Luis Ramírez

The accurate and efficient numerical solution of the Riemann problem is the basis of Godunov-type schemes. Approximate Riemann solvers are widely used for their efficiency, although they exhibit inaccuracies and instabilities in challenging regimes such as strong rarefactions or near-vacuum conditions. This work explores the use of deep neural networks (NNs) to address these limitations. We present two distinct data-driven frameworks: first, a NN-based solver trained to predict the exact solution of the Riemann problem, and second, a high-performance hybrid scheme. The hybrid approach uses the standard Harten–Lax–van Leer-contact (HLLC) Riemann solver as the main solver, enhanced with a computationally inexpensive, physics-based detector that identifies interfaces where the HLLC solution is likely to be inaccurate or to fail. At these interfaces, the scheme selectively uses the pretrained NN to ensure a more accurate solution. Through a series of benchmark tests, we show that the NN solver accurately reproduces the exact solution of the Riemann problem, but at a significant computational cost. In contrast, the proposed hybrid solver achieves a comparable level of accuracy to the NN solver, while it requires nearly the same computational cost as the standard HLLC solver.

In this study, a historical review of the Finite Element Method (FEM) and Molecular Dynamics (MD), widely used at the macro and nanoscale respectively is presented, emphasizing the actual parallelisms between their development and applications. After this historical introduction, where certain similarities between both methods are pointed out, different FEM-like methods are analyzed and compared as for first order analysis of structures at the nanoscale. Firstly, the Structural Mechanics (SM) approach is analyzed, where it is assumed that the use of Euler Bernoulli beam elements is equivalent to working directly from the force field. On the other hand, the Molecular Element Method (MEM), which provides the stiffness matrices directly from the potentials, is analyzed. Several analytical static cases are studied for the validation and comparison of both methods. Finally, it is shown that, other branch of methods such as Elastic Network Models (ENM) can be viewed as a particular sub-case of the MEM, or as truss-type finite elements. As an example, the analysis of SARS-CoV2 spikes vibrations is included, comparing with both experimental results and continuous models.

Given the small wavelengths and wide range of frequencies of the acoustic waves involved in Aeroacoustics problems, the use of very accurate, low-dissipative numerical schemes is the only valid option to accurately capture these phenomena. However, as the order of the scheme increases, the computational time also increases. In this work, we propose a new high-order flux reconstruction in the framework of finite volume (FV) schemes for linear problems. In particular, it is applied to solve the Linearized Euler Equations, which are widely used in the field of Computational Aeroacoustics. This new reconstruction is very efficient and well suited in the context of very high-order FV schemes, where the computation of high-order flux integrals are needed at cell edges/faces. Different benchmark test cases are carried out to analyze the accuracy and the efficiency of the proposed flux reconstruction. The proposed methodology preserves the accuracy while the computational time relatively reduces drastically as the order increases.

Mesh-based and particle methods were conceived as two different discretization strategies to solve partial differential equations. In the last two decades computational methods have diversified and a myriad of hybrid formulations that combine elements of these two approaches have been developed to solve Computational fluid dynamics problems. In this work we present a review about the meshless-FV family of methods, an analysis is carried out showing that the MLS-SPH-ALE method can be considered as a general formulation from which a set of particle-based methods can be recovered. Moreover, we show the relations between the MLS-SPH-ALE method and the finite volume method. The MLS-SPH-ALE method is a versatile particle-based method that was developed to circumvent the consistency issues of particle methods caused by the use of the kernel approximation. The MLS-SPH-ALE method is developed from the differential equation in ALE form using the partition unity property which is automatically fulfilled by the Moving Least Squares approximation.