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Compressible subsonic flow on a staggered grid Bonner, Michael Patrick 2007

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Compressible Subsonic Flow on a Staggered Grid by Michael Patrick Bonner B.Sc, California Polytechnic State University, San Luis Obispo, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Computer Science) The University Of British Columbia October, 2007 © Michael Patrick Bonner 2007 Abstract This work focuses on numerically modelling the dynamics of a single phase fluid at varying densities and pressures. We explore the potential of in-compressible flow simulation methods in modelling compressible flow, with an eye towards computer animation applications. The methods developed capture the interesting thermodynamic effects of compressible flow, and re-duce to the standard Marker and Cell incompressible flow Poisson matrix in the incompressible limit. The method works well in modelling flows in the subsonic range that normal incompressible techniques do not capture and where compressible methods are inefficient. We have also investigated adapting these techniques to granular elastic-plastic flow. ii Table of Contents Abstract ii Table of Contents iii List of Figures v 1 Introduction - Background on Fluid and Material Simula-tion 1 1.1 Incompressible Flow 3 1.2 Compressible Flow 4 1.3 Elasto-Plastic Flow 7 2 Review of Incompressible Flow Simulation 10 2.1 The Marker and Cell (MAC) Grid 10 2.2 Operator Splitting 11 2.3 Semi-Lagrangian Advection 12 2.4 Operator Discretization 12 2.5 Boundary Conditions 13 2.6 Finite Difference Solution 14 3 Compressible Flow on Staggered Grids 16 3.1 ICE 16 3.2 A Simplified ICE Technique 17 3.3 Adding Simple Thermodynamics 20 3.4 A Change in Primary Variables 23 3.5 Boundary Conditions and Grid Aligned Solids . . . . . . . . 27 3.6 The Incompressible Limit 27 3.7 Implementation and Results 28 4 Towards Eulerian Granular Flow 30 4.1 Complementarity 31 4.2 Finite Difference Solution 33 iii Table of Contents 4.3 Implementation 34 Bibliography 36 List of Figures 2.1 A 2D Marker and Cell (MAC) cell with density and pressure defined at the cell center and the vertical component of veloc-ity denned on horizontal faces and the horizontal component of velocity denned on the vertical cell faces 11 2.2 An illustration of semi-Lagrangian advection of a horizontal velocity unknown 13 3.1 Results obtained using the method presented in section 3.3. A heat source is located in the middle of the domain in between two walls 23 3.2 The figure on the left uses the upwinding scheme in equation 3.28, while the figure on the right uses the zipped scheme (eq. 3.32)for the and the upwinded terms for the v(pu) terms. 24 3.3 Illustration of the cells and vectors involved in the upwinding scheme used in equations 3.31 and 3.6 25 3.4 Results of a simulation run in which a circular area of higher density (one order of magnitude) is brought to equilibrium. . 28 4.1 Plots showing sand settle under gravity. 35 v Chapter 1 Introduction - Background on Fluid and Material Simulation This work focuses on modelling the dynamics of a single fluid flow. How fast a fluid is flowing determines how a fluid is modeled. Scientists use a dimensionless number called the Mach number, defined as M = - (1.1) c to quantify the speed of the flow (v) relative to the speed of sound (c) in the fluid. Depending on the Mach number desired to reproduce, completely different solution techniques tracking different properties may be required. We can use the Mach number to classify three regions of flow. Flows with M < 1 are subsonic, M = 1 sonic, and flows with a Mach number greater than one are supersonic. Fluids begin to behave radically different when ob-jects in the fluid or parts of the fluid approach or exceed the speed of sound in the fluid. The partial differential equations that govern the behaviour of each flow region may loosely be classified as elliptic, parabolic, and hy-perbolic. For this work we are primarily interested in subsonic flow and elliptical PDEs, but it is important to note that higher speed flows require drastically different solution techniques. For this work we are concerned with a further delineation of subsonic flow, that between incompressible flow and compressible flow. Typically, flows with Mach numbers less than 0.3 can be modeled as incompressible, while flows with Mach numbers greater than 0.3 must be modeled as compressible. However, this is merely a rule of thumb; compressible effects have been noted with Mach numbers as low as 0.1, but it depends on changes in pressure and density relative to the speed of sound [13]. Although flows and solution methods can differ drastically they all start with the same underlying Navier-Stokes equations, defined continuously for a differential element. We begin with conservation of momentum, written 1 Chapter 1. Introduction - Background on Fluid and Material Simulation in non-conservative velocity form as du + - V p V • o~v — Fj = 0, P P (1.2) the continuity equation (1.3) and conservation of energy p(— + u • VE) - V • {KhVT)+pW • u = 0. (1.4) Here u is the velocity vector, p is density, p is pressure, av is the viscous stress tensor, F represents body forces such as gravity, E is internal energy, and Kh is a heat conduction coefficient. Note that this system is not closed; an equation of state that relates two or more of the variables—such as pressure and density or pressure and temperature—like the ideal gas law, is needed. In many cases, certain terms in the equations can be neglected because their physical and/or numerical effects are negligible; later we will be dropping the viscous stress term and heat conduction term in particular, since numerical viscosity from the treatment of advection terms (u • Vu or u • V.E) will dominate them. Equations 1.2 through 1.4 are described in a continuous setting. In order to solve them we must discretize our computational domain in a discrete way and solve the equations at points in our domain. There are two ways to discretize the domain. The Lagrangian approach attaches a mesh or unstructured particles to the fluid, i.e. the discrete elements move with the fluid velocity. In the Eulerian approach the domain is voxelized into fixed discrete elements and fluid is free to move between them: fluid properties are tracked at nodes between cells or at cell centers. Lagrangian models are not as popular for fluids as they are for solids. Obviously a fluid can move and distort a great deal—if you imagine a mesh moving with a fluid, the mesh will become entangled beyond recognition. For this reason regular staggered or collocated grids, irregular grids, or adaptive grids are usually used to model fluids in an Eulerian setting. We will see in the subsequent sections that the type of grid used is also dependent on the type of flow being modeled. Since we are interested in producing data sets to be used in production computer graphics environments, one of our most stringent constraints is time. We seek visually believable simulations as quickly as possible. In 2 Chapter 1. Introduction - Background on Fluid and Material Simulation contrast, engineers and physicists care that the simulation is correct, not that it just looks correct. So long as the results "look" correct, the accuracy that physicists and engineers obsess over is often sacrificed for improvements in efficiency and run time. We also need a complete representation of our domain; physicists and engineers will often take advantage of symmetry in order to reduce the problem size, figuring why simulate the entire domain if a subset will do? Also, the time frame in which important physics take place is usually not macroscopic as it is in a movie; nobody wants to animate something that is hardly perceivable. Therefore, scientific simulations usu-ally span a much shorter time span than those needed for computer graphics applications which may run for several seconds. Conversely, physicists may be interested most in the first several milliseconds of an explosion. We will now introduce the different flow regimes we are interested in and the modelling techniques that are used to model a particular flow regime. 