Smart cloud collocation: a unified workflow from CAD to enhanced solutions

Doctoral thesis (2022)

Computer Aided Design (CAD) software packages are used in the industry to design mechanical systems. Then, calculations are often performed using simulation software packages to improve the quality of the ... [more ▼]

Computer Aided Design (CAD) software packages are used in the industry to design mechanical systems. Then, calculations are often performed using simulation software packages to improve the quality of the design. To speed up the development costs, companies and research centers have been trying to ease the integration of the computation phase in the design phase. The collocation methods have the potential of easing such integration thanks to their meshless nature. The geometry discretization step which is a key element of all computational method is simplified compared to mesh-based methods such as the finite element method. We propose in this thesis a unified workflow that allows the solution of engineering problems defined by partial differential equations (PDEs) directly from input CAD files. The scheme is based on point collocation methods and proposed techniques to enhance the solution. We introduce the idea of “smart clouds”. Smart clouds refer to point cloud discretizations that are aware of the exact CAD geometry, appropriate to solve a defined problem using a point collocation method and that contain information used to improve locally the solution. We introduce a unified node selection algorithm based on a generalization of the visibility criterion. The proposed algorithm leads to a significant reduction of the error for concave problems and does not have any drawback for convex problems. The point collocation methods rely on many parameters. We select in this thesis parameters for the Generalized Finite Difference (GFD) method and the Discretization-Corrected Particle Strength Exchange (DC PSE) method that we deem appropriate for most problems from the field of linear elasticity. We also show that solution improvement techniques, based on the use of Voronoi diagrams or on a stabilization of the PDE, do not lead to a reduction of the error for all of the considered benchmark problems. These methods shall therefore be used with care. We propose two types of a posteriori error indicators that both succeed in identifying the areas of the domain where the error is the greatest: a ZZ-type and a residual-type error indicator. We couple these indicators to a h-adaptive refinement scheme and show that the approach is effective. Finally, we show the performance of Algebraic Multigrid (AMG) preconditions on the solution of linear systems compared to other preconditioning/solution methods. This family of preconditioners necessitates the selection of a large number of parameters. We assess the impact of some of them on the solution time for a 3D problem from the field of linear elasticity. Despite the performance of AMG preconditions, ILU preconditioners may be preferred thanks to their ease of usage and robustness to lead to a convergence of the solution. [less ▲]

Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions

in Archives of Computational Methods in Engineering (2019)

We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these ... [more ▼]

We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these methods as well as all data files including input files, boundary conditions, point distribution and solution fields, so as to facilitate future benchmarking of new methods. We review traditional methods and recent ones which appeared in the last decade. We aim to help newcomers to the field understand the main characteristics of these methods and to provide sufficient information to both simplify implementation and benchmarking of new methods. Some of the examples are chosen within a subset of problems where collocation is traditionally known to perform sub-par, namely when the solution sought is non-smooth, i.e. contains discontinuities, singularities or sharp gradients. For such problems and other simpler ones with smooth solutions, we study in depth the influence of the weight function, correction function, and the number of nodes in a given support. We also propose new stabilization approaches to improve the accuracy of the numerical methods. In particular, we experiment with the use of a Voronoi diagram for weight computation, collocation method stabilization approaches, and support node selection for problems with singular solutions. With an appropriate selection of the above-mentioned parameters, the resulting collocation methods are compared to the moving least-squares method (and variations thereof), the radial basis function finite difference method and the finite element method. Extensive tests involving two and three dimensional problems indicate that the methods perform well in terms of efficiency (accuracy versus computational time), even for non-smooth solutions. [less ▲]