Operationalizing Engineering Models Of Steady-State Equations Into Efficient Simulation Programs
Nicolas Rouquette · University of Southern California Digital Library · 2017
We characterize a tractable class of feedback decomposition problems to operationalize steady-state, lumped-parameter engineering models of algebraic equations into high-performance, numerical simulation software. An algebraic ordering graph captures how the parameter of the model algebraically depends on values of other model parameters. Using graph matching techniques, we construct a necessary and sufficient validity test for a set of n equations and n parameters in the sense that there exists a program to numerically compute a solution to these equations. The sufficiency of this test constitutes a useful and tractable refinement of the conventional condition that there be as many equations as unknown parameters. We operationalize this validity test into a prescriptive matching algorithm for constructing valid algebraic orderings. Our approach accounts for limitations in the symbolic solvability of algebraic equations and modeler-imposed restrictions. Parameters are allowed to be implicitly constrained (by all equations that refer to it), properly constrained (by a unique equation), or over-constrained (by multiple equations). Equations are allowed to properly constrain a unique parameter or over-constrain the values of all its parameters. Interpreting each path through the dependency graph as a sequence of numerical computations and each cycle as a potential feedback loop, we show how to exploit domain-specific properties of physical and algebraic feedback loops to make a hierarchical feedback analysis tractable. Tractability stems from exploiting the sparseness inherent to lumped-parameter modeling and the causality typical of hydro-thermal models. Decomposability is guaranteed when each feedback loop is identical to a strongly-connected sub-component of the algebraic ordering graph and the collection of all feedback loops must be mutually disjoint at each level of the hierarchy. Experimental results substantiate that better, faster, and cheaper equation-solving programs result from an inherent reduction of the number of parameters solved simultaneously. Comprehensive modeling and numerical simulation is made of a two-phase ammonia thermal controller. Further empirical evidence of the robustness of our approach is provided by comparing our models to Biswas' model and through a preliminary study of quantitative diagnosis.