CC BY
17Magazine of Civil Engineering. 2020. 98(6). Article No. 9812
Magazine of Civil Engineering issn
2712-8172
journal homepage: http://engstroy.spbstu.ru/
DOI: 10.18720/MCE.98.12
Improved gradient projection method for parametric optimisation of bar structures
V.V. Yurchenko*a, I.D. Peleshkob
a Kyiv National University of Construction and Architecture, Kyiv, Ukraine b Lviv Polytechnic National University, Lviv, Ukraine * E-mail: vitalinay@rambler.ru
Keywords: parametric optimization, bar structures, nonlinear programming, gradient projection method, finite element analysis
Abstract. Numerical optimization and the finite element method have been developed together to make possible the emergence of structural optimization as a potential design tool. The main research goal of this paper is the development of mathematical support and a numerical algorithm to solve parametric optimization problems of structures with orientation on software implementation in a computer-aided design system. The paper considers parametric optimization problems for bar structures formulated as nonlinear programming ones. 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. Equivalent Householder transformations of the resolving equations of the method have been proposed. They increase numerical efficiency of the algorithm developed based on the considered method. Additionally, proposed improvement for the gradient projection method also consists of equivalent Givens transformations of the resolving equations. They ensure acceleration of the iterative searching process in the specified cases described by the paper due to decreasing the amount of calculations. The comparison of the optimization results of truss structures presented by the paper confirms the validity of the optimum solutions obtained using proposed improvement of the gradient projection method. The efficiency of the proposed improvement of the gradient projection method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints.
1. Introduction
Over the past 50 years, numerical optimization and the 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 [1]. 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 [2].
Applied optimum design problems for bar structures in some cases are formulated as parametric optimization problems, namely as searching problems for unknown structural parameters, which provide an extreme value of the specified objective function in the feasible region defined by the specified constraints. In this case, structural optimization is performed 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.
Kibkalo et al. in the paper [3] formulated a parametric optimization problem for thin-walled bar structures and considered methods to solve them. The searching for the optimum solution has been performed by varying the structural parameters providing the required load-carrying capacity of structural members and the minimum value of manufacturing costs.
Alekseytsev has described the process of developing a parametrical-optimization algorithm for steel trusses in the paper [4]. Parametric optimization has been performed taking into account strength, stability and stiffness constraints formulated for all truss members. The objective function has been formulated depending
Yurchenko, V.V., Peleshko, I.D. Improved gradient projection method for parametric optimisation of bar structures. Magazine of Civil Engineering. 2020. 98(6). Article No. 9812. DOI: 10.18720/MCE.98.12
This work is licensed under a CC BY-NC 4.0
Magazine of Civil Engineering, 98(6), 2020
on the specific manufacturing of the truss panel joints in term of the manufacturing cost calculated based on the labor costs and materials used.
Serpik et al. in the paper [5] developed an algorithm for parametric optimization of steel flat rod systems. The optimization problem has been formulated as a structural weight minimization problem taking into account strength and displacement constraints, as well as overall stability constraints. The cross-sectional dimensions of the truss members and the coordinates of the truss panel joints have been considered as design variables. The structural analysis of internal forces and displacements for considered structures has been performed using the finite element method. An iterative procedure for searching for optimum solution has been proposed in [6].
Sergeyev et al. in the paper [7] formulated a parametric optimization problem with constraints on faultless operation probability of bar structures with random defects. The weight of the bar structures has been considered as the objective function. Initial global imperfections have been considered as small independent random variables distributed according to normal distribution law, as well as buckling load value has been also considered as a random variable.
The mathematical model of the parametric optimization problem of structures includes a set of design variables, an objective function, as well as constraints, which reflect generally non-linear dependences between them [8]. If the objective function and constraints of the mathematical model are continuously differentiable functions, as well as the search space is smooth, then the parametric optimization problems are successfully solved using gradient projection non-linear methods [9]. The gradient projection methods operate with the first derivatives or gradients only both of the objective function and constraints. The methods are based on the iterative construction of such a sequence of the approximations of design variables that provides convergence to the optimum solution (optimum values of the structural parameters) [10].
Additionally, a sensitivity analysis is a useful optional feature that could be used in scope of the numerical algorithms developed based on the gradients methods [11]. Thus, in the paper [12] Sergeyev et al. formulated a parametric optimization problem of linearly elastic space frame structures taking into account the stress and multiple natural frequency constraints. The cross-sectional parameters of structural members as well as node positions of the bar structures has been considered as design variables. The sensitivity analysis of multiple frequencies has been performed using analytic differentiation with respect to the design variables. The optimal design of the structure has been obtained by solving a sequence of quadratic programming problems.
Although many papers are published on the parametric optimization of structures, the development of a general computer program for the design and optimization of structures according to specified design codes remains an actual task. Therefore, the main research goal of this paper is the development of mathematical support and a numerical algorithm to solve parametric optimization problems of structures with orientation on software implementation in a computer-aided design system.
