Raphael T. Haftka

Approximation concepts for optimum structural design — a review

This paper reviews the basic approximation concepts used in structural optimization. It also discusses some of the most recent developments in that area since the introduction of approximation concepts in the mid-seventies. The paper distinguishes between local, medium-range and global approximations; it covers function approximations and problem approximations. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function approximations and use of intermediate variable and response quantities. It appears also that some new methodologies emerge which could greatly benefit from the introduction of new computer architectures.

Structural shape optimization — a survey

Abstract This paper is a survey of structural shape optimization with an emphasis on techniques dealing with shape optimization of the boundaries of two- and three-dimensional bodies. Attention is focused on the special problems of structural shape optimization which are due to a finite element model which must change during the optimization process. These problems include the requirement for sophisticated automated mesh generation techniques and careful choice of design variables. They also include special problems in obtaining sufficiently accurate sensitivity derivatives.

Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm

Various genetic parameters including the population size, the probability of mutation, and the probability of crossover are optimized by numerical experiments. A new genetic operator-permutation-is proposed and shown to be effective in reducing the cost of the genetic search. Results are obtained for a graphite-epoxy plate, first when only the buckling load is considered, and then when constraints on ply contiguity and strain failure are added. The influence on the genetic search of the penalty parameter enforcing the contiguity constraint is studied

Sensitivity analysis of discrete structural systems

Methods for calculating sensitivity derivatives for discrete structural systems are surveyed, primarily covering literature published during the past two decades. Methods are described for calculating derivatives of static displacements and stresses, eigenvalues and eigenvectors, transient structural response, and derivatives of optimum structural designs with respect to problem parameters. The survey is focused on publications developed in nonstructural fields such as electronics, controls, and physical chemistry which are directly applicable to structural problems. Most notable among the nonstructural-based methods are the adjoint variable technique from control theory, and the Greens function and FAST methods from physical chemistry.

Recent developments in structural sensitivity analysis

The present paper reviews recent developments in two major areas of structural sensitivity analysis: sensitivity of static and transient response; and sensitivity of vibration and buckling eigenproblems. Recent developments from the standpoint of computational cost, accuracy, and ease of implementation are presented.In the area of static response, current interest is focused on sensitivity to shape variation and sensitivity of nonlinear response. Two general approaches are used for computing sensitivities: differentiation of the continuum equations followed by discretization, and the reverse approach of discretization followed by differentiation. It is shown that the choice of methods has important accuracy and implementation implications.In the area of eigenproblem sensitivity, there is a great deal of interest and significant progress in sensitivity of problems with repeated eigenvalues. The paper raises the issue of differentiability and continuity that is inherent to the repeated eigenvalue case.

Preliminary design of composite wings for buckling, strength and displacement constraints

An unstiffened panel buckling constraint for balanced, symmetric laminated composites is included on the global design level in a mathematical programming structural optimization procedure for designing wing structures. Constraints are introduced by penalty functions, and Newtons method based on approximate second derivatives of the penalty terms is used as the search algorithm to obtain minimum-mass designs. Constraint approximations used during the optimization process contribute to the computational efficiency of the procedure. A criterion is developed that identifies the appropriate conservative form of the constraint approximations that are used with the optimization procedure. Minimum-mass design results are obtained for a multispar high-aspect-ratio wing subjected to material strength, minimum-gage, displacement, panel buckling and twist constraints. The material systems considered for the examples are all graphite-epoxy, graphite-epoxy with boron-epoxy spar caps, and all aluminum. The composite material designs are shown to have an advantage over the aluminum designs since they can often satisfy additional constraints with only small mass increases.

