A Generalized Finite Difference-Based Fragile Points Method for Heat Conduction Problems in Non-Homogeneous Media
Abstract
This paper presents an enhanced formulation of the Fragile Points Method (FPM), a truly meshless approach for efficiently modeling implicit interfaces in two-dimensional differential equations involving non-homogeneous materials. The proposed framework eliminates the need for specialized numerical integration techniques and provides a systematic mathematical foundation for solving interface problems. Discontinuities in both primary and secondary variables across interfaces are naturally handled through the inherently discontinuous shape functions of FPM. Unlike conventional Galerkin methods, FPM employs simple, local, point-based polynomial trial and test functions constructed via a generalized finite difference approach. These discontinuous functions bypass the continuity requirements of standard Galerkin frameworks. To address the resulting inconsistency due to discontinuities, we incorporate numerical flux corrections inspired by the discontinuous Galerkin method. The proposed method is validated through several benchmark problems, demonstrating its efficiency and robustness.
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BibTeX
@misc{smriti2026generalized,
title = {{A Generalized Finite Difference-Based Fragile Points Method for Heat Conduction Problems in Non-Homogeneous Media}},
author = {Smriti and Sundararajan Natarajan and Abhinash Malla},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06891},
eprint = {2609.06891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06891},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Generalized Finite Difference-Based Fragile Points Method for Heat Conduction Problems in Non-Homogeneous Media
AU - Smriti
AU - Sundararajan Natarajan
AU - Abhinash Malla
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06891
UR - https://arxiv.org/abs/2609.06891
AB - This paper presents an enhanced formulation of the Fragile Points Method (FPM), a truly meshless approach for efficiently modeling implicit interfaces in two-dimensional differential equations involving non-homogeneous materials. The proposed framework eliminates the need for specialized numerical integration techniques and provides a systematic mathematical foundation for solving interface problems. Discontinuities in both primary and secondary variables across interfaces are naturally handled through the inherently discontinuous shape functions of FPM. Unlike conventional Galerkin methods, FPM employs simple, local, point-based polynomial trial and test functions constructed via a generalized finite difference approach. These discontinuous functions bypass the continuity requirements of standard Galerkin frameworks. To address the resulting inconsistency due to discontinuities, we incorporate numerical flux corrections inspired by the discontinuous Galerkin method. The proposed method is validated through several benchmark problems, demonstrating its efficiency and robustness.
ER -