Fluid Simulation and Reaction-Diffusion Textures on Surfaces Maria Andrade Luiz Velho (supervisor) Technical Report TR Relatório Técnico

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1 Laboratório VISGRAF Instituto de Matemática Pura e Aplicada Fluid Simulation and Reaction-Diffusion Textures on Surfaces Maria Andrade Luiz Velho (supervisor) Technical Report TR-0-0 Relatório Técnico February Fevereiro The contents of this report are the sole responsibility of the authors. O conteúdo do presente relatório é de única responsabilidade dos autores.

2 Fluid Simulation and Reaction - Diffusion Textures on Surface Maria Andrade mcosta@impa.br Supervisor: Luiz Velho February 0 Abstract: In this note we show how to simulate fluids on surfaces using Navier- Stokes equations and generate texture on surfaces with the reaction-diffusion model due to Gray and Scott. The parametrization of Catmull-Clark surfaces is used to obtain a good discretization of the operators defined on surfaces which implies nice results on simulation and textures. Introduction Firstly, to simulate fluid (see the figure ) on surface we need to get a parametrization for this surface, one good candidate is a Catmull-Clark surface, which is smooth almost everywhere [5]. Moreover, it is necessary to compare vectors in different points on the surface to evaluate velocity, force, etc. Figure : Simulation of fluid on closed surfaces. To solve the Navier-Stokes equations on surfaces we follow these steps: take a parametrization of a Catmull-Clark surface; calculate a metric that depends on its tangent vectors and use this metric to get the necessary differential operators; solve the Navier- Stokes equations on a discretization of the surface, dealing with the transition of the fluid between the patches that form the surface. This research on fluid simulation on surfaces was performed togheter with Leonardo Carvalho and Dalia Bonilla based on [8] during the Trimester Program on Computacional Manifolds and Applications, see mcosta/fluidsurf/

3 In the second part of this note it is shown how to generate texture on a surface with the reaction diffusion model due to Gray and Scott. More precisely, to solve reactiondiffusion sytems on surfaces the basics steps used here are: use the parametrization of a Catmull-Clark surface and calculate only the Laplacian operator defined on this surface. In this case it is simpler than the simulation fluid, because it is not necessary to calculate the gradient and divergent operators. Previous Work In recent years, many researches were developed to use the Navier-Stokes equations for fluid simulation. In particular, these investigations have contributed for many areas. For example, special effects industry. In this way, Jos Stam [6] had proposed a stable algoritm called stable-fluid that solves the Navier-Stokes equations for three-dimensional fluids, which is fast, stable and it is the basis to simulate smoke, water and fire, but this process is dissipative. Fedkiw, R. et al. [7] did a change in the discretization, which improved the problem with dissipation. More recently, algorithms were developed to simulate fluids on arbitrary surfaces [8]. In this case, one important point is to get information about the metric defined on surfaces, because to solve Navier-Stokes equations we need to calculate differential operators: gradient, Laplacian and divergent, which depend on the metric and if this is ignored, then there will be visible artefacts (see figure ). Figure : Ignores the metric (left) [8]. In the case of reaction-diffusion systems, Turing [] in 95 had described at the first time a chemical mechanism for pattern formation which is called reaction-diffusion. Greg Turk [4] used this mechanism to generate textures on surfaces. Moreover, Sanderson et al. [9] used many reaction-diffusion models for textures synthesis. Bajaj, C. et al. [] gave an approach to solve reaction-diffusion sytem on surfaces using a Galerkin based finite element methods.

