Computer simulation of physical processes

BACKGROUND
This invention relates to computer simulation of physical processes, such as fluid flow.
High Reynolds number flow has been simulated by generating discretized solutions of the Navier-Stokes differential equations by performing high-precision floating point arithmetic operations at each of many discrete spatial locations on variables representing the macroscopic physical quantities (e.g., density, temperature, flow velocity). More recently, the differential equation approach has been replaced with what is generally known as lattice gas (or cellular) automata, in which the macroscopic-level simulation provided by solving the Navier-Stokes equations is replaced by a microscopic-level model that performs operations on particles moving between sites on a lattice.
The traditional lattice gas simulation assumes a limited number of particles at each lattice site, with the particles being represented by a short vector of bits. Each bit represents a particle moving in a particular direction. For example, one bit in the vector might represent the presence (when set to 1) or absence (when set to 0) of a particle moving along a particular direction. Such a vector might have six bits, with, for example, the values 110000 indicating two particles moving in opposite directions along the X axis, and no particles moving along the Y and Z axes. A set of collision rules governs the behavior of collisions between particles at each site (e.g., a 110000 vector might become a 001100 vector, indicating that a collision between the two particles moving along the X axis produced two particles moving away along the Y axis). The rules are implemented by supplying the state vector to a lookup table, which performs a permutation on the bits (e.g., transforming the 110000 to 001100). Particles are then moved to adjoining sites (e.g., the two particles moving along the Y axis would be moved to neighboring sites to the left and right along the Y axis).
Molvig et al. taught an improved lattice gas technique in which, among other things, many more bits were added to the state vector at each lattice site (e.g., 54 bits for subsonic flow) to provide variation in particle energy and movement direction, and collision rules involving subsets of the full state vector were employed. Molvig et al., PCT/US91/04930; Molvig et al., "Removing the Discreteness Artifacts in 3D Lattice-Gas Fluids", Proceedings of the Workshop on Discrete Kinetic Theory, Lattice Gas Dynamics, and Foundations of Hydrodynamics, World Scientific Publishing Co., Pte., Ltd., Singapore (1989); Molvig et al., "Multi-species Lattice-Gas Automata for Realistic Fluid Dynamics", Springer Proceedings in Physics, Vol. 46, Cellular Automata and Modeling of Complex Physical Systems, Springer-Verlag Berlin, Heidelberg (1990) (all hereby incorporated by reference). These improvements and others taught by Molvig et al. produced the first practical lattice-gas computer system. Discreteness artifacts that had made earlier lattice gas models inaccurate at modeling fluid flow were eliminated.
Chen et al. taught an improved simulation technique in U.S. Pat. No. 5,594,671, "Computer System For Simulating Physical Processes Using Multiple-Integer State Vectors" (which is hereby incorporated by reference). Instead of the lattice gas model in which at each lattice site, or voxel (these two terms are used interchangeably throughout this document), there is at most a single particle in any momentum state (e.g., at most a single particle moving in a particular direction with a particular energy), the system used a multi-particle technique in which, at each voxel, multiple particles could exist at each of multiple states (e.g., in an eight-bit implementation, 0-255 particles could be moving in a particular direction). The state vector, instead of being a set of bits, was a set of integers (e.g., a set of eight-bit bytes providing integers in the range of 0 to 255), each of which represented the number of particles in a given state. Thus, instead of being limited to a single particle moving in each direction at each momentum state, the system had the flexibility to model multiple particles moving in each direction at each momentum state.
Chen et al.'s use of integer state vectors made possible much greater flexibility in microscopic modeling of physical processes because much more variety was possible in the collision rules that operated on the new integer state vectors. The multi-particle technique provided a way of achieving the so-called microscopic Maxwell-Boltzmann statistics that are characteristic of many fluids.
The Chen et al. system also provided a way of simulating the interaction between fluid particles and solid objects using a new "slip" technique that extended the simulation only to the outer surface of the boundary layer around a solid object, and not through the boundary layer to the surface of the solid object. At the outer surface of the boundary layer, the collision rules governing interactions between particles and the surface allowed particles to retain tangential momentum.
Chen et al. employed both "slip" and "bounce back" collision techniques in combination to simulate surfaces with a range of skin friction, from the very high skin friction of pure "bounce back" to the very low skin friction provided by "slip". Varying fractions of the particles were treated with "bounce back" rules, and the remainder were treated with "slip" rules. The multi-particle model of Chen et al. accommodated arbitrary angular orientation of the solid boundary with respect to the lattice by allowing use of a weighted average of multiple outgoing states to assure that the average momentum of the outgoing particles was in a direction closely approximating true specular reflection.
Chen et al. described techniques for preserving energy, mass and momentum normal to the solid boundary. Momentum normal to the solid boundary was preserved using a "pushing/pulling" technique that compared the overall incoming normal momentum to the overall outgoing normal momentum and recorded the normal surplus or deficit (i.e., the amount of normal momentum that had to be made up in some way that did not introduce artifacts into the simulation). Chen et al. then used a set of pushing/pulling rules to drive the normal surplus toward zero. Particles were moved from certain out-states to other out-states so that only normal momentum was affected.
Changes in energy were accommodated by a "cooling" (or heating) technique that used a total energy counter to keep track of an energy surplus (or deficit) and cooling/heating rules to drive the surplus toward zero. Similarly, "dieting" rules were used to remove any surplus mass that accumulated as the result of one or more of the collision rules.
SUMMARY
The invention provides an improved technique for simulating the interaction between fluid particles and solid boundaries in a computer system for simulating physical processes. The technique models skin friction at the outer surface of the boundary layer in a way that permits accurate simulation of the separation of fluid flow from the solid object. In particular, the technique automatically determines where a flow is becoming separated and increases the skin friction at that point by modifying a friction parameter based on changes in pressure along the surface.
The improved technique is related to techniques described by Chen et al. in U.S. application Ser. No. 08/566,032, entitled "Computer System for Simulating Physical Processes" (which is hereby incorporated by reference). In the techniques described by Chen et al., the surface of a boundary is represented as a collection of facets that are unrestricted by voxel boundaries, and may be arbitrarily oriented and sized relative to the lattice geometry of the voxels. This permits a continuously varying surface to be simulated with precision that is significantly greater than the precision of the lattice.
A lattice gas is traditionally pictured as a set of discrete points distributed in a lattice structure that overlies a continuum. Particles of the lattice gas are point masses that propagate between lattice sites. The entire system is discretized in both space and time, and no physical meaning is assigned to non-integer space and time values. This model of a lattice gas has the virtue of being simple to envision and implement, but is not easily translated to a continual real physics system.
For the techniques described by Chen et al., an alternate physical picture of the lattice gas is formed by assuming that the mass associated with each particle of the lattice gas is uniformly distributed throughout the volume of the voxel occupied by the particle. Propagation of particles is viewed as movement of this distributed mass in the distance and direction indicated by the velocity of the particle. When particles are moving between adjacent fluid voxels, this distributed mass view of the lattice gas is equivalent to the traditional view of the lattice gas. However, the distributed mass view of the lattice gas offers the considerable advantage that it permits the particles of the lattice gas to interact with a surface that is arbitrarily oriented relative to the structure of the lattice.
In one aspect, generally, the invention features storing in a memory state vectors for multiple voxels and a representation for at least one surface. The state vectors include multiple entries that correspond to elements at particular momentum states of multiple possible momentum states at a voxel. Interaction operations that model interactions between elements of different momentum states are performed on the state vectors, and surface interaction operations that model interactions between a surface and elements at one or more voxels near the facet are performed on the representations of the surface. Finally, move operations that reflect movement of elements to new voxels are performed on the state vectors. The surface interaction operations retain at least a portion of tangential momentum of elements moving along the surface, with the portion that is retained corresponding to a friction parameter. The friction parameter is varied based on changes in pressure near the surface.
Embodiments of the invention may include one or more of the following features. The friction parameter may be varied based on a pressure gradient near the surface in a direction parallel to tangential flow along the surface, where the pressure gradient is based on a weighted average of pressure changes in voxels that directly interact with the surface.
The surface may be sized and oriented independently of the size and orientation of the voxels. By permitting the surface to be oriented and sized arbitrarily relative to the orientation and size of the voxels, the invention permits interactions with a surface to be simulated with increased resolution relative to the size of the voxels. This, in turn, permits highly accurate simulations of physical processes such as fluid flow.
In some instances, only a portion of a voxel may interact with a surface. When this occurs and the friction parameter is varied based on a weighted average of changes in pressure in voxels that directly interact with the surface, a contribution of a voxel to the weighted average is weighted by the portion of the voxel that interacts with the surface.
The entries of the state vectors may represent a density of elements per unit volume in a particular momentum state of a voxel. As discussed above, this distributed mass view of the state vectors offers considerable advantages.
In addition, as described by Chen et al., the entries of the state vectors may have integer values, and the interaction operations may be performed using integer values. However, to permit arbitrary orientation and sizing of the surface facets, the surface interaction operations may be performed using values representative of real numbers, such as floating point values. This hybrid approach permits the processing efficiency offered by integer operations to be obtained simultaneously with the structural flexibility offered by real numbers. To ensure that system accuracy will not be impaired by inaccuracies introduced by real number calculations, exact conservation of mass, momentum and energy is enforced as simulated elements are moved from a surface facet to a voxel. Voxels that are intersected by a surface (and occupied by the region underlying the surface) may be viewed as partial voxels.