1.1 Incompressible Flow If the relative speeds within a flow are low enough (typically Mach number less than 0.3), thermodynamic effects and density changes due to changes in pressure become negligible. If density is constant and mass is conserved so is volume. This condition is expressed mathematically as the divergence of velocity is zero V-u = 0 (1.5) Essentially what goes into a differential volume must exit it simultaneously. Coupling this equation with conservation of momentum 1.2 makes the sys-tem fully determined, without need of the energy equation or an equation of state, and yields extremely efficient simulations. Pioneering research in the computational modelling of incompressible flow began in the 1950's. The primitive variable approach1 favoured in computer graphics was pioneered by Harlow and Welch who developed the Marker and Cell method (MAC) in 1965 [25], Chorin who developed the artificial compressibility method in 1967 [12], Patankar and Spalding who developed the semi-implicit method for pressure linked equations (SIMPLE) in 1972 [41], and Issa who devel-oped the pressure implicit with splitting of operators (PISA) method in 1985 [29]. The computer graphics industry has primarily focused on the work of 1 Primitive variable refers to using velocity and pressure as the chief unknowns, as opposed to other techniques such as vorticity methods; readers can consult [13] for a more in-depth consideration. 3 Chapter 1. Introduction - Background on Fluid and Material Simulation Harlow and Welch and the use of MAC grids. You will see in the subse-quent chapter that it is ideally suited for efficient solutions of incompressible flow problems. MAC grids store the field variables in a staggered fashion, whereas collocated grids store all data at cell centers. Collocated methods for incompressible flow that use central differences are sensitive to checker-board instabilities. This problem can be dealt with by using complicated filtering schemes, but they have the disadvantage of adding additional nu-merical dissipation on coarse grids. Therefore, collocated methods are not popularly used for applications in computer graphics. MAC methods typi-cally treat the advection term explicitly but the pressure, term as an implicit constraint—and thus can use time steps restricted by the material velocity only, not the speed of sound (which, for low Mach numbers, is obviously desirable). A vast amount of research in graphics has started with simple incom-pressible flow and tweaked it with the goal that some physical process which is normally difficult to model can be approximated: this is the general spirit behind this work. Starting with Foster and Metaxes in 1996 [22], the amount of graphics research in modelling the Navier-Stokes with incompressible flow has included semi-Lagrangian advection [47], vorticity confinement [18, 39], surface tracking [17], [16], [5], surface tension [31], surface flow [48], dy-namic meshes [30], fire [39], viscosity [10, 44], viscoelastic modelling [23], coupling with rigid and deformable solids [24], efficient implementations [31], [28], smooth-particle hydrodynamics ([36, 43], irregular boundaries [7], mul-tiple fluid flow [32], large bodies of water [33], and of course, explosions [20, 38, 45, 51] , and sand [8, 53]. 1.2 Compressible Flow-While in low speed flows, temperature, energy, and changes in density may be insignificant, and thus efficient incompressible flow solvers can be used, for higher speed scenarios more complex compressible methods must be used. Consider a small element of fluid, v, under pressure, p, from all sides. If we increase the pressure by a tiny amount, dp, the element will compress by a corresponding amount, dp. The compressibility of a fluid is defined as r = Compressibility measures how much a static material deforms subjected to a unit of pressure. Under a change in pressure the corresponding change in density is then dp = prdp (1.6) 4 Chapter 1. Introduction - Background on Fluid and Material Simulation Compressibility is a material property; water has a value of 5 x 10~10 while air has a value of 1 x 10~5 [1]. Air is easier to compress than water—not surprising. Things become interesting as the fluid is subjected to forces or loads and begins to move. It can be seen plainly in equation 1.6 that if large pressure changes are present, than the change in density is significant. This is usually the case for flows with Mach speeds greater than 0.3, which in air corresponds to a flow speed of 100^- Further flow classification begins within the envelope of compressible flow: subsonic flow (0.3 > M > 0.8), transonic flow (0.8 > M > 1.2), supersonic flow (1.2 > M > 5), and hyper-sonic (M > 5). Each region warrants special consideration and may benefit from specialized methods. Typically in all cases, in order to accurately capture the interesting effects of compressible flow, the conservative variables p, pu, and pE, are used—in fact it's generally considered essential to use these variables, and the conservative form of the equations, to capture shock waves at transonic and higher speeds. In contrast, incompressible methods track the non-conservative variables p and v; while the focus of this thesis will be on modelling the compressible effects present in sub-sonic flow that incompressible simulators cannot approximate, we will find that the non-conservative variables will do fine. In order to take into account the compressibility and capture the vari-ations in density we now need all of the equations listed in the previous chapter (1.2 to 1.4)2 and an equation of state, such as the ideal gas law (pv = nRT), that relates pressure, density, and temperature3. As airplanes approached the speed of sound and as scientists began to model explosions accurately computational fluid dynamics has evolved from modelling incompressible flows to modelling fully compressible fluid dynam-ics. In supersonic flows containing shocks, information about conditions downstream of a shock cannot propagate back upstream past the shock; this effect produces large gradients in energy and pressure. Understandably higher resolution solution techniques are needed. Currently, two methods for solving the hyperbolic partial differential equation conservation laws in-volved in compressible flow are commonly used. The first and most common are Riemann solvers. Riemann solvers capture shock formation and evolu-tion by solving simplified scalar conservation laws with constants for x < 0 and x > 0 (the shock face or front is located at x = 0). In order to reduce the PDEs to separate scalar equations the variables must be collocated so that 2Though as noted modified to use the conservative variables, and written in "conser-vation law" form. 3This is the traditional approach. We will see later other approaches exist, although they may not be as physically accurate. 5 Chapter 1. Introduction - Background on Fluid and Material Simulation the flux matrix can be diagonalized. The second and less popular approach, but still extensively researched, is epitomized by the Lax-Friedrichs method. Variables are again collocated but every other time step is staggered by half a grid cell. The staggering allows central differences to be used without concerns about stability, so long as a CFL-like condition is obeyed. Lax-Friedrichs methods avoid Riemann problems and do not require the matrix to be diagonalized. The latter method is closest in spirit to the approach taken in this thesis. As mentioned before, incompressible flow methods employ operator split-ting to separate an explicit and implicit part of the solution. This allows much larger time steps to be used which means simulations can be produced quicker. This is a huge advantage in a production computer graphics en-vironment. In contrast, the Riemann solvers and Lax-Friedrichs methods use fully explicit methods. In supersonic flows the time step restrictions are from the material velocity, not from the speed of sound. There is no reason to treat the advection and pressure separately. However, since these compressible methods are explicit, the conservation equations are very stiff in the low speed limit. The usual solution for stiffness is to treat pressure implicitly, but because the spatial discretization is highly non-linear and po-tentially non-smooth, doing this for a Riemann solver is extremely difficult. Lax-Friedrichs type methods also do not fare well with this flow regime and have a similar stiffness problem—but since the space and time discretiza-tions are tightly integrated in staggering, there is no obvious way to make these methods implicit. On the other hand, incompressible solvers cannot model flows where the divergence of velocity is not zero. Another concern when modelling compressible flow is that a region of the flow will be moving slow enough to be theoretically incompressible. When the difference between flow speeds and the speed of sound is very large it can lead to a poorly con-ditioned system which may not converge using iterative procedures. This problem has been extensively researched in [42], [11], and elsewhere. In fact flowfield-dependent variation methods have been developed motivated by the difficulty in modelling very low velocities on one side of a shock and the very high velocities on the other side of a shock. In contrast to the large amount of work done in the computer graphics community on incompressible flow, compressible flow and the conservative Navier-Stokes system has received little attention. A surprising fact noting the prevalence of explosions in movies and video games. Yngve el al. [52] have produced the most notable work modelling the full compressibility of a shocked fluid. The work used two simplifying assumptions used by engineers and although they did capture the effects of temperature and the 6 Chapter 1. Introduction - Background on Fluid and Material Simulation conservation of internal energy, the method is inefficient as the flow speed decreases [20]. The methods developed for this thesis model both low and relatively high speed flows and capture some of the visual effects of compressible flow. Because neither standard compressible or incompressible techniques han-dle flow speeds in the sub-sonic range above M = 0.3 we have aimed our research at this flow regime and are not primarily concerned with conserva-tion of energy and momentum or with modelling shock creation and prop-agation. Specifically, we will never attempt to conserve energy, instead we make simplifying assumptions that allow us to forgo energy conservation and to provide a robust and quick method. 