4 Subdivision Surfaces In computer graphics it is usual to create a coarse polygon mesh that aproximates the shape of a desired surface. To obtain the smooth surfaces we subdivide each polygonal face into smaller faces that better approximate the smooth surface and in the limit of subdision we get the smooth surface. In this work it is used the Catmull-Clark subdivision (see figure 3), which is a generalization of bi-cubic uniform B-spline for arbitrary meshes. This process generates limit surfaces that are C continuos everywhere except at extradordinary vertices where they are C continous, in particular at each point on a surface the tangent plane can be defined. Figure 3: Example of a Catmull-Clark subdivision surface ( c wikipedia). One vertice is called extraordinary if its valence in not four. For example, in figure 4 the color blue corresponds to the regions that have extraordinary vertices. Figure 4: Paramettrization of a Catmull-Clark surface [5]. 3

5 Operators on Surfaces In order to solve the differential equations on a surface we need to calculate some differential operators on the surface. In our case, we need the Laplacian, gradient and divergent operators [3]. Given a parametrization X(x, x ) of a surface, we obtain the X X metric gij = h x, i, i, j =,, the matrix of the metric g = (gij ) and the inverse of i xj g g g, g = g = and g =, the metric is denoted by g = (g ij ), where g = G G G here G = det(gij ) = g g g > 0. The differential operators Laplacian, gradient and divergent on the surface are, respectively, given by ϕ= G xi 3 i,j ϕ Gg xj ϕ, ϕ = g and u = j x G i,j i Gu. xi Discretization We have used a parametrization of the Catmull-Clark subdivision surface, as described by Jos Stam [5], where the surface is divided into patches Xp (x, x ) : Ωp R3, here Ωp = [0, ] [0, ], p =... P (see the figure 5). Figure 5: On the left, surface divided into patches. The colors indicate the coordinate system of each patch. On the right, the scalar functions are calculated at the cell centers and for vectors functions the first and second coordiantes are defined at the squares and diamonds, respectively. Note that in the Euclidean case we have G P= =i,jg = g and g = 0. i,j The Einstein s notation means ai b cj = i,j ai b cj. 4

6 Each domain is divided in a uniform grid with (N + ) (N + ) cells, where the N N inner cells correpond to the regions of each patch ( white cells in figure 5) and the outer cells correspond to regions of neighbor patches (gray cells), for each domain we need to update the neighbor cell it is important to know the orientation of each neighbor patch. We associate a label for each edge ( see figure 6). 3 Figure 6: Boundaries and label each edge. 0 The base mesh can be supposed to be made up only by quads, if it is not can be refined once. To simulate fluid through the patches we need to determine the transition function between neighbor patches. The transition function φ depends only on the number given by t k = e k, e j = (4 + e k (e j + )%4)%4 for k and j {0,,, 3}. The transition functions between patches depends only on four functions: φ 0 = id, φ = R π/ φ 0, φ = R π/ φ and φ 3 = R π/ φ 3, where R π/ is the rotation of 90 in counterclockwise. To update the boundary values for scalar functions we calculate the transition number and then we determine the transition function calculate in the corresponding domain, as shown in figure 8. The figure 7 showns an example of the transition function. In this case, t = 3. So, the boundary is calculated by ϕ(n +, j) = ϕ (φ 3 (, j)) Figure 7: Example of transition function. 5

7 Figure 8: Boundary values for scalar functions. For vector fields, we need to calculate the change of coordinates given by the derivatives of the transitions functions, the new coordinate systems are given by the matrices: ( ) ( ) ( ) ( M t0 =, M t =, M t =, M t3 = ) 0. 0 So, we apply the following rules, see figure 9, to update the neighbor values for vector fields: Figure 9: Boundary values for vector fields. 6

8 4 Navier-Stokes Equations A fluid is a velocity field satisfying the Navier-Stokes equations: u = (u )u + ν u + f, t u = 0 where u is the velocity, ν is the viscosity and f is an external force. We can add a scalar field representing a density moving through the velocity field satisfying: ρ t = (u )ρ + κ ρ + S where ρ is the density, κ is a diffusion rate and S is source of density. Solution of Navier-Stokes Equations Temam [] had proved that we can solve the Navier-Stokes equations in four steps. u 0 = force u == diffuse u == advect u 3 ==== project In the first step the force is added, after the diffusion is solved, then the advection and finally the projection, in other words we solve each one of these equations, sequentially u 4. Adding forces - First, we solve this equation u t = f which is discretized by u = u 0 + tf. So, we just sum the values of the external forces to the current velocity, see figure 0. Figure 0: Forces added to a velocity field on the surface. 7

