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Опір матеріалів і теорія споруд/Strength of Materials and Theory of Structures. 2023. № 110
Peleshko I.D., Yurchenko V.V.
178
UDC 624.04.012.4.044, 519.853
PARAMETRIC OPTIMIZATION OF BAR STRUCTURES WITH
DISCRETE AND CONTINUOUS DESIGN VARIABLES USING
IMPROVED GRADIENT PROJECTION METHOD
I.D. Peleshko1,
Candidate of Technical Science, Associate Professor
V.V. Yurchenko2,
Doctor of Technical Science, Professor
1Lviv Polytechnic National University
St. Bandery, 12, Lviv, 79013
2Kyiv National University of Construction and Architecture
Povitroflotskyj av., 31, Kyiv, 03680
DOI: 10.32347/2410-2547.2023.110.178-198
The paper considers a parametric optimization problem for the bar structures formulated as
nonlinear programming task, where the purpose function and non-linear constraints of the
mathematical model are continuously differentiable functions. The method of the objective
function gradient projection onto the active constraints surface with simultaneous correction of the
constraints violations has been used to solve the parametric optimization problem. A discretization
technique for the design variables that should vary discretely has been proposed. The discretization
of the optimal design solution obtained in the continuous space of the design variables is
performed by the purposefully selecting discrete points around the point of the continuous
optimum. The comparison of the optimization results presented by the paper demonstrates that
improved gradient method together with proposed discretization technique for the discrete design
variables converges to better solutions comparing to meta -heuristic algorithms.
Keywords: shape optimization, bar structures, nonlinear programming, design code
constraints, gradient projection method, optimization software, finite element method.
Introduction. Over the past 50 years, numerical optimization and finite
element method have individually made significant advances and have
together been developed to make possible the emergence of structural
optimization as a potential design tool [4, 6, 21]. In recent years, great efforts
have been also devoted to integrate optimization procedures into the CAD
facilities. With these new developments, lots of computer packages are now
able to solve relatively complicated industrial design problems using different
structural optimization techniques [9, 10].
Applied optimum design problems for the bar structures in some cases are
formulated as parametric optimization problems, namely as searching problems
for unknown structural parameters that provide an extreme value of the specified
purpose function in the feasible region defined by the specified constraints [24].
In this case structural optimization performs by variation of the structural
parameters when the structural topology, cross-section types and node type
connections of the bars, the support conditions of the bar system, as well as
loading patterns and load design values are prescribed and constants.
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Mathematical model of the parametric optimization problem of the structures
includes the set of design variables, the objective function, as well as constraints
reflected in general case non-linear interdependences between them [5, 15].
Although many papers are published on the parametric optimization of the
structures, the development of a general computer program for the design and
optimization of building structures according to specified design codes remains
an actual task. Therefore, the main research goal is the development of
mathematical support and numerical algorithm to solve parametric
optimization problems of the building structures with orientation on software
implementation in a computer-aided design system.
One of the effective methods to solve parametric optimization problems for
building structures is the gradient projection non-linear methods since the
purpose function and non-linear constraints of the presented mathematical
model are continuously differentiable functions, as well as the search space is
smooth. Thus, the method of objective function gradient projection onto the
active constraints surface with simultaneous correction of the constraints
violations has been successfully used for parametric optimization of cross-
sectional dimensions for cold-formed steel structural members [1], steel trusses
[18], as well as lattice portal frames [23].
When applying gradient projection methods, the search for the optimum
point is performed in a continuous space of the design variables only. If a
nonlinear programming problem is solved where some (or all) design variables
vary discretely (for example, according to a defined set of the possible values),
then after obtaining a continuous optimal solution, the question of its
discretization arises. That is why, the following research tasks are states: to
propose a discretization technique for the design variables that vary discretely
allowing using the gradient projection methods to parametric optimization
problems with mixed (discrete and continuous) design variables; to
demonstrate the effectiveness of the proposed discretization technique by
comparing obtained optimization, as well as the results presented by the
literature and widely used for testing.
Parametric optimization problem formulation for bar structures. Let us
consider a parametric optimization problem of a structure consists of the bar
members, which can be formulated as presented below: to find optimum
values for geometrical parameters of the structure, bar’s cross-section sizes and
initial pre-stressing forces introduced into the redundant members of the bar
system, whose provide the extreme value of the determined optimality
criterion and satisfy all load-bearing capacities and stiffness requirements. We
assume, that the structural topology, cross-section types and node type
connections of the bars, the support conditions of the bar system, as well as
loading patterns and load design values are prescribed and constants.
The formulated parametric optimization problem can be stated as a non-
linear programming task in the following mathematical terms: to find unknown
structural parameters
T
X X
,
1,
X
N
, (1)
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providing the least value of the determined objective function:
* *
min
X
f f X f X
, (2)
in feasible region (search space)
defined by the following system of
constraints:
0 | 1,
IC
X X N
φ
, (3)
where
X
– vector of the design variables (unknown structural parameters);
X
N
– total number of the design variables;
,
f
– continuous functions of
the vector argument;
*
X
– optimum solution (the vector of optimum values of
the structural parameters);
*
f
– optimum value of the optimum criterion
(objective function);
IC
N
– number of constraints-inequalities
X
, which
define a feasible region in the design space
.
The vector of the design variables can include as components unknown
geometrical parameters of the structure
,
T
G G
X X
,
,
1,
X G
N
, as well as
unknown cross-sectional sizes of the structural members
,
T
CS CS
X X
,
,
1,
X CS
N
:
,
T
G CS
X X X
,
1,
X
N
, (4)
where
,
X CS
N is the total number of the unknown cross-sectional sizes of the
structural members,
,
X G
N is the total number of the unknown parameters of
the structural geometry or shape, , ,
X G X CS X
N N N
.
