The Graphics Simulation Of Incompressible Fluid Based On Orthogonal Grid

This paper mainly discusses the graphics simulation of incompressible fluid based on orthogonal grid. Animating fluids like water, smoke, and fire by physics-based sim-ulation is increasingly important in visual effects and is starting to make an impact in real-time games.In the first chapter, we state the process to derive the incompressible Navier-Stokes equations which can be used to describe the motion of incompressible fluid with low speed.In the second chapter, we introduce a method which can be used to solve the incom-pressible Navier-Stokes equation in computer graphics, called splitting method. This method has been proved to work very well in graphics. The solution of this equation can be splitted into four parts:the advection part, the gravity part, the viscous part, and the pressure/incompressible part. We introduce the concept of MAC grid which can be used to avoid the problem of non-trivial null space which is resulted from common central difference.In the third chapter, we discuss the algorithm for solving the advection part. In this chapter we will use the general fluid variable q, because we may be not only interested in the advection of velocity u, we may be also interested in the advection of other fluid variables. The algorithm of advection is denoted as advect(u,△t; q), which means advecting variable q in the velocity field u during At."semi-Lagrangian" method has been used in the algorithm advect(u, At; q).In the fourth chapter, we discuss the algorithm for solving the viscous part. We discuss the fluid with high viscosity and the fluid with low viscosity separately. Because we can consider the equation of the viscous part as diffusion part, the algorithm to solve the viscous part is denoted as diffuse(u, At).In the fifth chapter, we discuss the algorithm of the pressure/incompressible part. The algorithm of this part is called project(△t, u), the algorithm calculate and apply the correct pressure which makes the divergence of velocity u to be zero. The algorith-m which has been used to solve the pressure equation is the so-called MICCG(0), the complexity of the algorithm is O (n1/4). We talk about the method to deal with the bound-ary condition more accurately, and the pressure equation derived by it is consistent with the equation derived by the classical methods.In the sixth chapter, we discuss a special boundary condition:the periodic bound-ary condition. Under this condition, by using the fast fourier transform, the above algo-rithm is simplified. In the seventh chapter, we show the results of our simulation.