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Structural and Multidisciplinary Optimization manuscript No.

(will be inserted by the editor)

Topology optimization of load-bearing capacity

Leyla Mourad ·Jeremy Bleyer ·Romain Mesnil ·Joanna Nseir ·

Karam Sab ·Wassim Raphael

Received: date / Accepted: date

Abstract The present work addresses the problem of

maximizing a structure load-bearing capacity subject to

given material strength properties and a material vol-

ume constraint. This problem can be viewed as an ex-

tension to limit analysis problems which consist in ﬁnd-

ing the maximum load capacity for a ﬁxed geometry. We

show that it is also closely linked to the problem of mini-

mizing the total volume under the constraint of carrying

a ﬁxed loading. Formulating these topology optimiza-

tion problems using a continuous ﬁeld representing a

ﬁctitious material density yields convex optimization

problems which can be solved eﬃciently using state-

of-the-art solvers used for limit analysis problems. We

further analyze these problems by discussing the choice

of the material strength criterion, especially when con-

sidering materials with asymmetric tensile/compressive

strengths. In particular, we advocate the use of a L1-

Rankine criterion which tends to promote uniaxial

stress ﬁelds as in truss-like structures. We show that

the considered problem is equivalent to a constrained

Michell truss problem. Finally, following the idea of

the SIMP method, the obtained continuous topology is

post-processed by an iterative procedure penalizing in-

termediate densities. Benchmark examples are ﬁrst con-

sidered to illustrate the method overall eﬃciency while

L. Mourad ·J. Bleyer (B)·R. Mesnil ·K. Sab

Laboratoire Navier, Ecole des Ponts ParisTech, Univ Gustave

Eiﬀel, CNRS

6-8 av. Blaise Pascal, Cit´e Descartes

77455 Champs-sur-Marne, France

Tel : +33 (0)1 64 15 37 43

E-mail: jeremy.bleyer@enpc.fr

L. Mourad ·J. Nseir ·W. Raphael

Universit´e Saint Joseph, Facult´e des sciences, Mar Roukos-

Dekwaneh, Lebanon

ﬁnal examples focus more particularly on no-tension

materials, illustrating how the method is able to repro-

duce known structural patterns of masonry-like struc-

tures. This paper is accompanied by a Python package

based on the FEniCS ﬁnite-element software library.

Keywords Topology optimization ·Limit Analysis ·

Bearing capacity ·Second-order cone programming ·

No-tension material ·Michell truss

Funding Information

This work is part of the PhD thesis of L. Mourad

who is supported by Universit´e Paris-Est and Univer-

sit´e Saint-Joseph.

1 Introduction

Construction materials used for primary structures

of building and infrastructures are responsible for

a consequent share of mankind’s greenhouse gases

emissions [26]. The reduction of weight through

structural optimization is therefore a challenge of

growing importance, even in the construction industry.

Building structures have to comply on limit state

design, which requires satisfying two principal criteria:

Ultimate Limit State (ULS) and Serviceability Limit

State (SLS). The two limit states diﬀer fundamentally

in their nature, since SLS criteria usually restrict the

structures to perform in the elastic range, whereas ULS

requires to check the structure maximal load-bearing

capacity with respect to material non-linearities.

When designing a structure with respect to ULS, it is

important to take into account stress redistributions

occurring during the material non-linearity phase.

2 Leyla Mourad et al.

This non-linearity is particularly stringent for usual

construction materials, like plain concrete which has

diﬀerent compressive and tensile strength.

Topology optimization is a mathematical problem

of optimal material distribution in a given domain. In

structural mechanics, topology optimization problems

can be classiﬁed as stiﬀness optimization (related to

SLS) and strength optimization (related to ULS).

The seminal work of Michell [34], which dealt with

fully-stressed continuous truss structures was in fact

a compliance optimization. Michell truss have been

extended in the middle of the twentieth century, either

to construct discrete solutions [38], or to consider

optimal plastic design of truss structures [39]. Bendsøe

and Kikuchi [7] were among the ﬁrst to introduce

layout optimization where the distribution of the

material within a domain is optimized rather that the

shape of the domain itself.

Stiﬀness-based optimization is now a well-

established ﬁeld of research, with numerous ap-

plications in diﬀerent industries [8]. Optimization

problems aiming at maximizing stiﬀness are classically

formulated as compliance minimization problems over

displacement and stress ﬁelds satisfying the equations

of linear elasticity (with a density-dependent elasticity

tensor C(ρ)), while the volume is bounded by a fraction

of the total volume.

The solutions to such elastic compliance mini-

mization problems result in densities continuously

distributed between 0 and 1, which are hard to

manufacture. Existing methods attempt at penalizing

intermediate densities towards a black and white design

and diﬀer in the expression of the stiﬀness tensor

C(ρ) [45]. Homogenization methods make an analogy

with perforated composite materials and replace the

layout optimization by a sizing problem of the eﬀective

properties of this homogenized material [2]. This

technique has been envisioned from the pioneering

work of [7] and used to prove existence of solution

on this relaxed problem. They are also compatible

with penalized problem formulations that converge

towards black and white designs [1]. The SIMP Method

(Solid Isotropic Material with Penalization) [8] uses a

ﬁctitious power law C(ρ) = C0·ρp. Although yielding

non-convex problems for p≥1, heuristic iterative

method and careful choice of power pallow to quickly

converge towards near-optimal black and white designs.

The non-convexity implies non-uniqueness of solutions,

which makes the penalized optimization problem sen-

sitive to initialization, as well as numerical scheme e.g.

the Method of Moving Asymptotes [48] or Optimal-

ity Criterion [43] which may thus yield diﬀerent results.

Stress-based optimization can be formulated by

analogy with classical compliance minimization prob-

lems by considering a stress density cost function

J(ρ) = RDσeﬀ dx where σeﬀ is an eﬀective stress (e.g.

the von Mises equivalent stress). The main advantage of

such a formulation is that classical methods, e.g. SIMP

or level-set methods [3] can readily be used. For that

reason, this formulation has been used in several appli-

cations. Pedersen showed that the result of stress opti-

mization and compliance optimization will diﬀer if the

strain energy is not consistent with the eﬀective stress

measure [36]. One of the main diﬃculties encountered

in stress optimization is the diﬃculty to include many

local stress constraints in the optimization problem as

well as the existence of a singularity problem at zero

densities [20].