Fig. 1. Unknown parameters of a structure considered as design variables
The specific technical-and-economic index (material weight, material cost,
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construction cost etc.) or another determined indicator can be considered as the
objective function taking into account the ability to formulate analytical
expression of the purpose function depending on the design variables
X
. In
some cases the material weight of the structure is considered as the objective
function Eq. (2) of the optimization problem:
* *
min ,
G CS
X
M M X M X X
. (5)
Load-bearing capacities constraints (strength and stability inequalities) for all
design sections of the structural members subjected to all design load case
combinations at the ultimate limit state as well as displacements constraints
(stiffness inequalities) for the specified nodes of the bar system subjected to all
design load case combinations at the serviceability limit state should be included
into the system of constraints Eq. (3). Additional requirements, whose describe
structural, technological and serviceability particularities of the building
structure under consideration, can be also included into the system Eq. (3).
Design internal forces in the bar structural members used in the strength and
stability inequalities of the system Eq. (3) are considered as state variables
depending on design variables
X
and can be calculated from the following
linear equations system of the finite element method:
, ,
,
G CS ULS k ULS k G
X X z p X
Κ
, 1,
ULS
LCC
k N, (6)
where
,
G CS
X XΚ
is the stiffness matrix of the finite element model of the
bar structure, which should be formed depending on the design variable
,
T
G CS
X X X
of the optimization problem Eqs. (1) – (3);
,
ULS k G
p X
is
the column-vector of the node’s external loads for
k
th design load case
combination corresponded to the ultimate limit state, which should be formed
depending on unknown (variable) parameters of the geometrical scheme
(shape)
G
X
of the considered bar structures;
,
ULS k
z
is the result column-vector
of the node displacements for
k
th design load case combination corresponded
to the ultimate limit state,
, , ,
ULS
ULS k k PS CS
z X XFEM
Ζ
;
ULS
LCC
N is the total
number of the design ultimate load case combinations. In this way, for each
i
th
design section of
j
th bar finite element subjected to
k
th ultimate design load
case combination the design internal forces (axial force
ijk
N X
, bending
moments
,y ijk
M X
,
,z ijk
M X
and corresponded shear forces
, ,
,
z ijk y ijk
Q X Q X
) can be calculated depending on node displacement
column-vector
,
ULS k
z
.
Node displacements of the bar structure used in stiffness inequalities of the
system Eq. (3) are also considered as state variables depending on design
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variables
X
and can be calculated from the following linear equations system
of the finite element method:
, ,
,
G CS SLS k SLS k G
X X z p X
Κ
, 1,
SLS
LCC
k N, (7)
where
,SLS k G
p X
is the column-vector of the node’s external loads for
k
th
design load case combination corresponded to the serviceability limit state,
which should be formed depending on unknown (variable) parameters of the
geometrical scheme (shape)
G
X
of the considered bar structures;
,
SLS k
z
is the
result column-vector of the node displacements for
k
th serviceability design
load case combination,
, , ,
SLS
SLS k k G CS
z X XFEM
Ζ
;
SLS
LCC
N is the total number of
the design serviceability load case combinations. For each
m
th node of the
finite element model subjected to
k
th serviceability design load case
combination the design vertical
,z mk
X
and horizontal
,x mk
X
displacements can be calculated depending on node displacement column-
vector
,
SLS k
z
.
The system of constraints Eq. (3) should cover strength and stability
constraints formulated for all structural members of the bar structure subjected
to all design load case combinations corresponded to the ultimate limit state. In
case of parametric optimization of truss structures as particular sub-case of the
bar structures the following normal stresses verifications should be included in
the system of constraints:
,
,
1 0
t jk
j CS t ult
N X
A X
;
1,
B
j N
, 1,
ULS
LCC
k N , (8)
,
,
1 0
c jk
j CS c ult
N X
A X
;
1,
B
j N
, 1,
ULS
LCC
k N , (9)
where
B
N
is the total number of the truss structural members;
j CS
A X
is the
cross-section area of
j
th structural member of the truss structure;
,
t ult
and
,
c ult
are the allowable tension and compression normal stresses respectively;
,t j k
N X
and
,c jk
N X
are the tension and compression axial force
respectively acting in
j
th structural member of the truss structure subjected to
k
th ultimate load case combination calculated from the linear equations system
of the finite element method Eq. (6). In case of statically indeterminate truss
structure the value of the axial force should be calculated depending on the
variable geometrical parameters of the structure
G
X
and variable cross-
sectional dimensions of the structural members
CS
X
.
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The following flexural and torsional-flexural buckling verifications should
be included in the system of constraints Eq. (3) formulated for all structural
members of the truss structure subjected to all ultimate load case combination,
namely:
,
, ,
1 0
,
c jk
y j G C S j CS c ult
N X
X X A X
;
1,
B
j N
, 1,
ULS
LCC
k N , (10)
,
, ,
1 0
,
c jk
z j G CS j CS c ult
N X
X X A X
;
1,
B
j N
, 1,
ULS
LCC
k N , (11)
,
, ,
1 0
,
c jk
c j G CS j CS c ult
N X
X X A X
;
1,
B
j N
, 1,
ULS
LCC
k N , (12)
where
,,
y j G CS
X X
and
,,
z j G CS
X X
are column’s stability factors
corresponded to flexural buckling relative to main axes of inertia and
calculated according to the specified design code depending on the design
lengths
, ,
ef y j
l,
, ,
ef z j
l, cross-section type and cross-section geometrical
properties for the
j
th structural member;
,,
c j G CS
X X
is the column’s
stability factor corresponded to torsional-flexural buckling and calculated
according to the specified design code depending on the design lengths
, ,ef y j G
l X
,
, ,
ef z j G
l X
,
, ,
ef T j G
l X
, cross-section type and cross-section
geometrical properties for
j
th structural member. The flexural buckling factors
,,
y j G CS
X X
and
,,
z j G CS
X X
, as well as torsional-flexural buckling
factor
,,
c j G CS
X X
should be calculated depending on the variable
geometrical parameters of the structure
G
X
and variable cross-sectional
dimensions of the structural members
CS
X
.
