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Present algorithms for the selection of design variables produce a single combination of design variables which results in a solution sequence of minimum difficulty. A new theory entitled solution mapping results in information about all optimal combinations of design variables. This theory gives detailed information about the structure of all underspecified systems of algebraic equations which do not contain persistent iteration. SCOPETypical systems of algebraic equations in steady state, macroscopic design problems have a few characteristics which are dominating factors in the difficulty of obtaining solutions. Usually the equations contain few variables (are sparse) and are nonlinear. Solution strategies for sparse systems of equations are, in general, easily obtained. Conversely, the nonlinearity of the equations increases the difficulty of obtaining a solution.A general characteristic of a system of algebraic design equations is that the number of variables is equal to or greater than the number of equations. When the number of variables is greater than the number of equations, a set of design variables must be assigned numerical values in order that the system be reduced to a determinant system with an equal number of equations and variables. Judicious selection of design variables can lead to a set of equations whose solution sequence encounters a minimum of difficulty.Solution sequences for systems of algebraic equations can be separated into three classes: one-to-one or acyclic, simultaneous, and iterative solution sequences. A one-toone, as opposed to simultaneous or iterative solution sequences, solves each equation, whether linear or nonlinear, in a one at a time technique and does not require any assumed solution points. Systems of equations which can be reduced to a determinant system with a one-to-one solution sequence are said to be without persistent iteration. Any system which cannot be reduced such that it has a one-to-one solution sequence is said to contain persistent iteration.For systems of algebraic equations without persistent iteration, all one-to-one solution sequences are considered as equal minimum difficulty solution strategies. This is a somewhat arbitrary objective function, since it might be less difficult to solve a set of linear simultaneous equations than a series of acyclic equations for nonlinear elements.Previous algorithms developed for the selection of design variables in systems of equations without persistent iteration are based on the pioneering work of Steward (1962) which introduced the concept of admissable output sets. An admissable output set has two properties: each equation contains exactly one output variable, and each variable appears as the output element of exactly one equation.The work by Lee et al. (1966) was one of the earliest attempts to base design variable selection on a mathematiical basis. This algorithm operates on a bipartite graph representation for a system of equations and operates only on systems of equations without persistent iteration. The...

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...

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