Ravi Samtaney

Mamdouh S. Mohamed, Anil N. Hirani, Ravi Samtaney

Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially on curved surfaces. This paper presents numerical evidences demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.

Fang Chen, Ravi Samtaney

Nonlinear wave interactions, such as shock refraction at an inclined density interface, in magnetohydrodynamic (MHD) lead to a plethora of wave patterns with myriad wave types. Identification of different types of MHD waves is an important and challenging task in such complex wave patterns. Moreover, owing to the multiplicity of solutions and their admissibility for different systems, especially for intermediate-type MHD shock waves, the identification of MHD wave types is complicated if one solely relies on the Rankine-Hugoniot jump conditions. MHD wave detection is further exacerbated by the unphysical smearing of discontinuous shock waves in numerical simulations. We present two MHD wave detection methods based on a convolutional neural network (CNN) which enables the classification of waves and identification of their locations. The first method separates the output into a regression (location prediction) and a classification problem assuming the number of waves for each training data is fixed. In the second method, the number of waves is not specified a priori and the algorithm, using only regression, predicts the waves' locations and classifies their types. The first fixed output model efficiently provides high precision and recall, the accuracy of the entire neural network achieved is up to 0.99, and the classification accuracy of some waves approaches unity. The second detection model has relatively lower performance, with more sensitivity to the setting of parameters, such as the number of grid cells N_{grid} and the thresholds of confidence score and class probability, etc. The proposed two methods demonstrate very strong potential to be applied for MHD wave detection in some complex wave structures and interactions.