1.3 Elasto-Plastic Flow Another type of flow we are interested in, elasto-plastic flow, isn't actually fluid, flow. In particular we are interested in the flow of sand. The dynamic behaviour of sand under load resembles the behaviour of a fluid, but, unlike a fluid and more like a classic elastic-plastic material such as steel, sand stops flowing—it has a strength associated with it. Classic elastic-plastic materials will yield or deform under load. If the strength of the material is not exceeded, the deformation or strain, is recoverable: upon unloading the material will return to its initial configuration. If the material's strength is exceeded, it deforms plastically—plastic strain is not recoverable. If we con-sider a pile of sand we know that if it is too steep it will fail and flow—some sort of sand inter-grain strength has been exceeded—and it will continue to flow until it reaches a stable configuration. Once the grains move, that "deformation" is not recoverable: they will not return to their original con-figuration. This fact makes classic plasticity an obvious choice for modelling the dynamics of granular materials—clearly, regular fluids do not behave this way. Elastic-plastic techniques like those in [37] were developed by adapting known methods for dealing with classic problems in solid mechan-ics to things that flow like sand. Unfortunately, the deformations seen in typical solid mechanics problems are relatively small, and thus numerical work has focused on mesh-based Lagrangian techniques—which suffer from entanglement and require re-meshing in order to handle realistic sand flows. For this reason, Eulerian techniques, which are underdeveloped, are very attractive for sand. Continuum methods that calculate the dynamics of granular materials are very similar to the compressible and incompressible fluid dynamic models 7 Chapter 1. Introduction - Background on Fluid and Material Simulation mentioned in the previous sections4. Instead of solving for v and p or p, pu, and pE, they solve for v, e (strain), and a (stress). The models used consist of partial differential equation conservative laws for mass, energy and momentum. The most prominent differences with the Navier-Stokes system of equations we have seen are present in the conservation of momentum, which now relate stress and strain. One models used is Tij=cr(6ij + k^f-]) where Vitj = - ( | | + §|) is the strain rate tensor5, \V\2 = X^ 2 j> k is & material constant, and 5 is the angle of internal friction. This equation, and others like it, essentially replaces the viscous term in the Navier-Stokes equation with a shear rate independent term. This rate independence makes these equations generally more difficult to solve than the Navier-Stokes equa-tion. These conservative laws are combined with a yield condition such as Mohr-Coulomb or Drucker-Prager and a flow rule. Normal plastic materi-als flow in a direction in strain-space that corresponds with the gradient of the yield condition (a so-called "associated" flow rule); however, doing this would make the dilation part of plastic strain to be non-zero, an unphysical case for granular dynamics, and thus a more complicated "non-associative" flow rule must be used. Another popular technique for modelling elastic-plastic flow of sand is to use a discrete element method (DEM). DEMs use Newton's law to model the interaction of individual grains or clumps of sand. The simulations progress by detecting "contacts" between particles and then calculating the resultant forces. The proper and efficient detection of contacts is key to the success of discrete element methods. Even with an optimal technique, DEMs track individual particles; with sand this can means tracking the interaction of millions of grains of sand, clearly, a daunting computational task. In applications where knowing the exact location of particles is paramount this expense is worthwhile; this is not the case with computer graphics and DEM methods are often too costly. Bell et. al. used discrete elements in [8] specifically aimed at producing effects for movies, but their sand grains appear too large, most likely the effect of trying to reduce the problem to a manageable size. 4 At least, they are similar to fluids when considering low-speed, high-density flows; in contrast rapid "gaseous" flows are modeled using kinetic theory methods. 5 The negative sign in the strain rate tensor is there because sand spreads apart when subjected to a tensile load. 8 Chapter 1. Introduction - Background on Fluid and Material Simulation Both elastic-plastic and discrete element techniques require more re-sources and are significantly more computationally complex than incom-pressible simulation methods. Treating the dynamics of flowing sand using an incompressible flow technique is an attractive alternative; the work of Zhu and Bridson [53] was in this spirit. Although the method is more ef-ficient than competing techniques and does produce realistic simulations in many cases, for some basic granular physics their results are incorrect. In particular, the effects of internal friction are not accounted for in the pres-sure calculation and could lead to inconsistencies when pressure is corrected to account for zero divergence. Additionally, cohesion is handled incorrectly; when subjected to a tensile load the grains of sand do not come apart as in real life. In Chapter Four we introduce a method that attempts to improve Zhu's work and ultimately move toward a quick and accurate Eulerian granular dynamics model. We have introduced three types of fluid dynamics; modelling each tradi-tionally requires wildly different techniques. In order to capture the physics of compressible subsonic and elastic-plastic flow we have modified the sim-plest and most efficient techniques, those used to model incompressible flow—while maintaining the ability to accurately and efficiently model in-compressible flow. 9 Chapter 2 Review of Incompressible Flow Simulation We mentioned before that if the relative speed of flow is low enough temper-ature and pressure effects on density can be neglected. Essentially, density can be regarded as constant throughout our fluid. This allows us to subject our system to the following constraint: V - u = 0 (2.1) V- is the divergence operator. This equation simply states that all points in our velocity field must be divergence-free. In simpler terms, what goes into a point must go out and vice versa. Assuming that a flow is incompressible allows density to be treated as a constant—this eliminates the need to use the continuity equation. Further-more, as we will see in the following sections viscosity can also be neglected. This essentially leads us with equation 1.2 and 2.1; we will see how these equations are combined to model fluid dynamics efficiently 2.1 The Marker and Cell (MAC) Grid The Navier-Stokes system of equations are defined continuously for all points within a volume of fluid. Practically, this is not solvable; it becomes nec-essary to solve the equations using finite differences defined on a Eulerian grid where our domain is divided into a finite set of voxels of equal volume. The Navier-Stokes PDEs are sampled at cell centers and cell faces in the domain. For reasons that will be clear later we will use a Marker and Cell (MAC) grid [25]. A two-dimensional MAC grid cell (figure 2.1) defines the x component of a cell's velocity vector on the vertical faces of the cell and the y component on the horizontal faces. Pressure and density are defined at the center of the MAC cells. We use a half index notation, so for cell (i, j) in two dimensions we have pjj andpij at the center, and u^i j, ui+i j, vij_i, 10 Chapter 2. Review of Incompressible Flow Simulation and vij+i normal to the faces. Most of the graphics research mentioned in the incompressible section of Chapter One uses a staggered MAC grid. Figure 2.1: A 2D Marker and Cell (MAC) cell with density and pressure defined at the cell center and the vertical component of velocity defined on horizontal faces and the horizontal component of velocity defined on the vertical cell faces. 