Abstract

This study thoroughly explains the process of implementing Adaptive Mesh Refinement into a Finite Element Analysis program. Using the error norm, a routine for improving an original mesh for an arbitrary 2D structural problem has been developed. By utilizing user-defined relative global error, an optimal mesh for a given level of approximation can be obtained in an iterative manner. The routine (called AMG v1.0) runs with the pre-/postprocessor GiD and the calculation module Calsef, but can easily be tailored to work with other FEA programs. AMG operates with two mesh optimality criteria, which have proven to refine the mesh in a non-oscillatory manner. The open structure, along with the Graphical User Interface, makes AMG suitable for educational and experimental purposes. For validation, six examples within the field of structural engineering have been solved, four of which have been compared favorably with analytical solutions.

The mesh optimality routines used are general, why AMG can be extended to handle more types of elements, 3D analysis, and to be applicable to other types of engineering problems. Above all AMG serves as a first step of implementing Adaptive Mesh Refinement procedures into GiD.

Abstract

In order to reduce the discretization error, in this paper, Richardson’s Extrapolation and Convergence Error Estimator were used to investigating the buckling problem convergence. The main objective was to verify the convergence order of the stepped column problem and to define a consistent moment of inertia at the point of variation of the cross-section. The variable of interest was the critical buckling load obtained by the Finite Difference Method. The convergent solution obtained errors less than 10^^-8, and this work showed that the best solution is not defined by excessive mesh refinement, but by the solution convergence analysis.

Abstract

Computational fluid dynamics is a cornerstone for the modern aerospace industry, providing important insights on aerodynamic analysis while reducing the need of expensive experiments and tests. Nevertheless, simulations of complex geometries are often performed on a discrete spatial domain too coarse to capture all relevant physical phenomena for the sake of lowering the computational cost. A consistent spatial discretization on so-called grids approximates the analytical solution of the partial differential equation with increasing number of discrete points. Such a grid refinement study is an expensive method to assess the general grid quality. This work shows that machine learning as a post-processing tool is capable of improving coarse grid simulations, even if not converged. The results show that three machine learning models varying in their complexity, namely the random forest, the neural network, and the graph neural network, are capable of finding patterns in coarse grid simulations. These patterns are used to predict the discretization error to approximate the field variables of interest of the corresponding fine grid simulation mapped onto the coarse grid. Initial training and testing is performed on the RAE2822 airfoil leading to corrected flow fields, improved surface integrals and coefficients, even when shocks are present. Additional tests are performed on the RAE5212 airfoil, showing the generalization limits of the trained models. The proposed method promises to reduce computational expenses while increasing the accuracy of the coarse grid results which works locally, e.g. it corrects the error for each cell individually and is therefore not restricted by the number of grid points. The presented results obtained by the machine learning models during post-processing are a promising baseline for more integrated developments, where the models will interact in a dynamic fashion with the flow solver to further improve coarse grid simulations.