2015/8/29 A Semi-Lagrangian CIP Fluid Solver Without Dimensional Splitting Doyub Kim Oh-Young Song Hyeong-Seok Ko presented by ho-young Lee EUROGRAPHICS 2008

Abstract USCIP : a new CIP method More stable, more accurate, less amount of computation compared to existing CIP solver Rich details of fluids CIP is a high-order fluid advection

Abstract Two shortcomings of CIP Makes the method suitable only for simulations with a tight CFL restriction CIP does not guarantee unconditional stability  introducing other undesirable feature This proposed method (USCIP) brings significant improvements in both accuracy and speed

Introduction Attempts for the accuracy of the advection Eulerian framework Monotonic cubic spline method CIP method (CIP, RCIP, MCIP) Back and force error compensation and correction(BFECC) Hybrid method (Eulerian and Largrangian framework) Particle levelset method Vortex particle Derivative particles

Introduction This paper develops a stable CIP method that does not employ dimensional splitting Original CIP Rational CIPMCIP StabilityUnstableMore stable than Origin CIP More stable than Rational CIP Computation time lower than MCIP high

Related Work “Visual simulation of smoke”, Fedkiw R., Stam J., Jensen H. W. Computer Graphics Monotonic cubic interpolation

Related Work CIP Methods “A universal solver for hyperbolic equations by cubic-polynomial interpolation”, Yabe T., Aoki T. Computer Physics Original CIP “Stable but non-dissipative water”, Song O.-Y., Shin H., Ko H.-S. ACM Trans Graph Monotonic CIP “Derivative particles for simulating detailed movements of fluids”, Song O.-Y., Kim D., Ko H.-S. IEEE Transactions on Visualization and Computer Graphics Octree data structure with CIP

Related Work Etc.. “Animation and rendering of complex water surfaces”, Enright D., Lossaso F., Fedkiw R. ACM Trans. Graph To achieve accurate surface tracking in liquid animation “Texure liquids based on the marker level set”, Mihalef V., Metaxas D., Sussman M. In Eurographics The marker level set method “Vortex particle method for smoke, water and explosions”, Selle A., Rasmussen N., Fedkiw R. ACM Trans. Graph Simulating fluids with swirls

Original CIP Method Key Idea Advects not only the physical quantities but also their derivatives The advection equation can be written as Differentiating equation (1) with respect to the spatial variable x gives

Original CIP Method The value is approximated with the cubic-spline interpolation

Original CIP Method 2D and 3D polynomials In 2D case

Original CIP Method 2D Coefficients

Original CIP Method Takes x and y directional derivatives Two upwind directions One starting point Not use the derivative information at farthest cell corner The method is accurate only when The back-tracked point falls near the starting point of the semi- Lagrangian advection

Original CIP Method Problem for simulations with large CFL numbers Stability is not guaranteed

Monotonic CIP Method To ensure stability Uses a modified version of the grid point derivatives Dimensional splitting

Monotonic CIP Method A single semi-Lagrangian access in 2D 6 cubic-spline interpolations Two along the x-axis for and One along the y-axis for and In 3D, 27 cubic-spline interpolations

Monotonic CIP Method Two drawback of MCIP method First, High computation time The computation time for MCIP is 60% higher than that of linear advection Second, Numerical error The split-CIP-interpolation requires second and third derivatives Must be calculated by central differencing This represents another source of numerical diffusion

Unsplit Semi-Lagrangian CIP Method To develop USCIP Go back to original 2D and 3D CIP polynomials Make necessary modifications Utilize all the derivative information for each cell 12 known values in a cell at the four corners at the four corners 2 additional terms

Unsplit Semi-Lagrangian CIP Method 2 extra terms The mismatch between The number of known values (12) and the number of terms (10) To overcome this mismatch Leat-squares solution Over-constrained problem Insert extra terms

Unsplit Semi-Lagrangian CIP Method Three principles for the two added terms Not create any asymmetry If is added, then must be added Contain both x and y Rotation and shearing The lowest order terms should be chosen To prevent any unnecessary wiggles The terms that pass all three criteria are and

Unsplit Semi-Lagrangian CIP Method To guarantee that the interpolated value will always be bounded by the grid point values A provision to keep the USCIP stable When the interpolated result is larger/smaller than the maximum/minimum of the cell node values, Replace the result with the maximum/minimum value Guarantees unconditional stability without over-stabilizing USCIP works on compact stencils No need to calculate high-order derivatives Reduce the computation time

Unsplit Semi-Lagrangian CIP Method USCIP requires fewer operations than MCIP Unsplit polynomial is more complicated But split-CIP involves multiple interpolations MCIP : 693 operations for a 3D interpolation USCIP : 296 operations for a 3D interpolation Only 43% of the total operation count needed for MCIP

Experimental Results Rigid Body Rotation of Zalesak’s Disk

Experimental Results Rising Smoke Passing Through Obstacles Generate realistic swirling of smoke Under complicated internal boundary conditions Without the assistance of vortex reinforcement mothods

Experimental Results Dropping a Bunny-shaped Water onto Still Water Generated complicated small-scale features Droplets Thin water sheets Small waves

Experimental Results Vorticity Preservation Test FLIP vs USCIP Noisy curl field

Conclusion Presented a new semi-Lagrangian CIP method Stable, fast, accurate result Two additional fourth-order terms Reflect all the derivative information Stored at the grid points The proposed technique ran more than Twice as fast as BFECC or MCIP Clearly less diffusive