Acta Polytechnica CTU Proceedings


doi:10.14311/APP.2020.26.0117
Acta Polytechnica CTU Proceedings 26:117–125, 2020 © Czech Technical University in Prague, 2020

available online at https://ojs.cvut.cz/ojs/index.php/app

ON OPTIMUM DESIGN OF FRAME STRUCTURES

Marek Tybureca, ∗, Jan Zemana, c, Martin Kružíkb, c, Didier Henriond, e

a Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7,
166 29 Prague 6, Czech Republic

b Czech Technical University in Prague, Faculty of Civil Engineering, Departement of Physics, Thákurova 7,
166 29 Prague 6, Czech Republic

c Czech Academy of Sciences, Institute of Information Theory and Automation, Department of Decision-Making
Theory, Pod vodárenskou věží 4, 182 08 Prague 8, Czech Republic

d University of Toulouse, LAAS-CNRS, 7 Avenue du Colonel Roche, 31400 Toulouse, France
e Czech Technical University in Prague, Faculty of Electrical Engineering, Department of Control Engineering,

Karlovo náměstí 13, 121 35 Prague 2, Czech Republic
∗ corresponding author: marek.tyburec@fsv.cvut.cz

Abstract. Optimization of frame structures is formulated as a non-convex optimization problem,
which is currently solved to local optimality. In this contribution, we investigate four optimization
approaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefi-
nite programming, and (iv) polynomial optimization. We show that polynomial optimization solves the
frame structure optimization to global optimality by building the (moment-sums-of-squares) hierarchy
of convex linear semidefinite programming problems, and it also provides guaranteed lower and upper
bounds on optimal design. Finally, we solve three sample optimization problems and conclude that the
local optimization approaches may indeed converge to local optima, without any solution quality mea-
sure, or even to infeasible points. These issues are readily overcome by using polynomial optimization,
which exhibits a finite convergence, at the prize of higher computational demands.

Keywords: Frame structures, global optimum, polynomial optimization, semidefinite programming,
topology optimization.

1. Introduction
Designing economical, efficient, and sustainable struc-
tures represents a major challenge of the contemporary
society. While structural engineers literally explore
designs that must satisfy the requirements of limit
states analysis, there is usually an infinite number
of such designs. Regardless of the properties of this
feasible design space, it is required to select only one,
the best design. This design quality is measured by
an objective function, which usually approximates
the expenses of production. Greatly simplified, one
such (most common) criterion considers maximization
of structural stiffness (while the amount of available
material is limited), or, equivalently, minimization of
structural volume (while requiring a certain structural
stiffness) [1].
Among the structural optimization problems the

greatest progress has been achieved so-far in optimiz-
ing (the cross-sectional areas of) truss structures [2].
This development can be attributed to convexity of
the feasible design space [3, Sections 1.3.5, 3.4.3, and
4.8], as the (axial) structural stiffness is an affine func-
tion of the cross-sectional areas. On the other hand,
when the bending stiffness comes into a consideration,
convexity does not hold in general. Quite surprisingly,
these problems are also much less studied, and to our
knowledge only local optimization algorithms have yet

been developed [4, 5].
In this contribution, we investigate the problem of

optimum design of frame structures. In particular, we
develop four different methods in Section 2: (i) general
non-linear optimization solved by the interior-point
method of fmincon, (ii) the first-order optimality
criteria (OC) method [1, 5], (iii) reformulation of
the problem into a non-linear semidefinite program
(NSDP) solved by PenLab [6], and (iv) a suitably
modified polynomial optmization (PO) problem (iii)
solved globally using polynomial optimization meth-
ods [7] and the Mosek [8] optimizer. We show that
the latter PO approach generates guaranteed lower
and upper bounds on the solution, providing a means
of assessing the design quality. Finally, Section 3 in-
troduces a set of three sample optimization problems
to compare the optimization approaches.

