AN IMPROVED GRADIENT-BASED METHOD TO SOLVE PARAMETRIC OPTIMISATION PROBLEMS OF THE BAR STRUCTURES

The paper considers parametric optimisation problems for the bar structures formulated as non-linear programming tasks. 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 optimisation 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 method under consideration. Additionally, proposed improvement for the gradient-based 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 optimisation results of truss structures presented by the paper confirms the validity of the optimum solutions obtained using proposed improvement of the gradient-based method. The efficiency of the propoced improvement of the gradient-based method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints.


Introduction.
Over the past 50 years, numerical optimisation and finite element method [7] have individually made significant advances and have together been developed to make possible the emergence of structural optimisation as a potential design tool. In recent years, great efforts have been also devoted to integrate optimisation 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 optimisation techniques.
Applied optimum design problems for the bar structures in some cases are formulated as parametric optimisation problems, namely as searching problems for unknown structural parameters, whose provide аn extreme value of the specified purpose function in the feasible region defined by the specified constraints. In this case structural optimisation 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. Besides, mathematical model of the parametric optimisation problem of the structures includes the set of design variables, the objective function, as well as constraints, whose reflect in general case non-linear interdependences between them [10].
In cases if the purpose 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-based non-linear methods [11]. The gradient-based methods operate with the first derivatives or gradients only both of the objective function and constraints. The methods are based on the iterative construction such sequence of the approximations of the design variables that provides the convergence to the optimum solution (optimum values of the structural parameters) [17].
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 [8].
Although many papers are published on the parametric optimization of the structures, the development of a general computer program for the design and optimisation of building structures according to specified design codes remains an actual task. Therefore, in this paper, a gradient-based method is considered as investigated object. The main research question is the development of mathematical support and numerical algorithm to solve parametric optimisation problems of the building structures with orientation on software implementation in a computer-aided design system.
1. Parametric optimisation problem formulation. Let us consider a parametric optimisation 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 optimisation problem can be stated as a nonlinear programming task in the following mathematical terms: to find unknown structural parameters , providing the least value of the determined objective function: in feasible region (search space)  defined by the following system of constraints:  1) can include as components unknown geometrical parameters of the structure, unknown cross-sectional sizes of the structural members, as well as 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 ability to formulate it 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). Additional requirements, whose describe structural, technological and serviceability particularities of the building structure under consideration, as well as constraints on the building functional volume can be also included into the system Eqs. (1.2) -(1.3).
2. An improved gradient-based method to solve the parametric optimisation problem. The parametric optimisation problem stated as nonlinear programming task by Eqs. (1.1) -(1.4) can be solved using a gradientbased 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 building structures [5,9].
The gradient-based method operates with the first derivatives or gradients only both of the objective function Eq. (1.1) and constraints Eqs. (1.2) -(1.3). The method is based on the iterative construction such sequence Eq. (2.1) of the approximations of the design variables Eq. (1.4) that provides the convergence to the optimum solution (optimum values of the structural parameters): , is the increment vector for the current values of the design variables t X  (see Fig. 2.1); t is the iteration's index.
Start point of the iterative searching process 0 t X   can be assigned as engineering's 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. Set of active constraints numbers A calculated for the current approximation t X  to the optimum solution (current design of the structure) is determined as: where  is small positive number introduced here in order to diminish the oscillations on movement alongside of the active constraints surface. Increment vector t X   for the current values of the design variables t X  can be determined by the following equation: The values of the constraint's violations for the current approximation t X  of the design variables are accumulated into the following vector: Let introduce into further consideration set L ,  L A , of the constraint's numbers, such that the gradients of the constraints at the current approximation to the optimum solution are linear-independent.