One of the effective methods to solve parametric optimization problems for structures is gradient projection methods, as shown by the review of scientific researches presented above. That is why, in this paper, a gradient projection method is considered as investigated object. The following research tasks are formulated: to propose an improvement of the gradient projection method that ensures the increase of the numerical efficiency of the algorithm developed based on the considered method, as well as the acceleration of the iterative searching process due to decreasing the amount of calculations.
Let us consider a parametric optimization problem of a structure consisting of bar members. It can be formulated as presented below: to find optimum values for geometrical parameters of the structure, bar's cross-section dimensions and initial pre-stressing forces introduced into the redundant members of the bar system, which 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
^ t -
following mathematical terms: to find unknown structural parameters X = {Xt} , i = 1,Nx , providing the least value of the determined objective function:
2. Methods
2.1. Parametric optimization problem formulation
(1.1)
in a feasible region (search space) 3 defined by the following system of constraints:
y (x) = {Wk(X) = 0 | k = 1^}; (1.2)
9 ( X) = {cf>v( X )< 0 | n = Nec +1, Njc } ; (1.3)
where X is the vector of the design variables (unknown structural parameters); f, y/K , p^ are the
^ *
continuous functions of the vector argument; X is the optimum solution or optimum point (the vector of
*
optimum values of the structural parameters); f is the optimum value of the optimum criterion (objective function); Nec is the number of constraints-equalities y/K('X), whose define hyperplanes of the feasible
solutions; Njc is the number of constraints-inequalities pv (X) , whose define a feasible region in the design space 3.
The vector of the design variables can consist of a set of unknown geometrical parameters of the structure, a set of unknown cross-sectional dimensions of the structural members, as well as a set of unknown initial pre-stressing forces introduced into the specified redundant members of the structure.
The specific technical-and-economic index (material weight, material cost, construction cost etc.) or another determined indicator can be considered as the objective function Eq. (1.1) taking into account the
ability to formulate its analytical expression as a function of design variables X .
Load-bearing capacities constraints (strength and stability inequalities) for all design sections of the structural members subjected to all design load 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 combinations at the serviceability limit state should be included into the system of constraints Eqs. (1.2) - (1.3). The design internal forces in the bar structural members used in the strength and stability inequalities of the system Eqs. (1.2) - (1.3) are considered as state variables depending on design variables
X and can be calculated from the linear equations system of the finite element method [13]. The node displacement of the bar system used in stiffness inequalities of the system Eqs. (1.2) - (1.3) are also
considered as state variables depending on design variables X and can be also calculated from the linear equations system of the finite element method [13]. Additional requirements, whose describe structural, technological and serviceability particularities of the considered structure, as well as constraints on the building functional volume can be also included into the system Eqs. (1.2) - (1.3).
2.2. An improved gradient projection method for solving the parametric optimization problem
The parametric optimization problem stated as non-linear programming task by Eqs. (1.1) - (1.3) can be solved using a gradient projection method. 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 occurred when optimum designing of the structures [14].
The gradient projection method operates with the first derivatives or gradients only of both the objective function Eq. (1.1) and constraints Eqs. (1.2) - (1.3). The method is based on the iterative construction of such
^ t -
sequence Eq. (2.1) of the approximations of design variables X = {Xl} , i = 1,Nx , that provides the convergence to the optimum solution (optimum values of the structural parameters):
Xk+1 = Xk +AXk , (2.1)
f \T --*
where Xk = {Xt} , i = 1, Nx is the current approximation to the optimum solution X that satisfies both
constraints-equalities Eq. (1.2) and constraints-inequalities Eq. (1.3) with the extreme value of the objective function Eq. (1.1); AXk = {AXl} , i = 1,Nx , is the increment vector for the current values of the design
variables Xk ; k is the iteration's index. The start point of the iterative searching process Xk=0 can be assigned as engineering estimation of the admissible design of the structure.
The active constraints only of constraints system Eqs. (1.2) - (1.3) should be considered at each iteration.
Magazine of Civil Engineering, 98(6), 2020
A set of active constraints numbers A calculated for the current approximation Xk to the optimum solution (current design of the structure) is determined as:
A = kun, k = {k \yK(xt)|>-s}, n = {n£C +n | $n{Xt)"-£} . (22)
where s is a small positive number introduced here in order to diminish the oscillations on movement alongside of the active constraints surface.
The increment vector AXf for the current values of the design variables Xk can be determined by the following equation:
AXk =AXk1_ + AXf, (2.3)
k k where AXk is the vector calculated subject to the condition of elimination the constraint's violations; AXf is
the vector determined taking into consideration the improvement of the objective function value. Vectors AXf
and AXk are directed parallel and perpendicularly accordingly to the subspace with the vectors basis of the
linear-independent constraint's gradients, such that:
T
(AXk) AXf = 0. (2.4)
The values of the constraint's violations for the current approximation Xk of the design variables are accumulated into the following vector:
V = [Wk{ X )vkg k ; (t>v( X )Vne n ) .
