A p-adaptive high-order mesh-free framework for fluid simulations in complex geometries

TL;DR

This paper introduces a p-adaptive, high-order mesh-free method for fluid simulations in complex geometries, dynamically adjusting polynomial order to improve accuracy and efficiency, demonstrated in compressible flow in porous media.

Contribution

It develops a novel p-adaptive framework with a new error-based refinement indicator for mesh-free methods, enhancing accuracy and reducing computational costs in fluid simulations.

Findings

Potential to save up to 50% computational costs

Effectively captures highly non-linear regions

Maintains high accuracy with adaptive refinement

Abstract

This paper presents a novel p-adaptive, high-order mesh-free framework for the accurate and efficient simulation of fluid flows in complex geometries. High-order differential operators are constructed locally for arbitrary node distributions using linear combinations of anisotropic basis functions, formulated to ensure the exact reproduction of polynomial fields up to the specified p order. A dynamic p-refinement strategy is developed to refine (increase) or de-refine (decrease) the polynomial order used to approximate derivatives at each node. A new refinement indicator for mesh-free methods is proposed, based on local error estimates of the Laplacian operator, and is incorporated into the solution procedure at minimal added computational cost. Based on this error indicator, a refinement criterion is established to locally adjust the polynomial order p for the solution. The proposed adaptive mesh-free scheme is then applied to a range of canonical PDEs, and its potential is demonstrated in two-dimensional simulations of a compressible reacting flow in porous media. For the test cases studied, the proposed method exhibits potential to save up to 50% of computational costs while maintaining the specified level of accuracy. The results confirm that the developed p-adaptive high-order mesh-free method effectively captures highly non-linear regions where high-order approximation is necessary and reduces computational costs compared to the non-adaptive method, preserving high accuracy and solution stability.