A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field $\mathbf{a}$ and a gauge variable $\phi$, $\mathbf{u} =\mathbf{a}+\nabla\phi$, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field $\mathbf{a}$ are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field $\mathbf{a}$ will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint ${\triangle t} / {\triangle x} \le C$. We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.

The accurate control of process parameters is particularly important for the growth of high quality structures during laser cladding. Melt pool size and cross sectional area are key process characteristics that can be controlled to allow the precise deposition of complex features. A feasible mathematical model is established to describe the interaction between laser and powder particles and to predict and control the cross sectional area. Also, a vision-based sensing system is proposed to detect the transient shape and size of the melt pool. The system focuses on the measuring of the geometric parameters of the melt pool by using a digital image processing method and thus giving researchers the parameters on-line, which gives us a necessary tool to gain insight into the cladding process and to optimize the material quality

Systematically generalizing planar geometric algorithms to manifold domains is of fundamental importance in computer aided design field. This paper proposes a novel theoretic framework, <i>geometric structure</i>, to conquer this problem. In order to discover the intrinsic geometric structures of general surfaces, we developed a theoretic rigorous and practical efficient method, <i>Discrete Variational Ricci flow</i>.

A Deuflhard-type exponential integrator sine pseudospectral (DEI-SP) method is proposed and analyzed for solving the generalized improved Boussinesq (GIBq) equation. The numerical scheme is based on a second-order exponential integrator for time integration and a sine pseudospectral discretization in space. Rigorous analysis and abundant experiments show that the method converges quadratically and spectrally in time and space, respectively. Finally the DEI-SP method is applied to investigate the complicated and interesting long-time dynamics of the GIBq equation.

Laser cladding has been widely used in direct laser fabrication and laser refabrication for metal parts. It is a high precision and complicated process due to multiple process parameters involved. These parameters are strongly tied to each other and their effects are still far from being completely understood. In order to make the optimal choice, a feasible mathematical model is established to describe the interaction between laser and powder particles and to predict and control the cross sectional area. Additionally, analysis concentrating on the effect of the various parameters on the cross sectional area is made. Results show that laser power, powder feed rate, scan speed and radius of focused powder stream are the main factors. Correlations between main process parameters and cross sectional area of an individual clad track have been found. The model also provides predictions of range of process parameters for melting powder to form specific cross sectional area, which will guide the experiment and greatly reduce the cost of the experimental investigations.