Let us introduce a set L , L e A , of the constraint's numbers, such that the gradients of the constraints
at the current approximation Xk to the optimum solution are linear-independent.
Component AXk is calculated from the equation presented below:
AXk = [V^k. (25)
r i dw -
where [V^J is the matrix that consists of components —— and —- , here i = 1,Nx , kgL , n e L ;
dXi dXi
fik is the column-vector that defines the design variables increment subject to the condition of elimination the constraint's violations. Vector ^ can be calculated as presented below.
In order to correct constraint's violations V, vector AXk to a first approximation should also satisfy Taylor's theorem for the continuously differentiable multivariable function in the vicinity of point Xf for each constraint from set L , namely:
-V = [V^JT AXk . (2.6)
With substitution of Eq. (2.5) into Eq. (2.6) we obtain the system of equations to determine column-vector fik:
[V^f [Vv]nk=-V . (2.7)
Component AXf is determined using the following equation:
AXf =£x p = {(Vf-[Vp]^|), (2.8)
where Vf is the vector of the objective function gradient in the current point (current approximation of the design variables) Xf ; p is the projection of the objective function gradient vector onto the active constraints surface in the current point Xf ; ^ is the column-vector that defines the design variable's increment subject
to the improvement of the objective function value. Column-vector can be calculated approximately using the least-square method by the following equation:
[V^H«V/, (2.9)
or from the equation presented below:
[Vpf^v^^vvfvf; (2.10)
where j is the step parameter, which can be calculated subject to the desired increment Af of the objective function on movement along the direction of the objective function anti-gradient. The increment Af can be assign as 5...25% from the current value of the objective function f (Xt):
<T _ - Af
(2.11)
A/ = f(vf ) Vf, £
(v/) V/
where in case of minimization Eq. (1.1) Af and j accordingly have negative values. The parameter j can be also calculated using the dependency presented below:
, = Af
j=WV- • <2'12)
that follows from the condition of attainment the desired increment of the objective function Af on the
movement along the direction of the objective function anti-gradient projection onto the active constraints surface. Step parameter j can be also selected as a result of numerical experiments performed for each type
of the structure individually [15, 16].
Fig. 2.1 presents a graphical illustration for step to the point Xk+1 depending on location of the current approximation X k in the two-dimension search space.
Using Eq. (2.5) and Eq. (2.8), Eq. (2.3) can be rewritten as presented below:
AXk =[W] li1+j{vf-[Vq>] ¿||), (2.13)
or
AXk =j vf + [Vv](£±-j £,,) , (2.14)
-» -»
where column-vectors and ^ are calculated using Eq. (2.7) and Eq. (2.9) or Eq. (2.10), respectively.
. objcctivc function / level curves
Figure 2.1. Step to the next point Xk+1 depending on location of the current approximation Xk for
two design variables Xj and X2 .
The linear-independent constraints of the system Eqs. (1.2) - (1.3) should be detected when
constructing the matrix of the active constraints gradients [V^] used by Eq. (2.7) and Eq. (2.9) or Eq. (2.10).
Selection of the linear-independent constraints can be performed based on the equivalent transformations of the resolving equations of the gradient projection method using the non-degenerate transformation matrix
H, such that the sub-diagonal elements of the matrix H [V^] equal to zero. Besides,
Magazine of Civil Engineering, 98(6), 2020
HT H = I ; (2.15)
H = H,x...xH,-x...xH2xH1; (2.16)
where I is the unit matrix; t is the total number of the linear-independent gradients of the active constraints, H is the transformation matrix, such that HT H, = I , at the same time the sub-diagonal element are equal
to zero in matrix H, xH;-ix...xH2 xH^[Vp] for column's numbers 1, i . Described conditions are satisfied by the orthogonal matrix of the elementary mapping (Householder's transformation) [17, 18].
Let us present here the following algorithm to form set L and to construct matrix H [Vp].
1. i = 0, L = 0 and [VO]o = [Vp] should be assumed, where [Vp] is the matrix that comprises
from the column-gradients of all active constraints. All columns of matrix [VO]o should be marked as 'not used' (or linear-independent).
2. i = i +1.
3. Among all 'not used' columns of matrix [VO], , whose correspond to the constraints-equalities Eq. (1.2), one jth column with extreme value of the specified criterion should be selected (e.g., the following
Nx 9X9
criterion Ij = £ gkj can be considered as such criterion, where gj are the jth column's components of
k=i
matrix [VO]i-i ). At the same time, all kth columns of matrix [VO], , for whose the following inequality
£k <Si met, should be marked as 'used', here Si is a small positive number. In case when no constraints-equalities exist or all constraints-equalities Eq. (1.2) are marked as 'used', the selection of jth column should
be performed among all 'not used' columns of matrix [VO],— , whose correspond to the constraints-
inequalities Eq. (1.3). If l2j < s1 , then generation of set L and matrix H[Vp] is finished.
