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

ABSTRACT

In the material point method (MPM), accurate enforcement of thermal boundary conditions, particularly convective heat flux boundaries, is crucial for the reliable simulation of thermally coupled multiphysics processes, such as natural heat‐convection phenomena on Earth. However, the MPM typically employs a fixed, regular background mesh, whereas material boundaries are often complex and evolve over time. As a result, the boundaries do not generally align with the mesh, making boundary tracking and the consistent application of boundary conditions challenging. This study introduces a virtual heat flux method (VHFM) that enables the simple and accurate imposition of nonconforming heat flux boundary conditions in the MPM. The core idea is to construct a virtual flux field that satisfies the prescribed boundary conditions and converts boundary integrals into volume integrals. This approach eliminates the need for explicit boundary tracking while maintaining ease of implementation and computational efficiency. A unified formulation of the virtual flux field is further developed, allowing the method to be readily extended to general Neumann boundary conditions. Through a series of numerical experiments, the proposed method is shown to achieve high accuracy and convergence across a wide range of nonconforming boundary configurations, including curved, moving, and evolving boundaries. The VHFM provides a potential way for Neumann thermal boundary imposition in thermo‐mechanical and other multiphysics MPM frameworks.