2.2 Operator Splitting In order to solve our PDEs on our grid we need to introduce the idea of operator splitting. Consider the continuity equation: pt + V • (pu) = 0. Expanding the V- operator yields pV-u+uVp. The idea of operator splitting is just substitution: solve for the uVp as pt = —uVp and then solve pt = p — pV • u. We do this with equation 1.2 and equation 1 by solving for the advection term first, that is ut = —u\7u. In the momentum equation it is advantageous to treat the advection part of the equation with an explicit technique, for reasons of efficiency, accuracy, and simplicity [9]. We will see in the subsequent sections that the advection algorithm commonly used behaves well for arbitrarily large time steps. Treating the pressure term in the momentum equation implicitly produces a well conditioned system. If in contrast the pressure term were to be dealt with explicitly, the CFL condition6 would require that the speed of sound be taken into account when determining a safe time step—doing so is extremely inefficient for low speed flow. 6The CFL condition will be defined later, but for now it is important to know that it is a restriction on the size of the time steps taken during the simulation. 11 Chapter 2. Review of Incompressible Flow Simulation 2.3 Semi-Lagrangian Advection In graphics a first order accurate7 technique called semi-Lagrangian advec-tion [47] is used to determine the advection term in our conservation of momentum equation. We are seeking the value or quantity, in this case u, that will be present at this quantity's current location at the end of the cur-rent time step. This is done by determining a velocity vector at our position x (interpolated from nearby u and v velocity components). The vector is then negated and multiplied by At. We trace back to the location (x — Atv) where we estimate the fluid came from in our domain, and then interpolate (linearly, bi-linearly, and tri-linearly in Id, 2d, and 3d) from nearby values to determine our final value (see figure 2.2 for an illustration of this process). This technique is stable with arbitrarily large time steps—a great quality for a computer graphics application. Unfortunately, this process involves aver-aging operations that have the effect of smoothing out high frequency (or sharp) features in our data set. In some situation this is acceptable because it can be shown to be mathematically similar to viscosity [9]. The excessive dissipation has a positive side effect: we are now justified in dropping the vV • Vtt term from our momentum equation. Semi-Lagrangian advection still damps out too much detail for many applications, which has prompted the research in more complex methods, e.g. [18, 39]. 2.4 Operator Discretization Before we consider solutions to our PDEs we need to look at the discretiza-tion of the gradient operator (Vp) and the divergence operator (V • u) on a MAC grid. Since our velocities are stored at cell faces and pressure at cell centers it is easy to solve for the pressure gradient across a cell face. If we had used a collocated grid we could have used a second order accurate central scheme such as Vp = ^ ' ^ A x ' - 1 • Unfortunately, this scheme ignores information stored at location i (were we are calculating Vp and ut) which can lead to poor numerical results—known as a checkerboard instability [9]. The staggered positions of our unknowns in the MAC grid yield a second order accurate central scheme without introducing inaccuracy or instability. Ultimately the gradient is approximated as Vp = Pi^~Pi • Similarly, if we consider our constraint V • u — 0 for cell i, this equation in two dimensions is 7 A second order accurate version has been developed, but the simple version is pre-sented here. 12 Chapter 2. Review of Incompressible Flow Simulation Figure 2.2: Illustration of semi-Lagrangian advection for a horizontal veloc-ity unknown. V-Mi = (Ui_i - ui+i + - vi+i)/Ax (2.2) which, like the gradient operator, is second-order accurate. It can be shown that if the divergence is calculated on a collocated grid the same problems present with the pressure gradient arise. 2.5 Boundary Conditions The most simple and most common boundary conditions used in computer graphics simulations are solid wall boundaries and free surface boundaries. In the inviscid case solid wall boundaries involve simply setting the normal velocity on a boundary to zero or u • h = 0. This is trivial for grid aligned solids on a M A C grid since velocities are stored perpendicular to cell faces. The issue of accurately handling curved un-grid aligned boundaries is more difficult, but recent work by Batty et. al. in [7], in which the pressure projection step is rephrased as a kinetic energy minimization, has proven to be accurate and efficient. In computer graphics, air is usually not modeled since it is so much lighter than water and has little effect on the dynamics of our fluid flow. Instead of actually calculating the pressures in the air it 13 Chapter 2. Review of Incompressible Flow Simulation is taken to be at atmospheric pressure, or, since only differences in pressure are important, simply zero. This is the free surface boundary condition. 2.6 Finite Difference Solution As was mentioned before, density is constant and the continuity equation can be neglected. If we use a forward Euler approximation to ut, substitute in u (section 2.3), and neglect viscosity (section 2.3), the conservation of momentum equation becomes8 un+l =ii+ -Vp (2.3) 9 Now applying the constraint V • u = 0 and re-arranging yields: V • un+1 = V • u + - V • Wp 9 iv-Vp = V - « (2.4) 9 We now have a system of linear equations in pressure, p, involving the Lapla-cian V - V or V 2 , the divergence of the gradient of pressure. Once discretized as described above, a single row in the linear system9 looks like S t 2 ( , N ' \ ~ PiJ-1 — Pi-l,j + 4 Pi j - Pi+lj + PiJ+1) Ax2pi ( U i i j - u i + i j + t J ^ . i - viJ+i)/Ax (2.5) All other entries in the ith row are zero—it is a sparse system of equations. The Laplacian operator discretized using our MAC grid is symmetric posi-tive semidefinite10, and in fact, if at least one pressure value is constrained to zero, positive definite. Moreover it is a diagonally dominant M-matrix, i.e. the off-diagonal entries are non-positive and the row-sums are non-negative. Large amounts of research have gone into solving this particular system and 8In order to ensure the final system is symmetric the equation has been divided through by P-9Note that we have multiplied both sides of 2.4 by —1 to make the matrix positive definite. A matrix is positive definite (PD) if x7 Ax > 0 V x € Rn. 1 0 A matrix is positive semidefinite (PSD) if xTAx > 0 V x € Rn • 14 Chapter 2. Review of Incompressible Flow Simulation systems like it very quickly. We solve for p and then correct the intermediate (advection only) velocities according to un+1 — u — Vp. The standard algorithm proceeds by advecting velocities and other quan-tities according to section 2.3; solving for pressure according to 2.5; and enforcing boundary conditions and updating secondary quantities with the new velocity field. As was seen in the previous sections the amount of computer graphics research in incompressible flow is staggering (as referenced in section 1.1). However, the question of modelling explosions well is still largely open and the question of modelling granular flow has been largely untouched. The following chapters will present methods that use a MAC grid, conservation of momentum, and rather than a divergence restraint, include conservation of mass, and an equation of state to close the Navier-Stokes system of equa-tions. We feel that this approach allows us to quickly and efficiently capture some important aspects of compressible and granular flow. 15 Chapter 3 Compressible Flow on Staggered Grids Our hope in modelling compressible flow on a staggered grid is to capture some of the visual effects of thermodynamics while maintaining the com-putational efficiency of incompressible methods. Explosions produce the dynamic loads required to make density changes visually important. Signifi-cant effort has been spent in the computer graphics community in modelling explosions; that effort has almost exclusively focused on modifying incom-pressible methods. Many production applications use heuristic rules; many physics based techniques replace or adapt a heuristic with more physically motivated heuristics. Starting with Reeves in 1983 [46] particle systems have been used exhaustively to model fire effects and explosions in com-puter graphics. Mazarak et. al. [35], Martins et. al. [34], and Neff [38] used analytical approximations to experimental data to model the blast wave as-sociated with an explosion. Feldman et. al. [20] model the blast wave using incompressible techniques but place a constraint on the divergence of veloc-ity, that is V • u = 0 becomes V • u = <j>. Rasmussen et. al. [45] use a tiled Kolmogorov velocity field combined with standard incompressible methods to produce appealing large explosions. The methods mentioned all require extensive artistic control and experienced users to achieve appealing results. Only Yngve et. al. [52], as mentioned in Chapter One, attempts to model the fully compressible Navier-stokes system of equations. Their method is inefficient compared to incompressible methods—our hope has been to pro-duce the most visually and physically accurate explosions in the computer graphics community. 