Component t X    is calculated from the equation presented below: In order to correct constraint's violations V , vector t X    to a first approximation should also satisfy Taylor's theorem for the continuously differentiable multivariable function in the vicinity of point t X  for each constraint from set L , namely: With substitution of Eq. (2.4) into the Eq. (2.5) we obtain the system of equations to determine column-vector    : is determined using the following equation: , (2.8) or from the equation presented below: where  is the step parameter, which can be calculated subject to the desired increment f  of the purpose function on movement along the direction of the purpose function anti-gradient. The increment f  can be assign as 5...25% from the current value of the objective function ( ) that follows from the condition of attainment the desired increment of the objective function f  on movement along the direction of the objective function anti-gradient projection onto the active constraints surface.
Step parameter  can be also selected as a result of numerical experiments performed for each type of the structure individually [6,13].
Using Eqs. (2.4) and (2.7), Eq. (2.2) can be rewritten as presented below: (2.14) where I is the unit matrix; t is the total number of the linear-independent gradients of the active constraints, i H is the transformation matrix, such that T i i  H H I , at the same time the sub-diagonal element are equal to zero in matrix   for column's numbers 1, i . Described conditions are satisfied by the orthogonal matrix of the elementary mapping (Householder's transformation) [18].
Let us present here the following algorithm to form set L and to construct is the matrix that comprises from the column-gradients of all active constraints.
All columns of matrix   0 Φ should be marked as 'not used' (or linearindependent). 2.
3. Among all 'not used' columns of matrix   1 i Φ , whose correspond to the constraints-equalities Eq. (1.2), one j th column with extreme value of the specified criterion should be selected (for example, the following criterion 2 2 can be considered as such criterion, where kj g are the j th column's components of matrix   1 i Φ ). At the same time all k th columns of matrix   1 i Φ , for whose the following inequality 2 met, should be marked as 'used', here 1  is the small positive number. In case when no constraints-equalities exist or all constraints-equalities Eq. (1.2) are marked as 'used', the selection of j th column should be performed among all 'not used' columns of matrix   1 i Φ , whose correspond to the constraints-inequalities ), there is a contradiction in the system of constraints Eq. (1.2) -(1.3). In other case, moving to the next step performs.
4. k th number of the constraint, that corresponds to the j th column number, should be included into set L , It is reasonable to execute the multiplication only for 'not used' columns. It should be noted, when using Householder's transformation matrix i H is not constructed evidently [18]. At the same time, matrix   i Φ may be constructed within the ranges of matrix   1 i Φ when no additional memory is needed.
Φ 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 is formed step-by-step. Besides, Eq. (2.6) and Eq. (2.8) can be rewritten as follow: . These columns correspond to those constraints from Eq. (1.3), for whose the following inequality satisfies: 0 Graphical illustration for the selection of the constraints-inequalities: graphical illustration: a -  return onto the active constraints surface from the feasible region  with simultaneous degradation of the objective function value perform (see Fig. 2.2, b). At the same time, in case of: Obvious method to calculate c and s for d th non-zero sub-diagonal element and for a th diagonal element is: The Givens matrix G may be calculated similarly to the matrix H using the following equation: can be rewritten as: (2.25) and the main resolving equation of the gradient-based method Eq. (2.12) and Eq. (2.13) can be rewritten as presented below:  4). So, the following convergence criterion of the iterative procedure Eq. (2.1) can be assign: where 1  is the small positive number. Taking into consideration Eq. (2.28) let formulate the following stop criteria in the iterative searching procedure Eq. (2.1).
Stop criterion 1: in case of the objective function gradient in the current approximation t X  of the design variables is close to zero value indicating on extreme character of the current approximation, as well as violated constraints are absent: where Σ is the set of the violated constraints numbers, Stop criterion 2: in case of the projection of the objective function gradient in the current approximation t X  onto the active constraints surface is close to zero value or objective function gradient is perpendicularly to the active constraints surface indicating impossible further improvement of the objective function value, as well as violated constraints are absent: Stop criterion 3: when in the current approximation t X  of the iterative searching procedure (2.2) the total number of the active constraints t equals to the number of design variables X N , as well as all active constraints are active (both not violated constraints and those ones for whose inequality Eq. (2.12) met): This stop criterion for the iteration process Eq. (2.1) corresponds to the case when the current approximation   , 1, .
3. Results and discussion. In order to estimate an efficiency of the new methods or algorithms, we should perform a comparison with alternative methods or algorithms presented by other authors using different optimisation techniques. Criteria to implement such comparison are described, i.e. by the papers [2,6]. Many of them, such as robustness, amount of functions calculations, requirements to the CPU 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 comparison of the optimisation results obtained using proposed improvement of the gradient-based method, as well as of the results presented by the literature and widely used for testing. Initial data and mathematical models of the parametric optimisation problems considered below were assumed as the same as described in the literature.