H[Vp] = [Vp],,— . In case of < s1 and i = 1 (i. e. L = 0 ), there is a contradiction in the system of constraints Eqs. (1.2) - (1.3). In other case, moving to the next step performs.
4. kth number of the constraint, that corresponds to the jth column number, should be included into set L , L ^ L + {k}.
5. Calculate [VO], = H, [VO], —. It is reasonable to execute the multiplication only for 'not used'
columns. It should be noted, when using Householder's transformation matrix H, is not constructed evidently [18]. At the same time, matrix [VO], can be constructed within the ranges of matrix [VO],— when no additional memory is needed.
6. If , = 1, then [Vp], = qj , where qj is jth column-vector of matrix [VO],. When , > 1 [Vp], is
constructed using extension of the matrix [Vp], — by the column-vector qj. jth column of matrix [VO], is selected as 'used', then moving to the step 2 performs.
Using Householder's transformations described above triangular structure of the nonzero elements of matrix H [Vp] is formed step-by-step. Besides, Eq. (2.7) and Eq. (2.9) can be rewritten as follow:
([Vp]T HT )( H[Vp])y51 = -V ; (2.17)
H[Vp]^, « HV/ . (2.18)
In order to calculate column-vectors /k and /, it is required only to perform forward and backward substitutions in Eq. (2.17) and Eq. (2.18).
To accelerate the convergence of the minimization algorithm presented above, hth columns should be excluded from matrix H[V^J. These columns correspond to those constraints from Eq. (1.3), for which the following inequality satisfies:
/kh-Zx/\h > 0. (2.19)
Actually, if /kh - %2/\\h > 0 , then the return onto the active constraints surface from the feasible region
3 is performed with simultaneous degradation of the objective function value (see Fig. 2.2, b). At the same time, in case of:
/kh -^i/\\h < 0 , (2.20) both the improvement of the objective function value and the return from the inadmissible region onto the active constraints surface are performed (see Fig. 2.2, a).
a b
Figure 2.2. The selection of the constraints-inequalities:
a - vih - < 0 ; b - ^ih- > 0.
When excluding h th columns from matrix H [Vp] corresponded to those constraints for whose
Eq. (2.13) is satisfied, the matrix (H[Vp])^ with a broken (non-triangular) structure of the non-zero
elements is obtained. The set L of the linear-independent active constraints numbers transforms into the set L d respectively. At the same time, the vector of the constraint's violations V reduced into the vector V
red
accordingly.
In order to restore triangular structure of the matrix ( H [Vp])^ with zero sub-diagonal elements, Givens transformations (Givens rotations) [19, 18] can be used. Givens transformations for the matrix (H [Vp])^ consist of construction such square matrix Gwz , for which corresponded wzth element of
matrix Gwz (H [Vp]) returns zero (see Fig. 2.3) [20]. Since c2 + s2 = 1 by definition, it follows:
( G wz ) G1
(2.21)
An obvious method to calculate c and s for dth non-zero sub-diagonal element and for ath diagonal
element is presented below:
c =
a
yla2 + d2
s =
d
Va2 + d2
(2.22)
G « (H [V^]L G - (H [V^IL
Figure 2.3. Scheme for Givens rotations (non-zero elements of the matrixes are hatched).
Magazine of Civil Engineering, 98(6), 2020
The Givens matrix G can be calculated similarly to the matrix H using the following equation:
G = G^x...xGf x...xG2 xGj. (2.23)
where y is the number of the Givens transformations. So, Givens transformations should be executed several
times (with different values z and w), while the matrix Gwz (H [V^]) has no all zero sub-diagonal elements (for example presented by Fig. 2.3, y = 5).
Taking into account Givens transformations, Eq. (2.17) and Eq. (2.18) for column-vectors , and
(^11 )red
'red
can be rewritten as:
([^ HT )red GTG(H [V^])red (^_)red = -Vd = (2.24)
G (H [V^])red fa )red ^ GHV? ; (2.25)
and the main resolving equation of the gradient projection method Eq. (2.13) and Eq. (2.14) can be rewritten as presented below:
^ = (H [V^])red (^red + 4V/ -(H [V^])red fa )red ) , <2.2«)
'red
or
AXk = {Vf + (H [V^])red ((¿^ed-^fa||)red ) ' (227)
The proposed improvement for the method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations consists of equivalent transformations of the resolving equations using Householder transformations. The transformations with matrix H presented by Eq. (2.24) and Eq. (2.25) of the resolving equations of the gradient projection method Eq. (2.7) and Eq. (2.9) increase the numerical efficiency of the algorithm developed based on the gradient projection method described above.
