1. Dimension Reduction of Combustion Chemistry Species reconstruction by Constrained Equilibrium Pre-Image Curve method (CE-PIC) Zhuyin (Laniu) Ren Steve Pope June 24 th, 2005

2. Outline Background and Introduction CE-PIC method Results

3. Dimension Reduction of combustion chemistry In combustion, the detailed chemical mechanism involves 50 – 1,000 species and 100-8,000 reactions In a numerical simulation, one often considers the following ODE system Dimension Reduction: Represent combustion chemistry in terms of n r reduced compositions r instead of by the n φ full composition φ Species reconstruction: Given a reduced composition r, provide a procedure to determine the corresponding full composition on the manifold φ m

4. Demonstration of CE-PIC method (H 2 /O system): Three Elements: H, O, N Six Species : H, O, H 2, OH, H 2 O, N 2 (Note: no O 2 ) Fixed T, p; Fixed elements 3 degrees of freedom Choose H 2 and O as the represented species

5. CE-PIC Method: Definition of the manifold Invariant manifold with constrained-equilibrium edge Reduced variables r; mapping from full composition Realizable region in r; convex polytope Consider point on boundary of realizable region in r Feasible region in full space; mapped to boundary point Constrained equilibrium composition; maximum entropy Manifold: generated by trajectories from the constrained-equilibrium composition corresponding to each boundary point in r

6. 1) CEM manifold

7. 2) CEM Boundary / Trajectory generated manifold: invariant manifold with constrained equilibrium edge From CEM boundary, we can construct a trajectory generated manifold

8. Criteria for dimension-reduction manifolds Existence, uniqueness and smoothness Ease of parameterization / independence of parameterization Attracting Invariance Realizability and consistency of conservation laws Nested hierarchy Ease of identification??

9. CE-PIC method How to perform species reconstruction? Given a reduced composition r, how to identify the corresponding point on the manifold Consider: The pre-image manifold (for the given r) The constrained equilibrium manifold (for all r) Their intersection – the constrained equilibrium pre-image curve Follow this pre-image curve to the boundary to identify the constrained-equilibrium pre-image point on the boundary From this boundary point, follow a trajectory forward until it intersects the feasible region

10. 3) CE-PIC method Given a reduced composition, we construct a pre-image curve, which is the intersection between the pre-image manifold and CEM manifold. The reconstructed point is taken to be the intersection between the feasible region and the trajectory, which is from the ending point of the pre-image curve (it is on the CEM boundary)

11. 3) CE-PIC method

12. Summary Identified an ideal manifold which is the trajectory generated manifold from the CEM boundary Invariant manifold with constrained equilibrium edge Construct the CE-PIC method to perform the species reconstruction locally => Work on the numerical implementation