Previous Design Points Research Articles

Enhanced Two-Point Exponential Approximation for Structural Optimization

This paper proposes a new approximation method called Enhanced Two-Point Exponential Approximation (ETPEA), which is derived based on Two-Point Exponential Approximation (TPEA). In TPEA, when the exponents of intervening variables cannot be calculated by fitting the sensitivities of the previous design point, the accuracy of TPEA is decreased. In such cases, ETPEA uses a substitution function to construct the approximate function, and the optimum result can be improved. Some typical structural optimization problems are tested, and the results indicate that the optimum design can be found accurately and quickly with the new approximation method. It is verified that the new approximation method is also efficient for large structural optimization problems.

EFFICIENT METHODS FOR STRUCTURAL OPTIMIZATION WITH FREQUENCY CONSTRAINTS USING HIGHER ORDER APPROXIMATIONS

Presented herein are four different methods for the optimum design of structures subject to multiple natural frequency constraints. During the optimization process the optimum cross-sectional dimensions of elements are determined. These methods are robust and efficient in terms of the number of eigenvalue analyses required, as well as the overall computational time for the optimum design. A new third order approximate function is presented for the structural response quantities, as functions of the cross-sectional properties, and four different methods for the optimum design are defined based on this approximate function. The main features of the proposed function are that only the diagonal terms of higher order derivative matrices are employed, and these derivatives are established by the available first order derivatives. The first order exact derivatives are obtained from a sensitivity analysis at the previous design points. We show that this approximate function creates high quality approximations of the structural responses, such as the frequencies. Examples are presented and the efficiency and quality of the proposed methods are discussed and compared.

이점 볼록 근사화 기법을 적용한 최적설계

In this paper, a new local two-point approximation method which is based on the exponential intervening variable is proposed. This new algorithm, called the Two-Point Convex Approximation(TPCA), use the function and design sensitivity information from the current and previous design points of the sequential approximate optimization to generate a sequence of convex, separable subproblems. This paper describes the derivation of the parameters associated with the approximation and the numerical solution procedure. In order to show the numerical performance of the proposed method, a sequential approximate optimizer is developed and applied to solve several typical design problems. These optimization results are compared with those of other optimizers. Numerical results obtained from the test examples demonstrate the effectiveness of the proposed method.

Optimum design of structures with discrete variables using higher order approximation

Some higher order function approximations are introduced during the optimization process with continuous and discrete design variables. Second and third order approximations are presented by employing only the diagonal terms of the matrices of the higher order derivatives. To reduce the computational cost, these derivatives are estimated by using the first order derivatives available from the previous design points. A hybrid form of the second and third order approximations is also presented for function approximation. It is observed that the hybrid form is more effective. To achieve the optimal discrete variables, a new penalty function is introduced. First the continuous optimization is carried out by means of penalty functions and then the results are used as the starting point for discrete optimization. In both the continuous and discrete optimization, the idea of various function approximations is employed to decrease the computation time required by the design process. Examples are solved and the effects of different proposed approximate techniques are discussed.

STRUCTURAL OPTIMIZATION USING A NEW LOCAL APPROXIMATION METHOD

A new method for solving structural optimization problems using a local function approximation algorithm is proposed. This new algorithm, called the Generalized Convex Approximation (GCA), uses the design sensitivity information from the current and previous design points to generate a sequence of convex, separable subproblems. The paper contains the derivation of the parameters associated with the approximation and the formulation of the approximated problem. Numerical results from standard test problems solved using this method are presented. It is observed that this algorithm generates local approximations which lead to faster convergence for structural optimization problems.

A Development of Two-Point Reciprocal Quadratic Approximation Mehtod for Configuration Optimization of Discrete Structures

The configuration optimization is a structural optimization method which includes the coordinates of a structure as well as the sectional properties in the design variable set. Effective reduction of the weight of discrete structures can be obrained by changing the geometry while satisfying stress, Ei;er bickling, displacement, and frequency constraints, etc. However, the nonlinearity due to the configuration variables may cause the difficulties of the convergence and expensive computational cost. An efficient approximation method for the configuration optimization has been developed to overcome the difficulties. The method approximates the constraint functions based onthe second-order Taylor series expansion with reciprocal design variables. The Hessian matrix is approzimated from the information on previous design points. The developed algotithms are coded and the examples are solved.

Optimization of geometrically nonlinear thin-walled structures using the multipoint approximation method

The present study concentrates on the optimization of geometrically nonlinear shell structures using the multipoint approximation approach. The latter is an iterative technique, which uses a succession of approximations for the implicit objective and constraint functions. These approximations are formulated by means of multiple regression analysis. In each iteration the technique enables the use of results gained at several previous design points. The approximate functions obtained are considered to be valid within a current subregion of the space of design variables defined by move limits. A geometrically nonlinear curved triangular thin shell element with the corner node displacements and the mid-side rotations as degrees of freedom is used for the FE analysis. The influence of initial shape imperfections on the optimum designs is investigated. Imperfections are considered as a shape distortion proportional to the lowest buckling modes of the perfect structure. Displacement, stress, and stability constraints are taken into account. To prevent finite element solutions from becoming unstable during the optimization process, a simple strategy for avoiding passage of stability points is applied. Some numerical examples are solved to show the practical use and efficiency of the technique presented.

Multiparameter structural optimization using FEM and multipoint explicit approximations

A unified approach to various problems of structural optimization, based on approximation concepts, is presented. The approach is concerned with the development of the iterative technique, which uses in each iteration the information gained at several previous design points (multipoint approximations) in order to better fit constraints and/or objective functions and to reduce the total number of FE analyses needed to solve the optimization problem. In each iteration, the subregion of the initial region in the space of design variables, defined by move limits, is chosen. In this subregion, several points (designs) are selected, for which response analyses and design sensitivity analyses are carried out using FEM. The explicit expressions are formulated using the weighted least-squares method. The explicit expressions obtained then replace initial problem functions. They are used as functions of a particular mathematical programming problem. Several particular forms of the explicit expressions are considered. The basic features of the presented approximations are shown by means of classical test examples, and the method is compared with other optimization techniques.

Impact of the Design Point on the Poloidal Field Coil System for the CIT (Compact ignition Tokamak)

The poloidal field (PF) coil system in a tokamak must provide the flux swing and the equilibrium and shaping field distributions required to create, maintain, and control the plasma during the discharge. The present design for CIT calls for an 11.8 MA plasma in a double null configuration with a major radius of 2.59 m, a minor radius of 0.795 m, and an elongation (at the 95% flux surface) of 2.0. The toroidal field at the major radius is 9 T. The central solenoid is self-supporting and the TF coils wedge under the Lorentz load.The previous design point called for the TF and PF to buck against each other in order to lower the stresses in both. That design called for an 11 MA, 10 T plasma with a major radius of 2.14 m and a minor radius of 0.66 m. The impact on the PF system of this change in the design point is assessed.