint aggregation techniques to handle the large number of localstress constraints within structural design problems [56, 1, 14, 35]. Constraint aggregation meth-ods form an equivalent global stress measure based on the local stress evaluated at points in thestructural domain. Aggregation techniques are used primarily because they alleviate the computa-tional cost of constraint gradient evaluation. Conventional stress-constrained design optimizationmethods use a reduced-space formulation (also called Nested ANalysis and Design (NAND) inthe MDO literature [22, 37]), which treats the state variables, u, as an implicit function of thedesign variables, x, through the governing equations such that u(x) = K(x)−1f. This formulationensures that the governing equations are satisfied exactly at every iteration. However, within thisreduced-space paradigm, sensitivity methods, such as the adjoint method, must be used to evalu-ate constraint gradients. As a result, the gradient evaluation cost scales strongly with the numberof stress constraints. Constraint aggregates reduce the computational cost of gradient evaluationby replacing the large number of local stress constraints with a small number of equivalent con-straints. The most common constraint aggregation techniques are the p-norm [14, 35, 24] and thediscrete Kreisselmeier–Steinhauser (KS) function, originally developed for control systems de-sign [33], and subsequently adapted to a wide range of structural [56, 1, 43, 30, 28] and multidis-ciplinary design optimization problems [38, 39, 32]. A key difficulty with constraint aggregation,however, is that it produces constraints that have high curvature, resulting in design optimizationproblems that are more difficult to solve. Specialized reduced-space methods that make use ofefficient Jacobian-vector products have recently been developed to address cost of constraint gra-dient evaluation [34, 23]. However, these methods have not reached the same maturity as classicalreduced-space techniques.

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An alternative to reduced-space methods, which are utilized in this paper, are full-space op-timization methods that search for both state and design variables simultaneously (these methodsare also called Simultaneous ANalysis and Design (SAND) or All-At-Once (AAO) methods inthe MDO literature [22, 37, 12]). Within our proposed full-space method, a logarithmic barrier isused to handle both the stress constraints and the variable-bound constraints. This approach buildson full-space methods such as the Lagrange–Newton–Krylov–Schur method of Biros and Ghattas[4, 5]. The advantages of our proposed full-space barrier method are as follows:

1. Through careful constraint and objective formulation, we control the convexity and concav-ity of the objective and constraints, leading to guaranteed mathematical properties of thesubproblems that arise in the full-space method.

2. Due to the use of a logarithmic barrier method, constraint aggregation techniques are notrequired, leading to sparse matrices with known sparsity structure that can be used in thesolution of the full-space subproblems. This is in contrast with reduced-space constraintaggregation methods which produce fully populated matrices.

These advantages enable the solution of large stress-constrained multimaterial design problems.The remainder of the paper is outlined as follows: In Section 2, we describe the formulation of

the material-selection and topology variables, the evaluation of the mass and element stiffness ma-trices, and the formulation of the stress constraints. In Section 3, we outline the full-space barriermethod and describe the essential aspects of the optimization algorithm. Finally, in Section 4, wepresent results from the full-space barrier method.

2 Multimaterial problem formulationIn this section, we describe the formulation of combined multimaterial and topology stress-constrainedmass minimization design problems. In practice, multimaterial and continuous thickness, or puremultimaterial problems could also be formulated using the approach described below through ap-propriate simplifications.

2.1 Variables and spatial filtersWe perform topology and multimaterial optimization using a combination of topology and material-selection variables [25]. For each element within the mesh, we allocate one topology variable andM material-selection variables, one for each of the candidate materials. We denote the topologyvariable as xi, with a single subscript, and the material-selection variables as xi j, with two sub-scripts, for elements i = 1, . . . ,ne, and for materials j = 1, . . . ,M. We group all of the topology andmaterial selection variables into the design vector x ∈ Rn, where n = (M+1)ne.

In a departure from previous work in multimaterial design optimization, we use inverse vari-ables such that 1/xi j = 1 indicates the selection of material j within element i, while 1/xi j � 1indicates the absence of material j within element i. In addition, we use inverse topology variablessuch that 1/xi = 1, and 1/xi � 1 indicate the presence and absence of element i, respectively.As will be shown below, the use of inverse variables leads to beneficial convexity and concavityproperties of the objective and constraints.

3

In addition to the use of inverse variables, we employ a conic spatial filter for both the materialand topology variables [7, 35]. As a result, the unpenalized thickness, ti, of element i is a weightedcombination of the design variables in adjacent elements within the filter

ti(x) = ∑k∈Fi

wik

xk,

where the sets of indices Fi denote the non-zero weights for element i. Note that the weightssatisfy wik ≥ 0 and

∑k∈Fi

wik = 1.

The material fraction variable, ti j(x), for material j in element i are computed in an analogousmanner

ti j(x) = ∑k∈Fi

wik

xk j.

Note that ti j(x) is the fraction of material j in element i. Throughout this work, we apply thesame filter to the thickness and material-selection variables. This is not required and the material-selection and thickness variables could be filtered independently. In the remainder of this paper,we drop the explicit dependence of ti and ti j on the design variables, x, for ease of presentation.

2.2 Mass and stiffness evaluationIn the proposed parametrization, the mass is evaluated as a function of both the topology variablesand the material selection variables

m(x) =ne

∑i=1

tiM

∑j=1

mi jti j =ne

∑i=1

[(∑

k∈Fi

wik

xk

)M

∑j=1

∑k∈Fi

mi jwik

xk j

], (1)

where mi j is the mass contributed by material j from element i. Expanding the expression (1), themass can be expressed as a sum of products of the form c/(xkxk j) which are convex functions whenc≥ 0, xk > 0, xk j > 0, which hold in this case. Therefore, the total mass (1) is a convex function.

The constitutive matrix for each element is parametrized using the Discrete Material and Thick-ness Optimization (DMTO) approach of Sørensen et al. [50]. In the DMTO method, the constitu-tive matrix of element i, given by Qi(x), is expressed as an interpolation of the candidate constitu-tive matrices Q j, for j = 1, . . . ,M as follows

Qi(x) = pt(ti;qt)M

∑j=1

p(ti j;q)Q j, (2)

where the functions pt(t;qt) and p(t;q) are penalization functions for the topology and materialselection variables, respectively. Within this work, we use RAMP penalization [52] and set

p(t,q) = pRAMP(t,q) =t

1+q(1− t),

and pt(t,qt) = pRAMP(t,qt). For discrete material selection problems, the material stiffness ma-trices, Q j, are given directly. For discrete ply-angle problems, the material stiffness matrices, Q j,

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are related through the rotation of the constitutive matrix from the material axis to the global axisgiven by

Q j = T(θ j)−1QmT(θ j)

−T , θ j ∈Θ = {θ1,θ2, . . . ,θM} (3)

where T(θ) is a transformation matrix for the components of stress [27, Ch. 2, pg. 74], and Qm isthe stiffness in the local material-aligned coordinate frame.