In some cases the system of stability constraints Eqs. (10) – (12) can be
simplified by considering the flexural Euler’s buckling verifications only. The
following flexural Euler’s buckling verifications can be included in the system
of constraints Eq. (3) formulated for all structural members of the truss
structure subjected to all ultimate load case combinations, namely:
,
,min,
1 0
,
c jk
j CS cr j G CS
N X
A X X X
;
1,
B
j N
, 1,
ULS
LCC
k N , (13)
where
,min,
cr j
is the minimum Euler’s buckling critical stresses calculated as
presented below:
,min, , , , ,
, min , , ,
cr j G CS cr y j G CS cr z j G CS
X X X X X X
, (14)
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2 2
,
, , 2
, ,
,
y j CS
cr z j G CS
ef y j G
E i X
X X l X
, (15)
2 2
,
, , 2
, ,
,z j CS
cr z j G CS
ef z j G
E i X
X X l X
, (16)
where
E
is the modulus of material elasticity;
,
y j CS
i X
and
,
z j CS
i X
are
the radiuses of inertia in the main planes of inertia calculated depending on the
variable cross-sectional dimensions of the structural members
CS
X
.
In case of equal design lengths in the main planes of inertia
, , , , ,ef j G ef y j G ef z j G
l X l X l X
, the minimum Euler’s buckling critical
stresses can be determined by the following equation:
2 2
min,
,min, 2
,
,j CS
cr j G CS
ef j G
E i X
X X l X
, (17)
where
min,
j CS
i X
is the minimum radius of inertia calculated depending on
the variable cross-sectional dimensions of the structural members
CS
X
. Taking
into account Eq. (17) the Euler’s buckling constraint-inequality Eq. (13) can be
rewritten as follow:
2
, ,
2 2
min,
1 0
c jk ef j G
j CS j CS
N X l X
E A X i X
(18)
or
2
, ,
2
1 0
c jk ef j G
j CS
N X l X
EA X
, (19)
where
is the factor determined depending on the cross-sectional type.
The system of constraints Eq. (3) should also cover the displacements
constraints (stiffness inequalities) for the specified nodes of the truss structure
subjected to all design load case combinations at the serviceability limit state.
The following horizontal and vertical displacements constraints should be
included into the system of constraints Eq. (3) formulated for nodes of the
truss structure subjected to all serviceability load case combination, namely:
,
,
1 0
x mk
ux m
X
;
1,
N
m N
; 1,
SLS
L
СC
k N , (20)
,
,
1 0
z mk
uz m
X
;
1,
N
m N
; 1,
SLS
L
СC
k N , (21)
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where
,x mk
X
and
,z mk
X
are the horizontal and vertical displacements
respectively for
m
th node of the truss structure subjected to
k
th serviceability
load case combination calculated from the linear equations system of the finite
element method Eq. (7);
,
ux m
and
,
uz m
are the allowable horizontal and vertical
displacements for
m
th structural node;
N
N
is the total node number in structure.
Additional requirements describing structural, technological and
serviceability particularities of the considered structure can be also included into
the system Eq. (3). In particular these requirements can be presented in the form
of constraints on lower and upper bound values for the design variables, namely:
1 0
L
X
X
;
1,
X
N
, (22)
1 0
U
X
X
;
1,
X
N
, (23)
where
L
X
and
U
X
are the lower and upper bounds for the
th design variable
X
.
Parametric optimization algorithm based on the gradient projection
method. The parametric optimization problem stated as non-linear
programming task by Eq. (4), Eq. (5), Eq. (8), Eq. (9), Eqs. (10) – (12) or
Eq. (19), Eqs. (20) – (23) can be successfully solved using gradient projection
non-linear methods since the purpose function and non-linear constraints of the
presented mathematical model are continuously differentiable functions, as
well as the search space is smooth. The method of objective function gradient
projection onto the active constraints surface with simultaneous correction of
the constraints violations ensures effective searching for solution of the non-
linear programming tasks [19]. The gradient projection method operates with
the first derivatives or gradients only of both the objective function Eq. (5) and
constraints Eq. (8), Eq. (9), Eqs. (10) – (12) or Eq. (19), Eqs. (20) – (23). The
method is based on the iterative construction of such sequence Eq. (24) of the
approximations of the design variables
T
X X
,
1,
X
N
, that provides
the convergence to the optimum solution (optimum values of the structural
parameters) [22]:
1
t t t
X X X
, (24)
where
, 1,
T
t X
X X N
is the current approximation to the optimum
solution
*
X
that satisfies constraints-inequalities Eq. (8), Eq. (9), Eqs. (10) –
(12) or Eq. (19), Eqs. (20) – (23) with the extreme value of the objective
function Eq. (5);
, 1,
T
t X
X X N
, is the increment vector for the
current values of the design variables
t
X
;
t
is the iteration’s index.
Let present the following numerical algorithm to solve the parametric
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optimization problem for truss structures formulated above.
Step 1. Describing an initial design (a set of design variables) and initial
data for structural optimization.
The design variable vector
,
T
k G CS
k
X X X
has been specified, where
k
is the iteration index,
0
k
. The structural topology, cross-section types and
node type connections of the bars, the support conditions of the bar system, as
well as loading patterns, load case combinations and load design values are
prescribed and constants.