2. Solution techniques to frame
optimization

2.1. Problem statement
Let us consider the problem of optimum design of
frame structures composed a finite number of nodes,
nn, and of admissible elements, ne, defining the so-
called ground structure [2]. The frame elements are
attributed with non-negative, and thus possibly zero,
given-shaped cross-sectional areas a. For simplicity,

117

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M. Tyburec, J. Zeman, M. Kružík, D. Henrion Acta Polytechnica CTU Proceedings

Ki(ai) =




Eiai
`i

0 0 −Eiai
`i

0 0
12EiIi
`3

i

6EiIi
`2

i
0 −12EiIi

`3
i

6EiIi
`2

i
4EiIi
`i

0 −6EiIi
`2

i

2EiIi
`i

sym.
Eiai
`i

0 0
12EiIi
`3

i
−6EiIi

`2
i

4EiIi
`i




(3)

we assume, in the following text, that the moments of
inertia of individual cross-sections Ii are polynomials
of degree two1, i.e., Ii = cI,ia2i for some cI,i > 0,
which includes all cross-sections with given aspect
ratios of all their components. Note that in such
a case, each cross-section is fully determined by its
area. Optimizing the cross-sectional areas a we search
the maximum stiff structures within the available
volume V of a linear-elastic material.

The structural stiffness is measured (inversely) by
the compliance c, work done by external forces, f (a),

c(a) = f (a)Tu = uTK(a)u, (1)

where u constitutes the generalized displacement vec-
tor, and K(a) is the symmetric positive semi-definite
stiffness matrix— a polynomial function of a,

K(a) =
ne∑
i=1

Ki(ai), (2)

assembled from contributions of individual elements
Ki(ai). The lower the compliance, the stiffer is the
structure with respect to the external forces.
In this contribution, we use the element stiffness

matrix of Euler–Bernoulli frame elements (3), with Ei
denoting the Young modulus. In (1), f (a) is the ex-
ternal force column vector—a linear function of a—to
allow for self-weight, assembled from the contributions
of elements f (ai),

f (a) =
ne∑
i=1

fi(ai). (4)

In the following text, we assume that ∀a > 0 : K(a) �
0, i.e., the structure is not a kinematic mechanism
and the stiffness matrix is positive definite (which
is denoted by “� 0”) for all positive cross-sectional
areas. Since K(a) has therefore the full rank, we also
have f (a) ∈ Im (K(a)), where Im(•) is the image of
•.

1The same procedure can be employed for higher-degree
polynomials, so that all cross-sectional parameters can be op-
timized concurrently. We restrict ourselves, however, to the
polynomials of degree two to maintain a simpler notation.

This optimization problem is formalized as

min
a,u

f (a)Tu (5a)

s.t. K(a)u = f (a), (5b)
`Ta ≤ V , (5c)

a ≥ 0, (5d)

with ` being the column vector of the frame elements
lengths. Notice that in general, (5) constitutes a non-
convex non-linear optimization problem because of the
bilinear objective function (5a) and the polynomial
equilibrium equality (5b) with possibly singular K(a).
On the other hand, the volume constraint (5c) and
the cross-sectional areas non-negativity constraint (5d)
are affine functions of the design variables, u and a,
and are in turn convex. The optimization problem
(5) can be solved to local optimality using standard
numerical optimization techniques. In particular, we
will solve this problem using the interior-point method
implemented in the fmincon function of Matlab.

2.2. Optimality criteria
The inherent difficulty of singularity of K(a) in (5b)
can be circumvented by assuming a small positive
lower-bound on the cross-sectional areas [1], which is
denoted by ε in this study. Consequently, the former
topology optimization problem (5) is effectively trans-
formed into the sizing one and the displacement field
u can be excluded from the design variables2. Notice
that in the limit when ε → 0, these problems are
equivalent, but the smaller ε the higher the condition
number of K(a), and thus it is more difficult to solve
(5b). Hence, we assume ε = 10−6 in this study.

Considering (5) with ε1 ≤ a, its Lagrangian func-
tion reads as

L(a, u,λ,µ,ν) = f (a)Tu + λT [f (a) − K(a)u]
+µ
(
`Ta −V

)
+ νT (ε1 − a) ,

(6)

with the Lagrange multipliers λ, µ, and ν; and 1 de-
noting the vector of all ones. In addition to the primal
feasibility (5b)–(5d), the Karush–Kuhn–Tucker condi-
tions require feasibility of the dual and complementary

2At the price of solving the equilibrium equation in each
iteration.