9 . Diffusion - Now, we solve the diffusion equation which is discretized by u t = ν u, u u t = ν u (I tν )u = u. This is a linear system that we solve using an iterative method (Jacobi) improved by a multigrid schemme. The figure shows the result after solve the diffusion. Figure : On the right after diffusion. where The Laplacian operator is discretized, see [8], as follows: ( ϕ) i,j = h ( G) i,j (D, i,j + D, i,j + D, i,j + D, i,j ), D, i,j = A i+,jd, A i+,j i,jd, i D,,j, i,j = B i+,jd, B i+,j i,jd, i,j, D, i,j = B i,j+ D, i,j+ B i,j D,, D, i,j i,j = C i,j+ D, i,j+ A = Gg,, B = Gg,, C = Gg, (see figure ). C i,j D,, i,j (i-, j+) (i, j+) (i+, j+) (i-, j) (i, j) (i+, j) (i-, j-) (i, j-) (i+, j-) Figure : For example D, i,j = ϕ i+,j ϕ i,j. 8

10 3. Advection - The advection equation is given by u 3 t = (u )u 3 = (ū R )u 3, here ū = (u g + u g, u g + u g ). Which is solved using a semi-lagrangian technique, where we calculate the trajectory of each point using ū to find its position in the moment t t ( see figure 4). x(t t) x(t) x(t t) = x(t) tū, u 3 (x(t)) = u (x(t t)). We denote the position in the time t t by x(t t) := (i, j). Then we calculate the velocity u(i, j): if (i < 0.5) u(i, j) = M t3 (u 3 (φ t3 (i + N, j))) if (i > N + 0.5) u(i, j) = M t (u (φ t (i N, j))) if (j < 0.5) u(i, j) = M t0 (u 0 (φ t0 (i, j + N))) if (j > N + 0.5) u(i, j) = M t (u (φ t (i, j N))) else u(i, j) = interpolation 9

11 Interpolation of velocities: If (j j 0.5) ( u (i, j) = bilerp i ī, j j + 0.5, u (ī, j 0.5), u (ī +, j 0.5), ) u (ī, j + 0.5), u (ī +, ī + 0.5) else ( u (i, j) = bilerp i ī, j j 0.5, u (ī, j + 0.5), u (ī +, j + 0.5), ) u (ī, j +.5), u (ī +, ī +.5) end if The same way for i. 4. Projection - The result of the last step is projected on a space whose divergent is 0 (see figure 3). Pu 3 = u 4, u 4 = 0. Figure 3: Projection. This operator can be defined using Theorem [Hemholtz-Hodge Decompositon Theorem ] Let W R n be a a fluid region and u : W R n a smooth vector field. There exists a unique decompostion of u such that, for smooth u s : W R n and u p : W R n, u = u s + u p, u s = 0 on W, u s n = 0 on W, u p ϕ on W for some smooth ϕ : W R. Proof: The proof of this theorem can be seen in [0]. 0

12 By the Hemholtz-Hodge decomposition theorem, we can write u = u s + u p = u s + ϕ. So, u = u s + ϕ = ϕ, where the divergence operator is discretized by here ( u) ij = ( G) i,j h (D i,j + D i,j), D i,j = ( G) i+,j(u ) i+,j ( G) i,j(u ) i,j Di,j = ( G) i,j+ (u ) i,j+ ( G) i,j (u ) i,j and finally we define u 4 = u 3 ϕ, where the gradient operator is discretized by ( ϕ) i+,j = h ( ϕ) i,j+ and h = /N is grid spacing. = h ( (g, i+,j)d, i+,j + (g, i+,j)d,, ) i+,j ) ( (g, )D, + (g, )D, i,j+ i,j+ i,j+ i,j+ Figure 4: On the left after advection and on the right after projection. The figure 4 compares the advection with the projection. For the density field we want to solve the equation This is done in three steps: ρ t = (u )ρ + κ ρ + S. ρ 0 ====== add source ρ ===== diffusion ρ ===== advection ρ 3