Initial data for optimization of the considered steel structure are design
strength for steel member (allowable stresses taken into account safety
factors), factors to define flexural design lengths
, ,
ef y j
l,
, ,
ef z j
l for all column
structural members; allowable values for horizontal and vertical displacements
,
ux l
and
,
uz l
of the specified nodes of the considered steel structure; lower
L
X
and upper
U
X
bounds for the design variables; as well as specified
objective function
( )
k
f X
.
Step 2. Calculation of the geometrical and design lengths for all structural
members.
The geometrical lengths
j
l
of all structural members are calculated based on
the node coordinates of the considered steel structure. The latter depend on the
unknown (variable) geometrical parameters of the structure
G
X
. The design
lengths
, ,
ef y j
l,
, ,
ef z j
l of all column structural members are calculated using
calculated geometrical lengths
j
l
and initial data relating to the design length
factors. The latter are constant during the iteration process presented below.
Variation of the geometrical lengths
j
l
and corresponded design lengths
, ,
ef y j
l,
, ,
ef z j
l on the further iterations has been performed based on the current values of
the variable (unknown) parameters
G
X
of the geometrical scheme.
Step 3. Calculation of the cross-section dimensions and geometrical
properties for all design cross-sections.
Geometrical properties of the design cross-sections (areas, moments of
inertia, elastic section moments, radiuses of inertia, etc.) have been calculated
depending on the current values of the unknown (variable) cross-section
dimensions
CS
X
.
Step 4. Linear structural analysis of the considered truss structure.
For each
m
th node of the finite element model subjected to
k
th
serviceability load case combination the displacements and rotations, as well
as the design horizontal
,x mk
X
and vertical
,z lk
X
displacements can be
calculated using the linear equations system of the finite element method
presented by Eq. (7).
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For each
i
th design section of
j
th structural member subjected to
k
th
ultimate load case combination the design internal forces can be calculated using
the linear equations system of the finite element method presented by Eq. (6).
Step 5. Calculation of the state variables (normal stresses, buckling factors
or buckling stresses etc.).
The value of the normal
,x ijk
X
stresses at the specified cross-section
point has been calculated depending on the axial force acting in
i
th design
section of
j
th structural member subjected to
k
th ultimate load case
combination as presented by the design code.
The flexural buckling factors
,,
y j G CS
X X
,
,,
z j G CS
X X
have been
calculated depending on the corresponded design lengths, cross-section type
and cross-section geometrical properties for the structural members according
to the considered design code.
Step 6. Verifications of the constraints and construction the set of active
constraints numbers
A
.
Verification the constraints Eq. (8), Eq. (9), Eqs. (10) – (12) or Eq. (19) has
been performed for all ultimate load case combinations and all design cross-
sections of all structural members. Verification the constraints Eqs. (20) – (21)
have been also conducted for all serviceability load case combinations and all
design structural nodes. Additional requirements Eqs. (22) – (23) on the lower
and upper bounds for the design variables have been also verified.
Step 7. Calculation the increment vector for the current design variables and
determination the improved approximation to the optimum solution. The
increment vector
k
X
for the current design variables values
k
X
has been
calculated according to resolving equations of the method of objective function
gradient projection onto the active constraints surface with simultaneous
correction of the constraints violations described by the paper [17]. The
improved approximation
1
k
X
to the optimum solution has been determined
according to Eq. (24).
Step 8. Stop criteria verification of iterative searching for the optimum
solution. If all constraints are satisfied with appropriate accuracy, as well as
one of the stop criteria described by the paper [17] is also satisfied, then
transition to the step 9 has been performed. In contrary case return to the step 1
has been conducted with
1
k k
.
Step 9. Discretization the optimum solution
k
X
obtained in the continuum
space of the design variables.
Step 10. Optimum parameters of the truss structure is
k
X
with the optimum
value of the objective function
( )
k
f X
.
Discretization technique for the design variables that vary discretely.
When moving along the direction of searching for the optimum point, hitting
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to the nodes of a discrete grid is associated with significant complications of
the optimization algorithm and may lead to deterioration of the optimization
process convergence. Therefore, when applying gradient methods, the search
for the optimum point is performed in a continuous space of the design
variables only. If a nonlinear programming problem is solved where some (or
all) design variables vary discretely (for example, according to a defined set of
the possible values), then after obtaining a continuous optimal solution, the
question of its discretization arises. The discretization of the optimal design
solution obtained in the continuous space of the design variables can be
performed by purposefully selecting discrete points around the point of the
continuous optimum.
Step 1. Let
*
X
is the optimum structural design obtained in the continuous
space of the design variables, moreover the design variable set covers as
continuous design variables
* *
,
T
C C
X X
, 1,
XC
N
, (for example, variable
parameters of the geometrical scheme of the truss structure), as well as design
variables which should vary discretely
* *
,
T
D D
X X
, 1,
XD
N
(for example,
variable size of the cross-section sizes of the structural members):
* * *
,
C D
X X X
, where
XC
N
is the total number of the continuous design
variables;
XD
N
is the total number of the design variables which should vary
discretely.
Step 2. For each design variable
*
i
X
,
* *
i D
X X
, two neighbor values can be
specified from the predefined set of the possible discrete values:
* *
,
L
i D i
X X
and
* *
,
U
i i D
X X
, where
*
,
L
i D
X
is the neighbor value on the left (lower) and
*
,
U
i D
X
is the neighbor value on the right (upper).
Step 3. Among all design variables
* *
,
T
D D
X X
, 1,
XD
N
, which
should vary discretely, the one
*
p
X
,
* *
p D
X X
, with the largest length of the
purpose function gradient is selected:
,*
,
max
p
D
f
X
, 1,
XD
N
; (25)
аnd further discretized at the level of neighbor lower discrete value:
* *
,
L
p p D
X X
. The design variables vector should be truncated accordingly:
* * *
p
X X X
. Total number
XD
N
of the design variables which should
vary discretely is decreased respectively:
1
XD XD
N N
.