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vol. 26/2020 On optimum design of frame structures

slackness,

µ ≥ 0, (7a)
µ
(
`Ta −V

)
= 0, (7b)

ν ≥ 0, (7c)
νT (ε1 − a) = 0, (7d)

together with the stationarity of the Lagrangian with
respect to u,

0 =
dL(a, u,λ,µ,ν)

du
= [f (a) − K(a)λ]T , (8)

and a. However, because the equation K(a)λ = f (a)
in (8) possesses a unique solution due to K(a) � 0,
we have λ = u. Using this observation, the necessary
first-order optimality conditions read as

0 =
∂L
∂ai

= 2uT
∂f (a)
∂ai

− uT
∂K(a)
∂ai

u + µ`i −νi. (9)

Conditions (9) and (7d) then imply that at the op-
timum, the frame elements with ai > ε have equal
constant energy

µ =
1
`i

uT
∂K(a)
∂ai

u −
2
`i

uT
∂f (a)
∂ai

. (10)

Aiming to satisfy (10), optimality criteria meth-
ods [1] build update schemes which increase stiffnesses
of elements with energies higher than µ, and, con-
versely, decrease stiffnesses of elements with the energy
lower than µ. The levels are balanced in an iterative
process, based on the value of µ. In each iteration,
the relative change of the design variables is bounded
by the move limit ζ, assumed as ζ = 0.2 in this paper.
Consequently, the fix-point update scheme reads as

a
(k+1)
i = max

{
max

{
(1 − ζ)a(k)i ,ε

}
,

a
(k)
i

[
b

(k)
i

]η} (11)
with the tuning parameter η = 0.3 and with

b
(k)
i =

uT
∂K(a)
∂ai

u − 2uT
∂f (a)
∂ai

µ`i
. (12)

Clearly, if ∀i ∈ {1, . . . ,ne} : b
(k)
i = 1, we reach

a local minimum as (10) is satisfied.
Combination of (11), (12) with (5c) allows us to

write the (current) volume V = `Ta(k+1) as a contin-
uous function of the multiplier µ. It can be seen from
(12) that V is a non-increasing function of µ. In fact,
strict decreasing occurs when εI < a. Consequently,
the bisection algorithm is used to find µ such that the
volume constraint is satisfied.

2.3. Nonlinear semidefinite programming
In this section, we describe another approach to elim-
inate the displacement field variables u from the op-
timization problem formulation. First, let us rewrite

(5) as

min
a,c

c (13a)

s.t. c− f (a)TK(a)†f (a) = 0, (13b)
`Ta ≤ V , (13c)

a ≥ 0, (13d)

where K(a)† denotes the Moore-Penrose pseudo-
inverse of K(a). Because we require that ∀a > 0 :
K(a) � 0, the (possible) singularity of K(a) is caused
exclusively by zero rows and columns belonging to
the degrees of freedom without any attached finite
element (or, equivalently, ai = 0 for all attached el-
ements in that node). This assumption allows us to
partition the stiffness matrix into a positive definite
principal submatrix K̂(a) and zero blocks, so that the
pseudo-inverse equals

K(a)† =
(

K̂(a)−1 0
0T 0

)
. (14)

Using the same partitioning, the force vector f (a)
is split into

(
f̂ (a)T 0T

)T
, in which the term 0T

appears due to the original assumption that f (a) ∈
Im(K(a)). Consequently, we see that (13b) can be
rewritten to

c− f̂ (a)TK̂(a)−1f̂ (a) = 0, (15)

and (13) is therefore equivalent to (5).
From (14) we have K†(a) � 0, so that c ≥ 0

based on (13b). Moreover, iff f̂ (a) 6= 0, we have both
f (a)TK(a)†f (a) > 0 and c > 0. Because we minimize
c (13a) and f (a)TK(a)†f (a) is bounded from below,
(13b) can be simplified to the one-sided inequality

c− f (a)TK(a)†f (a) ≥ 0. (16)

To use the generalized Schur complement lemma, e.g.,
[9, Theorem 16.1], we further need to show that[

I − K(a)K(a)†
]

f (a) = 0 (17)

holds, with I denoting the identity matrix. Indeed,
(17) is always satisfied because f (a) ∈ Im(K(a)), so
that we can substitute f (a) by K(a)v, where v is a
vector of coefficients of the linear combination. Then,
(17) is equivalent to[

