Approximate Jacobian Eigenanalysis For Unstructured Mesh Optimization of Finite-Volume Simulations

Abstract:

This paper presents a novel method for approximate eigenanalysis of large linear systems, with a specific focus on unstable eigenvalues in the Lyapunov sense. The method utilizes residual vector analysis, principal component analysis, and dynamic mode decomposition to identify unstable solution modes efficiently. Outlier detection models are employed to find the problematic cells on the unstructured mesh in the case of residual vector and principal component analysis while the dynamic mode decomposition points directly to the control volumes of interest without such help. By extracting and assembling corresponding rows of the Jacobian matrix, the large system is projected on a new matrix with significantly fewer degrees of freedom, leading to improved computational efficiency. The eigenvalue problem is then solved on this smaller matrix, and the obtained results are utilized for mesh optimization, enhancing the local stability of the dynamic system. By addressing the challenges of computational complexity and automation faced by previous methods, this novel approach offers a comprehensive and automated solution for the eigenanalysis of large linear systems. The potential impact of this method extends to various fields, providing a more efficient eigenanalysis process and opening new avenues for exploring and optimizing complex systems.