13 . Addition of density sources - We first solve this equation ρ t = S, which is discretized by ρ = ρ 0 + S t (see figure 5 after add source). Figure 5: From the initial density (on the left), we add sources (on the right).. Diffusion of density field - Next, we solve the diffusion ρ t = κ ρ, which is discretized by (I tκ )ρ = ρ (see figure 6). And solve using the same schemme used for the velocity field. Figure 6: On the left before diffusion and on the right after diffusion. 3. Advection of density field - Finally, the advection equation is given by ρ t = (u )ρ = (ū R )ρ, where ū = (u g + u g, u g + u g ).

14 In the same way that we analyze the boundary conditions for the velocity, now we calculate the value of density field in the boundary of domain. if (i < 0.5) ρ(i, j) = ρ 3 (φ t3 (i + N, j)) if (i > N + 0.5) ρ(i, j) = ρ (φ t (i N, j)) if (j < 0.5) ρ(i, j) = ρ 0 (φ t0 (i, j + N)) if (j > N + 0.5) ρ(i, j) = ρ (φ t (i, j N)) else ρ(i, j) = interpolation Interpolation of density ( ρ(i, j) = bilerp i ī, j j, ρ(ī, j), ρ(ī +, j), ) ρ(ī, j + ), ρ(ī +, j + ) Example Let f g = Gρu g be the gravity force (see figure 8 and 9), where G is a constant and u g are the coordinates of the projection d T = d (d n)n of d in the plane tangent. Figure 7: [8]. 3

15 Define a = d T t, b = d T s and c = t s. Then we get u g = a bc c u g = b cu g. Figure 8: Stanford Bunny initial (on the left) and after simulation (on the right). Figure 9: Model fertilily initial (on the left) and after simulation (on the right). For this example, we read result meshes of the work Mixed-Integer Quadrangulation []. 4

16 Example Another example is a simulation of a force applied by a finger moving on each surface patch. In our examples the finger follows a circular trajectory defined on the parameter space of each patch (see figure 0). Figure 0: Plane initial (on the left) and after appply the finger force (on the right). 5 Reaction-Diffusion Systems The other application of the discretization of differential operators on surfaces that we have studied is the solution of reaction-diffusion systems. This kind of systems can be applied in chemistry, but it can also be found for example in biology and ecology. In our case, they are used to obtain textures on surfaces. Reaction-Diffusion systems are defined by these nonlinear partial differential equations: a = F (a, b) + ra a, t b = G(a, b) + rb b, t where a and b are substances distributed in space, F and G are functions that control the production rate of a and b, the coefficients ra and rb are the diffusion rates, and is the Laplacian operator. These substances are affected by two processes: local chemical reactions, which means that the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. We consider the reaction-diffusion model developed by Gray and Scott, which is defined by F (a, b) = ab + f ( a) and G(a, b) = ab (f + k)b, where f and k are real parameters. 5

17 For implementation, we used the structure described in the first part where we have the parametrization of Catmull-Clark and the Laplacian operator. To solve the Gray- Scott reaction diffusion system, we have used the parameters k, f, r a and r b that Greg Turk 3 had suggested. Since this model is quite sensitive to its initial conditions, we firstly initialize the concentration values in the cell centers. After this, we use the Laplacian operator calculated before and we solve the reaction-diffusion system using the Euler method as the next algorithm shows. Data: r a, r b, f, k, t. Result: The concentratrions a and b. for l to N do end for c to N do Compute the Laplacian operator of concentrations a and b. //compute the new rate of change of a and b da(c, l) = a b b + f ( a) + r a Laplacian(a, c, l); db(c, l) = a b b (f + k) b + r b Laplacian(b, c, l); end //effect change for l to N do end for c to N do a(c, l)+ = ( t da(c, l)); b(c, l)+ = ( t db(c, l)); end Algorithm : Reaction diffusion system. Note that in this algorithm we do not update the neighbor concentrations, but this brings problems, because the concentrations in the patches influence each others (see figure ). So, we update the neighbor concentrations in the same way that we did for the density. 3 See turk 6

