Structural analysis and solution of systems of algebraic design equations

Abstract

A new method for expressing the structure of a system of equations is developed using a type of occurrence matrix entitled the functionality matrix. The functionality matrix indicates not only the Occurrence of variables in equations but also the functional form in which they occur. Since the difficulty of solving an equation for a variable is related to its functional form, analysis of the functionality matrix provides explicit information on the difficulty of solution of the equation.A methodology for the solution of design problems by digital computers is described. This methodology operates on the functionality matrix which describes the set of design equations. Algorithms using this methodology interact and guide the designer in an efficient selection of design variables and redundant equations. SCOPEDesign equations arc predominantly algebraic, often nonlinear, and commonly sparse in that each equation contains only a few of the system variables. The major difference between a simulation study and a design study is the parameters specified. The input, output, or combinations of input and output specifications may be known in a design study. Specified parameters may change from one solution to another. Equipment parameters and unspecified input and output parameters are the result of the calculations. All input and equipment parameters must be specified in a simulation study and the output parameters are calculated. The structure for simulation studies is set. This allows simulation studies to be formulated into a modular approach (Motard et al., 1975).Analysis of the characteristics of design problems demonstrates that a need exists for a methodology for analysis of systems of design equations. This methodology must be oriented a t the equation, not modular, level since the calculation path is not fixed for a design problem. This publication reports the development of a method for analyzing systems of equations which characterize design problems. Techniques for selecting design variables and redundant equations are given and methods for obtaining the solution to the system of equations efficiently are presented. CONCLUSIONS AND SIGNIFICANCEA new, flexible methodology is developed for the structural analysis and solution of large systems of algebraic equations. Application is made for the solution of design problems. The development which links the structural analysis with problem solution is the functionality matrix. The functionality matrix allows for the structural analysis and solution of systems of unordered, underconstraincd systems of equations which may contain redundant equations.Efficiency in the solution of large systems of algebraic problems is gained by the use of the solution methodology presented in this work. No ordering of the equations or logic is necessary in programming the solution of a system of equations. STRUCTURAL ANALYSISStructural analysis is the study of the interrelationships and interactions among the various components that form a system. The goal is the attainment of the simp...

On the Optimum Choice of Decision Variables for Equation-Oriented Global Optimization

“…Several authors have recognized that by specifying different sets for the decision variables, nonlinear sets of different numerical difficulty will occur. The pioneering work of Lee et al for the selection of design variables, using bipartite graph structuring, works very well if an acyclic information structure can be obtained but fails otherwise. , Rudd and Watson 18 present this algorithm in a more familiar form of an occurrence matrix manipulation. Book and Ramirez 23 developed an algorithm, which produces the entire solution set for the decision variables, but only for systems which may be described by an acyclic information structure.…”

“…In practice, the algorithm is applied in a series of steps, where the designer guesses a structurally valid decision variable set, obtains the solution path, and guesses another set of decision variables if the output subsystems are singular or nearly singular. Thus, this algorithm has the merit of avoiding solution paths, which may be either indeterminate or very ill-conditioned, but on a single pass produces a unique combination of decision and output/tearing variables …”

“…Although these criteria are easy to apply, they do not often produce a unique set of decision variables, even for small-scale problems; viz., many sets that meet these criteria are possible. However, some sets produce a sequential solution for the state variables, which is generally, but not always, a better path than forcing a simultaneous solution. , A sequential solution may, however, not be possible, even with small-scale problems, because of a recycled flow of information.…”

“…It is clear that the structure of the system equations, viz., their ordering or grouping, may have a strong impact on the solution process. ,− Several techniques are available to reduce the effort in solving the nonlinear system, viz., eliminate variables in equations with only one variable and move as many nonlinearities as possible from the equality constraints to the objective function. ,, In any case, the efficiency of these processes is highly dependent on the structure of the system equations. Because this structure may change drastically by specifying different independent variables for the degrees of freedom, this feature may be exploited in order to efficiently solve systems of nonlinear equations, and this is the subject of the present paper.…”

“…Because this structure may change drastically by specifying different independent variables for the degrees of freedom, this feature may be exploited in order to efficiently solve systems of nonlinear equations, and this is the subject of the present paper. Although this topic has been the subject of significant past studies, − it remains an important one that is, to the authors' knowledge, not yet resolved satisfactorily in the literature.…”