A surface may be represented as a collection of facets that are sized and oriented independently of the size and orientation of the voxels and, in combination, represent at least one surface. In this case, the surface interaction operations model interactions between a facet and elements of at least one voxel near the facet. The elements may represent particles of a fluid and the facets may represent at least one surface over which the fluid flows.
For each facet, the surface interaction operations may include gathering elements from a first set of one or more voxels that interact with the facet, modelling interactions between the gathered elements and the facet to produce a set of surface interacted elements, and scattering the surface interacted elements to a second set of one or more voxels that interact with the facet. Typically, the surface interaction operations occur as a component of the operations by which elements are moved from voxel to voxel. The gather and scatter steps account for the arbitrary orientation of the facets. This permits the modelling step to be performed without regard to that orientation. Typically, the first set of voxels is identical to the second set of voxels.
When a portion of a voxel is occupied by one or more facets, or portions thereof, a factor indicative of the portion of the voxel that is not occupied by any facets (or material underlying a facet) is stored with the state vector for the facet. This factor permits operations to be performed on an occupied voxel without further regard to the degree to which the voxel is occupied. For example, the factor for a voxel may be considered when gathering elements from the voxel.
The elements obtained in the gather step for a particular momentum state may be identified as the elements occupying a region projected from the facet in the opposite direction of the velocity of the momentum state. Accordingly, the first set of voxels may include one or more voxels that overlap at least a portion of a region defined by a vector dot product of a vector representing a normal direction relative to the facet with a vector representing the velocity of the momentum state for which elements are being gathered. In particular, the first set of voxels may include all voxels that at least partially overlap a region projected from the facet and may have a volume that equals the vector dot product multiplied by a surface area of the facet.
The elements obtained in the gather step also may include elements scattered from one or more other facets. When a surface includes a non-convex feature such as an interior corner, a region projected from a facet may be partially occupied by another facet. To account for this situation, elements are also gathered from appropriately oriented facets.
Typically, interactions between the gathered elements and the facet are modelled so that the set of surface interacted elements satisfies one or more boundary conditions. The boundary conditions may include tangential momentum conditions. When the surface represented by a facet has no surface friction, the tangential momentum condition is satisfied when a combined momentum of the gathered elements that is tangential to the facet equals a combined momentum of the surface interacted elements that is tangential to the facet. When the surface includes friction, the tangential momentum condition is satisfied when a difference between the combined tangential momentum of the gathered elements and the combined tangential momentum of the surface interacted elements is related to the level of surface friction of the surface (e.g., when the surface is completely non-slip, the combined tangential momentum of the surface interacted elements is zero).
The boundary conditions also may include a normal momentum condition that is satisfied when a difference between a combined momentum of the gathered elements that is normal to the facet and a combined momentum of the surface interacted elements that is normal to the facet equals a pressure in a region defined by one or more voxels of the first set of voxels.
The boundary conditions also may require conservation of mass and energy at the boundary. These conditions are satisfied when a combined mass of the gathered elements equals a combined mass of the surface interacted elements, and a combined energy of the gathered elements equals a combined energy of the surface interacted elements.
When a state vector includes integers representative of particular momentum states and the surface interaction operations are performed using values representative of real numbers, care must be taken to ensure that the integer values of the state vectors do not overflow or underflow when the surface interacted elements are scattered to the voxels. For this purpose, procedures that conserve the mass, momentum and energy of a voxel are employed whenever truncation is necessary to avoid overflow or underflow.
In another aspect, generally, the invention features simulating a physical process by storing in a memory state vectors for multiple voxels, with the state vectors including entries that correspond to particular momentum states of multiple possible momentum states at a voxel. Thereafter, operations are performed on the state vectors to model interactions between elements of different momentum states and movement of elements relative to the voxels, with at least some of the operations being performed using floating point numbers and some of the operations being performed using integers. Transitions between floating point and integer values may be made in a way that preserves physical properties of the physical process such as mass and energy. The entries of the state vectors may be integers representing a density of elements per unit volume in a particular momentum state. In addition, a representation of at least one surface that is sized and oriented independently of the size and orientation of the voxels may be stored in memory, and the operations performed using floating point numbers may be the operations that model interactions of the elements with the surface.
The invention may be implemented as part of the PowerFLOW.TM. software product available from Exa Corporation of Lexington, Mass. The PowerFLOW.TM. product is a software-only implementation of Exa's DIGITAL PHYSICS.TM. technology. The PowerFLOW.TM. product has been implemented on Ultra.TM. Workstations and Ultra Enterprise.TM. systems available from Sun Microsystems. However, the techniques described here are not limited to any particular hardware or software configuration; they may find applicability in any computing or processing environment that may be used for simulating a physical process. The techniques may be implemented in hardware or software, or a combination of the two. Preferably, the techniques are implemented in computer programs executing on programmable computers that each include a processor, a storage medium readable by the processor (including volatile and non-volatile memory and/or storage elements), at least one input device, and one or more output devices. Program code is applied to data entered using the input device to perform the functions described and to generate output information. The output information is applied to one or more output devices.
Each program is preferably implemented in a high level procedural or object oriented programming language to communicate with a computer system. However, the programs can be implemented in assembly or machine language, if desired. In any case, the language may be a compiled or interpreted language.
Each such computer program is preferably stored on a storage medium or device (e.g., CD-ROM, hard disk or magnetic diskette) that is readable by a general or special purpose programmable computer for configuring and operating the computer when the storage medium or device is read by the computer to perform the procedures described in this document. The system may also be considered to be implemented as a computer-readable storage medium, configured with a computer program, where the storage medium so configured causes a computer to operate in a specific and predefined manner.
The disclosures of U.S. application Ser. No. 08/409,685, filed Mar. 23, 1995; U.S. application Ser. No. 08/255,409, filed Jun. 8, 1994; and U.S. Pat. No. 5,377,129 are all hereby incorporated by reference.
Other features and advantages of the invention will be apparent from the following detailed description, including the drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a flow chart of a procedure followed by a physical process simulation system.
FIG. 2 is a perspective view of a microblock.
FIGS. 3A and 3B are illustrations of lattice structures used by the system of FIG. 1.
FIGS. 4 and 5 illustrate variable resolution techniques.
FIG. 6 illustrates regions affected by a facet of a surface.
FIG. 7 illustrates movement of particles from a voxel to a surface.
FIG. 8 illustrates movement of particles from a surface to a surface.
FIG. 9 is a flow chart of a procedure for performing surface dynamics.
FIG. 10 illustrates an interface between voxels of different sizes.
FIG. 11 is a flow chart of a procedure for simulating interactions with facets under variable resolution conditions.
FIG. 12 illustrates a boundary layer for flow along a surface.
FIG. 13 is a flow chart of a procedure for generating a friction coefficient.

DESCRIPTION
A. Model Simulation Space
Referring to FIG. 1, a physical process simulation system operates according to a procedure 100 to simulate a physical process such as fluid flow. Prior to the simulation, a simulation space is modelled as a collection of voxels (step 102). Typically, the simulation space is generated using a computer-aided-design (CAD) program. For example, a CAD program could be used to draw an automobile positioned in a wind tunnel. Thereafter, data produced by the CAD program is processed to add a lattice structure having appropriate resolution and to account for objects and surfaces within the simulation space.
The resolution of the lattice may be selected based on the Reynolds number of the system being simulated. The Reynolds number is related to the viscosity (.nu.) of the flow, the characteristic length (L) of an object in the flow, and the characteristic velocity (u) of the flow:
Re=uL/.nu..
The characteristic length of an object represents large scale features of the object. For example, if flow around an automobile were being simulated, the height of the automobile might be considered to be the characteristic length. When flow around small regions of an object (e.g., the side mirror of an automobile) is of interest, the resolution of the simulation may be increased, or areas of increased resolution may be employed around the regions of interest. The dimensions of the voxels decrease as the resolution of the lattice increases.
The state space is represented as N.sub.i (x, t), where N.sub.i represents the number of elements, or particles, per unit volume in state i (i.e., the density of particles in state i) at a lattice site denoted by the three-dimensional vector x at a time t. For a known time increment, the number of particles is referred to simply as N.sub.i (x). The combination of all states of a lattice site is denoted as N(x).
The number of states is determined by the number of possible velocity vectors within each energy level. The velocity vectors consist of integer linear speeds in a space having four dimensions: x, y, z and w. The fourth dimension, w, is projected back onto three-dimensional space and thus does not indicate an actual velocity in the three-dimensional lattice. For subsonic mono-species flows, i ranges from 0 to 53. The number of states is increased for transonic flows or multiple-species simulations.
Each state i represents a different velocity vector at a specific energy level (i.e., energy level zero, one or two). The velocity c.sub.i of each state is indicated with its "speed" in each of the four dimensions as follows:
C.sub.i =(C.sub.x, C.sub.y, C.sub.z, C.sub.w).
The energy level zero state represents stopped particles that are not moving in any dimension, i.e. C.sub.stopped =(0, 0, 0, 0). Energy level one states represent particles having a +/-1 speed in two of the four dimensions and a zero speed in the other two dimensions. Energy level two states represent particles having either a +/-1 speed in all four dimensions, or a +/-2 speed in one of the four dimensions and a zero speed in the other three dimensions.