18 Figure : Reaction diffusion sytems without update neighbors concentrations (on the left) and with update neighbors concentrations (on the right.) 6 Conclusion In this work, we had simulated fluids and applied textures on surfaces using the Navier-Stokes equations and the Gray-Scott model, respectively. The first case is more complicated, because we need to solve the Navier-Stokes equations in four steps, which involve the Laplacian, divergent and gradient operators. For future works, we want to improve the numerical instability caused by the Euler method using to solve the reaction - diffusion system. Moreover, we begin to study other models to reaction - diffusion to apply different textures on surfaces. Acknowledgements The author would like to thank Luiz Velho, Leonardo Carvalho and Dalia Bonilla for their constructive remarks and incentive. This work is partially financed by CAPES/INCTMat. Some Results. Simulation of fluids examples: Figure : This example compares the results of the simulation without and with the treatment of neighbour patches, respectively. 7

19 Figure 3: This example shows the result of the simulation without the treatment of neighbour patches. The fluid seems to vanish from one patch to another. Figure 4: Fluids simulation with the treatment of neighbour patches. Figure 5: For this example, we read result meshes of the work Mixed-Integer Quadrangulation []. 8

20 . Reaction-diffusion examples: Figure 6: We use ra = 0.08, rb = 0.5ra, f = 0.056, k = Figure 7: We use ra = 0.08, rb = 0.5ra, f = 0.08, k = Figure 8: We use ra = 0.08, rb = 0.5ra, f = 0.08, k =

21 Figure 9: We use ra = 0.08, rb = 0.5ra, f = 0.035, k = Figure 30: We use ra = 0.08, rb = 0.5ra, f = 0.038, k =

22 References [] Turing, A. The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society B, August 4, 95. Vol. 37, [] Temam, R. Sur l approximation de la solution des équations de navier-stokes par la méthodes des pas fractionnaires. Arch. Rat. Mech. Anal. 33, [3] Aris, R. Vectors, Tensors and the Basic Equations of Fluid Mechanics. 989, Dover, New York. [4] Turk, G. Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion. In Computer Graphics (SIGGRAPH 9), [5] Stam, J. Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values. In SIGGRAPH 98 Conference Proceedings, Annual Conference Series, July 998, [6] Stam, J. Stable Fluids. In SIGGRAPH 99 Conference Proceedings, Annual Conference Series, August 999, -8. [7] Fedkiw, R., Stam, J. and Jensen, H.W. Visual Simulation of Smoke. In SIGGRAPH 00 Conference Proceedings, Annual Conference Series, August 00, 5-. [8] Stam, J. Flows on Surfaces of Arbitrary Topology. ACM Transactions On Graphics (TOG), Volume, Issue 3 (July 003) : Proceedings of SIGGRAPH 003, [9] Sanderson, A. R., Kirby, R. M., Johnson, C. R. and Yang L. Advanced Reaction- Diffusion Models for Texture Synthesis. J. Graphics Tools 006,, [0] Macedo, I. On the Simaulation of Fluids for Computer Graphics. Master s thesis, IMPA - Instituto de Matematica Pura e Aplicada, December 007. [] Bajaj, C., Zhang, Y. and Xu, G. Physically-based Surface Texture Synthesis Using a Coupled Finite Element System. GMP 08 Proceedings of the 5th international conference on Advances in geometric modeling and processing.springer-verlag Berlin. [] Bommes, D., Zimmer H. and Kobbelt, L. Mixed-Integer Quadrangulation. ACM Transactions On Graphics (TOG),8(3), 009: Proceedings of SIGGRAPH 009.