Additionally, the proposed improvement for the gradient projection method includes equivalent transformations of the resolving equations using Givens rotations. The transformations with matrix G presented by Eq. (2.24) and Eq. (2.25) ensure acceleration of the iterative searching process Eq. (2.1) in case when Eq. (2.19) takes into account due to decreasing the amount of calculations.
It should be noted that the lengths of the gradient vectors for the objective function Eq. (1.1), as well as for constraints Eqs. (1.2) - (1.3), remain as they were in scope of the proposed equivalent transformations ensuring the dependability of the optimization algorithm.
The determination the convergence criterion is the final question when using the iterative searching for the optimum point Eq. (2.1) described above. Considering the geometrical content of the gradient steepest
descent method, we can assume that at the permissible point Xk the component of the increment vector Axf for the design variables should be vanish, AXj| ^ 0, in case of approximation to the optimum solution
of the non-linear programming task presented by Eqs. (1.1) - (1.3). So, the following convergence criterion of the iterative procedure Eq. (2.1) can be assigned:
AXlk
where is a small positive number.
2 , (2.28)
Taking into consideration Eq. (2.28), let us formulate the following stop criteria in the iterative searching procedure of Eq. (2.1).
Stop criterion 1: when the objective function gradient in the current approximation Xk is close to zero indicating on extreme character of the current approximation, as well as there are no violated constraints:
" L = 0,
- (2.29)
-s>Vf >
where L is the set of the violated constraints numbers, L =
Vs (Xk) >e-' ts (Xk )>e};
Stop criterion 2: when the projection of the objective function gradient in the current approximation Xk
onto the active constraints surface is close to zero (objective function gradient is perpendicular to the active constraints surface) indicating impossible further improvement of the objective function value, as well as there are no violated constraints:
- L = 0;
(2.30)
-s > p > +s;
Stop criterion 3: when in the current approximation Xk of the iterative searching procedure Eq. (2.3) the total number of the active constraints t equals to the number of design variables Nx , as well as all active constraints are s -active (both not violated constraints and those ones for whose inequality Eq. (2.13) met):
" L = 0;
t = Nx ; (2.31)
f -£xV\\f < ° Vf e L.
This stop criterion for the iteration process Eq. (2.1) corresponds to the case when the current approximation Xk =(X^) ,i = 1,Nx , to the optimum solution is located at the intersection of the
constraints (i.e., vertex). In this case, no correction of the constraints violations is needed and further improvement of the objective function value is not possible.
Stop criterion 4: when the objective function values within two consecutive iterations are the same with acceptable accuracy subject to the absence of the violated constraints:
" L = 0;
(2.32)
f ( Xk-1)« f ( Xk ).
3. Results and Discussion
In order to estimate an efficiency of the new methods 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 [15] and Crowder et al. [21]. 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 method of objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations presented above will be based on the comparison of the optimization results obtained using proposed improvement of the gradient projection method, 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.
3.1. Parametric optimization of a three-bar truss
Optimization of a three-bar truss (see Fig. 3.1) has been firstly solved by Schmit [22] using a non-linear programming method. Besides, the task has been also considered by Haug and Arora [15]. The parametric optimization problem is formulated as searching for optimum cross-sectional areas b , ^2 and b3 of the truss
bars providing the least value of the truss weight subject to normal stresses and flexural stability constraints, as well as displacements and eigenvalue constraints. The load cases for the considered truss are presented in Table 3.1. Initial data for optimization of the truss are shown in Table 3.2.
Table 3.1. Load cases for considered truss.
Load case j 1 2 3
Oj, ' 45 90 135
Pj x103 [pound-force] 40 30 20
Pj [kN] 177.9897 133.4922 88.9948
Figure 3.1. Three-bar truss.
Table 3.2. Initial data for optimization of the truss.
Unit weight of the truss material pg 0.1 pound/inch3 = 0.027154 N/cm3
Modulus of elasticity E 107 pound/inch2=6.8971*l06 N/cm2
Allowable stresses value cr^for the 1st and 3rd truss members 5000 pound-force/inch2=3.4486 kN/cm2
Allowable stresses value c2[ for the 2nd truss member 2000 pound-force/inch2=1.3794 kN/cm2
Non-dimensional factor /3 used to calculate second moment area of 1.0
inertia for each truss member, Ii = /b7
Ultimate vertical za and horizontal xa displacements of the truss nodes 0.005 inch=0.0127 cm
Lower limit value for eigenvalue Z0 = 1.872 108
The objective function can be written as presented below:
^ min;
¥0 = pgi ( +b2 + b3V2 )
(3.1)
where b , ^ and b3 are cross-sectional areas of the truss bars; l is the truss height, l = 25.4 cm (see Fig. 3.1). Let us formulate strength constraints for each truss members for all load cases as follows:
¥
Nj
3(i -1) + j
u * a
-1 < 0 ;
(3.2)
where N { is the axial force for 7 th truss member subjected to jth load case, i = 1,3 , j = 1,3 . Besides, let include to the system of constraints the inequalities for the positive values of the design variables:
y/9+i = -bi < 0 ; (3.3)
Flexural buckling constraints for all truss members can be written using Hooke's law as presented
below:
¥12+3( i-1)+j
( xj+ zJ ) l
n2ßbl
-1 < 0 ;
(3.4)
where x4, z4 are linear displacements for node 4 of the truss subjected to jth load case along the directions
of 0 x and 0 z axes respectively. The constraints on the minimum values of the eigenvalues can be written analytically using calculation results of the eigenvalues stability problem for the considered truss:
i42p i Z0
fh + b3 - ^
b
-V2+1
¥22 =
f f b - b3 2 1
3E b + b3 +yf2 -. + 2
b2 V I b2 J
-1 < 0 .