Strictly speaking, these methods do not however op-

timize for load-bearing capacity (ULS design), as they

typically consider stress ﬁelds which are solution of an

elasticity problem. Finally, some works included fully

non-linear elasto-plastic computations inside a topol-

ogy optimization procedure [4, 5, 32, 49, 51], resulting

in extremely high computational cost. The present

work aims thus at exploring another formulation that

optimize with respect to ULS, using yield design theory.

Yield design theory [40, 41] has been used in many

areas of civil engineering to design structures based on

the sole compatibility between the notions of equilib-

rium on one hand and resistance on the other hand.

Typical ﬁelds of application include geotechnical prob-

lems (soil slope stability, footing bearing capacity) [18],

design of reinforced-concrete structures [17, 35], espe-

cially using strut-and-tie methods [42] or rigid-block

stone or masonry structures, such as Heyman’s works

[23, 24] assuming inﬁnite compressive strength and zero

tensile strength for the material.

Yield design theory can be conceived as the ex-

tension of classical limit analysis theorems [25] in the

case of perfect plasticity which provide lower and up-

per bound approaches bracketing the structure ultimate

load. The former relies for instance on ﬁnding a stress

ﬁeld which should be statically admissible with a given

loading and verify the local material strength crite-

rion. Yield design/limit analysis approaches can now

be eﬃciently solved numerically by formulating them

as convex (more precisely conic) optimization problems

[12, 29, 31, 46, 50] and solved using dedicated solvers.

A noteworthy implementation of the lower bound

theorem for the design of masonry structures was intro-

duced with thrust network analysis (TNA), which was

Topology optimization of load-bearing capacity 3

presented by Block and Ochsendorf [13, 14]. TNA essen-

tially relies on the computation of statically admissible

membrane force ﬁelds satisfying Heyman’s hypotheses

using the force-density method. In recent studies, the

Block Research Group from ETH Z¨urich has worked

over constructing dry-stack no-tension structures to

show that those structures are capable of putting in

place a complex shape that surpasses beyond simple

walls, or traditional vault geometries [15, 33].

Only few works concentrated on optimization with

respect to the structure ultimate limit state. One

can mention the work of Damkilde and Krenk [19]

which determined an optimized material distribution

in reinforced concrete slabs in bending. More recently,

similar ideas have been used in [21, 22, 27] to propose

strength-based topology optimization of von Mises

plastic materials using limit analysis formulations. Our

contribution aims at providing a general formulation

of topology optimization in such a context by relying

on the concepts of yield design/limit analysis theory

and on convex optimization tools.

The manuscript is organized as follows: section 2 for-

mulates the problem of maximizing the structure load

bearing capacity for a predeﬁned volume constraint.

The problem is initially formulated as a mixed-integer

programming and parallels with a volume-minimization

problem are drawn. Section 3 present both problem re-

laxations using a continuous pseudo-density ﬁeld. The

properties of the obtained convex problems are then

detailed. Section 4 presents usual material strength cri-

teria in civil engineering, especially regarding tension/

compression asymmetric materials. Section 5 presents

the discretization and resolution procedure for solving

the convex problems. A penalization procedure is also

proposed. Finally, the eﬃciency of the proposed formu-

lation is illustrated in section 6 through various exam-

ples.

2 Maximizing the load-bearing capacity of a

structure

2.1 A brief review of limit analysis theory

In this section, we recall the general concepts of limit

analysis (or yield design) theory for structures with a

known geometry Ω. When considering perfectly plastic

materials, the limit analysis theory provides a direct

characterization of the structure load-bearing capacity

corresponding to global plastic collapse. Note that the

load-bearing capacity diﬀers from the elastic limit of

a structure (obtained when considering strength con-

straints for an elastic solution) because of the struc-

ture’s ability to redistribute loads when its constitutive

materials oﬀer suﬃcient ductility. As a result, the stress

ﬁeld associated with the structure collapse may be quite

diﬀerent from the elastic solution. It is therefore ex-

pected that optimized design obtained from stress ﬁelds

at incipient collapse may be quite diﬀerent from opti-

mized design obtained in the elastic regime.

The load-bearing capacity is obtained by ﬁnding the

maximum load ampliﬁcation factor λfor which there

exists an internal stress ﬁeld σwhich can balance the

loading and still comply with a strength criterion σ∈G

at every point in Ωwhere Gis a convex set containing

0. More precisely, the limit load Λ+can be found as the

solution to the following convex optimization problem:

Λ+= max

λ,σλ

s.t. div σ+f= 0 in Ω

σ·n=λTon ∂ΩT

σ∈Gin Ω

(1)

in which we considered the body force fto be ﬁxed

whereas we look for the maximal value of the refer-

ence surface tractions Tacting on some part ∂ΩTof

the boundary. Also note that the local balance equa-

tion div σ+f= 0 is to be understood in the sense of

distributions i.e. σ·nmust be continuous.

As regards numerical resolution of problem (1), it

turns out that many usual strength criteria can be for-

mulated using second-order cone constraints [9, 31].

When combined with dedicated ﬁnite element dis-

cretization, the discrete counterpart of (1) falls into

the class of second-order cone programs (SOCP) [30].

Such convex optimization problems are particularly

important because they can be solved eﬃciently us-

ing interior-point algorithms implemented in dedicated

solvers such as Mosek, CPLEX, etc.

2.2 Extending limit analysis to topology optimization

Building upon the concepts of limit analysis, we

now aim at ﬁnding an optimized structure Ω⊆ D

contained in a computational domain Dwith maxi-

mum load-bearing capacity for a given material vol-

ume fraction constraint η. Problem (1) written on the

unknown geometry Ωcan be re-expressed on the com-

putational domain Dby introducing an additional op-

timization variable in the form of a characteristic func-

4 Leyla Mourad et al.

tion ρ(x)∈ {0,1}with ρ(x) = 1 when x∈Ω. Such a

problem reads as:

λ+= max

λ,σ,ρ λ

s.t. div σ+ρf= 0 in D

σ·n=λTon ∂DT

σ∈ρG in D

RDρdx ≤η|D|

ρ∈ {0,1}

(2)

in which |D| denotes the volume of Dand ∂DTthe

boundary part on which tensile loads are applied. In

the above problem, the main diﬀerence with respect to

(1) is that the strength domain Gis replaced by an

homothetic domain ρG. In particular, σ(x)∈ρ(x)G

enforces that σ(x)∈Gfor x∈Ω(i.e. where ρ(x) = 1)

and σ(x) = 0 for x/∈Ω(i.e. where ρ(x) = 0), provided

that Gis bounded. As a result, if ρis known, (2) is

indeed equivalent to (1) formulated on Ω.

