Optimal design of geometrically nonlinear shells of revolution with using the mixed finite element method

The article is concerned with a methodology of optimal design of geometrically nonlinear (flexible) shells of revolution of minimum weight with strength, stability and strain constraints. The problem of optimal design with constraints is reduced to the problem of unconstrained minimization using the penalty functions method. Stress-strain state of shell is determined within the geometrically nonlinear deformation theory. A special feature of the methodology is the use of a mixed finite-element formulation based on the Galerkin method. Test problems for determining the optimal form and thickness distribution of a shell of minimum weight are considered. The validity of the results obtained using the developed methodology is analyzed, and the efficiency of various optimization algorithms is compared to solve the set problem. The developed methodology has demonstrated the possibility and accuracy of finding the optimal solution.


Introduction
The proposed methodology of optimal design of shells allows designing the shells with variable thickness, e.g., ribbed shells, shells with a discrete inner layer, etc. In this paper the shell is regarded as structurally orthotropic. Stress-strain state of shell for the solution of optimization problem was determined using the Galerkin method [1] in mixed finite element formulation [2,3]. This method allows to do without the construction procedure of the functional of problem and to solve equations of problem in the form in which they are written. The authors were obtained in an explicit form of the matrix and vectors for the finite element shell. Their advantage is that the numerical integration is not used, which improves the accuracy of calculations. Another of the advantages of using the mixed formulation (as indicated by, e.g., Streng [2]) -reduction of the condition number of the matrix of the finite element, in comparison with the classical method of finite elements in displacements, which also positively effect on the accuracy of the calculation results. The methodology for determining the stress-strain state of geometrically nonlinear shells of revolution is described in detail in [3][4][5].

Description of the methodology
The problem of optimal design of the shell was considered as the problem of shell volume minimization with strength, stability and displacement constraints: (1) Such parameters as shell shape parameters, thickness distribution, etc. can be selected as components of the vector of the design variables x . In equation (1) , , The values are determined at each step of the optimization algorithm using the Galerkin method in mixed finite element formulation [4 -6]. The problem of optimal design with constraints is reduced to the problem of unconstrained minimization using the penalty functions method. Penalty function k r takes the form proposed by Mischke [6]: where, is the objective function of the problem with constraints; с The problem of optimal design in form as in equation (3) can be solved any of classical algorithms of unconstrained minimization (e.g. method of steepest descent, grid search method, method of conjugate gradients, etc.).

Test problems
To check the validity of the results obtained using the developed methodology and to compare the effectiveness of different algorithms of unconstrained optimization, two test problems of optimal design of a shallow axisymmetric shell of revolution were solved. nodes along the shell generatrix was calculated by formula: where Z is the shell shape parameter,  is the radial coordinate. The 1 st test problem is to determine the shape and thickness of the shell of minimum weight (design variables are Z and h) under the assumption that the shell thickness is the same over its entire surface. The 2 nd test problem is to determine the optimum thickness distribution for a minimal weight shell (design variables are h and h1) under the assumption that the thickness varies linearly from the centre (h1) to the edges (h) and does not change in the circumferential direction.

Checking the results
The diagrams of changes of objective and penalty functions depending on the node parameter values were plotted (see     When studying the shape of the boundary of the permissible values domain with regards to stability, minor irregularities associated with the inaccuracy of critical load determination algorithm were found. They can cause difficulties when determining an optimal solution. Therefore, to obtain reliable results search for an optimal solution should be started from several points, and then the best solution should be selected.

Analysis of efficiency of optimization algorithms
A comparative analysis of the effectiveness of different optimization algorithms used to solve fist problem have been carried out ( [7][8][9][10][11]). Such methods as the grid (multidimensional continuous search) with the exclusion of domains, the steepest descent method with a variable step, a method based on a combination of gradient and random search [9 -11] have been considered. Grid search method with the exclusion of domains was used with the objective function (see equation 3) without iterative refinement with the constant penalty parameter k r = 1,000,000 and parameter s, which is equal to the mean of function ) (x V beyond the boundaries of feasible solutions. Search domain is divided into 4 parts along each of the coordinates. Grid search method with the exclusion of domains has proved to be a reliable one for searching an optimal solution; it also requires a small number of calculations (384 calculations of the objective function were done in the problem considered) when searching is carried put with a small number of design variables. However, with an increase in the number of design variables a significant increase in the number of calculations of the objective function should be expected which hinders the use of this method in such cases. Application of the steepest descent method with a variable step and parameter s=0 (objective function without a barrier on the boundary of feasible solutions), has shown that in most cases it does not lead to finding an optimal solution. This is due to a ravine surface shape of the objective function (see equation 3). The algorithm usually 'loops' on the boundary of permissible values domain. The method, based on a combination of gradient and random search, proved to be a reliable one when solving problems when in equation 3 s=0. To obtain sufficiently accurate results 412 calculations of the objective function were done. This value is higher than the one, which was done for the grid search method with the exclusion of domains. However, with an increase in the number of design variables, in contrast to the grid search method, there is no significant increase in the number of calculations of the objective function. Therefore, it can be recommended for solving problems with the use of the developed optimal design method with a large number of design variables.

Conclusions
Methodology of optimal design of geometrically nonlinear shells of revolution of minimum weight with strength, stability and strain constraints is developed. Stress and strain state of the shell is determined by Galerkin method in the mixed finite element definition. Based on this methodology, algorithms for determining the optimal form and thickness distribution of a shell of minimum weight are obtained. The results of solving two test problems confirmed the possibility and accuracy of finding the optimal solution. The analysis of the effectiveness of different optimization algorithms to solve the set problem is given. The selected algorithm based on a combination of gradient and random search, allows to effectively searching the optimal solution for a large number of design parameters.

Acknowledgement
This work was supported by the grant of the President of the Russian Federation #МК-9203.2016.8