Efficient Meshless Approximation of Partial Differential Equations Using Spatially Dependent Node Regularity

Core Concepts

The total computational cost of the meshless approximation is reduced by using a less robust but computationally more efficient meshless setup on regular nodes and a more expensive, stable setup on scattered nodes.

Abstract

The paper proposes a hybrid regular-scattered meshless discretization method to efficiently solve partial differential equations. The key highlights are:

A dimension-independent node placing algorithm (HyNP) is presented that covers regions near geometric details with scattered nodes and the rest of the domain with regular nodes, supporting variable node density (h-refinement).

The proposed hybrid regular-scattered meshless discretization is demonstrated by solving non-linear natural convection problems in 2D and 3D, as well as a 3D contact problem (Boussinesq's problem).

The hybrid discretization reduces the total computational cost compared to a fully scattered meshless discretization, while maintaining accuracy. This is achieved by using a computationally more efficient meshless setup on regular nodes and a more expensive, stable setup on scattered nodes.

The performance of the HyNP algorithm is evaluated in terms of separation distance and maximal empty sphere radius, showing that the transition from regular to scattered nodes does not significantly affect the discretization quality.

Detailed analyses are provided on the computational efficiency and accuracy of the numerical solutions obtained using the spatially-variable node regularity, including the impact of different levels of h-refinement aggressiveness and scattered layer widths.

Stats

The total number of discretization nodes for the natural convection problem in the irregular 2D domain was around 95,000 for the hybrid approach and 83,500 for the fully scattered approach.
The execution time for the natural convection problem in the irregular 2D domain was around 7.2 hours for the hybrid approach and 14.2 hours for the fully scattered approach.
The total number of discretization nodes for the natural convection problem in the irregular 3D domain was around 88,725 for the hybrid approach, 96,800 for the regular approach, and 81,218 for the fully scattered approach.
The execution time for the natural convection problem in the irregular 3D domain was around 5.1 hours for the hybrid approach, 3.1 hours for the regular approach, and 7.9 hours for the fully scattered approach.

Quotes

"The total computational cost of the meshless approximation is reduced by using a less robust but computationally more efficient meshless setup on regular nodes and a more expensive, stable setup on scattered nodes."
"A dimension independent node placing algorithm that covers regions near geometric details with scattered nodes and the rest of the domain with regular nodes and supports variable node density (h-refinement) is presented."

Key Insights Distilled From

Spatially dependent node regularity in meshless approximation of partial differential equations

by Miha... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15345.pdf

Deeper Inquiries

How can the proposed hybrid regular-scattered meshless discretization be extended to handle time-dependent problems or multi-physics applications

The proposed hybrid regular-scattered meshless discretization can be extended to handle time-dependent problems or multi-physics applications by incorporating a time-stepping scheme into the solution procedure. For time-dependent problems, the discretization nodes can be updated at each time step based on the evolving solution field. This would involve recalculating the weights for the approximation of differential operators at each time step to account for the changing solution. Additionally, for multi-physics applications, the hybrid approach can be adapted to handle the coupling of different physical phenomena by incorporating additional equations and variables into the system. This would require extending the discretization method to accommodate the additional physics and ensuring consistency in the solution procedure across all coupled equations.

What are the potential limitations or drawbacks of the HyNP algorithm in terms of handling highly complex or irregular domain geometries

One potential limitation of the HyNP algorithm in handling highly complex or irregular domain geometries is the computational cost associated with generating and positioning the nodes. In cases where the domain has intricate features or irregular boundaries, the algorithm may require a large number of iterations to populate the domain with nodes, especially if a high level of refinement is needed. This can lead to increased computational overhead and longer processing times. Additionally, the algorithm's effectiveness may be limited in cases where the irregularities in the domain are too complex for the algorithm to efficiently determine the optimal node placement strategy. In such scenarios, manual intervention or additional preprocessing steps may be required to ensure an appropriate node distribution.

Could the spatially-varying node regularity concept be applied to other numerical methods beyond meshless approaches, such as finite element or finite volume methods, to achieve similar computational efficiency gains

The concept of spatially-varying node regularity can be applied to other numerical methods beyond meshless approaches, such as finite element or finite volume methods, to achieve similar computational efficiency gains. In finite element methods, for example, the node regularity can be adjusted based on the local geometry and solution requirements to optimize the accuracy and efficiency of the discretization. This can involve adapting the mesh refinement strategy to concentrate nodes in regions of interest while maintaining a coarser mesh in less critical areas. Similarly, in finite volume methods, the node regularity can be varied to improve the resolution of the solution in specific regions without increasing the overall computational cost. By incorporating spatially-varying node regularity concepts into these traditional numerical methods, researchers can enhance the computational efficiency and accuracy of simulations across a wide range of applications.