A Virtual Heat Flux Method for Simple and Accurate Neumann Thermal Boundary Imposition in the Material Point Method

TL;DR

The paper addresses the challenge of imposing Neumann-type thermal boundary conditions on nonconforming, evolving boundaries within the Material Point Method. It introduces the Virtual Heat Flux Method (VHFM), which uses a virtual domain and a volumetric representation of boundary flux to replace surface integrals, thereby avoiding boundary tracking or boundary particles. The method is unified for scalar, vector, and tensor Neumann conditions and is demonstrated to achieve accuracy comparable to nodal imposition in conforming cases and superior stability for nonconforming and moving boundaries, with second-order spatial convergence in 1D–3D benchmarks. VHFM's simplicity and robustness make it a promising tool for thermo-mechanical and multiphysics MPM applications, with clear pathways for extensions to coupled THMC problems, phase-change, and adaptive discretizations.

Abstract

In the Material Point Method (MPM), accurately imposing Neumann-type thermal boundary conditions, particularly convective heat flux boundaries, remains a significant challenge due to the inherent nonconformity between complex evolving material boundaries and the fixed background grid. This paper introduces a novel Virtual Heat Flux Method (VHFM) to overcome this limitation. The core idea is to construct a virtual flux field on an auxiliary domain surrounding the physical boundary, which exactly satisfies the prescribed boundary condition. This transforms the surface integral in the weak form into an equivalent, and easily computed, volumetric integral. Consequently, VHFM eliminates the need for explicit boundary tracking, specialized boundary particles, or complex surface reconstruction. A unified formulation is presented, demonstrating the method's straightforward extension to general scalar, vector, and tensor Neumann conditions. The accuracy, robustness, and convergence of VHFM are rigorously validated through a series of numerical benchmarks, including 1D transient analysis, 2D and 3D curved boundaries, and problems with large rotations and complex moving geometries. The results show that VHFM achieves accuracy comparable to conforming node-based imposition and significantly outperforms conventional particle-based approaches. Its simplicity, computational efficiency, and robustness make it an attractive solution for integrating accurate thermal boundary conditions into thermo-mechanical and other multiphysics MPM frameworks.

Figures (22)

• Figure 1: Illustration of material domains with varying geometries embedded in a regular background mesh. Except for the square domain in (a), all other geometries are partially or fully nonconforming with respect to the grid.
• Figure 2: Schematic illustration of the virtual domain $\bar{\Omega}$ surrounding the physical domain $\Omega$ and the associated virtual heat flux field.
• Figure 3: Conceptual illustration of surface nodes and surface particles. $P_1$: surface particle set; $P_2$: non-surface particle set; $N_1$: surface node set; $P_2$: non-surface node set.
• Figure 4: Transient heat transfer in a 1D semi-infinite Rod: (a) model geometry, initial and boundary conditions, (c) conforming boundary configuration, and (b) nonconforming boundary configuration.
• Figure 5: Transient heat transfer in a 1D semi-infinite Rod with both constant heat flux boundary (left column) and convective heat flux boundary (right column): (a-b) comparison of the temperature distributions at time instants of 0.1 s and 0.5 s obtained using MPM with VHFM and the analytical solution; (c-d) comparison of the absolute error for different boundary condition imposition methods. The zoomed-in views in (c) and (d) highlight the error near the origin, i.e., at the heat flux boundary, indicating that applying heat flux directly at the particle boundary results in significant errors near the boundary.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3