K(a) − K(a)K(a)†K(a)
]

v = 0, (18)

with the term in the square brackets always zero [9,
Lemma 14.1], as K(a) is symmetric.
Finally, application of the generalized Schur com-

plement lemma to (16) and (18) provides us with

min
a,c

c (19a)

s.t.
(

c −f (a)T
−f (a)T K(a)

)
� 0, (19b)

`Ta ≤ V , (19c)
a ≥ 0, (19d)

119



M. Tyburec, J. Zeman, M. Kružík, D. Henrion Acta Polytechnica CTU Proceedings

which is a non-linear semidefinite program equivalent
to both (13) and (5). Moreover, if we substitute c with
1/s, where 0 < s < ∞ is a measure of stiffness, (19)
is, after the application of the (standard) Schur com-
plement lemma, e.g., [9, Proposition 16.1], reducible
to

max
a,s

s (20a)

s.t. K(a) −sf (a)f (a)T � 0, (20b)
`Ta ≤ V , (20c)

a ≥ 0, (20d)

which is a useful reformulation when f is constant,
i.e., self-weight is not considered.

It shall be noted that due to the polynomial matrix
inequalities (19b) and (20b) both the optimization
problems are non-convex in general. They can still
be solved efficiently (to local optimality) using, e.g.,
augmented Lagrangian methods [6, 10]. In this contri-
bution, we solve these problems using the open-source
PenLab optimizer [6].

2.4. Polynomial optimization
Having introduced three different local approaches
to frame structure optimization, it is natural and
expected that one asks for a global optimization tech-
nique. In this section, we exploit the fact that al-
though (19) is non-convex it is indeed a polynomial
optimization (PO) problem at the same time, which al-
lows us to employ modern PO techniques, namely the
Lasserre hierarchy [7, 11], successively building tighter
and tighter convex outer semidefinite programming
(SDP) approximations called relaxations [7, Corollary
4.3].

To develop a more efficient formulation suitable for
PO, we first recognize that the design variables in (19)
are all bounded both from below and above:

0≤ai≤
V

`i
, ∀i ∈{1, . . . ,ne}, (21a)

0≤c ≤ ĉ. (21b)

While the lower bound in (21a) is caused by the non-
negativity of the cross-sectional areas (19d), the upper
bounds arise from the volume constraint (19c): none of
the structural elements can occupy larger volume than
V . In the compliance case (21b), the lower bound3 is
due to K(a) � 0, recall the discussion in Section 2.3,
and the upper bound is provided by an arbitrary
feasible design, e.g., the uniform cross-sectional areas

a =
V

1T`
1, (22)

which are used for the upper-bound computation in
this paper. Then, the compliance bound is computed
from

ĉ = f (a)TK(a)†f (a), (23)
3In fact the strict inequality 0 < c is satisfied in all non-

trivial optimization problems.

where K(a)† reduces to K(a)−1 in the case of (22).

To improve numerical stability of the solution pro-
cess, we state the optimization problem in terms of the
scaled cross-sectional areas asc and scaled compliance
csc instead of the original a and c, where the scaled
variables are defined in the [−1, 1] domain. Therefore,
we have

ai =
V (asc,i + 1)

2`i
(24)

and

c =
ĉ(csc + 1)

2
. (25)

Rewriting (21) in terms of asc,i and csc as unit ball
constraints

a2sc,i ≤ 1, ∀i ∈{1, . . . ,ne}, (26a)
c2sc ≤ 1, (26b)

we finally arrive at an equivalent formulation to (19),

min
asc,csc

0.5ĉ(csc + 1) (27a)

s.t.
(

0.5ĉ(csc + 1) −f (asc)T
−f (asc)T K(asc)

)
� 0, (27b)

2 −ne − 1Tasc ≥ 0, (27c)
1 −a2sc,i ≥ 0, (27d)

1 − c2sc ≥ 0. (27e)

In (27), the objective function (27a) and also the con-
straints (27b)–(27e) are all polynomial inequalities4
non-negative in the feasible region of (27), which is
therefore a semi-algebraic set.

4The entries in the PMI (27b) are polynomials, so that the
feasible region of the PMI is a semi-algebraic set. Alternatively,
we may require the roots of the characteristic polynomial of
(27b) to be non-negative. We refer to [12] for more details.