3.1 I C E Seeking inspiration for adapting incompressible techniques as well as the MAC grid towards modelling compressible flow led us, once again, to work by Harlow. In 1968 Harlow and Amsdan published "Numerical Calcula-16 Chapter 3. Compressible Flow on Staggered Grids tions of Almost Incompressible Flow" and refined this work in "A Numeri-cal Fluid Dynamics Calculation Method for All Flow Speeds" [26] published in 1972. In these two papers Harlow and Amsden described an Implicit Continuous-fluid Eulerian (ICE) technique for modelling single phase fluid dynamic problems. They start with the differential equations for conserva-tion of momentum and of mass, and solve them using central differences de-fined on a MAC grid. The momentum discretization includes both previous time step (multiplied by 1 — <j>) and forward time step pressures (multiplied by (j>). The scalar cf> (with values between 0 and 1) determines the time cen-tering of the pressure term. Similarly the continuity equation includes both previous and forward time values, this time weighted by 6 — 1 and 8. These two equations with unknown densities and velocities are combined to create a system for pressure that includes the product of the scalars (f> and 6. The resulting linear system is positive definite and reduces to a Poisson matrix as the speed of sound approaches infinity: the incompressible limit. This system is actually better conditioned than the incompressible Poisson ma-trix due to the presence of an additional pi term making it more diagonally dominant. ICE was unique because flows occurring over a wide range of Mach num-bers could be modeled. It incorporated the two scalars <j> and 6, that de-termine how explicit or implicit the solution technique is. If the two scalars are set to zero the method is fully explicit; if both scalars are 1. the method is fully implicit. In essence, changing the two independent of one another in between the two extremes becomes a knob a simulator can turn to tailor the outcome to their desires. Our experience in implementing the method presented was that the large number of parameters made the simulations fussy We felt many of the parameters were not necessary to meet our goals. ICE also supported arbitrary equations of state—we felt a simple linear EOS would suffice. In the end we did not feel there was a great need to have an arbitrary amount of explicitness versus implicitness. Instead, we developed a much simpler method that more closely resembles classical computer graph-ics incompressible flow methods. Advection and pressure are fully separated into an explicit and implicit step; doing this allows any advection scheme to be used, further adding to the overall robustness of the method. 3.2 A Simplified ICE Technique Like Harlow and Amsden, our initial developments used velocity and density as our primary variables. In one dimension the method begins with the 17 Chapter 3. Compressible Flow on Staggered Grids conservation of mass and momentum we have seen before: dpu d{pu2) _ dp , . ~dt+^x~-~dx~ [ 6 ' p = c2p (3.3) Equation 3.2 is an equation of state (EOS), where c is the speed of sound. The EOS is needed to close the system, as equations 3.1 and 3.2 are not enough to fully determine the variables present. In one dimension we begin by expressing continuity explicitly as p?+1 = pf+^ + ^ K - i - \w+/>r+iK+i] (3-4) and velocity as At u. Notice that c2(pf — p"+1) is pressure via a linear equation of state (equation 3.2). The convective fluxes, (u")2 and (uf+1)2 (which are needed at cell centers), are upwinded as ( u2 j . if (u-i +ui+i) < 0 u2, x . if (u- i + ui+i) > 0 U7 = (3.6) This is a first order upwinding approach, but could easily be replaced by a higher resolution method or semi-Lagrangian formulas. This explicit step is limited by At = a (3.7) c + max(|u|) This is an expression of what is known as the Courant-Friedrichs-Lewy con-dition (CFL condition). The C F L condition states that the time step must be small enough in order to make the numerical domain of dependence of a solution point include the true physical domain of dependence, i.e. not miss any significant propagation of information. In our case this is partly dictated by fluid velocity: obviously we do not want anything to be advected 18 Chapter 3. Compressible Flow on Staggered Grids a distance greater than a cell length as we would lose the density associated with that cell. In addition, the sound speed c appears in the denominator of equation 3.7, since in this explicit treatment of pressure the CFL condi-tion dictates that sound waves must not travel further than a cell length in one time step. Using pn+1 instead of pn in 3.5 and solving for p implicitly removes the sound speed c from equation 3.7. Doing this is attractive for any flow where the flow speed is significantly slower than the speed of sound and is the approach taken in the next section. Solving for pressure implicitly requires the operator splitting technique presented in Chapter Two. The previous equation for un+1 becomes, as the pressure is now dealt with implicitly, an equation for an intermediate velocity value that only includes advection, u: = < i + -n ^ \ p u f - pUi+i 2} (3-8) The v? terms are upwinded as before in equation 3.6. Like velocity, we calculate an intermediate density that includes the advection term from the continuity equation ^ = P ? + A ^ [ ( p u ) - § - ( H i + l ] M where now the pi±i densities are upwinded as _ f pi+i if ui+i < 0 P i + 5 ~ \ Pi if ui+i > 0 Using these advected values the final velocity and density are <i= a<+i  + mh^^+1-^] (3'10) n+l - &t (^(Pi-l + Pi)^Ui-l ~\{pi + Pi+l)^Ui+i (3.11) where Au- i is simply un^l — u-,i from equation 3.10. After substituting the equation for Au into the equation for p™+1 and then substituting that into our constitutive relation p = c2p gives us our linear system to solve for 19 Chapter 3. Compressible Flow on Staggered Grids in pn+l. This system is tridiagonal in one dimension with non-zero bands in two and three dimensions. Moreover it is positive definite, and an M-matrix. From the layout of the matrix we can see that when the speed of sound goes to infinity (the incompressible limit) we are left with the incompressible Poisson matrix. If we let a = a single row of the matrix (in one dimension) looks like api-i + (1 + 2a)pi - api+i = c2pt (3-12) This system is identical to the linear system formed in the original ICE method without the dependence on <f> and 6. Again it reduces to the MAC matrix in the limit, but it is better conditioned because of the additional Pi term. 3.3 Adding Simple Thermodynamics The previous method performed well but we felt it could easily be extended in order to better capture thermodynamic expansion and contraction effects. We begin like most compressible flow simulators by using the equation of state p = nRT (the ideal gas law), instead of p — c2p which does not relate pressure to temperature. We assumed that viscosity and heat conduction were negligible in our flow and that the flow was isentropic: entropy is conserved. It can be shown [6] for an ideal isentropic gas flow that g p = e^p1 (3.13) where S is entropy and cv is the specific heat, and 7 is the heat capacity ratio (which is constant for an ideal gas, with 7 « 1.4 for air). Since entropy is conserved, the first term, e°v , is constant; for our purposes we will refer to it is as the entropy factor or F. Furthermore, since entropy remains constant the material derivative of F is zero—this means it can be advected like density in [19]. For air at regular conditions in an isentropic setting, the temperature can be shown to be: T = Fpf'1 (3.14) We manipulate the entropy factor field by adding heat sources during the calculation. The simplest heat source we used was to set F = Ttpafltt where Ttarget was a target temperature for a given cell. Temperatures were set to Ttarget for cells located within a heat source region of the domain. 