Using (2), the global stiffness matrix for the finite-element problem can then be written asfollows

K(x) =ne

∑i=1

Ki(Qi(x))

=ne

∑i=1

pt(ti;q)M

∑j=1

p(ti j;q)Ki(Q j)

= ∑i, j

pi j(x)Ki(Q j)

(4)

where Ki are the finite-element matrices for each of the ne finite-elements within the mesh, and wehave defined pi j(x) = pt(ti;q)p(ti j;q).

In addition to the stiffness parametrization (2), we also impose a constraint on the design vari-ables such that the inverse of the material-selection variables must sum to unity within each ele-ment. This constraint is written as follows

[cw(x)]i =M

∑j=1

1xi j−1 = 0, (5)

where cw(x) ∈ Rne is a vector containing the constraints from every element. Note that the con-straint (5) imposes a constraint on the material fraction variables, ti j, as follows

M

∑j=1

ti j =M

∑j=1

∑k∈Fi

wik

xk j= ∑

k∈Fi

wik

M

∑j=1

1xk j

= 1. (6)

Therefore, when the constraints (5) are satisfied, the material fraction variables satisfy a partitionof unity constraint.

2.3 Stress constraint formulationIn this work, we use failure criteria that take the following quadratic form

1− eT h− eT Ge≥ 0, (7)

where h ∈ R3 and G ∈ R3×3 are material-dependent constants and the matrix G is symmetricpositive semi-definite. The vector e ∈R3 contains the components of the engineering strain tensorin the global reference frame

e =

εxεyγxy

.

5

While the criterion (7) is written in terms of strain components, we still refer to it as a stressconstraint since its purpose is often to impose a local bound on stress.

Rather than using the element strain directly in the evaluation of the constraint (7), we use aspatially averaged strain from adjacent elements. Through numerical experiments, we found thatthe use of strain averaging made the proposed algorithm more robust, especially when using highlyanisotropic stiffness and strength properties. This averaging process is similar to the averagednodal stress computations performed by some finite-element tools during stress recovery [3, Ch4, pg. 256]. While the strain averaging introduces a non-physical length-scale dependence intothe optimization problem, this averaging length-scale can be linked directly to the mesh spacing,rather than a design feature length-scale. For convenience, in this work we have used the samespatial filter for the strain and for the design variables. In general, different filtering or averagingschemes for the design variables and strain could be used.

Many practical failure criteria can be formulated using the quadratic failure model (7). Withinthis paper, we use the von Mises and Tsai–Wu failure criteria [27, Ch. 2, pg. 114]. Other commonfailure criteria can fit this quadratic form, including the Tsai–Hill criterion. Furthermore, the max-imum strain or maximum stress criteria could also be formulated in this manner by setting G = 0(since G is only required to be semi-definite) and adding upper and lower bound constraint pairsfor each component of strain or stress, respectively.

For the von Mises stress criterion, h = 0, and G is

GvM =E2

σ2max(1−ν2)2

1−ν +ν2 −(1−4ν +ν2) 0−(1−4ν +ν2) 1−ν +ν2 0

0 0 34(1−ν)2

, (8)

where E and ν are the Young’s modulus and Poisson ratio, respectively, and where σmax is theupper bound on the von Mises stress. For the Tsai–Wu failure criteria, the coefficients are

hTW (θ) = T(θ)−1Qm

F1F20

, GTW (θ) = T(θ)−1Qm

F11 F12 0F12 F22 00 0 F66

QmT(θ)−T , (9)

where F1, F2, and F11, F22, F12, and F66 are the linear and quadratic Tsai–Wu coefficients for thestress in the local material-aligned axis. Here, T(θ) is the plane stress transformation matrix.

2.3.1 ε-relaxation constraints

In the context of multimaterial optimization, the stress constraint (7) cannot be imposed directlydue to the so-called stress singularity. As noted by Cheng and Guo [11] and Duysinx and Bendsøe[14] in the context of stress-constrained topology optimization, and Bruyneel and Duysinx [9] inthe context of stress-constrained laminate design, the stress singularity arises when a material ortopology variable vanishes entirely while its corresponding failure constraint is still active.

In this work, we use a stress constraint formulation that regularizes the stress singularity forvanishing material-selection and topology variables. This stress constraint formulation takes theform

gi j(x,u) = 1+εti+

εti j−2ε− eT

i h j− eTi G jei ≥ 0, (10)

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where ε is a small parameter and ei is the spatially averaged strain in element i. Note that whenboth the topology variable and material selection are active, such that ti = 1 and ti j = 1, then thefull constraint (7) is applied. However, as ti→ 0 or ti j→ 0, the value of the stress constraint (10)becomes large due to the ε/ti and ε/ti j terms, so that the constraint is satisfied regardless of thevalue of the original constraint (7). Within the optimization problem we use the parameter valueε = 0.1 and set an upper bound on the topology design variables of xi ≤ ε−2 = 100, which in turnresults in an effective lower bound on the thickness of ti ≥ ε2.

This formulation is similar to stress relaxation methods that apply a multiplicative factor di-rectly to the stress [35]. Multiplying through by the inverse of the multiplicative factor and rear-ranging yields the equivalence

tt + ε(1− t)

σ ≤ σmax ⇐⇒σ

σmax≤ 1+

εt− ε.

However, the proposed constraint (10) works naturally for both material and topology optimizationand ensures concavity of the stress constraints as described below.

An important feature of the constraint (10) is that it is a concave function of the state variables,u, and design variables, x. To observe the concavity with respect to u, note that the spatiallyaveraged strain in element i is a linear function of the displacements u, such that ei = Biu andthat G is positive semi-definite. To establish that the constraint is also concave in x requires moreanalysis. Clearly, the concavity of (10) can be established by examining the properties of thefunction

s =1ti=

1∑k∈Fi

wikxk

.

Here, we drop the notation k ∈Fi for simplicity and state that all sums are over Fi. The functions is concave if

vT (∇xxs)v = ∑j∑k

2s3(

wikvk

x2k

)(wi jv j

x2j

)−∑

k2s2 wikv2

k

x3k≤ 0,

for arbitrary v. This can be established by multiplying the previous expression by 1/s and notingthat the following inequality holds

∑j∑k

(wikvk

x2k

)(wi jv j

x2j

)−(

∑k

wikv2k

x3k

)(∑k

wik

xk

)≤ 0,

as a result of the Cauchy–Schwarz inequality, where (aT b)2 ≤ (aT a)(bT b), with

ak =

√wikvk

x3/2k

, bk =

√wik

xk.

