Current project
Multiscale Modeling Using Goal-oriented Adaptivity and Numerical Homogenization
Step and Flash Imprint Lithography (SFIL) is a novel imprint lithography process for fabricating microchips. The mechanical properties of polymers formed during the process are modeled using molecular statics. Numerical solution of such a molecular statics base model, which is assumed to describe the microstructure completely, is computationally very expensive. This is due to the problem size - on the order of millions of degrees of freedom (DOFs). Rapid variation in material properties, ill-conditioning, nonlinearity, and non-convexity make this problem even more challenging to solve.
To approximate the base model, we generate a sequence of coarse meshes and optimally homogenize the material properties on each mesh element. The obtained effective properties depend on the mesh, the load, and the local interpolation operator from coarse to fine mesh. This requires solution of a continuous-time Lyapunov equation on each element. Use of the adjoint solution provides local error estimates in the quantity of interest. The estimates indicate which elements should be h-refined for the next mesh.
Critical to the efficiency of the local homogenization is the computation of Moore-Penrose pseudoinverse of element stiffness matrices without using the Singular Value Decomposition. This is possible either by using Tikhonov regularization of the local operator or by using the knowledge of its null-space. Here are some lattices in equilibrium on homogenizing at different levels.
Here are some relevant files.
Previous projects
Numerical Solution of the Stefan Problem
The general problem of predicting rates of solidification and/or melting is known as the Stefan problem. The position of the moving boundary/interface is unknown a priori and this makes the problem nonlinear and not solvable analytically except in some simple one-dimensional cases.
In this project, the basis of the enthalpy model for multidimensional phase change problems is demonstrated and subsequent numerical applications are carried out. Its equivalence to the traditional formulation (without using enthalpy as another dependent variable) is proved. The model is verified using the analytical solutions of a generalized one-dimensional problem and a quasi-two-dimensional problem. The comparison shows excellent agreement with the analytical solution. Results are also presented for a two-dimensional transient laser melting problem.
The results show that change in volume upon phase change has minor effect on the rate of melting/solidification if Stefan number (roughly the ratio of sensible heat and latent heat) is small. Even for large Stefan numbers neglecting the volume change does not affect the heat transfer rate much.
