[ishan_ae]

I am trying to understand how Reduced Order Modelling can be used for CFD applications.

Full disclosure before I dive in; I got overwhelmed by a ton of linear algebra mathematics found in all my literature sources.

Before I start looking into higher complexity methods that aim to perform ROM with parameter changes, I want to perform ROM on a simple transient case( or a dynamic system) for flow over airfoil or backward facing step.

Information of an example CFD solution:

• Transient in nature with the number of Time Steps(T) ∈ [1, T]

• Numerical domain with a number of Nodes(N) ∈ [1, N] where (N>> T)

These are the steps which I have understood will go into a ROM setup:

• Step 1. Solve the setup with appropriate boundary conditions for T

• Step 2. Assemble, the scalar of interest(let’s say Pressure) in a matrix 'A' of size 'm x n'. Each element of the matrix, 'A_ij' signifies the value of Pressure at a time step 'i' for node 'j':

• Step 3. Perform Singular Value Decomposition on matrix 'A' to give three different components. The first component will contain modes.

• Step 4. Select the most energetic modes based on strength of singular values obtained from the SVD(second component of SVD).

• Step 5. Generate a matrix '∅' will contain the basis vectors or the most energetic modes. This matrix represents the lower order sub-space that has the capability to describe the entire CFD solution within the simulated time range:

• Step 6. Project the higher-order space contained in matrix 'A' onto lower order space '∅' using some projection method( Galerkin projection most probably).

I am stuck at Step 6. How should I move forward from there ?

[BIRAJ]

I am also stuck at Step 6. Do you have any idea how to do Step 6?

[eaglemarine]

Galerkin is a rather complex method to evaluate appropriate weight coefficients. I suggest you check out the 'Flux Matching Procedure' method or variants thereof, it uses the Moore-Penrose Psuedo-Inverse and is not too hard to implement, and works great for incompressible flows and problems of forced convection. I can send you a link to my research thesis for the algorithm, or you can instead look at work done by Rambo and Joshi on the Flux Matching Procedure. Hope this helps.

edit: algorithms operate on observation matrix of steady state CFD RANS solutions, transient may not be as effective - depending on extent of variation in flow features with time.

[abhijithmoni]

@Gamhain MacCionaoith

I'm struck at the projection stage as well. Please share the link for your thesis. Thanks in advance.

Best
Abhijith

[dhfang]

I may think Dynamic Mode Decomposition can be helpful to your step 6.