This establishes that the constraint (10) is indeed a concave function of x and u.

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2.4 Stress-constrained mass minimization formulationAssembling the components of the proposed formulation together, the complete stress-constrainedmass minimization problem takes the following form

minx,u

m(x)

governed by K(x)u = fsuch that 1≤ xi ≤ xu 1≤ xi j ≤ xu

cw(x) = 0gi j(x,u)≥ 0

(11)

This stress-constrained problem is non-convex. However, since the objective (1) is convex, andthe constraints (10) are concave, the non-convexity only enters through the governing equations,K(x)u = f, which are nonlinear in (x,u), and the material-selection constraints (5) 1.

As posed, the optimization problem (11) is a full-space (SAND) formulation. The reformu-lation to the reduced-space (NAND) would render the state variables nonlinear implicit functionsof the design variables, u(x), destroying the concavity properties of the original stress constraints.Instead, by using the full-space and retaining both x and u as independent variables, the under-lying mathematical structure of the stress constraints is preserved, leading to better control of thesubproblems that are formed within the full-space optimization algorithm.

3 Full-space barrier methodsIn this section, we present a full-space barrier method to solve the stress-constrained mass min-imization problem (11). The proposed method uses a classical interior-point barrier techniquewhich produces a sequence of equality-constrained subproblems [17, 15]. Each barrier subproblemis solved to a loose tolerance using an approximate and inexact Newton method that is describedbelow.

Within the barrier method, each barrier subproblem takes the form

minx,u

ϕ(x,u; µk)

such that K(x)u = fcw(x) = 0

(12)

where µk is the barrier parameter for the kth barrier problem, and ϕ(x,u; µk) is the barrier objective

ϕ(x,u; µ) =m(x)−µ ∑i, j

(lngi j(x,u)+ ln(xi j−1)+ ln(xu− xi j)

)−µ ∑

i(ln(xi−1)+ ln(xu− xi)) ,

(13)

where the logarithmic terms account for both the stress constraints (10), and the variable bounds,1≤ xi ≤ xu and 1≤ xi j ≤ xu, where xu is the upper bound. The barrier objective (13) is convex in

1Recall that an optimization problem is convex if the objective is convex and the constraints c(x)≥ 0 are concave,or equivalently −c(x)≤ 0 is convex.

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x and u. Note that convexity holds since − lnx is convex and non-increasing, for gi j > 0, and gi jare concave (see Boyd and Vandenberghe [6, Ch. 3, pg 84] for details). In addition to convexity,another advantage of the barrier method is that it eliminates the need to aggregate constraintsand select aggregation parameters. Instead, all stress constraints contribute directly to the barrierobjective (13), and the true feasible space is captured exactly as µk→ 0.

As the barrier parameter decreases to zero, µk → 0, the solution of the barrier problem (12)approaches the solution of the original optimization problem (11); for details see Forsgren et al.[17]. The proposed method uses the classical approach of reducing the barrier parameter by aconstant fraction after solving each barrier subproblem, until µk is sufficiently small. The barriermethod requires that all iterates must remain strictly feasible with respect to the inequality con-straints throughout the optimization such that 1 < xi,xi j < xu and gi j(x,u) > 0. A procedure forfinding a feasible starting point is described below. The feasibility of all subsequent iterates withrespect to the inequality constraints is guaranteed through a line search procedure.

The Lagrangian of the barrier problem (12) is

L (x,λ ,u,ψ; µ) = ϕ(x,u; µ)+λ T cw(x)+ψT (K(x)u− f), (14)

where the Lagrange multipliers λ and ψ are introduced to enforce the constraints on the materialselection variables (5) and the governing equations, respectively. The KKT conditions for thebarrier problem (12) are obtained by differentiating the Lagrangian with respect to the designvariables, x, displacement vector, u, and Lagrange multipliers, λ and ψ , which gives the followingsystem of equations

r(x,λ ,u,ψ; µ) =

rxrλrurψ

=

∇xϕ +AT

wλ +ATu ψ

cw(x)∇uϕ +Kψ

Ku− f

. (15)

Within this KKT system, the matrix Au is a linearization of the governing equations with respectto the design variables

Au =∂K(x)u

∂x.

In addition, the linearization of the material-selection constraints is given by Aw(x) = ∇xcw(x).At each iteration of the full-space barrier method, we find an inexact solution to an approximate

Newton system obtained by linearizing the KKT system (15). The approximate Newton systemtakes the form

D ATw AT

ψ +ET ATu

Aw 0 0 0Aψ +E 0 C K

Au 0 K 0

pxpλpupψ

=−

rxrλrurψ

. (16)

The linearization is exact except for the block term D≈ ∇xxL , which we modify to ensure that itis positive-definite. In this system of equations, the matrix Aψ is defined in a similar manner to thematrix Au as follows

Aψ =∂K(x)ψ

∂x.

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In addition, the C and E matrices are defined as follows

C = ∇uuϕ, E = ∇uxϕ.

As a result of the convexity of ϕ(x,u; µ), C is positive semi-definite. Instead of solving (15) to atight tolerance, we use an inexact solution [13] that we solve to a relative tolerance of εNK

rtol .To motivate the approximation used for D, consider the exact linearization of rx with respect

to x∇xxL = ∇xxϕ +∇xx(λ T cw(x))+∇xx(ψT K(x)u),

= ∇xxϕ +ne

∑i=1

λi∇xx[cw(x)]i +∑i, j

ψT Ki(Q j)u ∇xx pi j(x),

where we have used the definition of pi j(x) from (4). Note that the first term is positive-definitewhile the second two terms may be indefinite depending on the values of the material interpolationconstraint multipliers, λi, and the value of the products ψT Ki(Q j)u. To ensure a positive-definiteapproximation, we compute D as follows

D = ∇xxϕ +ne

∑i=1

max{0, λi} ∇xx[cw(x)]i +∑i, j

max{0, ψT Ki(Q j)u} ∇xx pi j(x).

Since the second two terms in this expression are positive semi-definite, and ∇xxϕ is positivedefinite, D is positive-definite.