Step 4. Searching for the optimum point with truncated design variable
vector
*
X
is performed. If at the same time (when * *
,
L
p p D
X X
) the optimum
solution does not exist, then the discretization of such variable should be
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performed at the level of neighbor upper discrete value ( * *
,
U
p p D
X X
) with the
further searching for the optimum point.
Third and fourth steps are carried out until the design variables which
should vary discretely
* *
,
T
D D
X X
, 1,
XD
N
, will be fully discretized,
namely while
0
XD
N
.
Geometry and cross-sectional optimization of a 18-bar cantilever truss.
A parametric optimization methodology presented above has been realized in
software OptCAD [18, 23]. This software provides solutions to a wide range of
problems, namely: (i) linear static analysis of bar structures; (ii) verification of
the load-bearing capacity of the structural members according to the specified
design code; (iii) searching for values of the structural parameters when
structure complies with design code requirements and designer’s criterions;
(iv) parametric optimization of the steel bar structures by the determined
criterion.
In order to estimate an efficiency of the new methods, techniques or
algorithms, a comparison with alternative methods or algorithms presented by
other authors using different optimization techniques should be performed.
Criteria to implement such comparison are described, e.g. by Haug & Arora
[13] and Crowder et al. [3]. Many of these criteria, such as robustness, amount
of functions calculations, requirements to the computer memory, numbers of
iterations etc. cannot be used due to lack of corresponded information in the
technical literature. Therefore, an efficiency estimation of the proposed
methodology for solving parametric optimization problems presented above
will be based on the comparison of the optimization results obtained using the
proposed numerical algorithm, as well as of the results presented by the
literature and widely used for testing. The initial data and mathematical models
of the parametric optimization problems considered below were assumed as
the same as described in the literature.
Figure 2 shows a 18-bar cantilever truss designed for the vertical loads
Р = 20 kips = 88.9644 kN (only one design load case combination). Initial data
for truss optimum design are: material density
= 0.1 lb/inch3 = 27.8014 ton/m3, coefficient of elasticity
E = 104 ksi = 6894.76 kN/cm2. The allowable displacements in the horizontal
and vertical direction for all nodes are limited by
δux = δuz = ±10 inch = ±254 mm. The absolute value of the allowable normal
stresses in tension and compression are σt,ult = σc,ult = 20 ksi = 137.895 N/mm2.
The geometry and cross-sectional optimization problem for 18-bar
cantilever truss has been formulated as searching for optimum values of the
coordinates for all nodes of the truss lower chord, as well as for optimum
values of the cross sectional areas for all truss members. Variable unknown
cross-sectional areas
CS
X
= {A1, A2, A3, A4}, as well as unknown horizontal
and vertical coordinates for all truss lower chord nodes
G
X
= {x3, z3, x5, z5, x7,
z7, x9, z9}, were considered as design variables Eq. (4). Table 1 presents the
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lower and upper bounds for the design variable and member grouping for the
variable cross-section areas.
Fig. 2. Design scheme of the 18-bar cantilever truss (with specified numbers of nodes
and numbers of bars)
Table 1
The lower and upper bounds for the design variable and member grouping for
the variable cross-section areas of the 18-bar cantilever truss problem
Design
variable Unit Lower
bound
Upper
bound
Truss
member’s
number
Type of design
variable
CS
X
A1 cm2 12.9032 140.3223 1,4,8,12,16
discrete with
step 1.6129 cm2
A2 cm2 12.9032 140.3223 2,6,10,14,18
discrete with
step 1.6129 cm2
A3 cm2 12.9032 140.3223 3,7,11,15
discrete with
step 1.6129 cm2
A4 cm2 12.9032 140.3223 5,9,13,17
discrete wi
th
step 1.6129 cm2
G
X
x
3
cm
1968.5
3111.5
–
continuous
z
3
cm
–
571.5
622.3
–
continuous
x
5
cm
1333.5
2476.5
–
continuous
z
5
cm
–
571.5
622.3
–
continuous
x
7
cm
698.5
1841.5
–
continuous
z
7
cm
571.5
622.3
–
continuous
x
9
cm
63.5
1206.5
–
continuous
z
9
cm
571.5
622.3
–
continuous
The optimum values of the design variables are presented by Table 2 (see
column 2). It should be noted that searching for the optimum point has been
firstly realized in the continuous design space.
The optimum continuous values for coordinates of the truss lower chord are
presented by Fig. 3. The optimum structural weight for the considered 18-bar
cantilever truss is 2043.852 kg. There are 10 active constraints in the optimum
point, namely the flexural Euler’s buckling verification Eq. (19) for 2nd, 6th, 7th,
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10th, 11th, 14th, 15th and 18th truss members (see Fig. 2), as well as tension
stress verifications Eq. (8) for 16th and 17th truss members.
Fig. 3. Optimum coordinates values for all nodes of the 18-bar cantilever truss lower chord
Fig. 4. Optimum truss layout for 18-bar cantilever truss (OptCAD screenshot)
The variable cross-sectional areas of the truss members have been further
discretized based on the proposed discretization technique presented above. At
the first step the purpose function gradient relative to the variable cross-section
areas of the truss structural members was {0.710440, 1.031737, 0.157964,
0.143727}, where the component corresponded to the variable А2 had the
maximum value. That is why, the variable A2 has been discretized firstly at the
level of neighbor value on the left (lower) A2 = 114.5159 cm2. Than searching
for the optimum point in the design space of the variable cross-sectional areas
of the truss structural members {A1, A3, A4}, as well as variable parameters of
the truss geometry {x3, z3, x5, z5, x7, z7, x9, z9} has been performed. As a result
the optimum truss design with structural weight 2043.905 kg has been
obtained (see 3rd column of Table 2).