(3.5)
Let also formulate displacements constraints for 4th truss node in the plane x0z :
w 22+j = -i --a- * 0 ; (36) xa
W25+j = xa -1 *0 ; (37)
W28+j =-1 - 4 *0 ; (38) z
W3i+j =4 -1 * 0. (39)
z
Starting from start values of the design variables b0 = (64.516, 32.258 , 32.258) 7 cm2 with truss
weight G0 = 116.602 N, an optimum solution b* =(57.4878, 12.4482, 27.4299)7 cm2 with the optimum
*
weight G = 91.383 N has been obtained. The comparison of the optimization results for the considered three-bar truss obtained by Haug and Arora [15] and in this article is presented in Table 3.3. Good correlation of obtained optimization results with the results of the other authors confirms the validity of the optimum solutions calculated using the proposed improvement of the gradient projection method.
The step-by-step characteristics of the iterative searching for optimum design of the three-bar truss are presented in Table 3.4. Eleven iterations have been performed. The iterative searching process for the optimum point was stopped due to the following stop criterion: increment of the design variables within two consecutive iterations was less than 0.0001, whereas there were no violated constraints.
Table 3.3. Comparison of the optimization results for the three-bar truss.
Start and optimum cross-section areas [cm2] for truss member
Source ---- Truss weight [N]
1 2 3
Start values by Haug and Arora [15] 64.516 32.258 32.258 116.602
Optimum values by Haug and Arora [15] 59.2257 13.9355 24.8387 91.588
Optimum values by this paper 57.4878 12.4482 27.4299 91.383
*truss member's numbers are indicated in Fig. 3.1
Table 3.4. Step-by-step characteristics of the iterative searching for optimum design of the three-bar truss.
C 1- o <u ~ -CI Current values of the design variables [cm2] Objective Numbers of the active constraints Maximum
2 E <D ^ bi b2 b3 function value [kN] violation of the constraints
0 64.5160 32.2580 32.2580 0.1166 - -
1 44.5160 22.2580 22.2580 0.0805 15 0.35835
2 60.7847 12.2580 12.2580 0.0797 10, 15, 18, 22, 24, 33, 34 0.46276
3 40.7847 14.8577 22.2580 0.0717 15, 18, 22, 26 0.42237
4 55.5763 15.8613 21.5250 0.0861 15, 22 0.10076
5 57.3934 13.0115 26.3367 0.0906 15, 22 0.01189
6 57.5871 12.5536 27.2368 0.0914 15, 22 0.00027
7 57.4967 12.4569 27.4140 0.0914 15, 22 9.67-10"6
8 57.4885 12.4489 27.4288 0.0914 15, 22 7.55-10"8
9 57.4878 12.4483 27.4299 0.0914 15, 22 1.43-10"9
10 57.4878 12.4483 27.4299 0.0914 15, 22 8.74-10"11
11 57.4878 12.4483 27.4299 0.0914 15, 22 6.55-10"12
3.2. Optimization of a ten-bar cantilever truss
A parametric optimization problem of a ten-bar cantilever truss (see Fig. 3.2) is widely used in the literature [15, 23, 24, 25] in order to compare different methods for solving optimization problems. The
Magazine of Civil Engineering, 98(6), 2020
parametric optimization problem is formulated as follows: to find unknown cross-sectional areas for each truss member b = (bi) , i = 1,10, with weight minimization of the truss subjected to stresses constraints in all truss bars, node displacements constraints, as well as constraints on the minimal cross-section areas. A start value b0 = 1.0 inch2=6.4516 cm2 was used as a start approximation for variable cross-sections areas for all bars of the considered truss.
The considered truss undergoes two load cases, as shown in Fig. 3.2 and presented in Table 3.5. Initial data for optimization of the truss are presented in Table 3.6.
Table 3.5. Load cases for ten-bar cantilever truss.