3.1 Solving the approximate Newton systemFollowing Biros and Ghattas [4, 5] we solve the approximate Newton system (16) at each iterationof the optimization algorithm using a preconditioned Krylov-subspace method. The matrix-vectorproducts required by the Krylov-subspace method are computed using a combination of exactmatrix-vector products, which are evaluated without storing the matrix explicitly in memory, andin-memory matrix-vector multiplications. Within this work, K and D are computed and storedexplicitly, while all other matrix-vector products are computed without explicit storage.

The preconditioner for the linear system (16) is based on block-elimination of a matrix obtainedby omitting certain block components from the exact linearization. The preconditioner presentedbelow is an extension of the P2 preconditioners described by Biros and Ghattas [4], modified withthe use of a null-space method to handle the material-selection constraint terms. Within this pa-per, the preconditioner uses an algebraic multigrid preconditioning method for each block andis therefore non-stationary. As a result, a flexible Krylov-subspace method is required. We useflexible GMRES [46, 45], the flexible variant of GMRES [47]. We note that other, more sophis-ticated Krylov subspace techniques have been developed with large-scale topology optimizationapplications in mind [53, 2].

The preconditioner is based on the following matrix obtained by discarding terms from thesystem (16) to obtain

M =

D AT

w 0 ATu

Aw 0 0 00 0 0 K

Au 0 K 0

.10

Approximate solutions of the form Mp = b are used for preconditioning. These solutions can beobtained by approximately solving three linear systems in sequence. The first step is to obtain theupdate for the Lagrange multipliers, pψ , by approximately solving

Kpψ = bu.

The next step is to obtain the update for the design variables and material-selection multipliers(px,pλ ), by approximately solving[

D ATw

Aw 0

][pxpλ

]=

[dxbλ

]=

[bx−Aupψ

bλ

]. (17)

Finally, the last step of the preconditioner is to find the update for the displacement variables, pu,by approximately solving

Kpu = bψ −Aupx.

To reduce computational cost of applying the preconditioner, a single application of a W-cycle ofmultigrid [54] is used in place of exact solutions of equations involving K. In addition, a null-spacemethod is used to approximately solve the system (17) that is described in the following section.

3.1.1 Null-space method

In this section, we present a null-space method for solving the linear system (17). The materialselection constraints (5) produce a Jacobian, Aw ∈Rne×ne(M+1) that has a structured block sparsitypattern. In particular, each material selection constraint only depends on variables in element i. Asa result, each of the ne, 1× (M+1) non-zero blocks in the Jacobian matrix, Aw, can be written asfollows

[Aw]ii =[

∂ [cw]i∂xi

∂ [cw]i∂xi1

∂ [cw]i∂xi2

∂ [cw]i∂xi3

. . .],

=[ai1 ai2 ai3 ai4 . . .

],

=

[0 −

(1

xi1

)2−(

1xi2

)2−(

1xi3

)2. . .

].

To use the null-space method, we first construct a basis for the null space of Aw, labeled Zw ∈Rne(M+1)×neM. Like Aw, the basis Zw is also highly structured with ne independent non-zero blocksof size (M + 1)×M. Each non-zero block in Zw is constructed by taking the index jmax of thelargest value of |ai j|, which corresponds is the smallest value of xi j. We then form Zw by takingthe (M+1)× (M+1) identity matrix, replacing the jmax row with the entries −ai j/ai jmax and thendeleting the jmax column. For instance, if jmax = 3, then this procedure would yield the followingblock

[Zw]ii =

1

1−ai2

ai3−ai4

ai3. . .

1. . .

.As a result of this construction, AwZw = 0 holds and no entries of Zw exceed unity. The solutionpx can be expressed as a linear combination of the basis Zw and the rows of Aw as follows

px = Zwuw +ATwvw.

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Substitution of this expression into the constraint Awpx = bλ yields the equation

AwATwvw = bλ .

Note that the matrix AwATw is diagonal with entries that can be easily evaluated

AwATw = diagi

{M+1

∑j=1

a2i j

}= diagi

{M

∑j=1

1x4

i j

}.

The components uw are obtained by substituting the expression for pw into the first equation inthe linear system (17) and pre-multiplying by ZT

w to obtain the following system of equations

ZTwDZwuw = ZT

w(dx−DATwvw). (18)

The block-sparsity of the matrix ZTwDZw is the same as the matrix D, where the block size is

reduced by one, such that ZTwDZw has blocks of size M×M. In this work, we approximately

solve the system of equations (18), with the linear system ZTwDZw, again using a single W-cycle of

an algebraic multigrid method. Once uw is obtained, px can be evaluated as px = Zwuw +ATwvw.

Finally, the update pλ can be obtained by solving the system

AwATwpλ = Aw(dx−Dpx)

This completes the solution of the linear system (17) using the null-space method.

3.2 Line search globalizationOnce an approximate solution of the Newton system (16) is obtained, the next design point is foundusing a line search technique. The line search globalizes the Newton method and it restricts theiterates to remain within the feasible space. The exact `2 merit function used within the line searchis given by

φ(x,u; γ) = ϕ(x,u; µ)+ γ||cw(x)||2 + γ||K(x)u− f||2, (19)

which is a function of the design variables, x, and the displacements, u, but not the Lagrangemultipliers. The penalty parameter γ is selected at each iteration to ensure a sufficiently negativedirectional derivative along the search direction [40]. A backtracking line search is used, whichseeks a point that satisfies the sufficient decrease condition

φ(x+αipx,u+αipu; γ)< φ(x,u; γ)+ c1αiDαφ(x,u; γ) (20)

where px and pu are the steps computed from the linearized KKT system (16), c1 = 10−4 is aconstant, and Dαφ denotes the derivative of the merit function with respect to α . The step length,αi is selected based on the backtracking formula, αi = 21−iαmax, where the maximum step length,αmax, satisfies 0 < αmax ≤ 1 and is chosen so that the step along the direction (px,pu) does notviolate the stress constraints or variable bounds.

To prevent the iterate from entering an infeasible region of the design space, the maximum linesearch step size, αmax, is selected based on a fraction-to-the-boundary rule. To find an accurateapproximation to the step length that violates the stress constraint boundary, we use a quadratic

12

approximation of the variation of each stress constraint along the search direction constructedusing a Taylor series expansion

gi j(x+αpx,u+αpu)≈ aα2 +bα + c = 0, (21)

where the coefficients are computed from the base point (x,u). This equation captures the vari-ation of the constraint (10) along the u-direction exactly. The smallest positive root of (21) is anapproximation of the maximum step along (px,pu) not violating the constraints gi j. A single pos-itive root, α+

i j , of the expansion (21), is guaranteed since c > 0, (because (x,u) is feasible), anda < 0 (because gi j is concave). The maximum step length, αmax, is computed such that

αmax = τ mini, j

α+i j (22)

where τ = 0.95. For the bound constraints 1 ≤ xi ≤ xu and 1 ≤ xi j ≤ xu, we impose the normalfraction-to-the-boundary rule [40, Ch. 19, pg. 567]. The step length from the approximation (21)typically produces a step to a feasible point, however, this is not guaranteed. On the rare occurrencewhen the scaled step does violate the stress constraints, we divide the maximum step length in half,αmax← αmax/2, until we find a feasible step.

