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

TL;DR

This paper presents a Virtual Heat Flux Method (VHFM) for accurately imposing Neumann thermal boundary conditions in the Material Point Method, eliminating complex boundary tracking and improving robustness and efficiency.

Contribution

The paper introduces VHFM, a novel approach that constructs a virtual flux field to enforce boundary conditions without explicit boundary tracking or surface reconstruction.

Findings

VHFM achieves accuracy comparable to conforming node-based methods.

The method outperforms traditional particle-based approaches in robustness.

Numerical benchmarks validate the method's convergence and efficiency.

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.