Load case Node number (Fig. 3.2) Point load along axis Point load along axis
number 0z , x 103 [pound-force] 0z [kN]
1 2 -100.0 -444.82
4 -100.0 -444.82
1 50.0 222.41
2 2 -150.0 -667.23
3 50.0 222.41
4 -150.0 -667.23
Figure 3.2. Ten-bar cantilever truss. Table 3.6. Initial data for optimization of the ten-bar truss.
Unit weight of the truss material pg
Modulus of elasticity E Non-dimensional factor /3 used to calculate second moment area of
2
inertia for each truss member, It = Lower limit value bL for cross-sectional areas for all truss bars Allowable stresses value for the all truss member ca _Ultimate vertical za displacements of the truss nodes_
0.1 pound/inch3=0.027143 N/cm3 107 pound/inch2=6.8948*106 N/cm2
1.0
0.10 inch2=0.64516 cm2 25000 pound-force/inch2=17.236 kN/cm2 ±2 inch= ± 5.08cm
Variable cross-section areas for each truss member b = (bi) , i = 1, 10, were considered as design variables. The objective function can be written as presented below:
if/0 = pgl( b1 + b2 + b3 + b4 + b5 + b6 +V2 (by + b8 + b 9 +b10 )) ^ min ; (3.10)
where l is the truss height, l = 914.4 cm (see Fig. 3.2). Constraints on lower limit value for variable cross-sectional areas for all truss bars are written as follows:
Wl = 1 - ^ < 0 . Stresses constraints can be formulated as presented below:
Kl
w10+i=f-a -1 < 0.
(3.11)
(3.12)
where N7 is the axial force in the ith truss member. Displacement constraints for the truss nodes are written as follows:
zj
^20+j =-1 --a < 0;
V24+j = -a -1 <0; z
where Xj, zj are linear displacements of jth truss node, j = 1,4.
(3.13)
(3.14)
Starting from the initial truss design with start weight G0 = 1.867 kN, an optimal solution with the
*
optimum weight of G = 22.514 kN has been obtained for the truss subjected to the first load case.
Additionally, starting from the initial truss design with the start weight of G0 = 1.867 kN, an optimal solution
*
with the optimum weight of G = 20.806 kN has been obtained for the truss subjected to the second load case. For both loaded cases, the iterative searching process for the optimum point was stopped due to the following stop criterion: the increment of the design variables within two consecutive iterations was less than 0.0001, as well as there were no violated constraints.
Table 3.7. Comparison of the optimization results for the ten-bar cantilever truss.
Bar Design variable Start values for . Optimal cross-section area for 1th truss member [cm2]
number, design for the first load case for the second load case
1 variables [cm2] Haug&Arora [15] This paper Haug&Arora [15] This paper
1 bi 6.4516 193.7480 196.7185 152.0255 151.8842
2 b2 6.4516 0.6452 0.6452 0.6452 0.6452
3 b3 6.4516 150.1545 149.6227 163.0771 163.1232
4 b4 6.4516 98.6192 98.1744 92.5353 92.7578
5 bs 6.4516 0.6452 0.6452 0.6452 0.6452
6 be 6.4516 3.5903 3.5536 12.7084 12.7079
7 by 6.4516 48.1825 48.0919 80.0063 79.8721
8 bs 6.4516 136.7610 135.7063 82.9031 82.7963
9 b9 6.4516 139.4707 138.8988 130.8707 131.1797
10 b10 6.4516 0.6452 0.6452 0.6452 0.6452
Truss weight [kN] 1.867 22.523 22.511 20.808 20.807
Number of active constraints Numbers of active constraints Modulus of the maximum violation in the constraints 4 2, 5, 10, 22 0.2710-4 5 2, 5, 10, 15, 21 6.3510-7 4 2, 10, 15, 22 0.1710-3 6 2, 5, 10, 15, 16, 22 1.1910-7
Table 3.8. Comparison of the optimization results for the ten-bar cantilever truss.
Optimum weight [kN]
Source of information Load case 1 Load case 2
Stresses constraints only All constraints Stresses constraints only All constraints
This paper 7.087 22.511 7.404 20.807
Haug & Arora [15] 7.089 22.523 7.411 20.808
Schmit & Miura [25] 7.089 22.591 7.407 20.811
Rizzi [24] 7.089 22.590 7.407 20.811
Dobbs & Nelson [23] 7.217 22.605 - 22.514
Table 3.7 presents comparison of the optimization results for considered ten-bar truss obtained by Haug and Arora [15] and in this article. Table 3.8 shows a comparison of the optimization results for the ten-bar cantilever truss obtained using the proposed improved method of objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations with optimization results presented by the literature [15, 23, 24, 25].
Good correlation of obtained optimization results with the results of the other authors confirms the validity of the optimum solutions calculated using proposed improvement of the gradient projection method. The efficiency of the proposed improvement of the gradient projection method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints. The deviations available in some presented results can be explained by using a numerical approach to the iterative searching with specified accuracy.