2.3 A related volume-minimization problem

Instead of maximizing the load-bearing capacity for

a given volume fraction constraint as in (2), one can

also attempt at minimizing the total material volume

under the condition that the structure can sustain (ac-

cording to limit analysis theory) a given load level λT,

with λnow having a prescribed value. Such a volume-

minimization problem has already been considered in

previous works [22, 27] and reads:

η−= min

σ,ρ

1

|D| ZD

ρdx

s.t. div σ+ρf= 0 in D

σ·n=λTon ∂DT

σ∈ρG in D

ρ∈ {0,1}

(3)

The relation between (2) and (3) will be further ex-

plored in the subsequent section when considering their

convex relaxations.

3 Convex continuous relaxation

3.1 Relaxed problems deﬁnitions

Both problems (2) and (3) are extremely diﬃcult to

solve in practice due to the binary constraint ρ∈ {0,1}.

Following the classical procedure of topology optimiza-

tion, both problems are relaxed by considering instead

a continuous constraint ρ∈[0; 1]:

λ+= max

λ,σ,ρ λ

s.t. div σ+ρf= 0 in D

σ·n=λTon ∂DT

σ∈ρG in D

RDρdx ≤η|D|

0≤ρ≤1

(LOAD-MAX)

and

η−= min

σ,ρ

1

|D| ZD

ρdx

s.t. div σ+ρf= 0 in D

σ·n=λTon ∂DT

σ∈ρG in D

0≤ρ≤1

(VOL-MIN)

It turns out that both relaxed problems (LOAD-MAX)

and (VOL-MIN) are convex problems. Indeed, all con-

straints are linear except for the strength constraint

σ∈ρG which is convex (see proof in Appendix A).

This property motivates the use of the homothetic

scaling for the density-dependent strength domain

σ∈G(ρ) = ρG. Indeed, it enables to enforce σ= 0

and σ∈Gin the two limit cases ρ= 0 and ρ= 1

respectively while the relaxed problem becomes convex

with respect to the pair (σ, ρ) of optimization variables,

a property which will be eﬃciently taken advantage of

in section 5.

Finally, let us point out that we will always solve the

above problems simultaneously for σand ρ, in a mono-

lithic fashion. Indeed, contrary to elastic-based topol-

ogy optimization problems, solving the above problems

for σonly, at ﬁxed ρ, is a diﬃcult problem, since it is,

in fact, a limit analysis problem. The coupled problem

therefore adds only an extra scalar optimization vari-

able compared to a standard limit analysis computation

and avoids the need of alternate minimization between

the problem on σand the problem on ρ.

3.2 Properties of relaxed problems solutions

In this paragraph and in Appendix B, we will con-

sider the optimal objective value λ+(η) of problem

(LOAD-MAX) as a function of the volume fraction pa-

rameter ηas well as the optimal objective value η−(λ)

of problem (VOL-MIN) as a function of the load factor

parameter λ. One can show that both functions are

Topology optimization of load-bearing capacity 5

non-decreasing functions with λ+(1) = Λ+being the

solution to the limit analysis problem (1). Similarly,

η−(λ) = +∞for λ > Λ+since (VOL-MIN) becomes in-

feasible in this case. Finally, one can also show that

problems (LOAD-MAX) and (VOL-MIN) are in fact equiv-

alent since λ+(η) and η−(λ) are inverses of each other,

see Appendix B for the proof. See also Figure 3 for more

details.

If we remove the upper bound constraint ρ≤1 in

problems (LOAD-MAX) and (VOL-MIN), then we can easily

show that λ+(η) = Cη and η−(λ) = 1

Cλwith Cbeing

a constant. This constant corresponds to C=dλ+

dη

η=0

when considering the initial formulations including the

upper bound constraint ρ≤1. Without the upper

bound constraint, problems (LOAD-MAX) and (VOL-MIN)

will therefore always give the same optimized density

ﬁeld ρup to a scaling factor, irrespective of the value

of the design load level λor the maximum volume frac-

tion η. In particular, both problems will always give a

solution for any value of ηor λ.

4 On the choice of the material strength

criterion

4.1 Strength criterion for asymmetric materials

Previous works on limit analysis-based volume min-

imization considered only either plane strain [27] or

plane stress [22] von Mises strength criteria. In the

present work, the formulation has been established for

any strength criterion Gwhich can be chosen depending

on the material under investigation.

For instance, most materials in civil engineering

exhibit diﬀerent strength properties in tension and

compression. Incorporating such tension/compression

asymmetry can be achieved using a Drucker-Prager,

Mohr-Coulomb or, more simply, a Rankine strength cri-

terion (see Figure 1). The latter can be expressed as:

σ∈GRankine ⇐⇒ −fc≤σI, σII , σI II ≤ft(4)

where σJwith J=I, II , II I denote the principal

stresses and fc(resp. ft) is the material compressive

(tensile) strength. In 3D conditions, the Rankine crite-

−1 0 1

σI

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

σII

von Mises

Rankine

L1-Rankine

(a) Symmetric strengths ft=fc= 1

(b) Asymmetric strengths fc= 5, ft= 1

Fig. 1: Strength criteria shapes in the principal stress

space

rion can be expressed using SDP constraints but for 2D

conditions these can be reduced to SOCP constraints:

σ∈GRankine ⇐⇒

(ft−σxx)(ft−σyy )≥σ2

xy

(fc+σxx)(fc+σyy )≥σ2

xy

−fc≤σxx, σyy ≤ft

(5)

4.2 A L1-Rankine criterion and its link to truss-like

designs

One objective of topology optimization amounts to

ﬁnding optimized structures which often exhibit truss-

like designs. Away from truss connections or supports,

the stress state is uniaxial in the truss members so that

−fc≤σI≤ftand σII = 0. This condition is equivalent

to saying that σmust be of rank 1 (L0Schatten norm).

Unfortunately, the induced set is non-convex. Similarly,

6 Leyla Mourad et al.

the Rankine criterion can be viewed as a L∞-norm

(spectral norm) on the principal stresses kσk∞≤f0for

similar tension and compression strengths fc=ft=f0.

