Peridynamic solution of free boundary problems

This study presents an application of the PeriDynamic (PD) differential operator to solve free boundary problems. In these initial value problems, the unknown location of the free boundary must be determined as part of the solution. Such problems are also known as the Stefan problem which describes the mathematical model where a phase-change occurs in the case of melting or solidification. The characteristic feature of the phase-change process is the presence of a moving phase-separation interface.  Their mathematical models are non-linear and difficult for exact solutions.  In general, numerical techniques are employed to construct the solution.  Although the finite difference approach appears to be straightforward, it becomes challenging due to the presence of a phase change or sharp gradients.  The peridynamic differential operator enables the construction of the local form of the governing equations in their nonlocal form with an internal parameter.  The solution process involves neither a derivative reduction process nor a special treatment due to the presence of a moving a two-phase interface.  Its accuracy and robustness are demonstrated by comparing with the previous analytical solutions that concern the melting and solidification processes.  The PD results agree well with those of the analytical solutions.