3.2.1 Avoiding the Maratos effect

Since the merit function (19) is exact and relies on the `2 norm of the constraint violation, itis susceptible to the Maratos effect [16]. The Maratos effect occurs when a step which makesgood progress towards the solution does not satisfy a sufficient decrease condition, leading to slowconvergence from unwanted line search iterations. To alleviate the Maratos effect, we implementa corrective step that is designed to reduce the constraint violation. This corrective step is usedonly when the line search fails due to an increase in the constraint violation on the first iteration.Otherwise no corrective step is computed.

To find a design point with lower constraint infeasibility, we compute a step in the designvariables, px, and a step in the displacements, pu, that are added to the point (x+αmaxpx,u+αmaxpu). To compute the corrective step in the design variable, we use a least-squares solution ofa linearization of the weighting constraints about the new point

Awpx + cw(x+αmaxpx) = 0,

where Aw = Aw(x+αmaxpx). This solution is designed to decrease the constraint violation and isgiven by

px =−ATw[AwAT

w]−1 cw(x+αmaxpx). (23)

Next, we compute a step pu that decreases the constraint violation from the equilibrium equa-tions, first evaluating the stiffness matrix K = K(x+αmaxpx), and then computing an approximatesolution of the equations

Kpu = f−(Ku+ Kpu

). (24)

Lastly, we safeguard the new step by computing the maximum step length to the boundary αmaxusing (21) and (22) such that the new point

x = x+αmaxpx + αmaxpx,

u = u+αmaxpu + αmaxpu,(25)

13

remains within the feasible design space. The provisional corrected point is accepted if it satisfiesthe sufficient decrease conditions, otherwise the backtracking line search is resumed.

3.2.2 Handling non-descent directions

Unfortunately, due to the use of inexact solutions, the step computation (17) does not alwaysproduce a descent direction for the merit function. This is detected when the line search penaltyparameter γ cannot be chosen to make the derivative, Dαφ(x,u; γ) negative.

Through numerical experiments, we have found that this failure occurs infrequently, typicallyimmediately following the reduction of the barrier parameter between subsequent barrier problems.Instead of searching in the opposite direction of the computed ascent direction, we temporarilyswitch to a line search based on the `2 norm of the residual, replacing the merit function (19) inthe line search with

φ2 = ||r(x+αpx,λ +αpλ ,u+αpu,ψ +αpψ)||2.

It can be shown that Dαφ2≈−||r||2. While the use of this merit function may lead to local maxima,we have found that in practice this approach often produces subsequent iterates that make progresstowards the solution.

3.3 Convergence criteriaThe proposed full-space barrier method is designed to quickly find an approximate solution of theoriginal stress-constrained mass minimization problem (11) that can then be used for further designstudies. Therefore, tightly converged solutions are not required. Within this paradigm, we focuson reaching a target barrier parameter, µtarget, and solving the barrier problem associated withµtarget to a loose tolerance. We use two convergence criteria within the optimization algorithm:a KKT-residual based criteria, to decide when to reduce the barrier parameter or terminate theoptimization, and a convergence criteria based on the change in the mass between iterations.

The KKT-residual criteria is based on the following normalized residual measure

E(x,λ ,u,ψ; µ) = max{||rx||∞, ||cw||∞,

||∇uϕ +Kψ||∞max{1,µ} , ||Ku− f||∞

}, (26)

where the third KKT residual norm is scaled by the barrier parameter when µ > 1. This scalingis applied since the derivative ∇uϕ is proportional to the barrier parameter, and as a result theLagrange multipliers, ψ , are also proportional to µ .

The convergence criteria applied to the mass is only evaluated after the target barrier parameterhas been reached and is based on the change in mass for the most recent optimization step. Ifthe step is a full unit step with α = 1, then we terminate the optimization algorithm if the massdecreased such that, m(x+px) ≤ m(x), and the decrease in mass was bounded from below witha relative tolerance m(x+px)≥ (1− εmass)m(x), where we have used εmass = 10−3. This conver-gence criteria corresponds to a 0.1% change in mass between iterations.

With the convergence criteria defined, we can now outline the complete full-space barrier al-gorithm. The full algorithm is shown in Figure 1. The barrier parameter is initialized at a valueof µ1 = 1000. Then, after each barrier subproblem has been solved approximately, such that

14

Given a feasible x and µtarget, set µ1 = 1000, θ = 0.25, u = 0, and k = 1Solve K(x)pu = f, set px = 0 and compute αmax from (22) and (21)Set u← αmaxpuSolve K(x)ψ =−∇uϕSet λ j = 1, j = 1, . . . ,newhile µk > µtarget do

while E(x,λ ,u,ψ; µk)≥ 10µk doInexactly solve (16) to obtain (px,pλ ,pu,pψ)Compute αmax using (22) and (21)if φ(x+αmaxpx,u+αmaxpu; γ)< φ(x,u; γ)+ c1αmaxDαφ(x,u; γ) then

Update (x,λ ,u,ψ)← (x,λ ,u,ψ)+αmax(px,pλ ,pu,pψ)continue

else if ||cw(x)||2 or ||K(x)u− f||2 increased from α = 0 to α = αmax thenCompute px and pu using (23) and (24)if φ(x, u; γ)< φ(x,u; γ)+ c1αmaxDαφ(x,u; γ) then

Update (x,u)← (x, u) and (λ ,ψ)← (λ ,ψ)+αmax(pλ ,pψ)continue

end ifend ifFind the smallest i = 1,2, . . . and αi = 21−iαmax satisfying sufficient decrease (20)if µk = µtarget and m(x+px)≤ m(x) and m(x+px)≥ (1− εmass)m(x) then

returnend ifUpdate (x,λ ,u,ψ)← (x,λ ,u,ψ)+αi(px,pλ ,pu,pψ)

end whileµk+1 = max{θ µk, µtarget}Update the multiplier estimates (λ ,ψ)← θ(λ ,ψ)k← k+1

end while

Figure 1: Full-space multimaterial optimization algorithm

E(x,λ ,u,ψ; µk) < 10µk, the barrier parameter is reduced by a constant fraction θ = 0.25 until atarget barrier parameter is met. After each barrier problem, the associated Lagrange multipliersare scaled by θ to match the barrier parameter reduction. Each barrier subproblem is solved ap-proximately within the inner loop using the inexact Newton method. A backtracking line searchwith a second-order corrective step is used to globalize the Newton method and also ensure thatthe iterates always remain feasible with respect to the stress and bound constraints. The efficiencyand robustness of this algorithm will be assessed in Section 4.

