PRISM -- Piece-Wise Reusable Implementation of Solution Mapping

Research Objectives

The study of combustion through numerical simulations involves computational fluid dynamics and computational chemical kinetics. The responsibility of the chemistry is to calculate the heat production and the change in concentration of each chemical species as time progresses. This is done by solving a system of ordinary differential equations (ODE) but is often computationally expensive, as the number of chemical species can reach into the hundreds. We have developed a procedure (PRISM) by which the solution of the ODE is parametrized by a set of algebraic polynomial equations in chemical composition space (an artificial space in which the concentration of each chemical species is plotted along each axis). As a reaction proceeds, it traces a trajectory through this space.

Additionally, over the duration of a flame simulation, it is likely that a particular set of concentrations and temperatures will occur repeatedly at different times and positions. Hence, it is tempting for economic reasons to store the outcome of a calculation and retrieve the result when required. Therefore, we have designed an approach in which we store the information in a data structure until needed again.

Computational Approach

We partition chemical composition space into hypercubes, each adjacent to one another. As the reaction trajectory proceeds through composition space, we calculate the polynomials for a hypercube when it is entered for the first time.

Given a virgin input point, we determine the hypercube in which it lies and proceed to determine the polynomial expressions. The hypercube can be quite large, 0.25 to 0.5 of an order of magnitude in concentration per side. To parametrize the response of the ODE solver, the solver itself needs to be called repeatedly to provide the final solution at various data points within the hypercube. The optimal placement of these points is determined by the use of a factorial design method known as Response Surface Theory, by which the number of points is kept relatively low. Using these few data points, we form a set of polynomials from which we can obtain the time evolution of any input point within the hypercube. We now place the hypercube and polynomial information into a data structure (a combination of binary tree and double-linked list) for future re-use. As we proceed with our chemistry calculation, we merely have to evaluate an algebraic polynomial instead of solving a system of differential equations.

Accomplishment

We have tested our PRISM method on three diverse combustion simulations, a zero-dimensional burn, a 1D laminar flame, and a 2D turbulent jet. These are time-evolving initial value problems. The accuracy of the method is excellent, and there is good agreement with the ODE solution even after 200 ms.

Significance

We have shown that it is possible to do a turbulent jet simulation using a parametrized substitute for the ODE solver, in a fraction (one-tenth) of the previous CPU time.