Modeling the growth and interaction of multiple dendrites in solidification using a level set method

A level set method is presented to study the growth and interaction of multiple dendrites in solidification. The method couples thermal and solute diffusion with propagation of multiple interfaces. A single signed distance function is used to track the solid-liquid interface with the aid of markers, the value of which is the orientation angle, for identification of different crystals. The problem of evolving multiple crystal interfaces is reduced to two tasks: (1) tracking one level set variable (signed distance function) and (2) determination of the marker for a newly solidified finite element nodal point. Tracking a single level set variable is implemented by solving the level set equation with interface velocity computed from an extended Stefan equation using the marker information (crystal orientation). Determination of the marker for a newly solidified finite element nodal point is implemented by using an algorithm modified from the fast marching technique. Both of these two steps are computationally efficient and the approach is suitable for incorporating effects of multiple crystals. Convergence and accuracy of this approach are demonstrated by using different grid spacings and comparing with results obtained from the multi-phase level set method. A parametric study is performed to investigate the effects of solidification speed and thermal gradient on the resulting solidification microstructure pattern. Numerical results of columnar-to-equiaxed transition (CET) qualitatively agree with an analytical estimation and are similar to previous numerical results obtained using a phase field method. A convergence study is performed to determine the appropriate grid spacing for numerical simulation. At lower surface tension, CET occurs at a lower thermal gradient for a giving solidification speed. Secondary dendrite formation is more apparent with lower surface tension. The differences and similarities between the three-dimensional and twodimensional growth results are analyzed. Randomness in crystal orientation and required under-cooling for nucleation are modeled and found to have a great effect on the microstructure pattern. The efficiency of the present approach is finally demonstrated with an example that includes the growth of hundreds of crystals with consideration of randomness effects. Preprint submitted to Elsevier Science 27 April 2007