This paper suggests the solution of geometric programs with degrees of difficulty by means of an auxiliary problem which is the reduced equivalent of the primal problem. The auxiliary problem is derived from the primal program by direct algebraic transformation and has a highly simplified convex separable structure. Using a condensation technique introduced by Duffin, it is shown that generalized polynomial programs can be solved via a sequence of approximating problems with a similar convex separable structure.

The Explicit Approximation Method for design optimization problems with implicit constraints is presented. This is a zero-order sequential approximation method. Only function Values of the implicit constraint are needed, and no gradient calculation is required. The basic idea is to use engineering knowledge instead of purely numerical gradient information to form an explicit approximation to the implicit constraint. Design examples are presented to show how good explicit approximations to the implicit constraints can be formed, and a good and feasible, if not theoretically optimum, design can be found very efficiently by the Explicit Approximation Method using only zero-order information.

The curvature function method for two-dimensional shape optimization under stress constraints is developed. This method uses curvatures along the boundary curve as the design variables. First it is shown that local curvature has a monotonic relation to stress. Based on this, a zero-order search direction can be defined to search for the optimum curvature function which achieves a fully stressed boundary. No sensitivity analysis is required, and the method is completely independent of the analysis techniques for calculating the stress. The resulting curve has C2 continuity if the curvature function is continuous. Three design examples are presented.

Definition of a "quality margin" concept permits putting product quality into an optimization model in the constraints instead of only in the objective function, as in Taguchi's method. This permits incorporating not only more than one quality attribute, but also other performance constraints, such as power or strength, into a single optimization model solvable by standard techniques, here monotonicity analysis These ideas are applied to an extended version of Taguchi's didactic electric circuit problem to obtain five very different designs. These illustrate the trade-offs between quality, power consumption, and cost characteristic of many consumer products

A convex decomposition method, called Alternating Sum of Volumes (ASV), uses convex hulls and set difference operations. ASV decomposition, however, may not converge, which severely limits the domain of geometric objects that the current method can handle. We investigate the cause of non-convergence and present a remedy; we propose a new convex decomposition called Alternating Sum of Volumes with Partitioning (ASVP) and prove its convergence. ASVP decomposition is a hierarchical volumetric representation which is obtained from the boundary information of the given object based on convexity. As an application, form feature recognition by ASVP decomposition is briefly discussed.

Taguchi’s robust circuit design problem can be formulated rigorously as an optimization problem. A necessary condition for optimality is that the control range be centered about the target value. This generates a constraint on the two design variables which cannot be solved for either variable. The present article shows that by approximating this unsolvable constraint with a simpler constraint that is solvable, one variable can be eliminated and the problem reduced to an unconstrained one in a single variable. Since this reduced objective turns out to be monotonic in the remaining design variable, its optimum value must be at the limit of its range. The corresponding optimum value of the other variable is then determined exactly from the true, not approximate, constraint. Since no model construction, experimentation, statistical analysis, or numerical iteration is needed, this procedure is recommended whenever the input-output relation is known to be a monotonic algebraic function.

To exploit convexity, a non-convex object can be represented by a boolean combination of convex components. A convex decomposition method of polyhedral objects uses convex hulls and set difference operations. This decomposition, however, may not converge. In this article, we formalize this decomposition method and find local cause of non-convergence.

Abstract A method is proposed for incorporating finite element stress analysis into the constraints of an optimization model. To reduce the number of computationally intensive finite element analyses, the more accurate FEA plate model is approximated by an algebraic beam model having an adjustable factor whose value is determined by comparing FEA stresses with the corresponding beam theory predictions. This factor compensates both for the inaccuracies of beam theory and the effect of stress concentration. The algebraic form is retained to permit application of powerful optimization techniques not applicable directly to finite element models. The optimization problem is thus reduced to the determination of the single factor by linear interpolation. When tested on Keith’s well-known welded cantilever problem, the method needs only three FEAs. Keith’s model is also shown to suffer from four errors, of which three are remedied here. Because of special problem structure, the resulting design is correct for three of the four design variables, but the length of the weld cannot be determined without a better weld stress model.

The well-known pictorial drafting technique of isometric drawing is here generalized in two ways, called “isoclinal” and “axial,” or collectively, “symmetric.” Although the isoclinal projection preserves the useful property of foreshortening adjacent edges equally, and the axial projection foreshortens adjacent normals equally, the directions for these projections usually differ from that for isometric projection. Formulas are derived for the isoclinal and axial directions and their foreshortening factors and rotation matrices. Although intended for three-dimensional computer-aided design, the symmetric projections also can be performed on a drawing board with a protractor. Graphic examples involving design of nonrectangular connectors for three skew structural members and adjacent plates in space are presented.

Abstract This paper discusses the problem of finding the intersection between two bicubic parametric patches. This is an important problem in Geometric Modeling since bicubic parametric patches are the most common design element in this field. A brief discussion of the existing approaches such as surface subdivision and curve tracing is given first. Next the algebraic solution to solve the intersection between cubic parametric curves is described in order to lay the foundations for an original algebraic method to solve the analog surface case.

In searching for an optimum by examining assignments of active constraints, one wishes to avoid assignments with no solution. A simple Maximal Activity Principle is stated to detect situations in which some subset of constraints has too few variables. The principle is illustrated on a well-known weldment design problem.

Error Linearization (EL), an iterative curve-fitting procedure recently proposed for designing minimum squared error four-bar function generating mechanisms, suffers from frequent instability. The cause seems to be the near singularity of a certain 3×3 matrix, which produces artificially large steps, usually toward designs with unrealistically short driver and follower. This degenerate case proves unfortunately to be the true global minimum. To bring this behavior under control, the coupler length, formerly regarded as an independent design variable, is made to depend on the driver and follower lengths. They are determined by solving a now well-conditioned 2×2 set of error linearization equations. In an example this Stabilized EL procedure (SEL) located five reasonable locally minimal designs which would have been missed by the unstabilized version.

Signomial programs are a special type of nonlinear programming problems which are especially useful in engineering design. This paper applies interval arithmetic, a generalization of ordinary arithmetic, to a dual equilibrium problem in signomial programming. Two constructive applications are considered. Application I involves uniqueness of local solutions; Application II involves existence and error bounds.

This paper treats a class of posynomial-like functions whose variables may appear also as exponents or in logarithms. It is shown that the resulting programs, called transcendental geometric programs, retain many useful properties of ordinary geometric programs, although the new class of problems need not have unique minima and cannot, in general, be transformed into convex programs. A duality theory, analogous to geometric programming duality, is formulated under somewhat more restrictive conditions. The dual constraints are not all linear, but the notion ofdegrees of difficulty is maintained in its geometric programming sense. One formulation of the dual program is shown to be a generalization of the chemical equilibrium problem where correction factors are added to account for nonideality. Some of the computational difficulties in solving transcendental programs are discussed briefly.