During the CMP process, a wafer is rotated and pressed face down against a rotating polishing pad. Polishing slurry is delivered on the top of pad continuously and forms a thin lubricating film between the wafer and the pad. In this study, a three-dimensional slurry flow model based on a generalized Reynolds equation is developed, which can apply to a rough pad with the compressibility of the pad, and the multi-grid method is used to reduce computational time. According to the force and moment balance equations, the tilted angles and the slurry film thickness can be evaluated. When the pad surface is rough, the squeeze term differentiated by time should be considered in this model due to the rotation of the pad. The influences of applied load, pad speed, wafer speed, pad compressibility, and surface roughness pattern on the tilted angles and the slurry film thickness are investigated. Results show that the variation of the tilted angles becomes more significant for the anisotropic than that for the isotropic during the rotation of the pad. And the slurry film thickness at the center of the wafer increases as applied load decreases or pad speed increases or wafer speed decreases or the compressibility of the pad increases.

We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.