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A Generalized Finite Difference-Based Fragile Points Method for Heat Conduction Problems in Non-Homogeneous Media
Smriti, Sundararajan Natarajan, Abhinash Malla
TL;DR
The paper addresses the lack of an FPM formulation for implicit interfaces in two-dimensional non-homogeneous materials. It develops a flux-corrected, meshless framework using discontinuous generalized-finite-difference shape functions and reports accurate, robust benchmark performance. The formulation is presented for steady heat conduction with specified function-space assumptions.
Problem
FPM had not previously been applied, to the authors’ knowledge, to implicit interfaces in two-dimensional differential equations within non-homogeneous materials.
Method
The method uses discontinuous pointwise trial and test functions from generalized finite differences, numerical flux corrections, and a piecewise heat-conduction formulation for heterogeneous media.
Results
Benchmark results demonstrate accurate and robust two-dimensional interface solutions with high accuracy and optimal convergence rates in the respective norms.
Takeaways & Limitations
FPM provides a meshless framework for modeling complex implicit interface geometries while naturally handling jumps in primary and secondary variables.
Takeaways & Limitations
The numerical formulation assumes trial and test functions drawn from H1-based spaces with prescribed Dirichlet conditions.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper extends FPM to implicit interfaces in two-dimensional non-homogeneous materials, addressing a stated gap in prior applications. It combines discontinuous point-based functions with numerical flux corrections and avoids specialized numerical integration techniques.
- 1. Introduction: The method targets implicit interfaces in two-dimensional differential equations for non-homogeneous materials, where interface discontinuities affect heat transfer and other physical processes.The authors identify this application as absent from prior FPM literature.
- 1. Introduction: FPM uses local polynomial trial and test functions constructed pointwise through generalized finite differences, with discontinuities at cell boundaries.Numerical flux corrections inspired by discontinuous Galerkin methods are used to enhance consistency.
- 1. Introduction: The framework establishes a mathematical treatment for both implicit and explicit interfaces as an alternative to finite element-based and mesh-free approaches.Its stated scope includes non-homogeneous materials and interface problems.
- 1. Introduction: The proposed formulation avoids specialized numerical integration techniques used in mesh-free methods and uses simple numerical integration for domain integrals.It also constructs interpolation shape functions without much computational complexity.
- 1. Introduction: The paper evaluates FPM accuracy, consistency, and robustness for interface problems through benchmark studies.The paper structure identifies benchmark demonstrations of global responses for different interface problems.
2. Governing equations
The governing problem is steady-state heat conduction in a two-region anisotropic heterogeneous domain with Dirichlet and Neumann boundaries and interface conditions. The formulation uses a piecewise material coefficient and finite-dimensional approximations over partitioned subdomains.
- 2. Governing equations: The domain consists of an inclusion Ω1 and surrounding matrix Ω2, with an external boundary divided into Dirichlet and Neumann portions.The regions are disjoint and together form Ω.
- 2. Governing equations: Steady-state heat conduction is governed by ∇·(β(x)∇u(x)) = Q(x), with prescribed temperature, heat flux, and continuity conditions at boundaries and the interface.The interface conditions impose continuity of temperature and normal flux.
- 2. Governing equations: The diffusion coefficient β(x) is piecewise constant across the material regions, while u(x) denotes temperature and Q(x) the internal heat source.The outward unit normal defines the boundary heat-flux condition.
- 2. Governing equations: Trial and test functions are selected from spaces Uh and Vh subject to the Dirichlet boundary conditions.The stated spaces use H1 regularity and impose u = gD on ΓD for the trial space and v = 0 on ΓD for the test space.
- 2. Governing equations: The domain is partitioned into non-overlapping elements, and the temperature is approximated by finite-dimensional nodal basis functions and unknown nodal temperatures.The weak form is then stated using a Bubnov-Galerkin formulation.
3. Point-based discontinuous trial and test functions for Fragile Points method
The Fragile Points Method constructs local, discontinuous polynomial trial and test functions over point-centered subdomains, using generalized finite differences to estimate unknown gradients from neighboring support points. These shape functions remain discontinuous across internal boundaries, requiring numerical correction when such inconsistencies appear.
- Domain partitioning: FPM partitions the domain into non-overlapping subdomains, each containing one internal point, supporting flexible treatment of complex geometries.Partitioning can use Voronoi, quadrilateral, triangular, or converted finite-element subdomains.
- Trial and test functions: Within each subdomain, FPM defines a simple local polynomial trial function from the solution value and gradient at its internal point.The formulation uses a 2D linear Taylor expansion centered at P0.
- Gradient construction: Generalized finite differences estimate the unknown gradient at P0 from neighboring support points selected by radius-based or nearest-neighbor strategies.The gradient is obtained by minimizing a weighted discrete L2 norm over the support points.