Although this set is convex, numerical examples of sec-

tion 6 will show that it has a tendency to promote biax-

ial stress states instead of uniaxial ones. A compromise

between the sparsity-inducing L0-norm and the convex

L∞-norm is the nuclear L1-norm: kσk1=|σI|+|σII | ≤

f0. Indeed, the L1-norm is the tightest convex relax-

ation norm of the L0-norm and has been used in many

applications for inducing sparse solutions [6] in com-

pressed sensing or image processing applications. As a

result, we advocate for the use of a L1-Rankine cri-

terion in order to promote sparse (i.e. as uniaxial as

possible) principal stress states at the optimum. In the

case of asymmetric tensile/compressive strengths this

L1-Rankine criterion reads as (see Figure 1 for a com-

parison of the criterion shapes in 2D):

σ∈GL1-Rankine ⇔X

J=I,I I,I II

max −σJ

fc

;σJ

ft≤1 (6)

We also refer to C for a second-order cone formulation

of the 2D L1-Rankine criterion.

We further justify this choice by relating it to

volume-optimal trusses studied by Michell [34]. Con-

tinuous volume-optimal 2D trusses have indeed been

characterized as ﬁnding a 2D stress state complying

with equilibrium conditions and minimizing the quan-

tity RD(|σI|+|σII |) dx in [2, 47]. We see that problem

(VOL-MIN) with a symmetric L1-Rankine criterion can

be written as:

min

σ,ρ

1

|D| ZD

ρdx

s.t. equilibrium

|σI|+|σII | ≤ ρf0in D

(7)

which is also equivalent to:

min

σ

1

f0|D| ZD

(|σI|+|σII |) dx

s.t. equilibrium

(8)

which is exactly the characterization of volume-optimal

continuous 2D trusses discussed in [28, 47]. A similar

equivalence can be obtained for asymmetric strengths.

Note that in the above problems, we removed the

upper bound condition ρ≤1 (see the discussion in

section 3.2) when establishing this connection. In par-

ticular, a constrained version of the Michell truss design

problem has also been proposed in [47] although not be-

ing completely equivalent to our formulation. However,

they share the similar feature of avoiding inﬁnitely large

truss member sections (and are thus unable to sustain

concentrated forces) but also of exhibiting a maximum

load level.

5 Numerical implementation and penalization

procedure

5.1 Conic representation

The fenics optim package [11] enables to formu-

late convex variational problems provided that the

involved convex functions admit a conic representa-

tion [10] i.e. can be expressed using linear equal-

ity/inequality constraints and conic constraints involv-

ing either the second-order Lorentz cone or the cone of

semi-deﬁnite matrices. In particular, the package can be

used to easily formulate and solve limit analysis prob-

lems [12]. If Gcan be formulated using second-order

cone constraints, the associated problems (LOAD-MAX)

and (VOL-MIN) will be SOCP problems. Similarly, if G

can be formulated using semi-deﬁnite constraints, prob-

lems (LOAD-MAX) and (VOL-MIN) will be Semi-Deﬁnite

Programming (SDP) problems.

5.2 Finite-element discretization

For both problems (LOAD-MAX) and (VOL-MIN), the

ﬁrst two constraints related to the equilibrium condi-

tion and traction boundary conditions are in fact re-

placed by their weak form:

ZD

σ:∇sudx = ZD

ρf·udx + Z∂DT

λT·u(9)

∀uregular enough and satisfying the ﬁxed displace-

ment boundary conditions on ∂D \ ∂DTand where ∇s

is the symmetrized gradient operator. This weak form

is discretized using discontinuous P1−Lagrange interpo-

lation for σand continuous P2−Lagrange interpolation

for uand P1for the density ﬁeld ρ. The discretized ver-

sion of the above equilibrium weak form would therefore

read as:

HΣ=Fρ+λT(10)

where Σis the vector of stress unknowns and ρthe vec-

tor of density unknowns, His the equilibrium matrix

arising from the left-hand side of (9), Fa matrix arising

from the ﬁrst right-hand side and Tis the nodal force

vectors corresponding to the boundary traction T.

Topology optimization of load-bearing capacity 7

The discrete (LOAD-MAX) problem therefore writes

as:

λ+= max

λ,Σ,ρλ

s.t. HΣ=Fρ+λT

Σi∈ρiG∀i

cTρ≤η

0≤ρ≤1

(11)

where Σi(resp. ρi) denotes the value of the stress ten-

sor σ(resp. pseudo-density ρ) at node iand cis the

vector obtained by discretizing the volume average op-

erator such that 1

|D| RDρdx = cTρ. Again, problem (11)

involves only linear equality or inequality constraints

except for the strength domain constraint Σi∈ρiG

which can be expressed using second-order cone con-

straints for the considered strength criteria.

Using the same notations, the discrete counterpart

to (VOL-MIN) reads as:

η−= min

Σ,ρcTρ

s.t. HΣ=Fρ+λT

Σi∈ρiG∀i

0≤ρ≤1

(12)

For more details on the discrete formulation of the

corresponding optimization problems, the reader can

also refer to [22].

5.3 A penalization procedure

As it will be seen in the next section, solution to

problems (LOAD-MAX) and (VOL-MIN) produce diﬀuse

density ﬁelds in general. In order to obtain a ﬁnal

truss-like design, some kind of penalization procedure

must be used based on the initial continuous solution.

Although it is not the main purpose of this paper, we

shortly describe a possible strategy, which has never

been done for plastic-based design to our knowledge.

We build upon the SIMP strategy by considering a

penalized strength criterion of the form σ∈ρpGwith

p≥1. We will follow a continuation procedure starting

from p= 1 to p=pmax >1 by solving a sequence of

convex optimization problems. More precisely, at iter-

ation n, we consider the following linearization of ρpn

around the previous density solution ρn−1:

ρpn≈ρpn

n−1+pnρpn−1

n−1(ρ−ρn−1) = an+bnρ(13)

where an= (1 −pn)ρpn

n−1and bn=pnρpn−1

n−1. We there-

fore replace the penalized non-convex strength criterion

constraint using the previous linear approximation as

follows:

σ∈(an+bnρ)G(14)

Again, this constraint is convex and problems

(LOAD-MAX) and (VOL-MIN) can be readily generalized

using this constraint. Note that the original formula-

tion is obtained for the particular case p1= 1, yielding

a1= 0 and b1= 1. The update rule for the penalization

exponent follows a heuristic which we found satisfying1,

namely:

pn+1 = 1.10.5+gnpn(15)

where gn=4

|D| ZD

ρn(1 −ρn) dx (16)

with gnrepresenting the average gray-level associated

with ρnor measure of non-discreteness as introduced

in [44].

Finally, implementing only the above-mentioned

procedure will lead to a well-known mesh dependency

issue when p > 1. This mesh-dependency is removed by

adding a slope-control constraint [37] of the form:

k∇ρk2≤1/` (17)

where `is a user-deﬁned minimal characteristic length.