20 Chapter 3. Compressible Flow on Staggered Grids Using the preceding simplifications we get the following equations11 (which are the non-conservative form of the Navier-Stokes system of equa-tions) u n+ 1 = u - ^ V p (3.15) P pn+l = p- AtpV • ^ + 1 (3.16) pn+l = RTpn+l (3_17) Notice that in this equation of state 3.17 the sound speed is RT—it depends on temperature. By using a "lagged" T calculated from intermediate explicit values of F and p, the equations are linear, but no longer uniform throughout the domain. The finite difference solution begins similarly by computing the interme-diate density and velocity values that account for advection: Pi,j = P?j ~ ^ « j ( V p ) * + <,-(Vp)„) (3.18) u i + h j = u?+hj - ^{ui+hj{Vu)x + vi+y(Vu)y) (3.19) where ufj and • are averaged to the cell centers from the neighbouring two face values and vi+i j is averaged from the neighbouring four vertical velocity values. The (Vp) x values are calculated as: ( V ' ) x _ t {Pli-Pi-^ i f ^ > o ( 3 - 2 0 ) Similarly the (Vu)x values from 3.19 are calculated as: f ( < 3 i •) if <0 Like density, our entropy factor (F) can be advected around in the same velocity field as: Where again, the "values are intermediate values that account for advection. 21 Chapter 3. Compressible Flow on Staggered Grids The temperature value used in the equation of state is calculated as Tjj = FupjT1. Since our EOS is a function of the temperature, using pn+1 in-stead of pn would produce a more difficult, although more accurate, non-linear problem to solve. We felt using pn and having a linear system in pressure, rather than a non-linear system in density (p7), was a reasonable and attractive simplification. In the explicit version, pn+1 is simply taken to be p and un+1 is updated using our equation of state 3.17, as: ,"+1 - „~, A i i - R-r- (Tijpij - Ti+1jpi+ij) (3.23) i + 5 "Ax In the implicit version a linear system in pressure is built by substituting „ » + ; = » , ^ ^ ( „ n + l _ „ » + l *"T 2 J into ^,3 r j /\x \ i - i j ij+i i j -^y v ' p^j1 in 3.25 is taken to be R ^ ^ . In order to ensure the system is symmet-ric we divide 3.25 by Atpij. A row in the resulting linear system looks like: At Ax2 Pi+lJ Pi-l,j Pi,j+1 Pi,j-1 1 Pi-l. + + At + + + — 5 u Pi+^3 PiJ-\ Pi,3 + : (un+l . - un+l . + vn+1 - vn+\) v » ~ 5 J »>J+2 h3~2 -)Pi,3 (3.26) The p. i i • values are averaged to the cell faces from the adjacent cells. This system is symmetric, positive definite, and reduces in the incompressible limit to the incompressible Poisson matrix. Although it is positive defi-nite the incorporation of the averaged densities into the matrix makes it more difficult to deal with than the previous system. The potential for ill-conditioning is now present. This leads to the following solution algorithm: • Advect p, entropy factor, and velocity according to 3.19-3.22 22 Chapter 3. Compressible Flow on Staggered Grids ii ii ii ii ir~ Figure 3.1: Results obtained using the method presented in section 3.3. A heat source is located in the middle of the domain in between two walls. The plots on top are of the density profile while the bottom plots depict the temperatures. The pressure wave can be seen in the density plot and, as expected, travels much faster than the plume of hotter, less dense air. • Add heat sources (increase entropy factor in a specified region) and calculate temperature as %j = Fijpjj1 • Determine the pressure according to 3.26 • Update p and velocity according to 3.25 and 3.24. Let Fn+1 = F Additional steps pertaining to boundary conditions and solids inside the domain have been neglected for simplicity and will be addressed in section 3.5. 3.4 A Change in Primary Variables The previous methods, although simple, are not guaranteed to conserve momentum. This fact prompted us to investigate a change in primary vari-ables from velocity to momentum, and using the conservative form of the Navier-Stokes equations to perfectly conserve momentum. In contrast to calculating fluxes when needed, velocities are now calculated by dividing the momentum by a density averaged between the two cells that share the face where the velocity is being calculated. In the presence of shocks this calculation would produce unphysical spikes in the velocity field because of the sharp gradients present when shocks occur. As before, our focus is on subsonic flow. Velocities are calculated without concern for the presence of sharp discontinuities in the density field. The method is presented in two dimensions because the upwinding terms require more attention. 23 Chapter 3. Compressible Flow on Staggered Grids Our explicit formulation begins as before by calculating densities as * J = Pfj + £«fi*)l+, - (Mly + (PV)1H - C H J H | ) (3-27) Since we store momentum values, pu directly on cell faces, it is possible to calculate equation 3.27 without any upwinding. Unfortunately, we found this gives rise to spurious oscillations, and so found the following upwind scheme for calculating (pu)i±ij and (pv)ij±i. ui+yPij if ui+y > 0 necessary. The velocity ui+1 in 3.28 is calculated as mentioned previously: ( H i + i , = l + ^ ' J • * f >n ( 3- 2 8) *** \{pij + Pi+lj) and has the previously mentioned problems in the presence of shocks. Figure 3.2: Results at t=0.005 for two simulations in which a circular region of higher density (one order of magnitude) is brought to equilibrium. The figure on the left uses the upwinding scheme in equation 3.28, while the figure on the right uses the zipped scheme (eq. 3.32) for the and the upwinded terms for the v(pu) terms. In contrast to the previous method where we would update velocity, we 24 Chapter 3. Compressible Flow on Staggered Grids now update our momentum in the horizontal direction with ~ At ({HI-1 (pu)i+if , f x n f v „ N { ( M ) ^ = A~X\~PY~~ ~ + {v(m)*kM - ^ W - * J (3.30) The (vpu)i+i . ± i values are calculated at the horizontally-aligned red points in 3.3 and are upwinded according to: ,x i W, - . i j . . if v-.i i < 0 p - 3 i ) where - + i is calculated as 2 ' 1 , J + ^ . The (pu)fj and (pw)2 + 1j terms are calculated at the vertically-aligned red points in 3.3 and are up-winded as in 3.6. We are considering the flux into the cell of size Ax by Ay centered around position (i + These flux locations are the red dots in figure 3.3, while the green arrows are the positions to which we average v values, and the blue arrows are the momentum values we use to upwind (pu)2 terms. The (vpu) and terms reduce to pu when multiplied by ^ . tH -^r f - H f - f — Figure 3.3: Illustration of the cells and vectors involved in the upwinding scheme used in equations 3.31 and 3.6. Harlow and Amsden describe a different way to calculate the (pu)2 terms, a form they called zip fluxes: (Hi- = K - ^ X ^ ) (3-32) 25 Chapter 3. Compressible Flow on Staggered Grids Similarly the v and the pu values in equation 3.31 can become Vij — (vij_ivij+i)5 and ((pu)i_ij(pu)i+i)2. However, this too gives rise to unstable oscillations without adding artificial viscosity, which we have tried to avoid. Using equation 3.2 the final step in our explicit formulation is to update our momentum values as: Ate 2 For the time being we ignored the pressure effects on density and simply let p n + l = p. As in the explicit formulation, we solve equation 3.27 and equation 3.30 that account for the advection of density and momentum. Our next time step density and momentum are: = A j - ^ ( ( A p « ) 1 _ i j - ( A p u ) i + i J + (Api ; )y_ r (Apt ; )y + i ) (3-34) = - £ (P&1,- - Pfr 1) (3.35) where (Apuj^ij = (pu)^!- — (pu)i_i • (from equation 3.35). As opposed to the previous technique from section 3.2 in which our equation for was substituted into our constitutive relationship, we now substitute pn+1 = c2pn+i m t o e q u a t ion 3.35 which is then substituted into equation 3.34. This produces a linear system in p™+1 as opposed to pn+1. With a = c 2 (- A | ) 2 a row in this linear system looks like - apij-i - api-ij + (1 + 4a)pjj - api+ij - j = Pi (3.36) Since p and pressure are linearly related, this system is identical to the one in the previous formulation. It is sparse, symmetric, and positive definite and reduces, in the limit, to the incompressible Poisson matrix. The solution algorithm is identical to the algorithm listed in section 3.2, only now the variables and upwinding procedures differ. 