Parametric optimisation of a three-bar truss.
Optimisation of a three-bar truss (see Fig. 3.1) has been firstly solved by Schmit L. A. [15] using a non-linear programming method. Besides, the task has been also considered by the authors of the paper [6].
A parametric optimisation problem was formulated as searching for optimum crosssectional areas 1 b , 2 b and 3 b 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. Load cases for truss under consideration are presented by Table 3.1.
where l is the truss height, 25.4 l  cm (see Fig. 3.1). Let formulate strength constraints for each truss members for all load cases as follows: where j i N is the axial force for i th truss member subjected to j th load case, . Besides, let include to the system of constraints the inequalities describing that the design variables should have positive values: (3.3) Flexural buckling constraints for all truss members can be written using Hooke law as presented below: where 4 4 , j j x z are linear displacements for 4 th node of the truss subjected to j th load case along the directions of 0x and 0z axes respectively. Constraints on the minimum values of the eigenvalues can be written analytically using calculation results of the eigenvalues stability problem for truss under consideration: Let also formulate displacements constraints for 4 th truss node in the plane 0 x z : 11 iterations have been performed. 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, as well as there were no violated constraints.

Optimisation 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 [3,6,14,16] in order to compare different methods for solving optimisation problems. The parametric optimisation problem is formulated as follows: to find unknown cross-sectional areas for each truss member   , 1,10 , with weight minimisation of the truss subjected to stresses constraints in all truss bars, node displacements constraints, as well as constraints on the minimal cross-section areas.
The truss under consideration is undergone for two load cases (see Fig. 3 where l is the truss height, 914.4 l  cm (see Fig. 3.2). Constraints on lower limit value for variable cross-sectional areas for all truss bars are written as follows: (3.11) Stresses constraints can be formulated as presented below: 10 1 0 12) where i N is the axial force in the i th truss member. Displacement constraints for the truss nodes are written as follows: (3.14) where , j j x z are linear displacements of j th truss node, 1, 4 j  . Starting from the initial truss design with start weight 0 1.867 G  kN optimal solution with optimum weight * 22.514 G  kN has been obtained for the truss subjected to the first load case. Additionally, starting from the initial truss design with start weight 0 1.867 G  kN optimal solution with optimum weight * 20.806 G  kN has been obtained for the truss subjected to the first load case. Comparison of the optimisation results for three-bar truss under consideration obtained by authors of the paper [6] and in this article is presented by Table 3.5.
For both loaded cases 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, as well as there were no violated constraints.
Comparison of the optimisation 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 optimisation results presented by the literature [3,6,14,16] are shown in Table 3.6.

Optimisation of a 24-bar translational tower.
Parametric optimization problem for a translational tower (see Fig. 3.3) has been considered by the paper [6]. The translation tower is subjected to 2 load cases (see Table 3.7). 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 tower under consideration (see Table 3.8). The parametric optimization problem is formulated as searching for optimum cross-sectional areas , 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.
Initial data for optimisation of the tower are as follows: unit weight of the tower material is 0. Comparison of the optimisation results for the translational tower is presented by Table 3.8. At the continuum optimum point there were 5 active constraints: 3 rd node displacement constraint of the tower along axis 0x for 1 st and 2 nd load cases, 3 rd node displacement constraint along axis 0z for 1 st load case, 4 th node displacement constraint along axis 0x for 2 nd load case, as well as 4 th node displacement constraint along axis 0z for 1 st load case. Internal axial forces at the optimum design of the translational tower are shown by Table 3.9.   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 6 1 10   , as well as there were no violated constraints (maximum value among constraint violations was 0.049 10 10   ). Conslusion. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been considered by the paper. Equivalent Householder transformations of the resolving equations of the method have been proposed. They increase numerical efficiency of the algorithm developed based on the method under consideration.
Additionally, proposed improvement for the gradient-based method also includes equivalent transformations (Givens rotations) of the resolving equations. They ensure acceleration of the iterative searching process in 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 optimisation algorithm.
The comparison of the optimisation results presented by the paper confirms the validity of the optimum solutions obtained using proposed improvement of the gradient-based method. Start values of the design variables have no influence on the optimum solution of the non-linear problem confirming in such way accuracy and validity of the optimum solutions obtained using the algorithm developed based on the presented improved gradient-based method. The efficiency of the propoced improvement of the gradient-based method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints. The deviations availabled in some presented results can be explained, on the one hand, by using a numerical approach to the iterative searching with specified accuracy (as in the optimisation of the ten-bar cantilever truss), on the other hand, by possible existence of several local optimum points (as in the optimisation of the translation tower).

AN IMPROVED GRADIENT-BASED METHOD TO SOLVE PARAMETRIC OPTIMISATION PROBLEMS OF THE BAR STRUCTURES
The paper considers parametric optimisation problems for the bar structures formulated as non-linear programming tasks. In the paper a gradient-based method is considered as investigated object. The main research question is the development of mathematical support and numerical algorithm to solve parametric optimisation problems of the building structures with orientation on software implementation in a computer-aided design system.
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 optimisation problem. Equivalent Householder transformations of the resolving equations of the method have been proposed by the paper. They increase numerical efficiency of the algorithm developed based on the method under consideration. Additionally, proposed improvement for the gradient-based 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. 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 optimisation algorithm.
The comparison of the optimisation results of truss structures presented by the paper confirms the validity of the optimum solutions obtained using proposed improvement of the gradient-based method. Start values of the design variables have no influence on the optimum solution of the nonlinear problem confirming in such way accuracy and validity of the optimum solutions obtained using the algorithm developed based on the presented improved gradient-based method. The efficiency of the propoced improvement of the gradient-based method has been also confirmed taking into account the number of iterations and absolute value of the maximum violation in the constraints. Keywords: bar system, parametric optimisation, non-linear programming task, gradient-based method, finite-element method.