Note again that this constraint is convex and directly

ﬁts into the second-order cone programming frame-

work. It can be added at a small extra cost without

impacting on the overall convergence of the interior-

point algorithm. Note that we include this constraint

only for the penalization phase when p > 1.

6 Illustrative applications

6.1 MBB beam

6.1.1 Initial convex solutions

We ﬁrst consider a MBB beam example (Figure 2)

of length l= 36 and height h= 6 with simple supports

on the left and roller supports on the right, a verti-

cal force of reference intensity P= 1 is applied at the

top. In the following, only one half of the model will be

represented, taking symmetry into account. Both sup-

ports and force are distributed over a small distance

1This heuristic was inspired by one we found in

http://www.cmap.polytechnique.fr/%7Eallaire/map562/

console.simp.edp

8 Leyla Mourad et al.

Fig. 2: A MBB beam example

Fig. 3: Comparison between load maximization and vol-

ume minimization for the L1-Rankine criterion. The

horizontal line corresponds to the standard limit analy-

sis solution Λ+and the oblique line of equation λ+(η) =

Cη to the solution of (LOAD-MAX) when removing the

upper bound constraint ρ≤1.

s= 0.5 to mitigate stress concentrations. The used

ﬁnite-element mesh consists of approximately 40,000 el-

ements.

We ﬁrst consider the case of symmetric ten-

sile/compressive strength fc=ft= 1 using either a

plane stress von Mises, Rankine or L1-Rankine strength

criterion. The load maximization problem (LOAD-MAX)

is solved for increasing prescribed volume fractions η

while the volume minimization problem (VOL-MIN) is

solved for increasing imposed load factor λ. The results

corresponding to the L1-Rankine criterion are reported

on Figure 3.

As expected, both problems yield the same solution

in terms of λ+(η). It should be noted that the volume

minimization problem is only relevant for load factors

less than the structure maximum load-bearing capacity

Λ+, the problem being unfeasible for larger imposed

loads. On the contrary, the load maximization problem

has a solution for any imposed volume fraction. The

most interesting feature is the existence of a plateau

where λ+(η) = Λ+for η≥ηmax =η−(Λ+). This can

be explained by the fact that, for a given geometry,

limit analysis solutions do not necessarily involve opti-

mal stress ﬁelds lying at the boundary of the strength

criterion everywhere in the domain. Some regions are

indeed only weakly stressed or even unstressed so that

a smaller strength criterion can be used. This leads to a

material distribution with signiﬁcantly lower total den-

sity while preserving the maximum load-bearing capac-

ity. Interestingly, in the present example the material

volume savings are signiﬁcant since ηmax ≈50%.

Solutions to the convex load maximization problems

have been represented in Figure 4 for η= 20% for the

three considered strength criteria. Obviously, the ob-

tained ﬁelds do not exhibit a strong truss-like pattern

which should be obtained through the use of a penal-

ization procedure. This will be investigated in section

6.1.2. For now, let us investigate the convex solutions.

First, roughly the same topologies are obtained for all

criteria despite small diﬀerences, especially in the cen-

tral diﬀuse region. Better insight can be gained when

looking at the corresponding principal stresses (Figure

5). It can be seen that biaxial stress states are essen-

tially present near the loading region, the supports or in

the central region connecting the top compressed strut

(in blue) and the bottom region in traction (in red).

In order to assess more quantitatively which strength

criterion yields stress ﬁelds close to being uniaxial, we

computed throughout the domain the value of the fol-

lowing angle in the principal stress state:

θ= arctan |σII |

|σI|(18)

For uniaxial stress ﬁelds, either σIor σI I = 0 so that

θ= 0◦or 90◦. Non-uniaxial stress ﬁelds will therefore

correspond to values 0◦< θ < 90◦. We therefore mea-

sure the amount of uniaxial stress ﬁelds in the obtained

solution by representing the frequency distribution of

θfor the three criteria. Figure 6a clearly indicates

that the Rankine criterion produces much more biaxial

stress states (near 45◦) than the two others, which can

well be explained by the criterion shape (see Figure 1a)

which favours stress states lying on the square vertices.

On this example, the diﬀerence between L1-Rankine

and von Mises criterion is not signiﬁcant. However, if we

change the way loading is applied by distributing it over

a larger region (s= 5), one can clearly see in Figure 6b

that the L1-Rankine criterion is more eﬃcient than the

von Mises criterion at promoting uniaxial stress states.

Topology optimization of load-bearing capacity 9

(a) L1-Rankine criterion (b) Rankine criterion (c) von Mises criterion

Fig. 4: Optimized topology design of the MBB example using non-penalized load maximization problem for η= 0.20

(a) L1-Rankine criterion (b) Rankine criterion (c) von Mises criterion

Fig. 5: Principal stresses of the MBB example using non-penalized load maximization problem for η= 0.20 (blue:

compression, red:traction)

(a) With a concentrated load (s= 0.5) (b) With a more diﬀused load (s= 5.)

Fig. 6: Frequency distribution of the θangle for the MBB beam example with η= 0.2

(a) L1-Rankine (LOAD-MAX) (b) Rankine (LOAD-MAX) (c) von Mises (LOAD-MAX)

(d) L1-Rankine (VOL-MIN) (e) Rankine (VOL-MIN) (f) von Mises (VOL-MIN)

Fig. 7: Optimized design of the MBB-beam using either penalized load maximization (top) or volume minimization

(bottom) with diﬀerent strength criteria

10 Leyla Mourad et al.

1

(a) Evolution of λ+during penalization for (LOAD-MAX)

1

(b) Evolution of η−during penalization for (VOL-MIN)

Fig. 8: Inﬂuence of the penalty procedure on the

optimal objective functions for the (LOAD-MAX) and

(VOL-MIN) problems

6.1.2 Penalized solutions

We now apply the penalization procedure described

in section 5.3 for the three strength criteria.We use a

slope control parameter `= 0.5, set pmax = 3 and run

the iterative procedure for 20 iterations which is usually

enough to obtain a converged penalized design.

