Previously presented method of calculating local average gradients for solving partial differential equations (PDEs) is enhanced by using interpolating grid-points and triangular grids. The interpolating mesh provides finer computational grid, which is then used for solving the PDE. The combined use of the finer interpolating grid together with the original sparser grid is a two-grid method. By comparing the previous application of rectilinear grid for diffusion from initial point concentration to the new triangular two grid method, it was found that the application of triangular two-grid method improves stability of the solution and it provides more rapid convergence to the correct analytical solution.

1 Abstract The numerical solution of partial differential equations is often performed on a numerical grid, where the grid points are used for estimating the partial derivatives. The grid can be fully static as in Eulerian type of solution method, or the grid points can move during the solution, which is the case in Lagrangian type of method. In the current article, a numerical solution method is presented , where the grid points are located on iso-contours of the two-dimensional field. The method calculates the local movement of the iso-contours according to an evolution equation described by the PDE, and the solution proceeds by moving the grid points towards the calculated direction. Additional stability is obtained by setting the grid points to move along the iso-contour line. To exemplify the application of the method, numerical examples are calculated for the two-dimensional diffusion equation. 2 Introduction In the previous study [1], the movement of a phase interface was simulated with level-set type method using a physical science based model which takes into account the transformation strains and carbon partitioning and diffusion. In that study the connection to the Allen-Cahn equation [1] was made, which connected it's solution to the level-set approach. As a extension of this idea, the connection between a general partial differential equation (PDE) with first order time derivative and the level set formulation is investigated in the current study. The current approach is closely connected with the level-set method [2], which is often applied for simulations involving phase boundary movement. Level set methods usually perform the solution of a partial differential equation (PDE) in a grid that is not based on the points contained on the isocontours. The level-set method can be used in any types of grids, even in completely Eu-lerian framework [3, 4], where the computational grid does not need to move.

The numerical solution of partial differential equations is often performed on a numerical grid, where the grid points are used for estimating the partial derivatives. The grid can be fully static as in Eulerian type of solution method, or the grid points can move during the solution, which is the case in Lagrangian type of method. In the current article, a numerical solution method is presented , where the grid points are located on iso-contours of the two-dimensional field. The method calculates the local movement of the iso-contours according to an evolution equation described by the PDE, and the solution proceeds by moving the grid points towards the calculated direction. Additional stability is obtained by setting the grid points to move along the iso-contour line. To exemplify the application of the method, numerical examples are calculated for the two-dimensional diffusion equation.

A novel mathematical formulation is presented for describing growth of phase in solid-to-solid phase transformations and it is applied for describing austenite to ferrite transformation. The formulation includes the effects of transformation eigenstrains, the local strains, as well as partitioning and diffusion. In the current approach the phase front is modelled as diffuse field, and its propagation is shown to be described by the advection equation, which reduces to the level-set equation when the transformation proceeds only to the interface normal direction. The propagation is considered as thermally activated process in the same way as in chemical reaction kinetics. In addition, connection to the Allen-Cahn equation is made. Numerical tests are conducted to check the mathematical model validity and to compare the current approach to sharp interface partitioning and diffusion model. The model operation is tested in isotropic two-dimensional plane strane condition for austenite to ferrite transformation, where the transformation produces isotropic expansion, and also for austenite to bainite transformation, where the transformation causes invariant plane strain condition. Growth into surrounding isotropic austenite, as well as growth of the phase which has nucleated on a grain boundary are tested for both ferrite and bainite formation.