Garret N. Vanderplaats

A NEW APPROXIMATION METHOD FOR STRESS CONSTRAINTS IN STRUCTURAL SYNTHESIS

A new approximation method for dealing with stress constraints in structural synthesis is presented. In previous methods, it has been common to create an approximation of the actual constraints with respect to the design variables and then use this with optimization to produce a new candidate design. In the present method, the finite-element nodal forces are, instead, approximated, and these are used to create an explicit, but often nonlinear, approximation to the original problem. The principal motivation is to create the best approximation possible, in order to reduce the number of detailed finite-element analyses needed to reach the optimum. The method is shown to be quite simple, while providing significant efficiency gains in the overall optimization task. Also, the method produces such high-quality approximations to the original problem that the need for move limits during the approximate optimization stage is greatly reduced. Examples are offered and compared with published results to demonstrate the efficiency and reliability of the proposed method. It is concluded that there are considerable gains yet to be achieved through continued careful investigation of approximation techniques.

A general-purpose optimization program for engineering design

Abstract A new general-purpose optimization program for engineering design is described. ADS (Automated Design Synthesis) is a FORTRAN program for nonlinear constrained (or unconstrained) function minimization. The optimization process is segmented into three levels: Strategy, Optimizer, and One-dimensional search. At each level, several options are available so that a total of nearly 100 possible combinations can be created. An example of available combinations is the Augmented Lagrange Multiplier method, using the BFGS variable metric unconstrained minimization together with polynomial interpolation for the one-dimensional search. Scaling is included to improve the numerical conditioning. The program is demonstrated with several engineering design examples.

Stiffness Optimization of Orthotropic Laminated Composites Using Lamination Parameters

The paper presents an efficient stiffness optimization approach on orthotropic laminated composites using lamination parameters. It is efficient to use lamination parameters as design variables since the stiffness components of laminated composites are expressed as a linear function of lamination parameters in the classical lamination theory. As an example of stiffness optimizations, the buckling optimization of orthotropic laminated cylindrical shells under combined loadings is treated using a mathematical programming method. The present approach shows good convergence behaviors for the optimum design and gives reliable optimization results.

An efficient feasible directions algorithm for design synthesis

A nonlinear optimization algorithm is described that combines the best features of the method of feasible directions and the generalized reduced gradient method. The algorithm uses the direction-finding subproblem from the method of feasible directions to find a search direction that is equivalent to that of the generalized reduced gradient method, but without the need to add a large number of slack variables associated with inequality constraints. This leads to a core-efficient algorithm for the solution of optimization problems with a large number of inequality constraints. Also, during the one-dimensional search, it is not necessary to separate the design space into dependent and independent variables using the present method. The concept of infrequent gradient calculations is introduced as a means of gaining further optimization efficiency. Finally, it is shown that, using the basic direction-finding algorithm contained in this method, the sensitivity of the optimum design with respect to problem parameters can be obtained without the need for second derivatives or Lagrange multipliers. The optimization algorithm and sensitivity analysis is demonstrated by numerical example.

Approximation method for configuration optimization of trusses

An efficient method for truss configuration optimization is presented. Elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints. Constraints on member stresses and Euler buckling are imposed, and multiple loading conditions are considered. The method presented here utilizes an approximate structural analysis based on first-order Taylor series expansions of the member forces. The force approximation succeeds in reducing the degree of coupling between the sizing and geometry design variables and thereby increases the accuracy of the approximation. A numerical optimizer uses information from the approximate structural analysis as it minimizes the weight of the truss.

Optimum design of trusses with discrete sizing and shape variables

The objective here is to present a method for optimizing truss structures with discrete design variables. The design variables are considered to be sizing variables as well as coordinates of joints. Both types of variables can be discrete simultaneously. Mixed continuous-discrete variables can also be considered. To increase the efficiency of the method, the structural responses, such as forces and displacements are approximated in each design cycle. The approximation of responses is carried out with respect to the design variables or their reciprocals. By substituting these approximate functions of the responses into the original design problem, an explicit high quality approximation is achieved, the solution of which does not require the detailed finite element analysis of the structure in each sub-optimization iteration. First it is assumed that all the design variables are continuous and a continuous variable optimization is performed. With the results of this step, the branch and bound method is employed on the same approximate problem to achieve a discrete solution. The numerical results indicate that the method is efficient and robust in terms of the required number of structural analyses. Several examples are presented to show the efficiency of the method.

Structural shape optimization with MSC/NASTRAN

Abstract A structural shape optimization capability has been added to the MSC/NASTRAN finite element program. The grid locations in the finite element model are changed using a reduced basis method. The base shape vectors, required for the design model, could be generated either using a mesh generator or a static finite element analysis. Recognizing that shape optimization technology is making steady progress, the program structure is designed to be flexible and modular to include additional capabilities in future versions.

ADS-1 : A New General-Purpose Optimization Program

Today, numerous programs are available which may be coupled with finite element analysis or other analysis techniques to perform the optimization function in the solution of structural synthesis problems. However, most of these codes include only one or two algorithms and many have not been tested on problems of significant size and complexity. There is, therefore, a need for a reliable, general-purpose, publicly available code, containing a variety of modern algorithms for use in structural synthesis as well as general engineering design. The ADS-1 program (Automated Design Synthesis: Version 1) was written in response to this need. The present investigation has the objective to present the capabilities of the ADS program and to demonstrate its application to structural synthesis. The ADS program solves the general nonlinear constrained optimization problem in the standard form. At each level of the optimization process, several options are available.

An Efficient Approximation Method for Structural Synthesis with Reference to Space Structures

A method is presented for the optimum design of structures which is very robust and efficient in terms of the number of required analyses of the structure. Some explicit approximation expressions are generated for the structural response quantities such as nodal displacements, forces and frequencies as functions of the cross-sectional properties. By substituting these expressions into the constraint equations, the design task becomes a non-linear programming problem which is an explicit problem in terms of the design variables. The solution of this problem gives the actual cross-sectional dimensions. The method is an iterative technique and the results indicate that the convergence to the optimal solution results indicate that the convergence to the optimal solution is very rapid. The robustness of the proposed method is due to the generation of explicit approximate relations for the structural response quantities, as in the past the design constraints were approximated. Also a high quality approximation is obtained for the internal forces directly without using the approximate values of the displacements. The quality of approximations is enhanced by expressing the structural responses, in particular, the frequencies with respect to the cross-sectional areas and second moment of inertias instead of using the cross-sectional dimensions. A double-layer grid and a grillage are chosen as test cases, the results of which are presented.

Structural optimization using beam elements

Techniques for handling beam elements in structural optimization are investigated. The complexities of evaluating the geometrical properties of a section, stress recovery, and design-variable selection are successfully handled by the concept of an element library, use of equivalent stress, and separation of sensitivity variables and design variables. Program implementation is based on the use of an existing finite-element analysis package and a numerical optimizer. The quality of linearization of the behavior constraint functions is also investigated.