If the convex load maximization (LOAD-MAX) and

volume minimization (VOL-MIN) problems are equiva-

lent for the initial solution, it will no longer be the

case during the penalization procedure. Indeed, start-

ing a penalizing sequence of (LOAD-MAX) problems with

η= 0.2 will usually decrease the computed optimal

load-bearing capacity λ+. Similarly, starting a penal-

izing sequence of (VOL-MIN) problems with a loading

corresponding to the initial λ+(η= 0.2) will usually in-

crease the computed optimal volume fraction η−. The

ﬁnal penalized solutions are therefore not equivalent

anymore since they correspond to diﬀerent loadings and

volume fractions. Figure 7 represents the obtained pe-

nalized design using either (LOAD-MAX) or (VOL-MIN) for

the three strength criteria. First, it can be observed

that the penalization procedure systematically fails in

the case of a Rankine strength criterion. This may be

attributed to the fact that this strength criterion has a

tendency to promote biaxial stress states over uniaxial

stress states as already mentioned when discussing Fig-

ure 6. The penalization procedure is however eﬃcient

for the L1-Rankine and von Mises for which truss-like

designs are indeed obtained. The designs are quite dif-

ferent between both criteria, especially as regards mem-

bers near the support and secondary members. Figure

8a (resp. 8b) shows the evolution of the computed load-

bearing capacity (resp. volume fraction) during the dif-

ferent steps of the penalization procedure. As men-

tioned before, the penalization procedure tends to de-

teriorate the value of the objective function, it is how-

ever moderate for the L1-Rankine and von Mises cri-

teria, contrary to the Rankine criterion. Interestingly,

the evolution of the objective function is very similar

between the L1-Rankine and von Mises criterion.

On Figure 9, we investigate the inﬂuence of the

slope control parameter `on the optimized solution.

As expected, a small parameter yields a design with

more members of shorter length. Increasing the value

of this parameter results in optimized topologies with

only a few members of larger length, their thickness is

also spread out over a wide region, resulting in a gradi-

ent of intermediate densities. This is further conﬁrmed

by the ﬁnal value of the measure of non-discreteness

gend at the end of the penalization procedure which

clearly increases for larger `. Conversely, the resulting

load-bearing capacity λ+decreases with a larger `, the

resulting topology being less optimized for larger val-

ues of `. Finally, Figure 10 illustrates the inﬂuence of

the imposed volume fraction ηon the ﬁnal optimized

topologies. As expected, the computed structures ex-

hibit thicker members but also additional secondary

members for larger values of η, resulting in a larger

load-bearing capacity.

6.2 Bridge example with asymmetric strengths

The second example considers the design of a bridge

structure (see Figure 11) with potentially asymmetric

tensile/compressive strengths ftand fcusing either the

Rankine or L1-Rankine criterion.

Figure 12 represents the evolution of the load-

bearing capacity ratio as a function of ηfor three

diﬀerent strength ratios: fc={0.1; 1; 10}with ft= 1.

One can observe that the obtained load-bearing

Topology optimization of load-bearing capacity 11

(a) `= 0.25: gend = 0.23, λ+= 0.297 (b) `= 0.5: gend = 0.29, λ+= 0.237 (c) `= 1: gend = 0.39, λ+= 0.154

Fig. 9: Inﬂuence of the slope control parameter `on the optimized topologies (LOAD-MAX with η= 0.2, L1-Rankine

criterion)

(a) η= 0.2: gend = 0.23, λ+= 0.297 (b) η= 0.3: gend = 0.30, λ+= 0.447 (c) η= 0.4: gend = 0.32, λ+= 0.628

Fig. 10: Inﬂuence of the imposed volume fraction ηon the optimized topologies (LOAD-MAX with `= 0.5, L1-Rankine

criterion)

5

10.5

Fig. 11: A bridge structure with a central uniformly

distributed loading T=−ey. Fixed supports are dis-

tributed over regions of length 0.1 at both extremities.

capacities obtained with both criteria are quite similar

for strongly asymmetric strengths whereas substantial

diﬀerences are observed only for the symmetric case

fc=ft. Although the stress domain corresponding to

the L1-Rankine criterion is much smaller than that of

Rankine criterion, the load-bearing capacities are not

so diﬀerent for the asymmetric cases, indicating that

the obtained solution are mostly uniaxial since both

criteria coincide for uniaxial stress states. This is a

further indication that the L1-Rankine criterion can

be used eﬃciently without being too conservative with

respect to the predicted load-bearing capacity. More-

over, if analyzing the obtained truss-like structures

with either the L1-Rankine or the Rankine strength

criterion, the diﬀerence in terms of limit load would be

even less than that of Figure 12.

Figure 13 represents the obtained principal stress

ﬁelds, either the initial unpenalized or the ﬁnal pe-

L1-Rankine

Rankine

Fig. 12: Evolution of the load-bearing capacity for dif-

ferent strength asymmetry ratios

nalized solution, in the case of a larger compressive

strength fc= 10, ft= 1 for the case η= 0.20 with

the L1-Rankine criterion. Conversely, Figure 14 rep-

resents the corresponding solutions for the case of a

smaller compressive strength fc= 0.1 and ft= 1. It

can ﬁrst be seen that the penalization procedure is very

eﬃcient even in these asymmetric cases and indeed pro-

duces truss-like patterns. The obtained patterns show a

clearly distinct topology depending on the value of the

compressive/tensile strength ratio.

12 Leyla Mourad et al.

(a) Initial unpenalized solution

(b) Final penalized solution

Fig. 13: Principal stress distributions for the bridge ex-

ample with ft= 1 and fc= 10 (blue: compression,

red:traction)

(a) Initial unpenalized solution

(b) Final penalized solution

Fig. 14: Principal stress distributions for the bridge ex-

ample with ft= 1 and fc= 0.1 (blue: compression,

red:traction)

6.3 Example of a no-tension material

We ﬁnish by illustrating the eﬃciency of our method

on the important case of no-tension materials ft= 0 (in

practice we still set some residual tension ft= 10−3fc

to avoid numerical issues). We investigate the case of

the domain represented in Figure 15 subject to ﬁxed

supports on its bottom and lateral sides and uniformly

distributed compressive T=−eyload on the top

boundary. We investigate two diﬀerent cases for the cen-

tral region width with either s= 0 or s= 0.3. We set

η= 0.2 and `= 0.1 for slope control.

1.5

1.0

1.0

1.8 1.8

1.4s s

Fig. 15: Geometrical domain for optimization of a no-

tension material

First, the principal stress distributions obtained

from the resolution of the unpenalized convex problem

for the case s= 0 are represented in Figure 16a. We

can remark that the obtained stress ﬁeld is indeed in a

purely compressive state with a localized uniaxial ﬁeld

in the inclined strut which is supported by the bottom

boundary. The central top region is subject to a uni-

formly distributed uniaxial stress state. We can high-

light that the obtained solution is therefore already ex-

tremely satisfying in terms of manufacturability. As a

result, there is little use here for a penalization pro-

cedure, which will essentially replace the top uniform

region by a series of columns and slightly modify the

main struts inclination (Figure 16b).