3.3. Optimization of a 24-bar transmission tower
Magazine of Civil Engineering, 98(6), 2020
A parametric optimization problem for a transmission tower (see Fig. 3.3) has been considered by Haug and Arora [15]. The transmission tower is subjected to 2 load cases, as shown in Table 3.9. The initial data for optimization of the tower are presented in Table 3.10. Taking into account the symmetry of the structural form, the vector of the design variables has been reduced to 7 variable cross-section areas for 25 structural members of the considered tower (see Table 3.11). The parametric optimization problem is formulated as
searching for optimum cross-sectional areas X = (X7 )T , i = 1,7 , of the tower structural members, whose
provide the least weight of the tower subjected to stresses constraints, node displacements constraints, as well as constraints on the minimal cross-section areas.
A start value Aq = 1.0 inch2=6.4516 cm2 was used as a start approximation for the variable cross-
sections areas for all members of the considered tower. The dimensions of the optimization problem comprised 7 design variables and 129 constraints.
Table 3.9. Load cases for the transmission tower.
Load case
Node
Point load along axis [kN]
number (Fig. 3.3) 0x 0 y 0z
1 2.2241 — —
1 2 2.2241 — —
3 4.4482 —22.241 44.4822
4 - —22.241 44.4822
2 3 4 — —22.241 —22.241 88.9644 —88.9644
The positive direction of the point loads coincides with the _positive direction of the corresponded axes_
Figure 3.3. Design model of the transmission tower.
Table 3.10. Initial data for optimization of the transmission tower.
Unit weight of the tower material pg 0.1 pound/inch3=0.027154 N/cm3
Modulus of elasticity E 107 pound/inch2=6.8971*106 N/cm2
Non-dimensional factor / used to calculate second moment area of 1.0
2 inertia for each tower structural member = pbt
Lower limit value AL for cross-sectional areas for all tower members 0.01 inch2=0.0645 cm2
Ultimate node displacements of the tower xa, ya, za 0.35 inch=0.889 cm
Allowable stresses value for the all tower member (Ja ± 40000 pound/inch2= ± 27.5885 kN/cm2
Table 3.11. Comparison of the optimization results for the transmission tower.
Design variable Tower structural Optimal cross-section areas for tower members [cm2]
members* Haug and Arora [15] This paper
A 1 0.0645 0.0655
2, 3, 4, 5 13.2103 13.1754
A3 6, 7, 8, 9 19.3322 19.3654
A4 10, 11, 12, 13 0.0645 0.0645
A5 14, 15, 16, 17 4.4213 4.4056
A6 18, 19, 20, 21 10.4626 10.4669
A 22, 23, 24, 25 17.2335 17.2343
Tower weight [kN] 2.4250 2.4245
bar's numbers are indicated in Fig. 3.3
At the continuum optimum point there were 5 active constraints: constraints on lower limit value for
variable cross-sectional area for 10th, 11th, 12th and 13th tower members; 3rd and 4th node displacement constraint of the tower along axis 0z for both load cases. The iterative searching process for the optimum point was stopped due to the following stop criterion: increment of the design variables within two consecutive
iterations was less than 1x 10-6, as well as there were no violated constraints.
The comparison of the optimization results for the transmission tower is presented by Table 3.11. Good correlation of obtained results with the results of the other authors confirms the validity of the optimum solutions calculated using the proposed improvement of the gradient projection method. Start values of the design variables have no influence on the optimum solutions for the considered non-linear problems, validating thus the accuracy of the obtained optimum solutions. The efficiency of the proposed improvement of the gradient projection method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints.
4. Conclusion
The paper considers parametric optimization problems for the bar structures formulated as nonlinear programming task. 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.
Equivalent Householder transformations of the resolving equations of the method have been proposed. They increase numerical efficiency of the algorithm developed based on the considered method.
Equivalent transformations (Givens rotations) of the resolving equations of the method have been also proposed. They ensure acceleration of the iterative searching process in the specified cases described by the paper due to decreasing the amount of calculations.
Lengths of the gradient vectors for objective function, as well as for constraints remain as they were in scope of the proposed equivalent transformations ensuring the reliability of the optimization algorithm.
In order to estimate an efficiency of the proposed improvement of the gradient projection method, a comparison of the obtained optimization results with the results presented by the literature and widely used for testing has been performed. Good correlation of obtained results with the results of the other authors confirms the validity of the optimum solutions calculated using the proposed improvement of the gradient projection method. The efficiency of the proposed improvement of the gradient projection method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints.
5. Acknowledgement
Authors would like to thank the two anonymous reviewers for their critically reading the manuscript and suggesting substantial improvements.
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Contacts:
Vitalina Yurchenko, vitalinay@rambler.ru Ivan Peleshko, ipeleshko@hotmail.com
© Yurchenko V.V., Peleshko I. D., 2020