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vol. 26/2020 On optimum design of frame structures

The optimization problem (27) can readily be solved
by building the (Lasserre) hierarchy of convex linear
SDP relaxations, with a monotonously converging
objective functions to the global optimum. The hier-
archy is generated by the GloptiPoly [13] software
package interfaced with the Yalmip [14] toolbox, and
the underlying SDP relaxations are solved by the
state-of-the-art Mosek optimizer [8].

2.4.1. Solution process
To simplify the notation, let us now denote the ob-
jective function polynomial (27a) by p0, and the con-
straining polynomials (27c) to (27e) by p1 to p3, re-
spectively. Further, let P4 denote the PMI (27b) and
let

x =
(
csc asc,1 . . . asc,ne

)T (28)
be the vector of the design variables. Moreover, let

br(x) =
(
1 x1 x2 . . . xne+1 x

2
1 x1x2 . . .

x1xne+1 x
2
2 x2x3 . . . x

2
ne+1 . . .

xr1 . . . x
r
ne+1

)T (29)
denote the polynomial space basis of the maximum de-
gree r. Then, we can express each of the polynomials
pj,j ∈{0, . . . , 3} as a linear combination

pj(x) =
|br (x)|∑
β=1

pα,j,β(xα)β (30)

of monomials

xα =
ne+1∏
m=1

xαmm ,

ne+1∑
m=1

αm ≤ r, (31)

indexed in the basis br(x). The vectors α associate
a non-negative integer number with each element in
x, and the vector pα,j contains coefficients of the
linear combinations of the monomials. Notice that∑ne+1
m=1 αm ≤ dj, where dj stands for the degree of the

polynomial pj. A similar approach is also applied to
the elements of P4 [12].

Further, introducing y, with its components yβ cor-
responding to a monomial in the basis br(x), we build
the Lasserre hierarchy of convex linear semidefinite
programming relaxations

min
y

|br (x)|∑
β=1

pα,0,βyβ (32a)

s.t. Mr(y) � 0, (32b)
Mr−dj (y) � 0, ∀j ∈{1, . . . , 4}, (32c)

and solve it successively with an increasing relaxation
order r until the globally optimal solution(s) are found.
For all our test cases, this convergence was always
finite.
In (32), the matrix Mr(y) is the moment matrix

of the r-th order, and Mr−dj is the (r − dj)-order
localization matrix associated with pj or Pj. For
more details about these matrices, we refer the reader
to [7, 12].

2.4.2. Recognizing global optimality
There are (at least) two ways to recognize whether
y∗r, a r-th order relaxation solution to (32), is globally
optimal.

The first (common) way is based a sufficient condi-
tion for global optimality of (32) [15, 16],

rank Mr(y∗r) = rank Mr−d(y
∗
r), (33)

in which y∗ constitutes the optimal solution in the
basis br(csc, asc), and rank Mr(y∗) is less or equal
to the number of these solutions [7]. The values of
the original design variables can be extracted using
Cholesky or singular value decompositions [16].
For the optimization problem (27), we introduce

another sufficient condition for global optimality. Let
ãr be the (optimized) values of the cross-sectional ar-
eas and let c̃r denote the compliance found in the r-th
order relaxation. Indeed, c̃r ≤ c∗ [7]. Moreover, be-
cause any feasible design a generates an upper bound
ĉr, recall Eq. (23), we have

c̃r ≤ c∗ ≤ ĉr. (34)

A globally optimal solution to (27) is found if there is
no gap,

ĉr − c̃r −→ 0. (35)

Note that (35) is substantially simpler to check than
(33). If (35) is not satisfied in the (current) relaxation
order r, we have at least the quality measure for the
current solution in hand.

3. Sample problems
In this section, we describe and solve three rather
small-scaled structural optimization problems, and
compare the four optimization approaches.

3.1. Cantilever beam
Consider a (simple) cantilever beam design problem,
as shown in Fig. 1a. The beam is discretized into ne
finite elements of equal lengths, each of them assigned
a prismatic square cross-section. The beam is loaded
with a skew force at its tip. Further, we assume
(dimensionless) E = 1 and V = 0.1.

It can be seen from Table 1 that the problem is
relatively “easy” to solve, as all of the tested algo-
rithms converge to the unique global optima (shown
in 1b–1e), proved by the PO approach (Figs. 1h–1j).
In fact, we include this optimization problem mainly
to show that PO scales unambiguously worse with the
problem dimension than the other methods, and that
the optimality criteria (OC) method is the fastest one.
Using OC, we can optimize the discretizations of 150
or 300 elements in 0.4 and 0.8 seconds (Figs. 1f and
1h).