Generating all of the possible permutations of the three energy levels gives a total of 49 possible states (one energy zero state, 24 energy one states, 24 energy two states). However, the subsonic flow state space maintains a total of six stopped states as opposed to one, giving a total of 54 states.
Each voxel (i.e., each lattice site) is represented by a state vector N(x). The state vector completely defines the status of the voxel and includes 54 multi-bit entries, each of which corresponds to an integer value. The 54 entries correspond to the six rest states, 24 directional vectors at the first energy level and 24 directional vectors at the second energy level. The six rest states are employed to ensure that there are a sufficient number of rest "slots". Of course, this same effect could be achieved by increasing the number of bits in the entry corresponding to the rest state in a 49 entry embodiment. By using multi-bit entries, the system can produce Maxwell-Boltzmann statistics for an achieved equilibrium state vector.
For processing efficiency, the voxels are grouped in 2.times.2.times.2 volumes called microblocks. The microblocks are organized to permit parallel processing of the voxels and to minimize the overhead associated with the data structure. A short-hand notation for the voxels in the microblock is defined as N.sub.i (n), where n represents the relative position of the lattice site within the microblock and n .di-elect cons.{0,1,2, . . . , 7}. A microblock is illustrated in FIG. 2.
Referring to FIGS. 3A and 3B, a surface S (FIG. 3A) is represented in the simulation space (FIG. 3B) as a collection of facets F.sub..alpha. :
S={F.sub..alpha. }
where .alpha. is an index that enumerates a particular facet. A facet is not restricted to the voxel boundaries, but is typically sized on the order of or slightly smaller than the size of the voxels adjacent to the facet so that the facet affects a relatively small number of voxels. Properties are assigned to the facets for the purpose of implementing surface dynamics. In particular, each facet F.sub..alpha. has a unit normal (n.sub..alpha.), a surface area (A.sub..alpha.), a center location (x.sub..alpha.), and a facet distribution function (N.sub.i (.alpha.)) that describes the surface dynamic properties of the facet.
Referring to FIG. 4, different levels of resolution may be used in different regions of the simulation space to improve processing efficiency. Typically, the region 150 around an object 155 is of the most interest and is therefore simulated with the highest resolution. Because the effect of viscosity decreases with distance from the object, decreasing levels of resolution (i.e., expanded voxel volumes) are employed to simulate regions 160, 165 that are spaced at increasing distances from the object 155. Similarly, as illustrated in FIG. 5, a lower level of resolution may be used to simulate a region 170 around less significant features of an object 175 while the highest level of resolution is used to simulate regions 180 around the most significant features (e.g., the leading and trailing surfaces) of the object 175. Outlying regions 185 are simulated using the lowest level of resolution and the largest voxels. Techniques for processing interactions between voxels of different sizes are discussed by Molvig et al. in U.S. Pat. No. 5,377,129, which is incorporated herein by reference, at col. 18, line 58 to col. 28, line 21. Techniques for processing interactions between voxels of different sizes and the facets of a surface are discussed below.
B. Identify Voxels Affected By Facets
Referring again to FIG. 1, once the simulation space has been modelled (step 102), voxels affected by one or more facets are identified (step 104). Voxels may be affected by facets in a number of ways. First, a voxel that is intersected by one or more facets is affected in that the voxel has a reduced volume relative to non-intersected voxels. This occurs because a facet, and material underlying the surface represented by the facet, occupies a portion of the voxel. A fractional factor P.sub.f (x) indicates the portion of the voxel that is unaffected by the facet (i.e., the portion that can be occupied by a fluid or other materials for which flow is being simulated). For non-intersected voxels, P.sub.f (x) equals one.
Voxels that interact with one or more facets by transferring particles to the facet or receiving particles from the facet are also identified as voxels affected by the facets. All voxels that are intersected by a facet will include at least one state that receives particles from the facet and at least one state that transfers particles to the facet. In most cases, additional voxels also will include such states.
Referring to FIG. 6, for each state i having a non-zero velocity vector c.sub.i, a facet F.sub..alpha. receives particles from, or transfers particles to, a region defined by a parallelepiped G.sub.i.alpha. having a height defined by the magnitude of the vector dot product of the velocity vector c.sub.i and the unit normal n.sub..alpha. of the facet (.vertline.c.sub.i .multidot.n.sub..alpha. .vertline.) and a base defined by the surface area A.sub..alpha. of the facet so that the volume V.sub.i.alpha. of the parallelepiped G.sub.i.alpha. equals:
V.sub.i.alpha. =.vertline.c.sub.i .multidot.n.sub..alpha. .vertline.A.sub..alpha..
The facet F.sub..alpha. receives particles from the volume V.sub.i.alpha. when the velocity vector of the state is directed toward the facet (c.sub.i .multidot.n.sub..alpha. <0), and transfers particles to the region when the velocity vector of the state is directed away from the facet (c.sub.i .multidot.n.sub..alpha. >0). As will be discussed below, this expression must be modified when another facet occupies a portion of the parallelepiped G.sub.i.alpha., a condition that could occur in the vicinity of non-convex features such as interior corners.
The parallelepiped G.sub.i.alpha. of a facet F.sub..alpha. may overlap portions or all of multiple voxels. The number of voxels or portions thereof is dependent on the size of the facet relative to the size of the voxels, the energy of the state, and the orientation of the facet relative to the lattice structure. The number of affected voxels increases with the size of the facet. Accordingly, the size of the facet, as noted above, is typically selected to be on the order of or smaller than the size of the voxels located near the facet.
The portion of a voxel N(x) overlapped by a parallelepiped G.sub.i.alpha. is defined as V.sub.i.alpha. (x). Using this term, the flux .GAMMA.i.alpha.(x) of state i particles that move between a voxel N(x) and a facet F.sub..alpha. equals the density of state i particles in the voxel (Ni(x)) multiplied by the volume of the region of overlap with the voxel (Vi.alpha.(x)):
.GAMMA..sub.i.alpha. (x)=N.sub.i (x)V.sub.i.alpha. (x).
When the parallelepiped G.sub.i.alpha. is intersected by one or more facets, the following condition is true: ##EQU1## where the first summation accounts for all voxels overlapped by G.sub.i.alpha. and the second term accounts for all facets that intersect G.sub.i.alpha.. When the parallelepiped G.sub.i.alpha. is not intersected by another facet, this expression reduces to: ##EQU2## C. Perform Simulation
Once the voxels that are affected by one or more facets are identified (step 104), a timer is initialized to begin the simulation (step 106). During each time increment of the simulation, movement of particles from voxel to voxel is simulated by an advection stage (steps 108-116) that accounts for interactions of the particles with surface facets. Next, a collision stage (step 118) simulates the interaction of particles within each voxel. Thereafter, the timer is incremented (step 120). If the incremented timer does not indicate that the simulation is complete (step 122), the advection and collision stages (steps 108-120) are repeated. If the incremented timer indicates that the simulation is complete (step 122), results of the simulation are stored and/or displayed (step 124).
1. Boundary Conditions For Surface
To correctly simulate interactions with a surface, each facet must meet four boundary conditions. First, the combined mass of particles received by a facet must equal the combined mass of particles transferred by the facet (i.e., the net mass flux to the facet must equal zero). Second, the combined energy of particles received by a facet must equal the combined energy of particles transferred by the facet (i.e., the net energy flux to the facet must equal zero). These two conditions may be satisfied by requiring the net mass flux at each energy level (i.e., energy levels one and two) to equal zero.
The other two boundary conditions are related to the net momentum of particles interacting with a facet. For a surface with no skin friction, referred to herein as a slip surface, the net tangential momentum flux must equal zero and the net normal momentum flux must equal the local pressure at the facet. Thus, the components of the combined received and transferred momentums that are perpendicular to the normal n.sub..alpha. of the facet (i.e., the tangential components) must be equal, while the difference between the components of the combined received and transferred momentums that are parallel to the normal n.sub..alpha. of the facet (i.e., the normal components) must equal the local pressure at the facet. For non-slip surfaces, friction of the surface reduces the combined tangential momentum of particles transferred by the facet relative to the combined tangential momentum of particles received by the facet by a factor that is related to the amount of friction.
2. Gather From Voxels to Facets
As a first step in simulating interaction between particles and a surface, particles are gathered from the voxels and provided to the facets (step 108). As noted above, the flux of state i particles between a voxel N(x) and a facet F.sub..alpha. is:
.GAMMA..sub.i.alpha. (x)=N.sub.i (x)V.sub.i.alpha. (x).
From this, for each state i directed toward a facet F.sub..alpha. (c.sub.i .multidot.n.sub..alpha. <0), the number of particles provided to the facet F.sub..alpha. by the voxels is: ##EQU3## Only voxels for which V.sub.i.alpha. (x) has a non-zero value must be summed. As noted above, the size of the facets are selected so that V.sub.i.alpha. (x) has a non-zero value for only a small number of voxels. Because V.sub.i.alpha. (x) and P.sub.f (x) may have non-integer values, .GAMMA..sub..alpha. (x) is stored and processed as a real number.