26 Chapter 3. Compressible Flow on Staggered Grids 3.5 Boundary Conditions and Grid Aligned Solids Both solid wall and periodic boundary conditions were used with the preced-ing methods. Values needed for periodic boundaries are taken from the other side of the domain when needed. The solid wall boundaries and interior grid aligned solids were modeled by skipping density calculations (setting them to zero) and setting the velocity or momentum values into and out of solid cells to be zero. For cells bordering on solid cells, the matrix entries in equa-tion 3.36 must account for the pressure in the wall equalling the pressure within the current cell. For example, for a cell on the top boundary, its corresponding row in the matrix becomes: The only difference being that the pij+i entry is gone and (1 + 4a)p;j has become (1 + 3ct)pij, because Pij+i = Pij-3.6 The Incompressible Limit As we saw in Chapter Two, enforcing the divergence of velocity to be zero results in the following linear system of equations: The systems shown in 3.12, 3.26, and 3.36 all reduce as the speed of sound approaches infinity to the incompressible Poisson matrix: the V - V matrix in 3.38. This means that these methods will model incompressible flows as well as the compressible flows they have been designed to model. Since we track variations in density using the continuity equation and use it to construct our linear systems, the right hand side vectors differ from the vectors in 3.38. The incompressible methods assume density to be constant-a reasonable, but incorrect, assumption. Changes in pressure can actually lead to changes in density, if for no other reason than numerical error. If an incompressible simulator did track these minuscule deviations in density space and included a drift correction term, the incompressible system would resemble the ones presented in the previous sections. In contrast, our models directly use the continuity equation to track and respond to changes in density in a fully physical way. - apij-i - api-ij + (1 + 3a)pij - api+i,j = p% (3.37) V • Vp = V • u (3.38) 27 Chapter 3. Compressible Flow on Staggered Grids Figure 3.4: Results of a simulation run in which a circular area of higher density (one order of magnitude) is brought to equilibrium. 3.7 Implementation and Results The preceding methods were implemented using C++ in two dimensions with both solid wall and periodic boundaries for a square domain that in-cluded solid objects. The momentum formulation was also implemented in three dimensions. A preconditioned conjugate gradient solver and a sparse matrix class were used to solve and represent the linear systems presented in 3.12, 3.26, and in 3.36. Solutions are usually found in a single iteration when a warm start is used. Figure 3.3 shows the results of a simulation that included the heat source defined in section 3.3. The heat source is located in between the walls in the center of the domain and ends at approximately three quarters of the wall height. It continually sets the temperature in the heat source region to 10001C The figure contains density profiles (top plots) and temperature profiles (bottom plots). The initial pressure wave can been seen in the density plots, and, as expected, travels much faster than the plume of hot air which rises out of the heat source. In figure 3.6 a circular region within the center of the domain with density one order of magnitude larger than the surrounding cells is brought to equi-librium. On the left portion of the domain is a wall. This simulation used the method presented in section 3.4 and the upwinding scheme for fluxes presented in 3.31 and 3.6 (as opposed to the zipped procedure). Notice the diffraction and reflection waves in the last frames of the figure. Although these results look fine, often the results obtained when using momentum val-ues directly in equation 3.27 as opposed to the upwinding scheme presented in 3.28 were incorrect. At other times the simulations simply crashed. Es-sentially, the upwinding scheme is, as expected, more stable as we begin to 28 Chapter 3. Compressible Flow on Staggered Grids push the limits of the simulation. For the same initial conditions, if we begin to increase the CFL factor from 0.1 to 1.0, the simulation using 3.27 will produce unphysical results at a lower CFL factor. Although, in the cases where this occurs we are also pushing the theoretical limits of the method, unphysical results occur when the Mach number approaches and exceeds 1.0. This failure is inconsequential since we freely admit our method will not produce physically accurate results for supersonic flow. The zipped fluxes presented in 3.32 are second order accurate, and with this accuracy comes less stability. As can be seen in figure 3.4, the results obtained using the zipped fluxes are more detailed, but these simulations are more likely to crash or produce small time-steps. Like the semi-lagrangian techniques, the first order accurate upwinding scheme adds diffusion and al-lows us to neglect viscosity. Incorporating viscosity would make the zipped method more stable, but it would add considerable computational cost. Har-low and Amsden favour the higher order schemes combined with viscosity because they favour more physically accurate simulations. We are willing to sacrifice physical accuracy for an improvement in run time. In practice, neither upwinded nor zipped fluxes worked better all of the time. As we changed the CFL factor and/or the initial density gradient, one method would produce more spherical results than the other, or in some cases, es-pecially when directly using the momentum values in equation 3.27, the simulation would become unstable and crash. 29 Chapter 4 Towards Eulerian Granular Flow Granular material dynamics, like those of sand, are essentially macroscopic fluid flows. At the microscopic level the molecules that make up a regular fluid collide and interact, transferring kinetic energy from one molecule to another. The time and length scales over which a fluid is modeled cover an enormous number of molecular interactions. Individual collisions are aver-aged over these significantly larger scales: modelling the flow as a continuum makes sense. Granular flows also behave in this way, but the time and length scales are significantly closer to the scales that are significant in individual grain interactions. Treating these flows as a continuum is not as straightfor-ward or as plausible; some phenomena, such as force "bridging" where force is transferred through discrete chains of grains in contact and not averaged through the volume, are out of reach of the continuum approach. That said, the continuum approach has been successfully applied in engineering to solve soil mechanics problems, for example, and appears to be the most promising approach12 to efficiently solve large-scale problems. We have all seen sand flow on a dune or a beach like it was water, yet sand always settles in a way that water never does. The primary reason for this response is the presence of friction between the individual grains of sand. As mentioned in Chapter Two, scientists who model granular flow often use mesh-free methods like discrete elements where the response of every single grain is tracked, or with Lagrangian mesh-based elasto-plastic continuum methods that are strictly limited to small amounts of flow. We feel that an Eulerian continuum approach would be an ideal third possibility for applications in computer graphics or as quick analysis tools for engineers. While this research was ultimately unsuccessful in effectively simulating granular material with an Eulerian continuum formulation, we here report on our first steps towards this goal. Rather than tackle the full problem 1 2 As opposed to the previously mentioned elasto-plastic Lagrangian or discrete element techniques 30 Chapter 4. Towards Eulerian Granular Flow with a Mohr-Coulomb constitutive law, which relates shear stress and strain rate in the continuum to ihter-particle friction, we restricted attention to just the pressure/density relationship of an idealized frictionless granular material. This is in line with the previous chapters' work on improved handling of pressure/density changes in low speed flow, but also exposes a key distinction of granular flow from fluid flow: the presence of inequality constraints, expressed as complementarity conditions. Our continuum model for sand is created by averaging properties onto a MAC grid. The density in a cell essentially measures the number of sand particles in a cell, and the pressure measures the normal contact force be-tween them. This approximation is obviously isotropic; direction and ori-entation of individual contacting particles is lost. The cells that represent an unsettled flowing region of sand may have density values less than the cells of a settled region of sand. As a simplified first approximation, when the density is less than the settled value, we take the pressure (the average normal contact force) to be zero. Similarly, if the sand in an area of our domain has settled, its density will exactly equal some ideal settled density and the pressure will be allowed to be positive to resist applied compressive load (e.g. from gravity). This relationship is known as a complementarity relationship; one quantity (pressure) is positive while the other (the differ-ence between current density and the settled density) is zero, but they are never positive at the same time. Mathematically this is expressed as This constraint replaces the divergence-free condition of incompressible flow (where p can be both positive and negative, but density is always held con-stant), or the equation of state relating pressure and density in compressible flow. 4.1 Complementarity Complementarity problems are a broad class of problems, with the simplest case being the linear complementarity