Interestingly, when increasing the central opening

width (case s= 0.3). The previous solution now be-

comes impossible to sustain since the struts would have

to kink to avoid the opening. Since this kink cannot be

supported by a no-tension material, the ﬁnal solution

introduces a secondary strut which will be supported

by the vertical boundaries in this case (Figure 17a).

Again, the obtained solution is remarkably well local-

ized and the penalization procedure only modiﬁes the

top uniform region (Figure 17b).

For the sake of comparison, we also used the Rank-

ine criterion for the same examples. The diﬀerent op-

timized pseudo-density ﬁelds are compared on Figures

19 and 20. Although being globally similar, especially

concerning the number and inclination of the structure

legs, both criteria yield diﬀerent optimized topologies

on the upper part of the domain serving to transmit the

distributed loading to the supporting arch. In particu-

lar, the Rankine criterion produces a secondary arch

and a horizontal member above the opening contrary

to the L1-Rankine criterion which results in an array

of columns. This might suggest that, although mem-

bers are in uniaxial stress states, connections between

Topology optimization of load-bearing capacity 13

(a) Initial unpenalized solution

(b) Final penalized solution

Fig. 16: Principal stress distributions for the no-tension

material example with s= 0

members experience biaxial stress states which are in-

ﬂuenced by the choice of the strength criterion and

may impact the ﬁnal optimized topologies. This aspect

should deserve a more thorough investigation in future

works.

Finally, the evolution of the load-bearing capac-

ity obtained from the initial unpenalized solution as

a function of the prescribed volume fraction ηhas been

represented in Figure 18 for both cases s= 0 and

s= 0.3. Again, the curves follow the same trend as

in Figure 3. Interestingly, the critical volume fraction

for which the maximum load-bearing capacity Λ+is

attained is here ηmax ≈0.25–0.3 for both situations,

further showing that important material savings can

be achieved through topology optimization without af-

fecting the structural load-bearing capacity. Finally, al-

though aesthetically pleasing, the obtained penalized

solutions tend to decrease the ﬁnal load-bearing capac-

ity by roughly 20% (starred symbols in Figure 18).

7 Conclusions

This article aimed at proposing a general framework

for unifying the concepts of topology optimization and

limit analysis/yield design theory, which is frequently

used in civil and structural engineering. Contrary to

(a) Initial unpenalized solution

(b) Final penalized solution

Fig. 17: Principal stress distributions for the no-tension

material example with s= 0.3

Fig. 18: Evolution of the initial load-bearing capacity

for the no-tension material example as a function of the

prescribed volume fraction. Starred symbols represent

the load-bearing capacity obtained at the end of the

penalization procedure.

14 Leyla Mourad et al.

(a) s= 0

(b) s= 0.3

Fig. 19: Optimized pseudo-density ﬁelds with the L1-

Rankine criterion

compliance-based topology optimization, geometry is

here optimized with respect to the structure maximum

load-bearing capacity for a ﬁxed material volume frac-

tion. An alternative approach consists in minimizing

the total material volume under the constraint of sup-

porting a ﬁxed loading under the material local strength

capacities. Both topology optimization problems are

relaxed into convex problems, which can be cast as

second-order programming problems for usual yield do-

mains in 2D. The proposed optimization scheme allows

optimizing for ultimate limit state (ULS) and can be

used in combination with classical optimization for ser-

viceability limit state (SLS). This is essential for civil

engineering applications, where material with diﬀerent

compressive and tensile strengths are commonly used.

The main results of this work can be summarized as

follows:

–both load maximization and volume minimization

convex formulations are in fact two sides of the same

coin, yielding the same solution;

–a small amount of material is often enough to sup-

port the load-bearing capacity obtained from classi-

cal limit analysis, oﬀering important potential sav-

ings;

(a) s= 0

(b) s= 0.3

Fig. 20: Optimized pseudo-density ﬁelds with the Rank-

ine criterion

–diﬀerent strength criteria can be used to model the

underlying material properties. We propose using

an original L1-Rankine criterion which oﬀers several

advantages:

–it can model materials with diﬀerent ten-

sile/compressive strength, contrary to a von

Mises criterion for instance;

–it provides a conceptual link with optimal truss

theory [34];

–it tends to promote more eﬃciently uniaxial

stress states, contrary to a more classical Rank-

ine criterion for instance;

–no-tension materials can be handled without diﬃ-

culty and, for such strongly asymmetric materials,

the convex formulation tend to produce manufac-

turable truss-like designs;

–for other cases where the obtained solution is con-

tinuous, we propose an eﬃcient penalization proce-

dure based on an iterative resolution of such convex

problems.

Interestingly, volume minimization with a stress

constraint is generally diﬀerent from the use of a stress-

dependent objective function with a volume constraint

when dealing with elastic materials. This picture is

clearly diﬀerent in limit analysis-based formulations as

Topology optimization of load-bearing capacity 15

we have shown. Although it can be proven mathemati-

cally, the fundamental origin of this equivalency is how-

ever still a bit elusive at this stage and should be inves-

tigated more deeply.

Future work will also be devoted to the general-

ization of the proposed formulation to bi-material op-

timization, e.g. reinforced concrete or ﬁber placement

in composite laminates. Similarly to compliance-based

optimization problems, the proposed framework should

also be extended to the case involving multiple loadings

as well as to the consideration of buckling constraints.

Finally, in our considered numerical examples we did

not consider any self-weight. The introduction of self-

weight is known to introduce some numerical diﬃcul-

ties in SIMP-like compliance optimization as discussed

in [16]. This point should also deserve more attention

in the future.

A Proof that σ∈ρG is a convex constraint

The constraint σ∈ρG is equivalent to σ/ρ ∈Gfor ρ > 0.

