Finite element implementation based on explicit, Galerkin and Crank-Nicolson methods to phase field theory for thermal- and surface- induced martensitic phase transformations

In this paper, a nonlinear finite element approach is used to solve the time-dependent phase field or Ginzburg-Landau (GL) equation for phase transformations (PTs) at the nanoscale. The utilized Helmholtz energy includes a third-degree polynomial of the phase variable which results in a nonlinear dependence on the order parameter in the GL equation. The method of weighted residuals is used to derive the corresponding finite element formulation. Using the divergence theorem, the Laplace term in the GL equation is reduced to the first-order term which can represent both the isolated and variable surface energy boundary conditions, allowing for surface-induced phenomena. In order to apply the Dirichlet boundary condition, i.e., constant phase order parameter, the penalty method is used. Quadrilateral elements have been used in the Cartesian coordinate system, and irregular elements for complex geometries are transformed to regular ones by using natural coordinates. Four-point Gaussian integral has been used for integral calculations. The alpha family and both the explicit and implicit methods (Crank-Nicolson and Galerkin methods) are used for time discretization. Thus, the Picard and Newton-Raphson methods are used to linearize the nonlinear equation and their computational efficiencies are compared. In the explicit method, in order to calculate the inverse of the stiffness matrix, the lumping technique has been used which transforms it into a modified diagonal matrix, and this significantly reduced the computation time. In the implicit methods of Picard and Newton-Raphson, both the exact and numerical approaches were used to calculate the tangential matrix for the unknown order parameter and no difference in results between them was found. The convergence for both the Picard and Newton-Raphson methods was studied and compared. Cubic to tetragonal, thermal- induced transformation in NiAl is considered. The order parameter profile and the width and velocity of austenite (A)-martensite (M) interface are calculated and verified with previous works. Examples of thermal- and surface-induced PTs in nanosized samples without hole and with it are studied, presenting planar and nonplanar interfaces. The developed algorithm and code present a proper tool to solve more advanced phase field problems for PTs at the nanoscale with complex geometries and including mechanics effects and with considering complex interactions.