Due to the nature of the full-space method, the governing equations are not satisfied to a tighttolerance at every iteration. At early iterations during the optimization, a full step in the statevariables, u, to satisfy the governing equations at fixed design variables, x, may cause the stressconstraints to be violated. However, we have found that at the optimized solution typically satisfies||Ku−f||∞/||f||∞∼O(10−3) with the original problem data. Furthermore, at the optimized design,a full step to satisfy the governing equations will typically not cause a stress constraint violation.

15

Property Value

ρ 2810 kg/m3

E 70 GPaν 0.3

σmax 300 MPa

(a) Aluminum alloy

Property Value Property Value

ρ 1265 kg/m3

E1 207 GPa E2 6 GPaG12 2.6,GPa ν12 0.25

Xt 1035 MPa Xc 689 MPaYt 41 MPa Yc 117 MPaS12 69 MPa

(b) Carbon/epoxy

Table 1: Material properties for the representative aluminum alloy and carbon/epoxy

3.4 Optimization problem scalingThe scaling of the problem data has a direct impact on the efficiency of the full-space barrier algo-rithm and the quality of the solution. The dependence of the solution on scaling can be illustratedby considering scaling parameters for the material stiffness data, αm, and the stress constraint data,αs, respectively. Using these scaling parameters, it can be shown that the optimum values of thedesign variable, x, and state variables, u, are not modified when the problem stiffness, stress con-straint and load data (Q j, h j, G j, f), are replaced with the scaled data (αmQ j, αsh j, α2

s G j, αm/αsf).While the constraint scaling does not modify the solution (x,u), it does have an impact on the it-erations and ease of solving the approximate Newton system. We have found that it is usuallynecessary to scale the problem using αm and αs to achieve better optimization performance.

4 ResultsIn this section, we present results from the application of the full-space barrier method describedabove to three design problems: a cantilever, an L-bracket, and the MBB beam [41]. The problemdomains and loading conditions for these problems are shown in Figure 2. The cantilever problemis discretized using a 128× 64 mesh with 8192 bilinear elements. The L-bracket problem uses adiscretization with 32 elements through the bracket arm, and 80 elements along each side of thebracket resulting in a total of 4096 bilinear elements. Finally, the MBB-beam is discretized usinga 144×48 mesh with 6912 bilinear elements.

We consider two types of material selection problems:

1. Material selection between representative aluminum alloy and carbon/epoxy oriented at 0◦

along the x-axis; and

2. Discrete ply-angle selection using the same orthotropic carbon/epoxy properties oriented atdifferent lamination angles, given by Θ = {−45◦, 0◦, 45◦, 90◦}.

The representative aluminum alloy and carbon/epoxy material properties for this problem are listedin Table 1a, and Table 1b, respectively. For all problems described here, we use αm = 1, andαs = 0.1 for the material and stress constraint scaling, respectively.

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200

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40

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Figure 2: Design domains and loading for the problems

4.1 Problem and algorithm parametersBefore presenting the results, we first provide details of the problem and algorithm parametersthat are common to all the cases described below. The full-space barrier algorithm shown in Fig-ure 1 is implemented using a combination of Fortran and Python. The computationally expensiveoperations, including matrix assembly, and matrix-vector product computations are performed inFortran, which are then wrapped in Python using the f2py tool [42].

In all problems, a stress constraint exclusion zone is used in the neighborhood of applied loads.This exclusion zone consists of a 2-element layer immediately adjacent to applied loads whereno stress constraints are imposed. For elements in the exclusion zone, the bound on the topologyvariables are modified such that the inverse of the topology variable, 1/xi, cannot fall below 0.9.The bounds on the corresponding material selection variables are not modified to give the optimizerfreedom to select the material at the point of load application.

17

The stress relaxation parameter, ε , has an impact on both the computational cost of the opti-mization and how closely the relaxed stress constraints match the true stress constraint boundary.A low value of ε makes it more difficult for the optimizer to exploit the ability to entirely removematerial from the design. We have found that a stress relaxation factor of ε = 0.1 often producesgood results. The spatial filter also has an impact on computational cost and the smoothness of thedesign space. We use the same conic filter for both the design variables and for the spatially aver-aged strains within the failure criteria. The conic filter is constructed with a radius of r = 2.2∆e,where ∆e is the element edge length. Larger filter radii produce smoother solutions with greateramounts of intermediate materials, while also increasing the cost of the evaluating the stress con-straints and their and their derivatives.

Within this work, we have exclusively used RAMP penalization [52]. We have found thatusing the lowest penalization necessary to achieve near 0-1 solutions typically produces the bestdesigns. For the topology penalization, we use a RAMP parameter value of qt = 5, and for thematerial selection penalization, we use a RAMP parameter value of q = 2. For the topology designvariables, we set an upper bound of xu = 100, which results in an effective thickness lower boundof ti ≥ 10−2 for the filtered topology thickness. For the material selection variables, we set anupper bound of xu = 1000, which results in an effective lower bound of ti j ≥ 10−3 for the filteredmaterial thickness variable.

Selecting an unbiased starting design point is important so that the solution does not convergetoo early towards an unwanted local minima. For the problems presented below, the starting pointis selected so that the initial unpenalized element thicknesses take a value of ti = 0.98, which isclose to their upper bound. This design point is selected so that the stress constraints and governingequations are nearly satisfied during the start-up procedure before the optimization shown in thealgorithm in Figure 1. For the material selection variables, we select xi j = M so that the materialfraction variables take a value of ti j = 1/M, which corresponds to an even weighting. Note that theinitial design point satisfies the material selection weighting constraints (5).

Within the Newton method, we use FGMRES(40) with a fixed relative solution tolerance εNKrtol .

The best solution tolerance varies with problem type. The lower the solution tolerance, the lesscomputational work must be done at each iteration. However, selecting too large a tolerance leadsto line search failures where either a non-descent direction is computed or a line search must selecta very small step length. We have found that εNK

rtol = 10−1 works well for the cantilever problem,but it is necessary to choose εNK

rtol = 10−3 for the MBB-beam and L-bracket problems. Finally, forthe cantilever and MBB-beam problems we use a target barrier parameter of µtarget = 1, while forthe L-bracket problems, we use a target of µtarget = 0.25.