3. Move From Facet to Facet
Next, particles are moved between facets (step 110). If the parallelepiped G.sub.i.alpha. for an incoming state (c.sub.i .multidot.n.sub..alpha. <0) of a facet F.sub..alpha. is intersected by another facet F.sub..beta., then a portion of the state i particles received by the facet F.sub..alpha. will come from the facet F.sub..beta.. In particular, facet F.sub..alpha. will receive a portion of the state i particles produced by facet F.sub..beta. during the previous time increment. This relationship is illustrated in FIG. 8, where a portion 800 of the parallelepiped G.sub.i.alpha. that is intersected by facet F.sub..beta. equals a portion 805 of the parallelepiped G.sub.i.beta. that is intersected by facet F.sub..alpha.. As noted above, the intersected portion is denoted as V.sub.i.alpha. (.beta.). Using this term, the flux of state i particles between a facet F.sub..beta. and a facet F.sub..alpha. may be described as:
.GAMMA..sub.i.alpha. (.beta.,t-1)=.GAMMA..sub.i (.beta.)V.sub.i.alpha. (.beta.)/V.sub.i.alpha.,
where .GAMMA..sub.i (.beta.,t-1) is a measure of the state i particles produced by the facet F.sub..beta. during the previous time increment. From this, for each state i directed toward a facet F.sub..alpha. (c.sub.i .multidot.n.sub..alpha. <0), the number of particles provided to the facet F.sub..alpha. by the other facets is: ##EQU4## and the total flux of state i particles into the facet is: ##EQU5##
The state vector N(.alpha.) for the facet, also referred to as a facet distribution function, has 54 entries corresponding to the 54 entries of the voxel state vectors. The input states of the facet distribution function N(.alpha.) are set equal to the flux of particles into those states divided by the volume V.sub.i.alpha. :
N.sub.i (.alpha.)=.GAMMA..sub.iIN (.alpha.)/V.sub.i.alpha.,
for c.sub.i .multidot.n.sub..alpha. <0.
The facet distribution function is a simulation tool for generating the output flux from a facet, and is not necessarily representative of actual particles. To generate an accurate output flux, values are assigned to the other states of the distribution function. Outward states are populated using the technique described above for populating the inward states:
N.sub.i (.alpha.)=.GAMMA..sub.iOTHER (.alpha.)/V.sub.i.alpha.,
for c.sub.i .multidot.n.sub..alpha. .gtoreq.0, wherein .GAMMA..sub.iOTHER (.alpha.) is determined using the technique described above for generating .GAMMA..sub.iIN (.alpha.), but applying the technique to states (c.sub.i .multidot.n.sub..alpha. .gtoreq.0) other than incoming states (c.sub.i .multidot.n.sub..alpha. <0)). In an alternative approach, .GAMMA..sub.iOTHER (.alpha.) may be generated using values of .GAMMA..sub.iOUT (.alpha.) from the previous time step so that:
.GAMMA..sub.iOTHER (.alpha.,t)=.GAMMA..sub.iOUT (.alpha.,t-1).
For parallel states (c.sub.i .multidot.n.sub..alpha. =0), both V.sub.i.alpha. and V.sub.i.alpha. (x) are zero. In the expression for N.sub.i (.alpha.), V.sub.i.alpha. (x) appears in the numerator (from the expression for .GAMMA..sub.iOTHER (.alpha.) and V.sub.i.alpha. appears in the denominator (from the expression for N.sub.i (.alpha.)). Accordingly, N.sub.i (.alpha.) for parallel states is determined as the limit of N.sub.i (.alpha.) as V.sub.i.alpha. and V.sub.i.alpha. (x) approach zero.
The values of states having zero velocity (i.e., rest states and states (0, 0, 0, 2) and (0, 0, 0, -2)) are initialized at the beginning of the simulation based on initial conditions for temperature and pressure. These values are then adjusted over time.
4. Perform Facet Surface Dynamics
Next, surface dynamics are performed for each facet to satisfy the four boundary conditions discussed above (step 112). A procedure for performing surface dynamics for a facet is illustrated in FIG. 9. Initially, the combined momentum normal to the facet F.sub..alpha. is determined (step 905) by determining the combined momentum P(.alpha.) of the particles at the facet as: ##EQU6## for all i. From this, the normal momentum P.sub.n (.alpha.) is determined as:
P.sub.n (.alpha.)=n.sub..alpha. .multidot.P(.alpha.).
This normal momentum is then eliminated using a pushing/pulling technique described by Chen et al (step 910) to produce N.sub.n- (.alpha.). According to this technique, particles are moved between states in a way that affects only normal momentum. The pushing/pulling technique is described by Chen et al. in U.S. Pat. No. 5,594,671, which is incorporated by reference.
Thereafter, the particles of N.sub.n- (.alpha.) are collided to produce a Boltzmann distribution N.sub.n-.beta. (.alpha.) (step 915). As described below with respect to performing fluid dynamics, a Boltzmann distribution may be achieved by applying a set of collision rules to N.sub.n- (.alpha.).
An outgoing flux distribution for the facet F.sub..alpha. is then determined (step 920) based on the incoming flux distribution and the Boltzmann distribution. First, the difference between the incoming flux distribution .GAMMA..sub.i (.alpha.) and the Boltzmann distribution is determined as:
.DELTA..GAMMA..sub.i (.alpha.)=.GAMMA..sub.iIN (.alpha.)-N.sub.n-.beta.i (.alpha.)V.sub.i.alpha..
Using this difference, the outgoing flux distribution is:
.GAMMA..sub.iOUT (.alpha.)=N.sub.n-.beta.i (.alpha.)V.sub.i.alpha. -.DELTA..GAMMA..sub.i* (.alpha.),
for n.sub..alpha. .multidot.c.sub.i >0 and where i* is the state having a direction opposite to state i. For example, if state i is (1, 1, 0, 0), then state i* is (-1, -1, 0, 0). To account for skin friction and other factors, the outgoing flux distribution may be further refined to: ##EQU7## for n.sub..alpha. .multidot.c.sub.i >0, where C.sub.f is a function of skin friction, t.sub.i.alpha. is a first tangential vector that is perpendicular to n.sub..alpha., t.sub.2.alpha., is a second tangential vector that is perpendicular to both n.sub..alpha. and t.sub.1.alpha., and .DELTA.N.sub.j,1 and .DELTA.N.sub.j,2 are distribution functions corresponding to the energy (j) of the state i and the indicated tangential vector. The distribution functions are determined according to: ##EQU8## where j equals 1 for energy level 1 states and 2 for energy level 2 states.
The functions of each term of the equation for .GAMMA..sub.iOUT (.alpha.) are as follows. The first and second terms enforce the normal momentum flux boundary condition to the extent that collisions have been effective in producing a Boltzmann distribution, but include a tangential momentum flux anomaly. The fourth and fifth terms correct for this anomaly, which may arise due to discreteness effects or non-Boltzmann structure due to insufficient collisions. Finally, the third term adds a specified amount of skin fraction to enforce a desired change in tangential momentum flux on the surface. Generation of the friction coefficient C.sub.f is described below. Note that all terms involving vector manipulations are geometric factors that may be calculated prior to beginning the simulation.
An alternative approach that employs floating point numbers may be used to generate .GAMMA..sub.iOUT (.alpha.). According to this approach, the normal momentum is determined as:
P.sub.n (.alpha.)=n.sub..alpha. .multidot.P(.alpha.).
From this, a tangential velocity is determined as:
u.sub.i (.alpha.)=(P(.alpha.)-P.sub.n (.alpha.)n.sub..alpha.)/.rho.,
where .rho. is the density of the facet distribution: ##EQU9## A temperature T of the facet distribution is then determined as: ##EQU10## where j equals 0, 1 or 2 and corresponds to the energy of the state i. The Boltzmann distribution is then determined using u.sub.i and T as: ##EQU11## where
<N>.sub.j.sup.(eq) (j=0, 1, 2)
is the equilibrium isotropic state population at energy level j: ##EQU12## and d.sub.0 is the number of rest states (i.e., six).
As before, the difference between the incoming flux distribution and the Boltzmann distribution is determined as:
.DELTA..GAMMA..sub.i (.alpha.)=.GAMMA..sub.iIN (.alpha.)-N.sub.n-.beta.i (.alpha.)V.sub.i.alpha..
The outgoing flux distribution then becomes:
.GAMMA..sub.iOUT (.alpha.)=N.sub.n-.beta.i (.alpha.)V.sub.i.alpha. -.DELTA..GAMMA..sub.i* (.alpha.)+C.sub.f (n.sub..alpha. .multidot.c.sub.i)�N.sub.n-.beta.i* (.alpha.)-N.sub.n-.beta.i (.alpha.)!V.sub.i.alpha.,
which corresponds to the first two lines of the outgoing flux distribution determined by the previous technique but does not require the correction for anomalous tangential flux.
Using either approach, the resulting flux-distributions satisfy all of the momentum flux conditions, namely: ##EQU13## where p.sub..alpha. is the equilibrium pressure at the facet F.sub..alpha. and is based on the averaged density and temperature values of the voxels that provide particles to the facet, and u.sub..alpha. is the average velocity at the facet.
