Discrete-Time Collocation Scheme For Time-Dependent Equations And Numerical Simulation Of The Mold

The collocation method is a numerical method which search for the approximation solution of the operator function by satisfying pure interpolation condition for about thirty years,and it is widely used for solving both engineering and computing mathematics due to its ease of implementation,high-order accuracy and no integrals need be not evaluated.For the incompressible viscous flow,it obeys to mass conservation and momentum conservation.Its mathematics form are Navier-Stokes equation and the continuity equation:This kind of equations are widely used in fluid mechanics,casting engineering,aerodynamics and so on. Casting is an old industry .in general,the casting process is pour the molten liquid state metal into casting and then .so the use of computer numerical analysis is first centered in velocity profile and pressure profile. In this way ,it is very important for the solution of the velocity profile and pressure profile in the filling mold.For this reason,the SOLA method is first given ,based on it,the volume of fluid(VOF) method is offered to solve the free surface and then the author summary the possible velocity boundary conditions and give the equations they should accord to. The advantage of doing this is it makes the amount of work greatly reduce for the complexity in the calculations and in the same time improve the calculation velocity and more practical.The thesis is divided into two chapters. In the first chapter the discrete-time collocation method is used to solve the equations above,of them the Navier-Stokes equation is solved with three times hermit spline collocation method .It is showed that for sufficient small time step-size the solution to the scheme is unique and the error analysis is given too.In the second chapter the discrete schemes of the basic equations such as continue equations and Navier-Stokes equation are first given.the SOLA method is used to solve velocity and pressure and then the volume of fluid(VOF) method is offered to solve the free surface and then the author summary the possible velocity boundary conditions and give the equations they should accord to.