problem (LCP). In its simplest form, the LCP can be stated as the goal of determining p E Rn such that 0<pl Psettled ~ P > 0 (4.1) P> o Ap + b > 0 pT(Ap + b) = 0 (4.2) (4.3) (4.4) 31 Chapter 4. Towards Eulerian Granular Flow which can also be written as before, 0 < p _L Ap + b > 0 [14]. To clar-ify, this can be thought of as a relationship where two quantities cannot be greater than zero at the same time. A vector p that satisfies the first two inequalities in 4.2 is said to be feasible. Another way of expressing this is to let w = Ap + b, then for all i = l , . . . ,n , PiWi = 0. Depending on the matrix A and vector b the L C P may have zero, one, or an infinite number of solutions. This complementarity relationship has been used to describe phenomena in a large number of fields including economics, civil engineer-ing, and computer graphics. LCPs are a specialized version of a mixed complementarity problem: a more general formulation that combines the previous complementarity problem, possible with a nonlinear relationship between the two non-negative quantities, with additional equations that do not involve complementarity[2, 49]. LCPs are not the only complementarity problem, but since we have a linear system we will only mention LCPs. The L C P can also be expressed as the following inequality quadratic program (QP): minimize xT(Ax + b) subject to Ax + b > 0 (4.5) subject to x > 0 If the matrix is positive definite the problem has a unique solution. If the matrix A is positive semidefinite, it can be shown that the problem has a solution (potentially non-unique). Using this expression of the L C P can be advantageous since QP problems have been extensively researched over a long period of time. Additionally, they do not necessarily suffer from the same shortcomings as competing L C P algorithms. Frequently in video games or in movie animation it is necessary to model the dynamic response of contacting rigid bodies. Rigid body contact (as it's known) can be formulated as a linear complementarity or mixed com-plementarity problem. If we consider two rigid bodies the idea is easy to conceptualize. When two bodies are not in contact there is no impulse (force integrated over time) between them, but the distance is greater than zero. When two bodies are in contact the force between them is positive, while the distance separating them is zero. If we solve for the position and velocity of our rigid bodies using the following scheme n vn+l = yn + ^ ^ + At g (4.6) i=0 xn+1 =xn + Atvn+1 (4.7) 32 Chapter 4. Towards Eulerian Granular Flow the following LCP can be formulated 0 < h _L \xi — Xj\ > 0 where \xi — Xj\ represents the distance between two rigid bodies13. A great deal of research [2, 3, 4, 49, 50] has begun with this simple idea. An obvious concern is that this analysis does not consider frictional forces that develop between two contacting bodies. Incorporating frictional constraints into our LCP is not as easy to visualize and is beyond the scope of this work. The interested reader can consult [2, 3, 27, 49] for details concerning frictional constraints. The simple argument begins with Coulomb friction. If the tangential friction impulse is less than a yield value IY = pl^ where p is the coefficient of friction and IN is the normal impulse between the two bodies, i.e. if IY — I > 0, then the two contacting bodies will not slide past each other. On the other hand, if I = IY, then the friction impulse I cannot exceed IY and there is tangential movement between the two bodies. This argument is analogous to our initial argument for approximating Mohr-Coulomb friction (the tensor stress/strain version of Coulomb's law) on an Eulerian grid phrased as a complementarity problem. 4.2 Finite Difference Solution The linear complementarity problem using pressure as the driving variable was initially built by discretizing, like before, the non-conservative momen-tum equation 1.2 and the continuity equation 1. We again employed operator splitting to separate the advection of density and velocity, but this time we used semi-Lagrangian advection to calculate p and u. The equations for density and velocity become: un+l . - un+l . vn+\-vn+\ ^ = -Pij - Atpitj( ^ + (4.8) n+1 _ - ^ (Pi+hL U i + h j - U i H j A x ^,3 ) (4-9) The equations for the next time step velocities were then substituted into the equation for the next time step density. This density was then subtracted from psettied to create the linear system in pressure for the complementarity problem p > 0 J. pmax — P > 0. After substitution a row in this system looks 1 3This is not the only way to formulate this general contact complementarity idea; a great deal of research has gone into this and no method has shown to be the optimal solution 33 Chapter 4. Towards Eulerian Granular Flow like: [ A i 2 fPi+ij Pi-ij Pij+i Pij-i _ Pij\ _ Pmax PiA A 9 I ~ "r _ ~r _ T _ ~ I L J&X*\Pi+y p._y piJ+i_ PiJ_i pijJ\ Here, as before, the p-.x • and 5- •. i values are averaged to cell faces from t i c 2 ) J ^ j J 1 1 ^ adjacent cell densities. The density that appears in equation 4.8 and 4.9 are the same if the density in equation 4.8 is brought inside the parentheses by using face averaged density. If this is done the densities cancel in the linear system. Similar to the method just outlined, we constructed the same system us-ing the initial velocity-based compressible discretization outlined in Chapter Three: ignore the constitutive relationship, substitute the equation for ve-locity into the equation for density, and a system that is identical to the one here is obtained. Since the advection is calculated in a different way, the right hand side vector of the problem is not the same and the two dis-cretizations produce different results. The linear system constructed in 4.10 is positive semidefinite; the +1 term in the diagonal (4a + l)pij from our previous systems is missing. As a result this system is singular, with a null-space corresponding to constant pressure values. Positive semidefinite (PSD) systems do not admit unique solutions to the LCP, but do have unique w = Ap + b vectors (known as the w-uniqueness property); that is, if pi and p2 are solutions to the LCP, then Api = Ap2 [14]. This result means that although multiple feasible pressure fields may exist, the density profile is unique. This singularity could be resolved by constraining one pressure (for instance the pressure value corresponding to the lowest density) to zero and thus eliminating it from the matrix sent to our LCP algorithms. A regularization approach, where a small amount is added along the diagonal, could also be used to improve the conditioning of the system. 4.3 Implementation The sand simulator using the LCP solution for pressures was implemented in Matlab. We attempted to solve the LCP presented in section 4.2 using the Path solver [15, 21], an implementation of Lemke's algorithm [14], and using a Gauss-Seidel method modified to use a clamped Newton step. For 34 Chapter 4. Towards Eulerian Granular Flow Figure 4.1: Plots showing sand settle under gravity. our application, the rich history and availability of robust QP methods is attractive, but primarily we need a method that scales well. The perfor-mance of Lemke degrades as the problem size increases, a trait we would have liked to have avoided by using a QP method. Also, Lemke's method can not take advantage of previous solutions (a warm start) as certain QP methods can. Despite its drawbacks Lemke's method is commonly used since, if the matrix is positive definite, it will find the solution if it exists. For this reason, we used it primarily with test problems (which were never passed). We assumed that the problem was poorly posed or an error existed if Lemke's algorithm failed to find a solution. Although the matrix characteristics point to the existence of a solution or solutions, the algorithms used frequently did not converge. The solutions also depended on the initial vector used; often a zero vector did not work. The system constructed from the method outline in section 3.2 behaved similarly. We suspect cells with densities approaching zero are to blame, but the question as to why the method failed is still an open one. Developing an algorithm that correctly solves this LCP requires that this question is answered. We do feel the idea is sound and could lead to significant improvement in quickly modelling granular dynamics, our stated goal. Unfortunately, we did not achieve this. Hopefully, spending additional time—specifically to develop a specialized QP algorithm for this problem—could lead to accurate and reliable solutions to this LCP. Once the pressure LCP is solved the framework can easily be extended to correctly model the physics between settled grains of sand since there are no restrictions placed on the divergence of velocity like in Zhu's work. 35 Bibliography [1] John D. Anderson. 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