To ensure that the mass and energy boundary conditions are met, the difference between the input energy and the output energy is measured for each energy level j as: ##EQU14## where the index j denotes the energy of the state i. This energy difference is then used to generate a difference term: ##EQU15## for c.sub.ji .multidot.n.sub..alpha. >0. This difference term is used to modify the outgoing flux so that the flux becomes:
.GAMMA..sub..alpha.jiOUT.sbsb.f =.GAMMA..sub..alpha.jiOUT +.delta..GAMMA..sub..alpha.ji
for c.sub.ji .multidot.n.sub..alpha. >0. This operation corrects the mass and energy flux while leaving the tangential momentum flux unaltered. This adjustment is small if the flow is approximately uniform in the neighborhood of the facet and near equilibrium. The resulting normal momentum flux, after the adjustment, is slightly altered to a value that is the equilibrium pressure based on the neighborhood mean properties plus a correction due to the non-uniformity or non-equilibrium properties of the neighborhood.
5. Move From Voxels to Voxels
Referring again to FIG. 1, particles are moved between voxels along the three-dimensional rectilinear lattice (step 114). This voxel to voxel movement is the only movement operation performed on voxels that do not interact with the facets (i.e., voxels that are not located near a surface). In typical simulations, voxels that are not located near enough to a surface to interact with the surface constitute a large majority of the voxels.
Each of the separate states represents particles moving along the lattice with integer speeds in each of the four dimensions: x, y, z and w. The integer speeds include: 0, +/-1, and +/-2. The sign of the speed indicates the direction in which a particle is moving along the corresponding axis. These linear speeds support simulation of particles with energy levels ranging from zero to four. Only energy levels zero to two are needed for subsonic flow simulations, while all five are needed for transonic flow simulations.
For voxels that do not interact with a surface, the move operation is computationally quite simple. The entire population of a state is moved from its current voxel to its destination voxel during every time increment. At the same time, the particles of the destination voxel are moved from that voxel to their own destination voxels. For example, an energy level 1 particle that is moving in the +1x and +1y direction (1, 1, 0, 0) is moved from its current voxel to one that is +1 over in the x direction and +1 up in the y direction. The particle ends up at its destination voxel with the same state it had before the move (1,1,0,0). Interactions within the voxel will likely change the particle count for that state based on local interactions with other particles and surfaces. If not, the particle will continue to move along the lattice at the same speed and direction.
It should be noted here that particles in the stopped states (energy level zero) do not get moved. In addition, the particles only get moved in three dimensions. Non-zero values in the w dimension of a state do not affect the determination of a lattice site to which the state's particles are to be moved. For example, an energy level one particle with a -1z and a +1w speed (0,0,-1,1) and an energy level one particle with a -1z and a -1w (0,0,-1,-1) speed would both move to a site that is -1 away in the z dimension. There are also two energy level two states that do not get moved at all: (0,0,0,2) and (0,0,0,-2).
The move operation becomes slightly more complicated for voxels that interact with one or more surfaces. As noted above, V.sub.i.alpha. (x) and P.sub.f (x) may have non-integer values. This can result in one or more fractional particles being transferred to a facet. Transfer of such fractional particles to a facet results in fractional particles remaining in the voxels. These fractional particles are transferred to a voxel occupied by the facet. For example, referring to FIG. 7, when a portion 700 of the state i particles for a voxel 705 is moved to a facet 710 (step 108), the remaining portion 715 is moved to a voxel 720 in which the facet 710 is located and from which particles of state i are directed to the facet 710. Thus, if the state population equalled 25 and V.sub.i.alpha. (x) equalled 0.25 (i.e., a quarter of the voxel intersects the parallelepiped G.sub.i.alpha.), then 6.25 particles would be moved to the facet F.sub..alpha. and 18.75 particles would be moved to the voxel occupied by the facet F.sub..alpha.. Because multiple facets could intersect a single voxel, the number of state i particles transferred to a voxel N(f) occupied by one or more facets is: ##EQU16## where N(x) is the source voxel. Because N.sub.i (f) may be a real number, N.sub.i (f) is temporarily stored in a buffer as a floating point number or a scaled integer until the advection stage is completed.
To store N.sub.i (f) as a scaled integer, N.sub.i (f) is determined as: ##EQU17## where scale is a constant having a value of 2.sup.16 and N.sub.si (x) and V.sub.si.alpha. (x) are, respectively, N.sub.i (x) and V.sub.i.alpha. (x) multiplied by scale. The random number rand takes on values between zero and one prior to scaling, and is then scaled up by scale to values between 0 and 2.sup.16 -1. The operator floor then produces an integer value. In particular, floor returns the largest integer that is less than or equal to its argument. For example, floor(2.3) equals 2 and floor(-2.3) equals -3.
6. Scatter From Facets to Voxels
Next, the outgoing particles from each facet are scattered to the voxels (step 116). Essentially, this step is the reverse of the gather step by which particles were moved from the voxels to the facets. The number of state i particles that move from a facet F.sub..alpha. to a voxel N(x) is: ##EQU18## where P.sub.f (x) accounts for the volume reduction of partial voxels. From this, for each state i, the total number of particles directed from the facets to a voxel N.sub.(x) is: ##EQU19## To maintain the entries of the state vectors for the voxels as integers (and to convert non-integer entries back to integer form), the entries are rounded to integer values:
N.sub.i (x, t.sub.S +)=floor�(scale(N.sub.i (x, t.sub.S -)+N.sub.iF-V (x))+rand)/scale!
where t.sub.S- is the time just before the scatter step, t.sub.S+ is the time just after the scatter step, and N.sub.i (x, t.sub.S -) is a floating point number. The random number rand takes on values between zero and one prior to scaling, and is then scaled up by scale to values between 0 and 2.sup.16 -1. Because the values are scaled up by scale, the division by scale is merely a matter of shifting the result of the multiplication and addition operations.
After scattering particles from the facets to the voxels, combining them with particles that have advected in from surrounding voxels, and integerizing the result, it is possible that certain directions in certain voxels may either underflow (become negative) or overflow (exceed 255 in an eight-bit implementation). This would result in either a gain or loss in mass, momentum and energy after these quantities are truncated to fit in the allowed range of values. To protect against such occurrences, the mass, momentum and energy that are out of bounds are accumulated prior to truncation of the offending state. For the energy to which the state belongs, an amount of mass equal to the value gained (due to underflow) or lost (due to overflow) is added back to randomly (or sequentially) selected states having the same energy and that are not themselves subject to overflow or underflow. The additional momentum resulting from this addition of mass and energy is accumulated and added to the momentum from the truncation. By only adding mass to the same energy states, both mass and energy are corrected when the mass counter reaches zero. Finally, the momentum is corrected using pushing/pulling techniques until the momentum accumulator is returned to zero.
7. Perform Fluid Dynamics
Finally, fluid dynamics are performed (step 118). This step may be referred to as microdynamics or intravoxel operations. Similarly, the advection procedure may be referred to as intervoxel operations. The microdynamics operations described below may also be used to collide particles at a facet to produce a Boltzmann distribution.
The microdynamics operations simulate the set of physical interactions that occur within a voxel by sequentially applying a series of binary, trinary, or "n"ary interaction rules (also referred to as collision rules) to the state vector of a voxel during each time increment. For example, in the system described herein, there are 276 collision rules, each of which is implemented in a separate collision stage. Since the collision rules are applied sequentially, the result of their application has a complex nonlinear dependence on the initial state values that cannot be written simply in terms of those values. While the collision rules can be implemented as binary, trinary, or "n"ary collision events, for simplicity and ease of understanding the following discussion will refer primarily to binary collision events.
There are two basic types of collisions: non-energy exchanging "self" collisions and energy exchanging collisions. Self collisions allow for particles to collide with each other, thus changing their velocity. Because the state occupied by a particle determines the particle's velocity, a change in a particle's velocity is accomplished by moving that particle into a different state. A binary collision rule describes a self collision with reference to two input states and two output states, and simulates the collision of one or more particles from each of the input states by moving the particles to the output states. Because the collision rules are bi-directional, the input and output states are determined at the time the collision takes place depending on the states' populations.
Energy exchanging collisions differ from self collisions in that the two outgoing particles are at different energy levels than the two incoming particles. For subsonic flows there are only three energy levels: zero (stopped), one and two. To conserve energy, the only possible energy exchanging collisions occur when one pair includes two energy level one particles and the other pair includes an energy level two particle and a stopped particle. Energy exchanging collisions do not happen at the same rate in both directions. Rather, they happen at a forward rate from the energy level one states and a backward rate from the energy two and stopped states. As will be discussed in more detail below, these collision rates are dependent on the temperature of the system.
Each collision rule, whether directed to self or energy exchanging collisions, operates on a subset of the states of the state vector for a single voxel or facet and simulates the net effect of collisions between particles in those states. For example, a binary collision rule operates on four states (i, j, k, l) and simulates the net effect of collisions between particles in the first two states (i, j) and collisions between particles in the second two states (k, l). Because a collision between a pair of particles in the first two states results in the velocities of those particles changing to the velocities corresponding to the second two states, and vice versa, a binary collision rule can be expressed as: ##EQU20## where .epsilon. denotes the physically infinitesimal time required to perform the collision rule. Similarly, a trinary collision rule can be expressed as: ##EQU21##
It should be appreciated that a state change resulting from application of a collision rule actually reflects a net state change. For example, when application of a collision rule results in four particles from each of states i and j moving to each of states k and l, this movement might actually represent collisions that resulted in, for example, fifty six particles from each of states i and j moving to each of states k and l, and fifty two particles from each of states k and l moving to each of states i and j.