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M. Tyburec, J. Zeman, M. Kružík, D. Henrion Acta Polytechnica CTU Proceedings

t [s] ĉ c̃ a1
fmincon 0.7 107.50 - 0.100
OC 0.1 107.50 - 0.100
NSDP 0.2 107.50 - 0.100
PO(1) 0.3 107.50 107.50 0.100

(a) 1 element

t [s] ĉ c̃ a1 a2 a3
fmincon 0.6 80.30 - 0.142 0.102 0.056
OC 0.1 80.30 - 0.142 0.102 0.056
NSDP 0.3 80.30 - 0.142 0.102 0.056
PO(1) 0.3 80.72 35.81 0.147 0.097 0.056
PO(2) 0.3 80.30 80.30 0.142 0.102 0.056

(b) 3 elements

t [s] ĉ c̃ a1 a2 a3 a4 a5
fmincon 0.4 77.19 - 0.151 0.128 0.103 0.075 0.043
OC 0.1 77.19 - 0.151 0.128 0.103 0.075 0.043
NSDP 0.5 77.19 - 0.151 0.128 0.103 0.075 0.043
PO(1) 0.3 79.09 24.72 0.151 0.123 0.096 0.073 0.058
PO(2) 0.7 77.37 76.34 0.154 0.131 0.103 0.072 0.040
PO(3) 26.8 77.19 77.19 0.151 0.128 0.103 0.075 0.043

(c) 5 elements

t [s] ĉ c̃ a1 a2 a3 a4 a5 a6 a7
fmincon 0.5 76.23 - 0.155 0.139 0.122 0.104 0.084 0.061 0.036
OC 0.1 76.23 - 0.155 0.139 0.122 0.104 0.084 0.061 0.036
NSDP 0.4 76.23 - 0.155 0.139 0.122 0.104 0.084 0.061 0.036
PO(1) 0.3 79.81 20.00 0.149 0.130 0.112 0.096 0.081 0.069 0.062
PO(2) 2.5 77.02 71.69 0.166 0.145 0.122 0.099 0.077 0.055 0.036
PO(3) 550.1 76.23 76.23 0.155 0.139 0.122 0.104 0.084 0.061 0.036

(d) 7 elements

Table 1. Comparison of the four optimization approaches on the design of the cantilever beam problem. Bold text
denotes the proven global optimum.

1
ne

1
ne

1 − 2
ne

1

1 2 ne nn

1 ne

1

30◦x

y

(a)

(b) c∗ = 107.50 (c) c∗ = 80.30 (d) c∗ = 77.19

(e) c∗ = 76.23 (f) c = 75.19 (g) c = 75.19

1 2
0

20
40
60
80

100
120

Relaxation order r

C
om

pl
ia
nc

e
c

ĉr
c̃r
c∗

(h)

1 2 3
0

20
40
60
80

100
120

Relaxation order r

C
om

pl
ia
nc

e
c

ĉr
c̃r
c∗

(i)

1 2 3
0

20
40
60
80

100
120

Relaxation order r

C
om

pl
ia
nc

e
c

ĉr
c̃r
c∗

(j)

Figure 1. Cantilever beam design problem. (a) shows the design domain, (b)–(e) are the optimal topologies for 1,
3, 5, and 7 elements; (f) and (g) display optimized topologies computed by OC with discretization by 150 and 300
elements. Figures (h)–(j) show the convergance of PO for 3, 5, and 7 elements.

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vol. 26/2020 On optimum design of frame structures

1 2

x

y

1 1

1

1

2

3

4 5 6 7

8

9 10

1 2 3

4 5 6

(a)

(b) c = 1042.20

(c) c∗ = 959.32

1 2
0

500

1,000

1,500

2,000

Relaxation order r

C
om

pl
ia
nc

e
c

ĉr
c̃r
c∗

(d)

Figure 2. 10-beam structure design problem. (a) shows the design domain. While all the tested local optimization
algorithms converged to the topology in (b), the global optimum design (c) obtained by PO possesses a clearly
different topology. Figure (d) shows the convergence of PO.