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Table 2
Optimum truss solution in continuous design space, as well as mixed design
space
Design variable
Optimum
solution in
continuous
space
Optimum solution in the mixed design space
Step 1 Step 2 Step 3 Step 4
1
2
3
4
5
6
A
1
,
cm
2
80.59012
80.58439
80.
645
80.645
80.645
A
2
,
cm
2
114.95256
114.5159
114.5159
114.5159
114.5159
A
3
,
cm
2
34.11417
34.68031
34.70766
35.4838
35.4838
A
4
,
cm
2
23.86916
23.94215
23.84594
23.84583
24.1935
x
3
, cm
2315.42400
2313.12283
2312.99463
2313.27053
2311.89259
z
3
, cm
471.462
99
468.63030
468.55013
468.72269
467.85908
x
5
, cm
1634.69376
1631.16983
1630.97784
1630.06180
1628.59515
z
5
, cm
374.30071
371.29859
371.25254
371.24689
369.69831
x
7
, cm
1051.21702
1048.09826
1047.93536
1046.4583
1045.74314
z
7
, cm
249.80907
247.41659
24
7.52573
247.72279
250.77424
x
9
,
cm
513.70153
511.86057
511.77234
511.77234
511.77235
z
9
, cm
78.10165
77.32506
77.70894
77.70894
77.70894
Weight, kg
2043.85212
2043.90517
2043.90744
2047.48540
2049.60990
Count of active
constraints 10 11 12 11 11
The m
aximum
constraint
violation
2.073×10-9
1.039×10-8
1.980×10-6
5.372×10-10
3.753×10-14
At the second step the purpose function gradient relative to the variable
cross-section areas of the truss structural members was {0.708205, 0.161196,
0.146550}, where the component corresponded to the variable А1 had the
maximum value. That is why, the variable A1 has been discretized secondly at
the level of nearest discrete value А1 = 80.645 cm2. Than searching for the
optimum point in the design space of the variable cross-sectional areas of the
truss structural members {A3, A4}, as well as variable parameters of the truss
geometry {x3, z3, x5, z5, x7, z7, x9, z9} has been performed. As a result the
optimum truss design with structural weight 2043.907 kg has been obtained
(see 4th column of Table 2).
At the third step the purpose function gradient relative to the variable cross-
section areas of the truss structural members was {0.161319, 0.145911}, where
the component corresponded to the variable А3 had the maximum value. That
is why, the variable A3 has been further discretized at the level of nearest
discrete value А3 = 35.4838 cm2. Than searching for the optimum point in the
design space of the variable cross-sectional area of the truss structural
members {A4}, as well as variable parameters of the truss geometry {x3, z3, x5,
z5, x7, z7, x9, z9} has been performed. As a result the optimum truss design with
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structural weight 2047.485 kg has been obtained (see 5th column of Table 2).
Finally, at the fourth step the variable A4 has been further discretized at the
level of nearest discrete value А4 = 24.1935 cm2. Than searching for the
optimum point in the design space of variable parameters of the truss geometry
only {x3, z3, x5, z5, x7, z7, x9, z9} has been performed.
Table 3
Optimization results for the 18-bar cantilever truss problem
Design
variable
Rajeev &
Krishna-
moorthy
[20]
Gholizadeh [8] Cheng
et al. [2]
Farqad
et al. [7]
This
study
GA
PSO
SCPSO
TLBO
АВС
GPM
A
1
, cm
2
80.645
77.4192
77.4192
80.6450
80.6450
80.6450
A
2
,
cm
2
104.8385
119.3546
111.2901
116.1290
114.5159
114.5159
A
3
,
cm
2
5
1.6128
33.8709
40.3225
33.8709
37.0967
35.4838
A
4
, cm
2
25.8064
29.0322
30.6451
24.1935
24.1935
24.1935
x
3
, cm
2265.426
2296.111
2293.402
2322.8910
2319.013
2311.893
z
3
, cm
369.062
471.883
443.789
479.5342
466.5487
467.8591
x
5
, cm
1550.924
1638.089
1607
.091
1644.2715
1632.4943
1628.595
z
5
,
cm
300.228
368.222
358.891
380.1948
365.4859
369.6983
x
7
, cm
978.916
1087.678
1034.116
1058.7507
1045.697
1045.743
z
7
, cm
184.15
255.428
218.270
257.3833
246.7550
250.7742
x
9
, cm
468.376
532.236
502.087
518.5791
51
0.3081
511.7724
z
9
, cm
59.436
61.912
50.316
80.4215
76.7564
77.7089
Weight, kg
2094.145
2090.608
2068.894
2053.280
2057.98
2049.61
A
1
, cm
2
80.6450
83.8708
79.0321
79.0321
80.6450
80.6450
A
2
, cm
2
117.7417
117.742
117.7417
117.7417
116.1290
114.5159
A
3
,
cm
2
35.4838
35.4838
30.6451
30.6451
33.8709
35.4838
A
4
,
cm
2
24.1935
19.3548
27.4193
27.4193
24.1935
24.1935
x
3
, cm
2369.82
2319.02
2328.926
2338.862
2324.5920
2311.893
z
3
, cm
477.52
462.28
487.6063
434.1165
478.6536
467.8591
x
5
, cm
1671.32
1645.92
166
1.729
1621.885
1645.1308
1628.595
z
5
, cm
375.92
386.08
396.494
355.270
378.2959
369.6983
x
7
, cm
1071.88
1059.18
1075.69
1039.917
1058.3781
1045.743
z
7
, cm
254
261.62
260.5303
233.106
255.5695
250.7742
x
9
, cm
520.7
518.16
527.0983
504.889
517.7244
511.7
724
z
9
, cm
81.28
99.06
72.5907
74.940
79.5078
77.7089
Weight, kg
2074.859
2062.89
2058.750
2065.259
2053.798
2049.61
As a result the optimum truss design with structural weight 2049.61 kg has
been obtained (see 6th column of Table 2).