To accurately simulate physical systems, each collision rule must conserve mass, momentum and energy. Because the rules simply move particles from one state to another, and do not create or destroy particles, they necessarily conserve mass. To ensure that a rule conserves momentum, the states affected by the rule are selected so that:
C.sub.i +C.sub.j =C.sub.k +C.sub.l.
Similarly, to ensure that a rule conserves energy, the states affected by the rule are selected so that:
C.sub.i.sup.2 +C.sub.j.sup.2 =C.sub.k.sup.2 +C.sub.l.sup.2.
Thus, the states i, j, k, and l are selected for each rule so that a pair of particles from states i and j has the same total momentum and energy as a pair of particles from states k and l. Because the incoming and outgoing pairs of states must always conserve mass, momentum and energy, not all possible quartets within the 54 states correspond to a "legal" collision rule.
For a particular collision rule, N.sub.scatt is determined by applying a collision operator to the states affected by the rule. The arithmetic/computational form of the collision operator determines the practicality of implementing the operator in a simulation system. The statistical properties of the collision operator determine the extent to which accurate behavior can be achieved during a simulation.
The preferred collision operator is one for which N.sub.scatt approaches zero as the number of particles in each state affected by a rule approaches its Boltzmann equilibrium value. For binary collisions, the preferred operator is the "multilinear rule", which has the following form:
N.sub.scatt =A.sub.j �R.sub.f *N.sub.i *N.sub.j -R.sub.b *N.sub.k *N.sub.l!
where A.sub.j is a collision coefficient and R.sub.f and R.sub.b are the forward and backward collision rates for energy exchanging collisions. The coefficients A.sub.j depend only on local temperature and density, while R.sub.f and R.sub.b depend only on local temperature. Since temperature and density do not change due to collisions, A.sub.j, R.sub.f and R.sub.b are constants for a particular time increment, and can be determined prior to all collisions.
The value of A.sub.j varies for each type of collision (self collisions between particles at energy level one, self collisions between particles at energy level two, and energy exchanging collisions) as shown below. If j=1, 2 or .di-elect cons. represents, respectively, self collisions at energy levels one or two, and energy exchanging collisions, then: ##EQU22## where
<N>.sub.j.sup.(eq) (j=0, 1, 2)
is the equilibrium isotropic state population at energy level j, as described above.
For energy exchanging collisions, states i and j are the energy level one states and states k and l are the energy level two and rest states so that R.sub.f /R.sub.b represents the rate of collisions from energy level one particles to energy level two and rest particles. R.sub.f /R.sub.b is defined as: ##EQU23## where d.sub.0 is the number of stopped particle states and equals six in the described embodiment. The temperature of the fluid, however, is not necessarily constant over the length of a simulation, especially for simulations involving heat transfer, and A.sub.j and R.sub.f /R.sub.b must be updated dynamically during the simulation to reflect changes in the local temperature. The temperature range supported for subsonic flows is between 1/3 and 2/3. For self collisions, R.sub.f /R.sub.b equals one.
As an example of a self collision, the following initial state is proposed:
C.sub.i =(1,0,1,0), N.sub.i =25,
C.sub.j =(1,0,-1,0), N.sub.j =40,
C.sub.k =(1,0,0,1), N.sub.k =53,
and
C.sub.l =(1,0,0,-1), N.sub.l =20.
As shown, i, j, k and l are selected so that the combined momentum of states i and j and of states k and l are two in the x dimension and zero in the y, z and w dimensions, and so that each state is an energy level one state. Assuming that the total density .rho. is 1656 and the temperature is 1/2, A.sub.l using the expression for <N>.sub.l.sup.(eq) so that <N>.sub.l.sup.(eq) equals .rho./48 and A.sub.l equals 12/.rho. or 1/138. For illustrative purposes, if the collision rule were applied using floating point numbers, the resulting state populations would have the following values:
N.sub.i =25-(-0.43478)=25.43478,
N.sub.j =40-(-0.43478)=40.43478,
N.sub.k =53+(-0.43478)=52.56522,
and
N.sub.l =20+(-0.43478)=19.56522.
These new values of N.sub.i, N.sub.j, N.sub.k and N.sub.l drive the equilibrium measure to zero: ##EQU24##
A potential for overflow or underflow of a state's particle count exists in the collision operation described above. An overflow would result in a loss of mass, momentum and energy and would occur if the sum of the state population and N.sub.scatt exceeded 255 in an eight-bit implementation. By contrast, an underflow would result in creation of mass, momentum and energy and would occur if the result of subtracting N.sub.scatt from the state population was less than zero. Because the conservation of mass, momentum and energy is paramount in the simulation environment, an exchange of particles is prevented if it would cause either an overflow or underflow in any of the states involved in the collision.
For non-energy exchanging collisions, N.sup.scatt is determined as:
N.sub.scatt =floor�A.sub.js *(N.sub.i *N.sub.j -N.sub.k *N.sub.l)+rand)/scale!
where A.sub.js is A.sub.j scaled by multiplication by a constant, scale, that has a value of 2.sup.16. The random number rand takes on values between zero and one prior to scaling, and is then scaled up by scale to values between 0 and 2.sup.16 -1. For energy exchanging collisions, N.sup.scatt is determined as:
N.sub.scatt =floor�(R.sub.f '*N.sub.i *N.sub.j -R.sub.b '*N.sub.k *N.sub.l)+rand)/scale!
where R.sub.f ' and R.sub.b ' are, respectively, R.sub.f and R.sub.b multiplied by A.sub..di-elect cons.s, which is A.sub..di-elect cons. multiplied by scale. Rand is employed to prevent the truncation that results from the floor operation from introducing statistical bias into the system. Use of rand ensures that this operation, which forces N.sub.scatt to take on integer values, will not statistically bias N.sub.scatt in a particular direction.
With this approach, the values of N.sub.i, N.sub.j, N.sub.k and N.sub.l for the example provided above would be set to (25, 40, 53, 20) with 56.522% probability and (26, 41, 52, 19) with 43.478% probability. Averaged over a large number of trials, the mean values for N.sub.i, N.sub.j, N.sub.k and N.sub.l would be (25.43478, 40.43478, 52.56522, 19.56522). Thus, though a single application of the rule does not necessarily drive the populations toward equilibrium, the statistical mean values of the populations over a large number of applications of the rule correspond to the equilibrium values for those populations.
The multilinear collision operator drives the simulated system to Boltzmann equilibrium. This means that repeated application of the collision rules drives the system to an equilibrium value where, for a given set of macroscopic conditions, the individual state populations, N.sub.i, take on known values as determined by the Boltzmann distribution: ##EQU25## where the factors in the exponent represent the conserved invariants of mass (m), momentum (mc.sub.i) and energy (mc.sub.i.sup.2), it is assumed that no additional ("spurious") invariants occur, and r.sub.i represents a weighting factor generated from R.sub.f and R.sub.b. Attainment of this distribution after each application of the collision rules will result in hydrodynamic behavior to some accuracy.
A simple way to derive this equilibrium is to assume that all collisions have occurred and have driven the interacting states to their individual equilibria. Thus, for a binary collision rule, the equilibrium is:
R.sub.f *N.sub.i *N.sub.j =R.sub.b *N.sub.k *N.sub.l
and, for the multilinear rule, N.sub.scatt equals zero. Taking logarithms of both sides in the preceding equation implies:
ln(r.sub.i.sup.-1 N.sub.i)+ln(r.sub.j.sup.-1 N.sub.j)=ln(r.sub.k.sup.-1 N.sub.k)+ln(r.sub.l.sup.-1 N.sub.l)
where
r.sub.i.sup.-1 r.sub.j.sup.-1 =R.sub.f
and
r.sub.k.sup.-1 r.sub.l.sup.-1 =R.sub.b.
This equation now must be satisfied for all pairs of pairs, (or pairs of trios, etc.) that are allowed under the collision rules. To satisfy the equation, lnr.sub.i.sup.-1 N.sub.i must be a sum of summational invariants of the collision process (i.e. quantities whose sum over particles are conserved) in the following form: ##EQU26## where I.sub.i.sup..alpha. is the .alpha.th invariant associated with state i, (e.g., a component of the momentum, m.sub.i c.sub.i, in the equation for N.sub.i above), and A.sub..alpha. are the coefficients associated with their corresponding invariants (e.g., .alpha., .beta. and .gamma. in the equation for N.sub.i above). As long as the mass, momentum and energy are the only conserved quantities (i.e., there are no spurious invariants), then the Boltzmann distribution results from this equation.
D. Variable Resolution
Referring to FIG. 10, variable resolution (as illustrated in FIGS. 4 and 5 and discussed above) employs voxels of different sizes, hereinafter referred to as coarse voxels 1000 and fine voxels 1005. (The following discussion refers to voxels having two different sizes; it should be appreciated that the techniques described may be applied to three or more different sizes of voxels to provide additional levels of resolution.) The interface between regions of coarse and fine voxels is referred to as a variable resolution (VR) interface 1010.