Improved Genetic Algorithm for the Design of Stiffened Composite Panels

The design of composite structures against buckling presents two major challenges to the designer. First, the problem of laminate stacking sequence design is discrete in nature, involving a small set of fiber orientations, which complicates the solution process. Therefore, the design of the stacking sequence is a combinatorial optimization problem which is suitable for genetic algorithms. Second, many local optima with comparable performance may be found. Most optimization algorithms find only a single optimum, while often a designer would want to obtain all the local optima with performance close to the global optimum. Genetic algorithms can easily find many near optimal solutions. However, they usually require very large computational costs. Previous work by the authors on the use of genetic algorithms for designing stiffened composite panels revealed both the above strength and weakness of the genetic algorithm. The present paper suggests several changes to the basic genetic algorithm developed previously, and demonstrates reduced computational cost and increased reliability of the algorithm due to these changes. Additionally, for a stiffened composite panel used in this study, designs lighter by about 4 percent compared to previous results were obtained.

An Approach to Structure/Control Simultaneous Optimization for Large Flexible Spacecraft

This paper presents an approach to the simultaneous optimal design of a structure and control system for large flexible spacecrafts based on realistic objective function and constraints. The weight or total cost of structure and control system is minimized subject to constraints on the magnitude of response to a given disturbance involving both rigid-body and elastic modes. A nested optimization technique is developed to solve the combined problem. As an example, simple beam-like spacecraft under a steady-state white-noise disturbance force is investigated and some results of optimization are presented. In the numerical examples, the stiffness distribution, location of controller, and control gains are optimized. Direct feedback control and linear quadratic optimal control laws are used with both inertial and noninertial disturbing force. It is shown that the total cost is sensitive to the overall structural stiffness, so that a simultaneous optimization of the structure and control system is indeed useful. Nomenclature [A] = plant matrix, Eq. (4) a = cross-sectional area of beam structure [B] = input matrix for control force, Eq. (5) b = a constant relating the bending rigidity with crosssectional area, Eq. (30) [ C] = damping matrix [D] = input matrix for disturbance force, Eq. (6) El = bending rigidity [F] =gain matrix, Eq. (7) (fd} = disturbance force [G] = closed-loop system matrix, Eq. (9) J =cost function or objective function, Eq. (19) [K ] = stiffness matrix L = half length of structure [M] =mass matrix mc — mass of a controller including its power source mN = normalizing mass ms = mass of half structure Nd = number of area design variables in half structure Nm = number of normal modes [P] = solution of the Riccati equation, Eq. (23) p(x,t) = disturbance force per unit length {q} = state vector, Eq. (3) [R{ ] = weighting matrix of the state, Eq. (10) [R2] = weighting matrix of the control force, Eq. (11) r = ratio of costs of controller and structure per unit mass [S] = output matrix t = time u, {u} = control force or control force vector V> [ V\ = intensity or intensity matrix of white-noise disturbance force per unit length {w) = discretized displacement x = coordinate along the structure with the origin at the center of spacecraft xc = location of controller

COMPOSITE LAMINATE DESIGN OPTIMIZATION BY GENETIC ALGORITHM WITH GENERALIZED ELITIST SELECTION

Abstract Genetic algorithms with elitist selection based on cloning a best single individual (SI) from one generation to the next are popular, but generalized elitist selection (GES) procedures have been proposed and tried in the past. The present paper explores several generalized elitist procedures for the design of composite laminates. It is shown that GES procedures are superior to an SI procedure for two types of problems. The first type involves many global optima, and the GES procedure can find several global optima more efficiently than the SI procedure. This may give a designer more design freedom. The second type of problem involves an isolated optimum surrounded by many designs with performance that is very close to optimal. It is shown that GES procedures can find the optimum and near optimal designs much more easily and reliably than the SI procedure.

Structural design under bounded uncertainty-optimization with anti-optimization

Abstract In many cases precise probabilistic data are not available on uncertainty in loads, but the magnitude of the uncertainty can be bound. This paper proposes a design approach for structural optimization with uncertain but bounded loads. The problem of identifying critical loads is formulated mathematically as an optimization problem in itself (called anti-optimization), so that the design problem is formulated as a two-level optimization. For linear structural analysis it is shown that the antioptimization part is limited to consideration of the vertices of the load-uncertainty domain. An example of a ten-bar truss is used to demonstrate that we cannot replace the anti-optimization process by considering the largest possible loads.