Let us consider any (σ1, ρ1) and (σ2, ρ2) such that ρi>0

and σi/ρi∈Gfor i= 1,2. Then, ∀λ∈[0; 1]:

(1 −λ)σ1+λσ2

(1 −λ)ρ1+λρ2

=(1 −λ)ρ1

(1 −λ)ρ1+λρ2

σ1

ρ1

+λρ2

(1 −λ)ρ1+λρ2

σ2

ρ2

= (1 −µ)σ1

ρ1

+µσ2

ρ2

(19)

where µ=λρ2

(1 −λ)ρ1+λρ2

∈[0; 1]. Owing to the convexity

of G, we obtain that (1 −λ)σ1+λσ2

(1 −λ)ρ1+λρ2

∈Gwhich proves that

the set {(σ, ρ) s.t. σ∈ρG}is convex. ut

B Proofs of solution properties of the relaxed

problems

Property 1:∀η, η0∈[0; 1] with η≤η0and η0>0:

η

η0λ+(η0)≤λ+(η)≤λ+(η0) (20)

Proof: Let σand ρ(resp. σ0and ρ0) the solutions to

(LOAD-MAX)(η) (resp. to (LOAD-MAX)(η0)). Then, (λ+(η),σ, ρ)

is feasible for problem (LOAD-MAX)(η0) so that λ+(η)≤λ(η0).

Besides, let us deﬁne b

λ=η

η0λ+(η0), b

σ=η

η0σ0and bρ=η

η0ρ0.

Then, since 0 ≤η

η0≤1, we have 0 ≤bρ≤ρ0≤1 and

1

|D| RDbρdx ≤η

η0

1

|D| RDρ0dx = η. Moreover, (b

λ, b

σ,bρ) verify

the equilibrium equations and we also have b

σ∈bρG. Thus,

we have a feasible point for problem (LOAD-MAX)(η) and one

therefore has b

λ≤λ+(η).

ut

Property 2:∀λ, λ0∈[0; Λ+]and λ≤λ0and λ0>0:

η−(λ)≤λ

λ0η−(λ0)≤η−(λ0) (21)

Proof: The proof follows the same ideas as Property 1 when

deﬁning (σ0, ρ0) solution to (VOL-MIN)(λ0) and b

σ=λ

λ0σ0

and bρ=λ

λ0ρ0. We then easily show that it is a feasible

point for (VOL-MIN)(λ) associated with an objective value

1

|D| RDbρdx = λ

λ0η−(λ0).

ut

Property 3:λ+(η)and η−(λ)are non-decreasing func-

tions. λ+(η)is continuous and η−(λ)is injective. Finally,

we also have:

ηΛ+≤λ+(η)∀η∈[0; 1] (22)

where Λ+=λ+(1) is the ultimate load factor of the limit

analysis problem.

Proof: The monotone property follows directly from Proposi-

tions 1 and 2. The continuity follows from Proposition 1 with

λ0=λ+and →0. Injectivity of η−(λ) follows from Propo-

sition 2 when assuming that η−(λ) = η−(λ0) then λ=λ0. Fi-

nally, the inequality follows from the particular case η0= 1 in

Proposition 1. Indeed, for problem (LOAD-MAX), one can take

ρ= 1 everywhere, yielding a classical limit analysis problem

with an ultimate load factor Λ+=λ+(1).

ut

Property 4:We have:

η−(λ+(η)) ≤η∀η∈[0; 1] (23)

λ+(η−(λ)) ≥λ∀λ∈[0; Λ+] (24)

Proof: Let (σ+, ρ+) (resp. (σ−, ρ−)) be a solution to

(LOAD-MAX)(η) (resp. (VOL-MIN)(λ)). Then, (σ+, ρ+) is a fea-

sible point for (VOL-MIN)(λ+(η)) associated with an objective

value 1

|D| RDbρdx = η. This proves the ﬁrst inequality.

Similarly, (σ−, ρ−) is a feasible point for (LOAD-MAX)(η−(λ))

for a load factor λ. This proves the second inequality.

ut

Property 5:For al l λ∈[0; Λ+],λ+(η−(λ)) = λi.e. λ+is

the (left) inverse of η−meaning that both problems (VOL-MIN)

and (LOAD-MAX)are in fact equivalent.

Proof: Let us write the ﬁrst inequality of Proposition 4 for

η=η−(λ) for λ∈[0; Λ+], then:

η−(λ+(η−(λ))) ≤η−(λ)

However, from the second inequality of Proposition 4,

λ+(η−(λ)) ≥λ. Using the fact that η−is non-decreasing

we also have that:

η−(λ)≤η−(λ+(η−(λ)))

Combining both results gives that η−(λ+(η−(λ))) = η−(λ)

i.e. λ+(η−(λ)) = λdue to η−being injective.

ut

16 Leyla Mourad et al.

C Conic formulation for the L1-Rankine

criterion in 2D

We consider the following isotropic criterion for 2D stress

tensors σ:

σ∈GL1-Rankine ⇔g(σI) + g(σII )≤1 (25)

where σI,I I are the principal stresses,

g(σ) = max σ

ft

;−σ

fcand ft, fcthe tensile and com-

pressive strengths respectively.

In 2D, we explicitly have that:

σI=σxx +σyy

2+1

2q(σxx −σyy )2+ 4σ2

xy

σII =σxx +σyy

2−1

2q(σxx −σyy )2+ 4σ2

xy

(26)

Introducing T=σxx +σyy and R=q(σxx −σyy )2+ 4σ2

xy,

we have that:

•If σI, σII >0:

σI+σII ≤ft⇔T≤ft(27)

•If σI, σII <0:

−σI−σII ≤fc⇔ −T≤fc(28)

•If σI>0 and σII <0:

σI

ft

−σII

fc

≤1⇔T1

2ft

−1

2fc+R1

2ft

+1

2fc≤1

⇔R≤2ftfc

fc+ft

−Tfc−ft

fc+ft(29)

Denoting by α=fc−ft

fc+ft

and f=2ftfc

fc+ft

, we ﬁnally

have the following conic formulation:

σ∈GL1-Rankine ⇔ − fc≤σxx +σyy ≤ft(30)

α α 0

1−1 0

0 0 2

σxx

σyy

σxy

+

X0

X1

X2

=

f

0

0

(31)

X0≥qX2

1+X2

2(32)

where the last condition is a quadratic conic constraint.

Finally, in the case of a density dependent criterion, we

have:

σ∈ρGL1-Rankine ⇔ − ρfc≤σxx +σyy ≤ρft(33)

α α 0

1−1 0

0 0 2

σxx

σyy

σxy

+

X0

X1

X2

=

ρf

0

0

(34)

X0≥qX2

1+X2

2(35)

Conﬂict of interest

The authors declare that they have no conﬂict of interest.

Replication of results

The Python code for implementing the topology opti-

mization and for reproducing the manuscript examples is

available as a supplementary material. This code relies on the

fenics optim Python package [11], itself relying on the FEn-

iCS ﬁnite-element software library https://fenicsproject.

org/ and the Mosek conic optimization solver https://www.

mosek.com/.

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