When variable resolution is employed at or near a surface, facets may interact with voxels on both sides of the VR interface. These facets are classified as VR interface facets 1015 (F.sub..alpha.IC) or VR fine facets 1020 (F.sub..alpha.IF). A VR interface facet 1015 is a facet positioned on the coarse side of the VR interface and having a coarse parallelepiped 1025 extending into a fine voxel. (A coarse parallelepiped is one for which c.sub.i is dimensioned according to the dimensions of a coarse voxel, while a fine parallelepiped is one for which c.sub.i is dimensioned according to the dimensions of a fine voxel.) A VR fine facet 1020 is a facet positioned on the fine side of the VR interface and having a fine parallelepiped 1030 extending into a coarse voxel. Processing related to interface facets may also involve interactions with coarse facets 1035 (F.sub..alpha.C) and fine facets 1040 (F.sub..alpha.F).
For both types of VR facets, surface dynamics are performed at the fine scale, and operate as described above. However, VR facets differ from other facets with respect to the way in which particles advect to and from the VR facets.
Interactions with VR facets are handled using a variable resolution procedure 1100 illustrated in FIG. 11. Most steps of this procedure are carried out using the comparable steps discussed above for interactions with non-VR facets. The procedure 1100 is performed during a coarse time step (i.e., a time period corresponding to a coarse voxel) that includes two phases that each correspond to a fine time step. The facet surface dynamics are performed during each fine time step. For this reason, a VR interface facet F.sub..alpha.IC is considered as two identically sized and oriented fine facets that are referred to, respectively, as a black facet F.sub..alpha.ICb and a red facet F.sub..alpha.ICr. The black facet F.sub..alpha.ICb is associated with the first fine time step within a coarse time step while the red facet F.sub..alpha.ICr is associated with the second fine time step within a coarse time step.
Initially, particles are moved (advected) between facets by a first surface-to-surface advection stage (step 1102). Particles are moved from black facets F.sub..alpha.ICb to coarse facets F.sub..beta.C with a weighting factor of V.sub.-.alpha..beta. that corresponds to the volume of the unblocked portion of the coarse parallelepiped (FIG. 10, 1025) that extends from a facet F.sub..alpha. and that lies behind a facet F.sub..beta. less the unblocked portion of the fine parallelepiped (FIG. 10, 1045) that extends from the facet F.sub..alpha. and that lies behind the facet F.sub..beta.. The magnitude of c.sub.i for a fine voxel is one half the magnitude of c.sub.i for a coarse voxel. As discussed above, the volume of a parallelepiped for a facet F.sub..alpha. is defined as:
V.sub.i.alpha. =.vertline.c.sub.i .multidot.n.sub..alpha. .vertline.A.sub..alpha..
Accordingly, because the surface area A.sub..alpha. of a facet does not change between coarse and fine parallelepipeds, and because the unit normal n.sub..alpha. always has a magnitude of one, the volume of a fine parallelepiped corresponding to a facet is one half the volume of the corresponding coarse parallelepiped for the facet.
Particles are moved from coarse facets F.sub..alpha.C to black facets F.sub..beta.ICb with a weighting factor of V.sub..alpha..beta. that corresponds to the volume of the unblocked portion of the fine parallelepiped that extends from a facet F.sub..alpha. and that lies behind a facet F.sub..beta..
Particles are moved from red facets F.sub..alpha.ICr to coarse facets F.sub..beta.C with a weighting factor of V.sub..alpha..beta., and from coarse facets F.sub..alpha.C to red facets F.sub..beta.ICr with a weighting factor of V.sub.-.alpha..beta..
Particles are moved from red facets F.sub..alpha.ICr to black facets F.sub..beta.ICb with a weighting factor of V.sub..alpha..beta.. In this stage, black-to-red advections do not occur. In addition, because the black and red facets represent consecutive time steps, black-to-black advections (or red-to-red advections) never occur. For similar reasons, particles in this stage are moved from red facets F.sub..alpha.ICr to fine facets F.sub..beta.IF or F.sub..beta.F with a weighting factor of V.sub..alpha..beta., and from fine facets F.sub..alpha.IF or F.sub..alpha.F to black facets F.sub..alpha.ICb with the same weighting factor.
Finally, particles are moved from fine facets F.sub..alpha.IF or F.sub..alpha.F to other fine facets F.sub..beta.IF or F.sub..beta.F with the same weighting factor, and from coarse facets F.sub..alpha.C to other coarse facets F.sub.C with a weighting factor of V.sub.C.alpha..beta. that corresponds to the volume of the unblocked portion of the coarse parallelepiped that extends from a facet F.sub..alpha. and that lies behind a facet F.sub..beta..
After particles are advected between surfaces, particles are gathered from the voxels in a first gather stage (steps 1104-1110). Particles are gathered for fine facets F.sub..alpha.F from fine voxels using fine parallelepipeds (step 1104), and for coarse facets F.sub..alpha.C from coarse voxels using coarse parallelepipeds (step 1106). Particles are then gathered for black facets F.sub..alpha.IRb and for VR fine facets F.sub..alpha.IF from both coarse and fine voxels using fine parallelepipeds (step 1108). Finally, particles are gathered for red facets F.sub..alpha.IRr from coarse voxels using the differences between coarse parallelepipeds and fine paralllelepipeds (step 1110).
Next, coarse voxels that interact with fine voxels or VR facets are exploded into a collection of fine voxels (step 1112). The states of a coarse voxel that will transmit particles to a fine voxel within a single coarse time step are exploded. For example, the appropriate states of a coarse voxel that is not intersected by a facet are exploded into eight fine voxels oriented like the microblock of FIG. 2. The appropriate states of coarse voxel that is intersected by one or more facets are exploded into a collection of complete and/or partial fine voxels corresponding to the portion of the coarse voxel that is not intersected by any facets. The particle densities N.sub.i (x) for a coarse voxel and the fine voxels resulting from the explosion thereof are equal, but the fine voxels may have fractional factors P.sub.f that differ from the fractional factor of the coarse voxel and from the fractional factors of the other fine voxels.
Thereafter, surface dynamics are performed for the fine facets F.sub..alpha.IF and F.sub..alpha.F (step 1114), and for the black facets F.sub..alpha.ICb (step 1116). Dynamics are performed using the procedure illustrated in FIG. 9 and discussed above.
Next, particles are moved between fine voxels (step 1118) including actual fine voxels and fine voxels resulting from the explosion of coarse voxels. Once the particles have been moved, particles are scattered from the fine facets F.sub..alpha.IF and F.sub..alpha.F to the fine voxels (step 1120).
Particles are also scattered from the black facets F.sub..alpha.ICb to the fine voxels (including the fine voxels that result from exploding a coarse voxel) (step 1122). Particles are scattered to a fine voxel if the voxel would have received particles at that time absent the presence of a surface. In particular, particles are scattered to a voxel N(x) when the voxel is an actual fine voxel (as opposed to a fine voxel resulting from the explosion of a coarse voxel), when a voxel N(x+c.sub.i) that is one velocity unit beyond the voxel N(x) is an actual fine voxel, or when the voxel N(x+c.sub.i) that is one velocity unit beyond the voxel N(x) is a fine voxel resulting from the explosion of a coarse voxel.
Finally, the first fine time step is completed by performing fluid dynamics on the fine voxels (step 1124). The voxels for which fluid dynamics are performed do not include the fine voxels that result from exploding a coarse voxel (step 1112).
The procedure 1100 implements similar steps during the second fine time step. Initially, particles are moved between surfaces in a second surface-to-surface advection stage (step 1126). Particles are advected from black facets to red facets, from black facets to fine facets, from fine facets to red facets, and from fine facets to fine facets.
After particles are advected between surfaces, particles are gathered from the voxels in a second gather stage (steps 1128-1130). Particles are gathered for red facets F.sub..alpha.IRr from fine voxels using fine parallelepipeds (step 1128). Particles also are gathered for fine facets F.sub..alpha.F and F.sub..alpha.IF from fine voxels using fine parallelepipeds (step 1130).
Thereafter, surface dynamics are performed for the fine facets F.sub..alpha.IF and F.sub..alpha.F (step 1132), for the coarse facets F.sub..alpha.C (step 1134), and for the red facets F.sub..alpha.ICr (step 1136) as discussed above.
Next, particles are moved between voxels using fine resolution (step 1138) so that particles are moved to and from fine voxels and fine voxels representative of coarse voxels. Particles are then moved between voxels using coarse resolution (step 1140) so that particles are moved to and from coarse voxels.
Next, in a combined step, particles are scattered from the facets to the voxels while the fine voxels that represent coarse voxels (i.e., the fine voxels resulting from exploding coarse voxels) are coalesced into coarse voxels (step 1142). In this combined step, particles are scattered from coarse facets to coarse voxels using coarse parallelepipeds, from fine facets to fine voxels using fine parallelepipeds, from red facets to fine or coarse voxels using fine parallelepipeds, and from black facets to coarse voxels using the differences between coarse parallelepipeds and find parallelepipeds. Finally, fluid dynamics are performed for the fine voxels and the coarse voxels (step 1144).
