# New Approach of Computing the Intrinsic Low Dimensional Manifold Reaction Space Via Complex Variables

Friday, October 15, 1999 - 2:00pm - 2:30pm

Keller 3-180

Nelson Butuk (Prairie View A&M University)

Most combustion fuels (e.g the hydrocarbon fuels) react via reaction mechanisms consisting of several hundreds of species participating in more than a thousand chemical elementary reactions. To model this mechanism with such a large number of species using CFD is computationally prohibitive. Hence, it is desired to reduce the number of species to be modeled to a low number that is suitable for CFD computations and yet be able to accurately represent the full detailed mechanism.

Several reduction techniques have been used to reduce the mechanism. The method presented here is the method of Intrinsic Low Dimensional Manifold (ILDM) also know as the Mass and Pope mechanism reduction method. This method is the most promising so far, because one does not require a prior knowledge of the reaction mechanism in order to develop the reduced mechanism. Purely mathematical techniques are used. New techniques of implementing ILDM, will be discussed in this presentation.

There are a number of difficulties when numerically implementing the ILDM approach. One of the major difficulty is the construction of the Jacobian Matrix required to perform the eigenvalue analysis. The Jacobians have to be constructed analytically, which is quite time consuming and tedious for large-scale reactions consisting of more than 20 species. Numerical computations using finite differencing is inaccurate due to the non-linear nature of the reactions and the inability to choose the appropriate step size, h, which is not known a priori. Recently, we have demonstrated an accurate numerical technique that can be used to construct the Jacobians and is step-size independent. In this procedure, one simply converts the relevant code to run using complex variables. The partial derivatives of the Jacobian Matrix are loaded into the imaginary part of the evaluated function and can be outputted appropriately. It will be shown that the complex variable approach is step size independent and is quite accurate. This approach of using complex variables, will make it possible to reduce large-scale reaction mechanism more efficiently.

The ILDM method also requires the construction of look-up tables representing the low dimensional manifold. Recently, we have successfully demonstrate the use of Neural Networks (NN) to represent the lookup table. Details of this approach will also be discussed. Using this approach, a single file is all that is required to store the network weights representing the entire low dimensional manifold. A simple subroutine can then be used as a link between this file and the CFD code.

Several reduction techniques have been used to reduce the mechanism. The method presented here is the method of Intrinsic Low Dimensional Manifold (ILDM) also know as the Mass and Pope mechanism reduction method. This method is the most promising so far, because one does not require a prior knowledge of the reaction mechanism in order to develop the reduced mechanism. Purely mathematical techniques are used. New techniques of implementing ILDM, will be discussed in this presentation.

There are a number of difficulties when numerically implementing the ILDM approach. One of the major difficulty is the construction of the Jacobian Matrix required to perform the eigenvalue analysis. The Jacobians have to be constructed analytically, which is quite time consuming and tedious for large-scale reactions consisting of more than 20 species. Numerical computations using finite differencing is inaccurate due to the non-linear nature of the reactions and the inability to choose the appropriate step size, h, which is not known a priori. Recently, we have demonstrated an accurate numerical technique that can be used to construct the Jacobians and is step-size independent. In this procedure, one simply converts the relevant code to run using complex variables. The partial derivatives of the Jacobian Matrix are loaded into the imaginary part of the evaluated function and can be outputted appropriately. It will be shown that the complex variable approach is step size independent and is quite accurate. This approach of using complex variables, will make it possible to reduce large-scale reaction mechanism more efficiently.

The ILDM method also requires the construction of look-up tables representing the low dimensional manifold. Recently, we have successfully demonstrate the use of Neural Networks (NN) to represent the lookup table. Details of this approach will also be discussed. Using this approach, a single file is all that is required to store the network weights representing the entire low dimensional manifold. A simple subroutine can then be used as a link between this file and the CFD code.