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The present work addresses the problem of maximizing a structure load-bearing capacity subject to given material strength properties and a material volume constraint. This problem can be viewed as an extension to limit analysis problems which consist in finding the maximum load capacity for a fixed geometry. We show that it is also closely linked to the problem of minimizing the total volume under the constraint of carrying a fixed loading. Formulating these topology optimization 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 considering 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 intermediate densities. Benchmark examples are first considered to illustrate the method overall efficiency while final examples focus more particularly on no-tension materials, illustrating how the method is able to reproduce known structural patterns of masonry-like structures. This paper is accompanied by a Python package based on the FEniCS finite-element software library.
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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 p1, 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 . 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 σG
at every point in where 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
σ·n=λTon ∂ΩT
σGin
(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 ⊆ 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 can 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. Such a
problem reads as:
λ+= max
λ,σλ
s.t. div σ+ρf= 0 in D
σ·n=λTon DT
σρG 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(i.e. where ρ(x) = 1)
and σ(x) = 0 for x/(i.e. where ρ(x) = 0), provided
that Gis bounded. As a result, if ρis known, (2) is
indeed equivalent to (1) formulated on .
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
1
|D| ZD
ρdx
s.t. div σ+ρf= 0 in D
σ·n=λTon DT
σρG 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,σλ
s.t. div σ+ρf= 0 in D
σ·n=λTon DT
σρG in D
RDρdx η|D|
0ρ1
(LOAD-MAX)
and
η= min
σ
1
|D| ZD
ρdx
s.t. div σ+ρf= 0 in D
σ·n=λTon DT
σρG in D
0ρ1
(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
σρG which is convex (see proof in Appendix A).
This property motivates the use of the homothetic
scaling for the density-dependent strength domain
σG(ρ) = ρG. Indeed, it enables to enforce σ= 0
and σGin 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 ρ1 in
problems (LOAD-MAX) and (VOL-MIN), then we can easily
show that λ+(η) = and η(λ) = 1
Cλwith Cbeing
a constant. This constant corresponds to C=+
η=0
when considering the initial formulations including the
upper bound constraint ρ1. 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:
σGRankine ⇒ −fcσI, 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:
σGRankine
(ftσxx)(ftσyy )σ2
xy
(fc+σxx)(fc+σyy )σ2
xy
fcσxx, 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
fcσIftand 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 kσkf0for
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):
σGL1-Rankine X
J=I,I I,I II
max σJ
fc
;σJ
ft1 (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
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 ρ1 (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 + ZDT
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 P1Lagrange interpo-
lation for σand continuous P2Lagrange 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
ΣiρiGi
cTρη
0ρ1
(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 ΣiρiG
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
ΣiρiGi
0ρ1
(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 σρpGwith
p1. 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 ρn1:
ρpnρpn
n1+pnρpn1
n1(ρρn1) = an+bnρ(13)
where an= (1 pn)ρpn
n1and bn=pnρpn1
n1. 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∇ρk21/` (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) =
to the solution of (LOAD-MAX) when removing the
upper bound constraint ρ1.
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 ηηmax =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= 0or 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 ηduring 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= 103fc
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 σρG is a convex constraint
The constraint σρG is equivalent to σGfor ρ > 0.
Let us consider any (σ1, ρ1) and (σ2, ρ2) such that ρi>0
and σiiGfor 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. σρG}is convex. ut
B Proofs of solution properties of the relaxed
problems
Property 1:η, η0[0; 1] with ηη0and 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
η01, we have 0 bρρ01 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
σbρG. Thus,
we have a feasible point for problem (LOAD-MAX)(η) and one
therefore has b
λλ+(η).
ut
Property 2:λ, λ0[0; Λ+]and λλ0and 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)) ηη[0; 1] (23)
λ+(η(λ)) λλ[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 ηmeaning 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 ηis 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 ηbeing 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:
σGL1-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
21
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 ftTft(27)
If σI, σII <0:
σIσII fc⇔ −Tfc(28)
If σI>0 and σII <0:
σI
ft
σII
fc
1T1
2ft
1
2fc+R1
2ft
+1
2fc1
R2ftfc
fc+ft
Tfcft
fc+ft(29)
Denoting by α=fcft
fc+ft
and f=2ftfc
fc+ft
, we finally
have the following conic formulation:
σGL1-Rankine fcσxx +σyy ft(30)
α α 0
11 0
0 0 2
σxx
σyy
σxy
+
X0
X1
X2
=
f
0
0
(31)
X0qX2
1+X2
2(32)
where the last condition is a quadratic conic constraint.
Finally, in the case of a density dependent criterion, we
have:
σρGL1-Rankine ρfcσxx +σyy ρft(33)
α α 0
11 0
0 0 2
σxx
σyy
σxy
+
X0
X1
X2
=
ρf
0
0
(34)
X0qX2
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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... Following the success of such methods in a limit analysis setting, some contributions explored their application to elastoplastic problems [33,34], including a recent extension toward non-convex finite-strain plasticity [19]. Topology optimization and plastic design of structures [54] have also been formulated either as LP programs for trusses [23] or generic conic programs for solids [43]. One can also mention free material optimization based on SDP formulations [32]. ...
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