Simon Praetorius, Florian Stenger

Dune-CurvedGrid - A Dune module for surface parametrization

Introduction

Dune-curvedgrid - a dune module for surface parametrization. Explore Dune-CurvedGrid, a module for implementing curved surface geometries within the Dune framework for grid-based discretizations, featuring element-local parametrization.

Abstract

In this paper we introduce and describe an implementation of curved (surface) geometries within the Dune framework for grid-based discretizations. Therefore, we employ the abstraction of geometries as local-functions bound to a grid element, and the abstraction of a grid as connectivity of elements together with a grid-function that can be localized to the elements to provide element local parametrizations of the curved surface.

Review

This paper introduces Dune-CurvedGrid, a novel module designed to integrate curved surface geometries into the Dune framework for grid-based discretizations. This work addresses a critical need in computational science and engineering: the accurate and efficient simulation of physical phenomena on non-planar domains, which are prevalent in numerous real-world applications. By enhancing Dune's capabilities to natively support complex, curved geometries, this module promises to significantly broaden the applicability and utility of this established numerical framework, making it a valuable architectural contribution to the scientific computing community. The proposed methodology is centered on an elegant abstraction for handling geometric complexity. It posits geometries as local functions intrinsically bound to individual grid elements. This is paired with a grid abstraction that combines the fundamental connectivity of elements with a specialized grid-function. This grid-function is designed to be localizable to each element, thereby providing element-specific parametrizations of the underlying curved surface. This approach suggests a highly flexible and robust design, effectively decomposing complex global surface definitions into manageable local representations—a strategy well-suited for integration within existing finite element and finite volume methodologies. The introduction of Dune-CurvedGrid signifies a substantial enhancement to the Dune framework's capabilities. To fully realize and demonstrate its impact, a comprehensive paper should elaborate on critical aspects such as the ease of integration with other Dune modules, the specific range and complexity of curved surfaces supported, and a thorough analysis of the performance implications of this new geometric abstraction. Furthermore, illustrating its application through concrete examples tackling challenging real-world problems, discussing the robustness of the parametrization under mesh refinement, and considering comparisons with alternative approaches for handling curved geometries would greatly strengthen the overall contribution. This module appears to be a highly promising development for researchers and engineers working with complex geometric domains within the Dune ecosystem.

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