- Discontinuous shape functions: The resulting shape function is defined locally from P0 and its support points, so neighboring subdomains can have different values at shared boundaries.This produces discontinuous shape functions because no continuity condition is imposed on internal boundaries.
- Trial and test functions: The test function uses the same shape function as the trial function, yielding local, point-based polynomial functions with piecewise-continuous behavior across the domain.The construction is described within the weak Galerkin formulation.
4. Fragile Points formulation for governing equation
The formulation derives point-based stiffness and weak forms for FPM, then adds numerical flux corrections to restore consistency for discontinuous trial and test functions. Interior penalty fluxes and interface-specific support modifications complete the implementation for non-homogeneous problems.
- 4.1. Concept of point stiffness: The weak form is integrated by parts, then trial and test functions are substituted to derive each subdomain’s point stiffness matrix.The construction uses the shape function N and its gradient B for subdomain E0.
- 4.1. Concept of point stiffness: Constant diffusion coefficients and constant B simplify point-stiffness integration to an area-based expression, after which local matrices are assembled globally.FPM assembles point stiffness matrices rather than finite-element element stiffness matrices and admits spatially varying anisotropic tensors.
- 4.1. Concept of point stiffness: Discontinuous Galerkin weak forms become inconsistent and inaccurate with discontinuous FPM trial and test functions, causing failure of the patch test.The issue persists in the one-dimensional steady-state heat-conduction example and is especially visible for randomly distributed points.
- 4.1. Concept of point stiffness: Figures 6 and 7 compare errors for 101 uniformly and randomly distributed points, respectively, in the one-dimensional test.The cited results motivate numerical flux corrections for the observed inconsistencies.
- 4.2. Concept of numerical fluxes: Numerical flux corrections adapt the discontinuous Galerkin idea to FPM through a weak form built from the governing equation, test function, and Gauss divergence theorem.The resulting formulation introduces a numerical flux approximation on subdomain boundaries.
- 4.2. Concept of numerical fluxes: The flux formulation rewrites boundary contributions using averages and jumps over internal, Dirichlet, and Neumann boundaries.The boundary set is defined as Γ̃ = Γh ∪ ΓD ∪ ΓN, with average and jump operators specified for scalar and vector quantities.
- 4.3. Interior Penalty (IP) Numerical Flux and Primal form: The method selects Interior Penalty fluxes, whose penalty parameter is positive and whose boundary-dependent length scale differs between one- and two-dimensional settings.Substitution of the Interior Penalty flux into the weak form yields the primal FPM formulation.
- 4.4. Numerical Implementation: The primal form is implemented as a matrix system assembled from global thermal conductivity, point-stiffness, Dirichlet-boundary, and load contributions.At material interfaces, neighboring support sets exclude one another so their discontinuous trial functions become independent.
5. Numerical examples
The framework is evaluated on one-dimensional and two-dimensional interface problems with straight, circular, and star-shaped geometries. Across these benchmarks, numerical solutions agree with analytical solutions and exhibit second-order L2 convergence where reported.
- Benchmark setup: The numerical-examples section evaluates accuracy and convergence on interfaces with straight, circular, and complex star-shaped geometries.Relative error in the L2 norm is used to assess convergence.
- Patch test: The 1D patch test achieves machine-precision accuracy for both uniformly and randomly distributed points.Linear trial and test functions and numerical fluxes are used with η = 10.
- Straight interface: The straight-interface problem tests zero and non-zero temperature jumps under prescribed boundary conditions and compares numerical results with analytical solutions.The cases use δ = 0 and δ = 2, with 64 points for the numerical discretization.
- Circular interface: The circular-interface results show good agreement with analytical temperature profiles for both zero and non-zero jump cases.The relative L2 error is reported to converge at rate 2 under mesh refinement.
- Star-shaped interface: For the star-shaped interface, the numerical and analytical temperature distributions closely agree, with relative L2 error converging at rate 2.The test uses a non-zero primary-variable jump and a penalty coefficient η = 10.
6. Conclusions
The paper concludes that FPM provides a meshless framework for two-dimensional interface problems in heterogeneous continua. Its benchmark validation covers complex interface geometries, while the authors identify extensions to graded media, moving boundaries, and three-dimensional applications.
- Main conclusion: FPM is presented as a robust and accurate meshless framework for two-dimensional interface problems in heterogeneous continua.Discontinuous generalized-finite-difference shape functions handle jumps in primary and secondary variables without additional constraints.
- Validation: The framework is validated on benchmark problems with complex interface geometries represented implicitly via level sets.The conclusion specifically highlights the flexibility of implicit interface representation.
- Implications: Numerical results report high accuracy and optimal convergence rates in the respective norms.The authors suggest extending the flux-based meshless framework to graded porous media with TPMS architectures.