使用波动方程式的拟真和互动式水面模拟课件

1、IntroductionWhat can fluid simulation do for us?What is the difficulty of the topic?ObjectiveReal-time and visual qualityInteract with usersHeight Field present water surface as a 2D function h(x,y) 2D array hi,jxyhhijHeight FieldPros: Process from 3D to 2D reduce computational timeCons: One locatio

2、n (x,y) maps to exactly one height no breaking wavesHello World!AlgorithmUse 2 arrays ANM , BNM of the same sizeInitialize Aij with arbitrary valuesInitialize Bij with 0loop forall i,j do Bi,j += (Ai-1,j + Ai+1,j + Ai,j-1 + Ai,j+1)/4- Ai,j; forall i,j do Bi,j *= 0.99; forall i,j do Ai,j += Bi,j; end

3、loopAlgorithmIt simulate the wave propagation phenomenontravel directionPros and ConsPros: 1). Simple 2). Computational time is independent of wave sourceCons: 1). Unnatural shape of wave propagation 2). Slow wave propagationdemoAddition of Sine WavesGoverning Equation h : height of water surface A

4、: amplitude : angular velocity t : time : phase : wave length f : frequencyAlgorithmsummate all sine wavesv = x f is a constant , and increases with tPhase increases constantly to animate wave propagationPros and ConsPros: Better look of shape of wave propagationCons: Computational time is proportio

5、nal to wave source numberdemoWave EquationGoverning Equationh: height of water surfacet: timec: velocityx: position of x directiony: position of y directionDampingWith dampingWhere k is damping factorFinite DifferenceWhere h is a insignificant and non-zero valueDiscrete Wave EquationDiscrete Wave EquationstabilityHow to choose ?Courant-Friedrichs-Lewy stability condition holds which says that information must not travel more than one grid cell per time step, i.e. c t xResultCPU : AMD Athlon 3500+ 2.2 GHZGraphics Card: Radeon