It is difficult to make a direct comparison between the computational costs of a reduced-spacemethod and the full-space barrier method proposed within this paper. In the results, we focus onusing FGMRES iterations as a metric for computational cost. More than 3/4 of the computationaltime of the full-space barrier method is consumed in FGMRES iterations. Each FGMRES iterationconsists of a matrix-vector product and preconditioning operations, where the preconditioner usesthree applications of a single W-cycle of multigrid: one within the null-space method, using thereduced D matrix, and two for the equations that involve the stiffness matrix. Reduced-space opti-mization methods perform a solution followed by a constraint gradient evaluation using the adjointmethod. Therefore, at least two full linear system solves are required at each iteration and morelinear solves are required if additional constraint aggregates are used. For the problems solvedhere, 15 iterations of multigrid is typically sufficient to obtain an accurate solution. Therefore, as

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Figure 3: Normalized mass and KKT residual history for the multimaterial cantilever problem

a rough comparison, the cost of 5 FGMRES iterations corresponds to the cost of a single iterationof a reduced-space method using a single stress-constraint aggregate.

4.2 Cantilever beam4.2.1 Multiple materials

In this problem, we consider material selection and topology optimization in the cantilever domainwith the representative aluminum and 0◦ carbon/epoxy material. The problem domain contains8192 elements, with one topology and two material selection variables per element, resulting in24 576 design variables.

Figure 3 shows the normalized mass and KKT residual measure history for this problem withthe solution parameters described above. The results are shown in terms of the number of FGMRESiterations, while the Newton iterations are marked on the top horizontal axis. The mass shownin Figure 3 is normalized by the initial structural mass which is 98% of the mass of the entirestructural domain composed of a material with the average candidate material density. Duringthe optimization, there is an initial large mass reduction during the first few Newton iterations,followed by a gradual increase in mass during which the KKT residual increases slightly. Afterroughly the first 400 FGMRES iterations, or 30 Newton iterations, the normalized mass and KKTresidual decrease rapidly and converge after an additional 300 FGMRES iterations. The largestmass reduction is accomplished between 400 and 500 FGMRES iterations. The entire optimizationrequires 706 FGMRES iterations and 72 Newton iterations.

Figure 4 shows the progression of the multimaterial cantilever designs during the optimiza-tion and the optimized values of the most critical stress constraints, computed as min j gi j for eachelement i within the domain. The color intensity within the design figures indicates the valueof the topology variable while the green and blue colors indicate selection of aluminum and car-bon/epoxy, respectively. The progression shows that there is considerable change in the designbetween 30 and 50 Newton iterations. The mass decreases by less than 6% between 50 Newtoniterations and the optimized design. The color scale in the stress constraint plot is truncated forstress constraints which exceed gi j > 2, so that the figure illustrates the most critical constraint

19

(a) 30th Newton iteration (b) 50th Newton iteration

(c) Optimized design (d) Optimized min j gi j

Figure 4: Multimaterial cantilever design history

values. The optimized stress constraint values show that the stresses are most critical where loadsare applied and where there is an interface between the aluminum and carbon/epoxy.

4.2.2 Ply-angle selection

In this problem, we consider ply-angle selection and topology optimization with representativecarbon/epoxy material oriented at the candidate angles, Θ = {−45◦, 0◦, 45◦, 90◦}. The prob-lem domain contains 8192 elements, with one topology and two material selection variables perelement resulting in 44 560 design variables.

Figure 5 shows the iteration history of the normalized mass and the KKT residual measurefor this problem. The full-space barrier method required 62 Newton iterations and 631 FGMRESiterations to obtain the optimized solution. The iteration history exhibits some similarity with themultimaterial cantilever problem, where initially the optimizer reduces the structural mass, fol-lowed by a gradual increase in mass, before the KKT residual decreases significantly. However,for the ply selection case, this start-up behavior consumes far fewer Newton and FGMRES itera-tions. The mass decreases by 3% from between 40 Newton iterations and the optimized design,requiring roughly 300 additional FGMRES iterations.

Figure 6 shows the progression of the orthotropic cantilever ply-angle designs during the opti-mization. The figures illustrate both the value of the thickness variables, ti, and the orientation ofthe material selection. The material selection angle is shown only if the material fraction variable,ti j, is larger than 0.75 such that ti j > 0.75. As shown, the solution converges to the outline ofthe final design within 20 Newton iterations, requiring roughly 150 FGMRES iterations. After 40Newton iterations, or 350 FGMRES iterations, the design has largely converged. There are only

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)

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Figure 5: Normalized mass and KKT residual history for the orthotropic ply angle selection can-tilever problem

small changes between the design after 40 Newton iterations and the optimized design. For the op-timized design, the stress constraints are most critical at the interface between members orientedat different angles. These interface regions are larger and have interesting geometric features thathelp reduce stress.

4.3 L-bracket4.3.1 Multiple materials

In this problem, we consider material selection and topology optimization within the L-bracketdomain using representative aluminum alloy and 0◦ carbon/epoxy material. The problem domaincontains 4096 elements, with one topology and two material selection variables per element, re-sulting in 12 288 total design variables.

Figure 7 shows the normalized structural mass and KKT residual measure over the course ofthe optimization. The full-space barrier algorithm required 30 Newton iterations and 1083 FGM-RES iterations to obtain the optimized solution. The larger ratio of FGMRES to Newton iterationsfor this problem is due to the tighter inexact Newton tolerance used for this case, εNK

rtol = 10−3,compared with the cantilever cases, where εNK

rtol = 10−1. In this case, the mass decreases mono-tonically over the entire optimization history, and the KKT residual measure does not increase asrapidly between iterations compared with the cantilever cases. This behavior can also be attributedto the tighter inexact Newton tolerance. Finally, note that the optimization algorithm terminateswhen the KKT residual measure criteria is satisfied.