t [s] ĉ c̃ a1 a3 a4 a5 a6 a7 a8 a9 a10
fmincon 1.0 1042.14 - 0.144 0.040 0.018 0.000 0.003 0.000 0.210 0.039 0.038
OC 0.6 1042.33 - 0.146 0.039 0.017 0.000 0.002 0.000 0.212 0.039 0.037
NSDP 1.0 1042.20 - 0.144 0.040 0.018 0.000 0.003 0.000 0.210 0.039 0.038
PO(1) 0.3 1429.31 443.20 0.103 0.082 0.000 0.029 0.000 0.000 0.121 0.095 0.069
PO(2) 5.2 959.32 959.31 0.070 0.186 0.000 0.043 0.000 0.098 0.000 0.064 0.000

Table 2. Comparison of the four optimization approaches on the design of 10-beam structure. Bold text denotes
proven global optimum. The cross-sectional area a2 is zero in all test cases.

1 2 3 4 5
1

1 2 3 4 5 6
2 2 2 2 2

10

(a)

tp

5tp

10
t p

(b)

1 2 3
0

500

1,000

1,500

2,000

2,500

Relaxation order r

C
om

pl
ia
nc

e
c

ĉr
c̃r
c∗

(c)

Figure 3. Girder beam design problem. (a) shows the design domain, (b) the cross-sectional shape, and (c) the
convergence of polynomial optimization.

t [s] ĉ c̃ a1 a2 a3 a4 a5
fmincon 0.4 - - 0.394 0.256 0.000 0.003 0.000
OC 0.1 1372.25 - 0.010 0.017 0.022 0.025 0.026
NSDP 2.0 - - 0.000 0.000 0.000 0.000 0.000
PO(1) 0.2 1456.75 297.34 0.007 0.015 0.022 0.027 0.029
PO(2) 1.0 1426.05 1286.44 0.007 0.016 0.023 0.026 0.028
PO(3) 37.7 1372.25 1372.25 0.010 0.017 0.022 0.025 0.026

Table 3. Comparison of the four optimization approaches on the design of the girder beam problem. Bold text
denotes the proven global optimum.

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M. Tyburec, J. Zeman, M. Kružík, D. Henrion Acta Polytechnica CTU Proceedings

3.2. 10-beam frame structure
Second, we consider the topology optimization prob-
lem of designing circular cross-sections of the 10-beam
structure shown in Fig. 2a. This structure is loaded
by two moments of magnitudes 1 and 2, placed at
the nodes 2 and 3 , respectively. Note that there is
no intersection between the elements 3 and 4 , and
between 6 and 7 . We assume the Young modulus
E = 1 and the volume bound V = 0.5.

Evaluating the four optimization methods, Table 2,
we see that while the local optimization methods con-
verged to nearby local minima of topologies shown in
Fig. 2b, PO found the global optimum of a clearly
different topology, Fig. 2c. Moreover, the global op-
timum possesses the compliance lower by 8%, which
was reached in the second relaxation, Fig. 2d.

3.3. I-shaped girder with self-weight
Last, we consider a simply-supported I-shaped girder
beam of the span 20, loaded by a uniform load 1
and self-weight. Due to the problem symmetry, we
consider only one half of the problem, Fig. 3a, and
discretize it using 5 finite elements of equal length.
This (half of the) girder beam is allowed to utilize at
most V = 0.2 material of the Young modulus E = 104
and of density ρ = 3. The beam cross-sections are
parameterized by the parameter tp, which denotes the
thicknesses of the flanges and web. The cross-sectional
area equals a(tp) = 18t2p. We assume that the upper
surfaces of the top flanges are aligned, so that we
have I(tp) = 696t4p = 58/27a(tp)2. Consequently, we
optimize ai of individual elements, rather than tp.
Table 3 reveals that for this specific problem the

OC converged to the global optimum. On the other
hand, the fmincon and non-linear semidefinite pro-
gramming approaches converged to an infeasible point.
Polynomial optimization exhibited convergence to the
global optimum in three relaxations, Fig. 3c. Notice,
moreover, that the designs found for r = {1, 2} were
of very high qualities.