The considered geometry and cross-sectional optimization problem for 18-
bar cantilever truss has been also solved in the papers [2, 7, 8, 11, 12, 14, 16,
20] using the different optimization methods and calculation techniques such
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as genetic algorithm (GA), improved genetic algorithm (IGA), Jaya algorithm
(JA), group search optimization (GSO), particle swarm optimization (PSO),
sequential cellular particle swarm optimization (SCPSO) as well as force
method and genetic algorithm (FMGA) and teaching learning-based
optimization (TLBO). Table 3 presents the results of the performed
optimization for the 18-bar cantilever truss. As you can see, the optimum truss
design obtained using the gradient projection method is better than the cited
references [2, 7, 8, 11, 12, 14, 16, 20].
Conclusion. The results of the presented study can be formulated as follow:
1. The paper considers geometry and cross-sectional sizes optimization
problems for the truss structures formulated as nonlinear programming task
with discrete and continuous design variables.
2. The method of the objective function gradient projection onto the active
constraints surface with simultaneous correction of the constraints violations
has been used to solve the presented parametric optimization problem in the
continuous design space.
3. A discretization technique for the design variables that vary discretely has
been proposed for parametric optimization problems stated as non-linear
programming task where the purpose function and non-linear constraints of the
mathematical model are continuously differentiable functions. The
discretization of the optimal design solution obtained in the continuous space
of the design variables is performed by the purposefully selecting discrete
points around the point of the continuous optimum.
4. The comparison of the optimization results presented by the paper
demonstrates that improved gradient method together with proposed
discretization technique for the discrete design variables converges to better
solutions comparing to metaheuristic algorithms (such as genetic algorithms,
improved genetic algorithms, Jaya algorithm, group search optimization,
particle swarm optimization, sequential cellular particle swarm optimization as
well as force method and genetic algorithm and teaching learning-based
optimization).
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Стаття надійшла 21.05.2023
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Пелешко І. Д., Юрченко В. В.
ПАРАМЕТРИЧНА ОПТИМІЗАЦІЯ СТЕРЖНЕВИХ КОНСТРУКЦІЙ ЗА
НАЯВНОСТІ ДИСКРЕТНИХ ТА НЕПЕРЕРВНИХ ЗМІННИХ ПРОЄКТУВАННЯ З
ВИКОРИСТАННЯМ ПОКРАЩЕНОГО МЕТОДУ ПРОЄКЦІЇ ГРАДІЄНТА
У статті розглядається задача параметричної оптимізації стержневих конструкцій,
сформульована як задача нелінійного програмування, у якій функція м ети та нелінійні
обмеження математичної моделі є неперервно диференційованими функціями змінних
проєктування. Для розв’язку сформульованої задачі оптимізації використовується метод
проекції градієнта функції мети на поверхню активних обмежень з одночасною ліквідацією
нев’язок у порушених обмеженнях. Запропонована методика дискретизації змінних
проєктування, які повинні варіювати дискретно. Дискретизація оптимального розв’язку,
отриманого у неперервному просторі змінних проєктування, виконується за допомогою
цілеспрямованого відбору дискретних точ ок в околі точки неперервного оптимуму. У статті
представлено порівняння результатів оптимізаційних розрахунків, яке засвідчило, що
покращений метод проєкції градієнта разом із запропонованою методикою дискретизації
змінних проєктування, що повинні варіювати дискретно, забезпечує з біжність до кращого
розв’язку задачі порівняно до мета-еврістичних алгоритмів.
Ключові слова: оптимізація форми, стержневі конструкції, нелінійне програмування,
нормативні обмеження, метод п роекції градієнта, програмне забезпечення, метод
скінченних елементів.
Peleshko I.D., Yurchenko V.V.
PARAMETRIC OPTIMIZATION OF BAR STRUCTURES WITH DISCRETE AND
CONTINUOUS DESIGN VARIABLES USING IMPROVED GRADIENT PROJECTION
METHOD
The paper considers a parametric optimization problem for the bar structures formulated as
nonlinear programming task, where the purpose function and non-linear constraints of the
mathematical model are continuously differentiable functions of the design variables. The method
of the objective function gradient projection onto the active constraints surface with simultaneou s
correction of the constraints violations has been u sed to solve the parametric optimization
problem. A discretization technique for the design variables that should vary discretely has been
proposed. The discretization of the optimal design solution obtained in the continuous space of the
design variables is performed by the purposefully selecting discrete points around the point of the
continuous optimum. The comparison of the optimization results presented by the paper
demonstrates that improved gradient method together with proposed discretization technique for
the discrete design variables converges to better solutions of the problem comparing to the meta-
heuristic algorithms.
Keywords: shape optimization, bar structures, nonlinear programming, design code
constraints, gradient projection method, optimization software, finite element method.
Пелешко И. Д., Юрченко В. В.