Figure 8 shows the progression of the multimaterial L-bracket designs over the course of theoptimization and the optimized values of the most critical stress constraints. Again, the colorintensity indicates the value of the topology variable while the green and blue colors indicatealuminum and carbon/epoxy, respectively. Figure 8 also shows the most critical stress constraintvalue, min j gi j, for each element in the domain. The optimized design consists of two verticalmembers supporting a truss structure that transmits the moment and vertical load produced by theapplied load. Members aligned along the x direction utilize 0◦ material, while all other connecting

21

(a) 20th Newton iteration (b) 40th Newton iteration

(c) Optimized design (d) Optimized min j gi j

Figure 6: Orthotropic ply-angle selection cantilever design history

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Figure 7: Normalized mass and KKT residual history for the multimaterial L-bracket problem

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(c) Optimized design (d) Optimized min j gi j

Figure 8: Multimaterial L-bracket design history

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Figure 9: Normalized mass and KKT residual history for the orthotropic ply-angle selection L-bracket problem

members utilize the representative aluminum alloy material. The outline of the optimized designcan be clearly seen after 20 Newton iterations, corresponding to just under 700 FGMRES iterations.

4.3.2 Ply-angle selection

In this problem, we consider ply-angle selection and topology optimization for the L-bracket do-main using the representative carbon/epoxy material oriented at the candidate angles, Θ = {−45◦,0◦, 45◦, 90◦}. The problem domain contains 4096 elements, with one topology and four materialselection variables per element, resulting in 20 480 design variables.

Figure 9 shows the normalized mass and KKT residual measure history for the ply-angle se-lection L-bracket problem. The proposed full-space optimization algorithm requires 33 Newtoniterations with 999 FGMRES iterations to find the optimized design. As in the multimaterialL-bracket problem, the mass decreases monotonically over the course of the optimization, andalgorithm terminates with KKT residual measure criteria.

Figure 10 shows the progression of the designs over the course of the optimization and themost critical stress constraint values at the optimized design. The design of the bracket is similarto the multimaterial bracket, but contains two parallel 45◦ members connecting across the cornerof the domain. This configuration reduces the stress at the re-entrant corner and reduces the shearin the cross-member. In this design, some of the members are composed of intermediate-thicknessmaterial with thicker interfaces between members. The intermediate thickness material is lighterand still satisfies the stress constraints. Finally, note that the stress constraints on the compressionside of truss structure are more critical than the stress constraints on the tension side.

4.4 MBB beam4.4.1 Multiple materials

In this problem, we consider material selection and topology optimization within the MBB-beamproblem domain with representative aluminum and 0◦ carbon/epoxy material. The problem do-

24

(a) 10th Newton iteration (b) 20th Newton iteration

(c) Optimized design (d) Optimized min j gi j

Figure 10: Orthotropic ply-angle selection L-bracket design history

main contains 6912 elements, with one topology and two material selection variables per element,resulting in 20 736 design variables.

Figure 11 shows the normalized mass and KKT residual measure over the course of the opti-mization. The proposed algorithm requires 47 Newton iterations and 1550 FGMRES iterations toconverge to a solution. Note that the problem exits with the mass convergence criteria. The massdecreases monotonically over the course of the optimization, with 95% of the decrease in massover the first 27 Newton iterations, requiring just over 750 FGMRES iterations. This constitutesroughly half of the computational time consumed by the algorithm.

Figure 12 shows the designs during the optimization after the 20 and 30 Newton iterations,respectively, as well as the optimized design and the most critical stress constraint values at theoptimized design point. As in the previous multimaterial examples, the structural members ori-ented at 0◦ with respect to the x-axis are composed of the carbon-epoxy composite. The remainingmembers which are not oriented along the x-axis are composed of the representative aluminumalloy.

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Figure 11: Normalized mass and KKT residual history for the multimaterial MBB-beam problem

(a) 20th Newton iteration (b) 30th Newton iteration

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Figure 12: Multimaterial MBB-beam design history

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Figure 13: Normalized mass and KKT residual history for the orthotropic ply-angle selectionMBB-beam problem

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(a) 20th Newton iteration (b) 30th Newton iteration

(c) Optimized design (d) Optimized min j gi j

Figure 14: Orthotropic ply-angle selection MBB-beam design history

4.4.2 Ply-angle selection

In this problem, we consider ply-angle selection and topology optimization within the MBB-beamdesign domain using a representative carbon/epoxy material oriented at the candidate angles, Θ ={−45◦, 0◦, 45◦, 90◦}. The problem domain contains 6912 elements, with one topology and twomaterial selection variables per element, resulting in 34 560 design variables.

Figure 13 shows the history of the normalized mass and KKT residual over the course of theoptimization. The entire optimization requires 48 Newton iterations with 1598 FGMRES itera-tions. The overall cost of the optimization, in terms of iterations, is very similar to the multima-terial MBB-beam problem presented above. As in the L-bracket problems, the normalized massdecreases monotonically throughout the optimization. However, in this case, the KKT residualremains higher for a larger fraction of the optimization history, and does not track as closely withthe reduction in mass as the previous examples. Over 95% of the mass reduction is achievedwithin the first 23 Newton iterations, which requires 632 FGMRES iterations and less than half thecomputational time.

Figure 14 shows the progression of the designs after 20 and 30 Newton iterations and at theoptimized design, as well as the most critical stress constraints in each element, found by evaluatingmin j gi j for each element i. The design has essentially converged at the 30th Newton iteration wherethe remainder of the iterations results in less than a 1% mass reduction.

The optimized design consists of a truss structure where the cross-members oriented at −45◦

are under compression, while the member at 45◦ is in tension. In addition, the top horizontalmember is under compression, while the bottom horizontal member is under tension. Due to thepoorer strength of the carbon/epoxy material under compression, the −45◦ members and the tophorizontal member are either split or are larger than the remaining members in tension. Anotherfeature of this design is the interface between the −45◦ cross-member and 0◦ bottom horizontalmember, which has been designed primarily using 0◦ material. We note that this design feature islikely due to the higher transverse compressive strength of the material compared to the in-planeshear strength.

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5 ConclusionsIn this paper, we have presented a full-space barrier method that is tailored for stress-constrainedmass minimization of discrete material problems. We have demonstrated the advantages of the full-space approach in the context of stress-constrained design optimization: first, the stress constraintsare provably concave, which facilitates the construction of convex optimization subproblems; sec-ond, the full-space approach avoids the use of constraint aggregation. Each barrier subproblem issolved using an inexact Newton method with an approximate linearization of the KKT system. Ateach iteration of the Newton method, the approximate linearization is solved using preconditionedflexible GMRES. The preconditioner is constructed to take advantage of the sparsity of the discretematerial optimization weighting constraints using a null-space method. Results demonstrating theapplication of the proposed algorithm were shown for a cantilever problem, an L-bracket problem,and the MBB-beam problem. The optimization problems required between 30 and 72 Newton iter-ations, with between 631 and 1550 FGMRES iterations. Future work will concentrate on applyingthis method to large-scale design problems to test the scalability of the proposed algorithm.

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