4. Conclusions
In this contribution, we investigated four topology
optimization techniques for the design of frame struc-
tures with a given aspect-ratios all components of their
cross-sections. Three of the optimization techniques—
general non-linear formulation solved by the fmincon
solver, optimality criteria, and non-linear semidefinite
programming—provide a local solution to the consid-
ered optimization problem, and in turn converge to
a local optimum in general. Unfortunately, the NSDP
and fmincon techniques also converged to infeasible
points. From the local optimization approaches, the
optimality criteria method seems to be the most effi-
cient one.

On the other hand, the last technique — polynomial
optimization — allows to solve the problem to proven
global optimality, while generating both lower and

upper bounds in each of the relaxation order. Com-
pared to the local approaches, the designs obtained in
lower relaxation orders are of a known quality with
respect to the optimum. However, finding a proven
global optimum requires a considerably higher compu-
tational resources than solving the local optimization
formulations. In all the test cases, the convergence of
the PO hierarchy was finite.

List of symbols
0 Column vector or matrix of all zeros
1 Column vector or matrix of all ones
a Cross-sectional areas column vector
asc Scaled cross-sectional areas column vector
ai Cross-sectional area of the element i
asc,i Scaled cross-sectional area of the element i
âr Optimized cross-sectional areas at the relaxation r
bi Auxiliary variable associated with the element i
br Polynomial space basis of maximum degree r
c Compliance (external work)
c̃ Lower bound on compliance
ĉ Upper bound on compliance
c∗ Globally optimal compliance
csc Scaled compliance
cI,i Positive constant
dj Degree of the polynomial j
E Young modulus
f Generalized force column vector
fi Generalized force column vector of element i
f̂ Generalized force column vector associated with K̂
i Element index
j Polynomial index
Ii Moment of inertia of the element i
I Identity matrix
k Iteration number
K Stiffness matrix
K̂ Positive definite principal submatrix of K
Ki Stiffness matrix of the element i
L Lagrangian function
` Column vector of element lengths
`i Length of the element i
m Index of x
Mr Moment matrix of the order r
Mr−dj Localization matrix of the order r −dj
ne Number of elements
nn Number of nodes
pj Polynomial inequality
Pj Polynomial matrix inequality
pα,j Coefficients of linear combinations of monomials
r Relaxation order, maximum polynomial degree
s Inverse of c
tp Parameter, web and flanges thickness
t Time
u Generalized displacement column vector
v Column vector of coefficients of linear combination
V Volume

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vol. 26/2020 On optimum design of frame structures

V Volume upper bound
x Design variables column vector
xα Monomial indexed by α
yβ Moment associated with the basis br
y Column vector of moments
y∗ Optimal column vector of moments
yr Column vector of moments at r-th relaxation
α Vector of integers
β br index
η Tuning parameter, 0.3 in this study
ζ Move limit, 0.2 in this study
ε Small positive number, 10−6 in this study
λ Lagrange multiplier column vector
µ Lagrange multiplier
ν Lagrange multiplier column vector

Acknowledgements
The work of Marek Tyburec was supported by the Grant
Agency of the CTU in Prague, SGS19/033/OHK1/1T/11.
Martin Kružík and Jan Zeman acknowledge the financial
support by the European Regional Development Fund
through the project CZ.02.1.01/0.0/0.0/16-019/0000778.

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https://doi.org/10.1007/978-3-662-05086-6
https://doi.org/10.1137/1.9780898718829
https://doi.org/10.1016/S0045-7825(99)00359-X
https://doi.org/10.1007/978-94-009-1161-1
1311.5240
https://doi.org/10.1017/CBO9781107447226
https://doi.org/10.1007/978-1-4419-9961-0
https://doi.org/10.1080/1055678031000098773
https://doi.org/10.1137/S1052623400366802
https://doi.org/10.1109/TAC.2005.863494
https://doi.org/10.1080/10556780802699201
https://doi.org/10.1090/S0002-9947-00-02472-7
https://doi.org/10.1007/10997703_15

	Acta Polytechnica CTU Proceedings 26:118–126, 2020
	1 Introduction
	2 Solution techniques to frame optimization
	2.1 Problem statement
	2.2 Optimality criteria
	2.3 Nonlinear semidefinite programming
	2.4 Polynomial optimization
	2.4.1 Solution process
	2.4.2 Recognizing global optimality


	3 Sample problems
	3.1 Cantilever beam
	3.2 10-beam frame structure
	3.3 I-shaped girder with self-weight

	4 Conclusions
	List of symbols
	Acknowledgements
	References