Smart cloud collocation: geometry-aware adaptivity directly from CAD

TL;DR

This paper introduces a geometry-aware collocation scheme using smart clouds that simplifies transforming CAD geometries into computational models with adaptive refinement, applicable to various PDEs and geometries.

Contribution

It presents a novel integrated collocation scheme based on smart clouds that directly converts CAD geometries into point models with adaptive refinement, streamlining the discretization process.

Findings

Applicable to 2D and 3D geometries

Works with any PDE and point collocation method

Demonstrated on linear elasticity problems

Abstract

Computer Aided Design (CAD) is widely used in the creation and optimization of various industrial systems and processes. Transforming a CAD geometry into a computational discretization that be used to solve PDEs requires care and a deep knowledge of the selected computational method. In this article, we present a novel integrated collocation scheme based on smart clouds. It allows us to transform a CAD geometry into a complete point collocation model, aware of the base geometry, with minimum effort. For this process, only the geometry of the domain, in the form of a STEP file, and the boundary conditions are needed. We also introduce an adaptive refinement process for the resultant smart cloud using an \textit{a posteriori} error indication. The scheme can be applied to any 2D or 3D geometry, to any PDE and can be applied to most point collocation approaches. We illustrate this with the meshfree Generalized Finite Difference (GFD) method applied to steady linear elasticity problems. We further show that each step of this process, from the initial discretization to the refinement strategy, is connected and is affected by the approach selected in the previous step, thus requiring an integrated scheme where the whole solution process should be considered at once.