ПАРАМЕТРИЧЕСКАЯ ОПТИМИЗАЦИЯ СТЕРЖНЕВЫХ КОНСТРУКЦИЙ ПРИ
НАЛИЧИИ ДИСКРЕТНЫХ И НЕПРЕРЫВНЫХ ПЕРЕМЕННЫХ
ПРОЕКТИРОВАНИЯ С ИСПОЛЬЗОВАНИЕМ УЛУЧШЕННОГО МЕТОДА
ПРОЕКЦИИ ГРАДИЕНТА
В статье рассматривается задача параметрической оптимиза ции стержневых
конструкций, сформулированная как задача нелинейного программирования, в которой
функция цели и нелинейные ограничения математической модели являются непрерывно
дифференцируемыми функциями переменных проектирования. Для решения
сформулированной задачи оптимизации используется метод проекции градиента функции
цели на поверхность активных ограничений при одновременной ликвидации невязок в
нарушенных ограничениях. П редложена методика дискретизации переменных
проектирования, которые должны варьировать дискретно. Дискретизация оптимального
решения, полученного в непрерывном пространстве п еременных проектирования,
выполняется при помощи целенаправленного отбора дискретных точек в окрестности точки
непрерывного оптимума. В статье представлено сравнени е результатов оптимизационных
расчетов, показывающее, что улучшенный метод проекции градиента вместе с
предложенной методикой дискретизации переменных проектирования, которые должны
варьировать дискретно, об еспечивает сходимость к лучшему решению задачи по сравнению
с мета-эвристическими алг оритмами.
ISSN2410-2547
Опір матеріалів і теорія споруд/Strength of Materials and Theory of Structures. 2023. № 110
197
Ключевые слова: оптимизация формы, стержневые конструкции, нелинейно е
программирование, нормативные ограничения, метод проекции градиента, программное
обеспечение, метод конечных элементов.
УДК 624.04.012.4.044, 519.853
Пелешко І.Д., Юрченко В.В. Параметрична оптимізація стержневих конструкцій за
наявності дискретних та неперервних змінних проєктування з використанням
покращеного методу проєкціі градієнта // Опір матеріалів і теорія споруд: наук.-тех.
збірн. – К.: КНУБА, 2023. – Вип. 110. – С. 178 – 198.
У статті розглядається задача параметричної оптимізації стержневих конструкцій,
сформульована як задача нелінійного програмування, у якій функція мети та нелінійні
обмеження математичної моделі є неперервно диференційованими функціями змінних
проєктування. Для розв’язку сформульованої задачі оптимізації використовується метод
проекції градієнта функції мети на поверхню активних обмежень з одночасною ліквідацією
нев’язок у порушених обмеженнях. Запропонована методика дискретизації змінних
проєктування, які повинні варіювати дискретно. Представлено порівняння результатів
оптимізаційних розрахунків, яке засвідчило, що покращений метод проєкції градієнта
разом із запропонованою методикою дискретизації змінних проєктування, що повинні
варіювати дискретно, забезпечує збіжність до кращого розв’язку задачі порівняно до
мета-еврістичних алгоритмів.
Табл. 3. Іл. 3. Бібліог. 24 назв.
UDC 624.04.012.4.044, 519.853
Peleshko I. D., Yurchenko V. V. Parametric optimization of bar structures with discrete and
continuous design variables using improved gradient projection method // Strength of
Materials and Theory of Structures: Scientific-and-technical collected articles – Kyiv: KNUBA,
2023. – Issue 110. – P. 178-198.
The paper considers a parametric optimization problem for the bar structures formulated as
nonlinear programming task, where the purpose function and non-linear constraints of the
mathematical model are continuously differentiable functions of the design variables. The method
of the objective function gradient projection onto the active constraints surface with simultaneous
correction of the constraints violations has been used to solve the parametric optimization
problem. A discretization technique for the design variables that should vary discretely has been
proposed. Presented comparison of the optimization results demonstrates that improved gradient
method together with proposed discretization technique for the discrete design variables
converges to better solutions of the problem comparing to the meta-heuristic algorithms.
Tabl. 3. Figs. 3. Refs. 24.
УДК 624.04.012.4.044, 519.853
Пелешко И. Д., Юрченко В. В. Параметрическая оптимизация стержневых конструкций
при наличии дискретных и непрерывных переменных проектирования с
использованием улучшенного метода проекции градиента // Сопротивление материалов
и теория сооружений: науч.- тех. сборн. – К.: КНУСА, 2023. – Вып. 110. – С. 178-198.
В статье рассматривается задача параметрической оптимизации стержневых
конструкций, сформулированная как задача нелинейного программирования, в которой
функция цели и нелинейные ограничения математической модели являются непрерывно
дифференцируемыми функциями переменных проектирования. Для решения
сформулированной задачи оптимизации используется метод проекции градиента функции
цели на поверхность активных ограничений при одновременной ликвидации невязок в
нарушенных ограничениях. Предложена методика дискретизации переменных
проектирования, которые должны варьировать дискретно. Представлено сравнение
результатов оптимизационных расчетов, показывающее, что улучшенный метод проекции
градиента вместе с предложенной методикой дискретизации переменных
проектирования, которые должны варьировать дискретно, обеспечивает сходимость к
лучшему решению задачи по сравнению с мета-эвристическими алгоритмами.
Табл. 3. Ил. 3. Библиог. 24 назв.
ISSN2410-2547
Опір матеріалів і теорія споруд/Strength of Materials and Theory of Structures. 2023. № 110
198
Автор: кандидат технічних наук, доцент кафедри будівельного виробництва Пелешко Іван
Дмитрович
Адреса робоча: 79013 Україна, м. Львів, вул. Ст. Бандери 12, Національний університет
«Львівська політехніка»
Робочий тел.: +38 (032) 258-25-41
Мобільний тел.: +38(098)41-57-517
E-mail: ipeleshko@polynet.lviv.ua
SCOPUS ID: 25637832500
ORCID ID: https://orcid.org/0000-0001-7028-9653
Автор: доктор технічних наук, професор кафедри металевих та дерев’яних конструкцій
Юрченко Віталіна Віталіївна
Адреса робоча: 03680 Україна, м. Київ, Повітрофлотський пр. 31, Київський національний
університет будівництва і архітектури
Робочий тел.: +38(044)249-71-91
Мобільний тел.: +38(063)89-26-491
E-mail: vitalina@scadsoft.com
SCOPUS ID: 25637856200
ORCID ID: https://orcid.org/0000-0003-4513-809X