E. Viscosity Reduction
The viscosity of the simulation may be modified using the over-relaxation technique described by Traub et al. in "Viscosity Reduction in Physical Process Simulation", U.S. application Ser. No. 08/255,409, filed Jun. 8, 1994, which is incorporated herein by reference. Viscosity is a measure of a fluid's resistance to a shear force (i.e., a force which acts parallel to the direction of fluid flow). In an actual fluid, viscosity results from interactions between neighboring particles in the fluid that cause the velocities of the particles to gravitate toward an average value. In a lattice system, viscosity results from interactions between particles positioned in specific voxels that cause the net velocity of the particles positioned in a voxel to gravitate toward the net velocity of the particles positioned in neighboring voxels. Because each voxel in a lattice system represents a region of simulated space that is substantially larger than the physical space that would be occupied by an actual particle, the viscosity resulting from interactions between voxels is substantially greater than that resulting from molecular particle interactions in real fluids (i.e., the "averaging" resulting from each voxel interaction affects a substantially larger region of space than that resulting from each molecular particle interaction).
Viscosity in a lattice system can be reduced by increasing the density of the lattice (i.e., by decreasing the quantity of simulated space that is represented by each voxel), and can also be reduced through use of over-relaxation. Viscosity, .nu., can be expressed in terms of .omega., the relaxation parameter: ##EQU27## where T is the temperature of the fluid. Thus, for example, relative to a relaxation parameter of one (.nu.=T/2), a relaxation parameter of 1.8 (.nu.=T/18) will reduce the viscosity in the lattice by a factor of nine.
Use of over-relaxation effectively increases the density of the lattice. Use of over-relaxation therefore has a dramatic effect on the processing necessary to simulate a physical system with a particular resolution (or the resolution with which a particular processor can simulate a physical system). For example, a tenfold increase in the effective density of a three dimensional lattice reduces the processing required to simulate a physical system with the lattice to a particular level of resolution by a factor of almost ten thousand (i.e., ten cubed less the additional processing required to implement over-relaxation and multiplied by a tenfold decrease in the time required to simulate a fluid of a given velocity).
To change the viscosity of the simulated physical process, the system performs viscosity modification operations on the state vectors. These operations are typically performed after the fluid dynamics operations and apply a set of rules that are similar to, or the same as, the rules applied during the fluid dynamics operations. Where the same rules are applied, the rules modify the state vectors by a first amount during the fluid dynamics operations and a second amount during the viscosity modification operations, where the first amount is related to the second amount by a relaxation parameter. Because the rules used in the interaction operations conserve mass, momentum and energy, this approach ensures that these properties will be conserved during the viscosity modification operations.
In one implementation, the viscosity of the lattice system is reduced by using a relaxation parameter having a value greater than one and less than two. As the relaxation parameter approaches two, the viscosity of the simulated system approaches zero and the system becomes unstable. Viscosity, which is essentially a form of friction, tends to damp out fluctuations in the system. Thus, instability occurs when there is no viscosity because these fluctuations are allowed to spread unchecked through the system. It has been found that instability can generally be avoided by using a relaxation parameter that is less than or equal to 1.9.
F. Modelling the Skin Friction Coefficient
As noted above, shear stress at a surface may be approximated by flow within the boundary layer above the surface. Referring to FIG. 12, a skin friction coefficient C.sub.f may be used to simulate the resistance to tangential flow 1200 at the location 1205 within the boundary layer 1210, which is a distance y.sub.S above the surface 1215.
For sufficiently-large Reynolds numbers, the velocity gradient near a surface may be expressed as: ##EQU28## where u is the flow rate or velocity, y is the distance above the surface, .kappa. is the Von Karman constant, which has a value of about 0.4, and u.sub..tau. is a friction velocity that is defined so that:
.tau..sub.w =.rho.u.sub..tau..sup.2,
where .tau..sub.w is the shear stress at the wall and .rho. is the particle density.
The above expression for the velocity gradient is a result of two assumptions: that the only relevant length scale is y, and that the shear stress is constant near the surface. This ensures that u.sub..tau. is the only relevant velocity scale. Consequently, the shear stress .tau. within the boundary layer also may be expressed as: ##EQU29## where C.sub.f is the skin friction coefficient and U.sub.S is the instantaneous velocity at a distance y.sub.S above the surface.
Combining the two expressions for the shear stress .tau..sub.w produces: ##EQU30## Similarly, integrating the expression for the velocity gradient to a height y.sub.S above the surface results in the so-called logarithmic velocity profile: ##EQU31## where B is an empirical constant having a value of approximately 5 and .nu. is the viscosity.
Combining these two expressions results in: ##EQU32## The terms u.sub..tau. and y.sub.S may be eliminated by matching the lattice viscosity .nu..sub.lat to the turbulent eddy viscosity .nu..sub.T at the height y.sub.S, where the expression for .nu..sub.T in the logarithmic part of the boundary layer is:
.nu..sub.T =.kappa.u.sub..tau. y.sub.S =.nu..sub.lat.
After eliminating y.sub.S and u.sub..tau., the friction coefficient may be expressed as: ##EQU33## Re is the flow Reynolds number, and Re.sub.lat is the Reynolds number of the lattice, based on the freestream lattice velocity, the lattice viscosity, and the number of voxels along the characteristic length. For the above analysis to be valid, the location of y.sub.S must be in the logarithmic region. Moreover, to ensure that all of the momentum in the flow is simulated, it is desirable that y.sub.S also be of the order of the momentum boundary layer displacement thickness, which implies a requirement on the resolution in the boundary layer. In practice, excellent results are obtained for simulations where the resolution is much coarser than the appropriate momentum boundary layer thickness.
The thickness of the region in which flow rates increase according to a logarithmic profile goes to zero at a separation point 1220 at which the flow separates from the surface. As a result, y.sub.S should approach zero at the separation point. However, the opposite occurs for the friction coefficient expression provided above, which indicates that the expression is inaccurate at the separation point.
An adverse pressure gradient in which the pressure increases in the direction of flow (dp/dx>0) occurs prior to the separation point 1220. As previously noted, the velocity on the surface equals zero. At the separation point, the adverse pressure gradient results in a change in the sign of the velocity gradient relative to the normal of the surface. In physical terms, this means that there is a point 1225 above the surface at which flow on either side of this point is going in opposite directions so that the velocity at that point equals zero (u.sub..tau. =0). Since the lattice viscosity .nu..sub.lat has a constant value, the expression
.nu..sub.T =.kappa.u.sub..tau. y.sub.S .nu..sub.lat
implies that y.sub.S approaches infinity as u.sub..tau. approaches zero.
This problem may be resolved by introducing another length scale that can be used to ensure that y.sub.S is within the logarithmic region. The presence of an adverse pressure gradient allows the definition of another length scale as: ##EQU34## with the previous length defined from viscosity matching being redefined as y.sub..nu. : ##EQU35## As dp/dx increases in magnitude, y.sub.p decreases and eventually reaches a point where y.sub.P <y.sub..nu.. The logarithmic region is valid only for y<y.sub.P. Therefore, a new y.sub.S may be defined such that y.sub.S approximates y.sub..nu. when Y.sub..nu. <y.sub.P and y.sub.P when y.sub.P <y.sub..nu. : ##EQU36## which is then used in the expression for C.sub.f. If the pressure gradient is favorable (dp/dx<0), then the additional term is ignored and y.sub.S equals y.sub..nu..
To ensure that the viscosity is consistent with the new y.sub.S, the local .nu..sub.lat in the fluid may be adjusted so that: ##EQU37## In practice, acceptable results have been produced without this adjustment.
Consequently, the friction coefficient may be expressed as: ##EQU38## where C.sub.f,0 is the value of C.sub.f when dp/dx equals zero, or: ##EQU39## For ease of processing, the values of C.sub.f may be clamped so that C.sub.f,min equals C.sub.f,0 and C.sub.f,max equals the value of C.sub.f when the logarithmic term equals zero (2/B.sup.2, which is approximately 0.08).
Derivation of this coefficient is also described by Anagnost et al. in "DIGITAL PHYSICS Analysis of the Morel Body in Ground Proximity", SAE Technical Paper Series, no. 970139, presented at the SAE International Conference and Exposition, Detroit, Mich., Feb. 24-27, 1997, which is incorporated by reference. This paper also describes a simulation of flow around a Morel body using the techniques described above.
G. Generating the Skin Friction Coefficient
Referring to FIG. 13, the skin friction coefficient may be determined for each time increment according to a procedure 1300. According to the expression discussed above, the friction coefficient C.sub.f.sup..alpha. at a facet varies based on the density at the facet (.rho..sup..alpha.), the velocity at the facet (U.sub.S.sup..alpha.), and the pressure gradient (dp.sup..alpha. /dx) at the facet in the direction of the velocity at the facet. As such, prior to determining the skin friction coefficient for a facet, the density at the facet is determined (step 1305). The density at the facet (.rho..sup..alpha.m) may be measured from the distribution at the facet as: ##EQU40## To reduce effects of high-frequency fluctuations, .rho..sup..alpha. (t), the density used in generating the friction coefficient for a time step t, is determined as:
.rho..sup..alpha. (t)=(1-w.sub.r).rho..sup..alpha. (t-1)+w.sub.r .rho..sup..alpha.m (t),
where w.sub.r is an under-relaxation factor having a value between 0 and 1. Generally, the density at the facet is determined as part of performing facet surface dynamics (step 112 of FIG. 1).
The velocity at the facet is determined in a similar manner (step 1310). First, the velocity at the facet (U.sub.S.sup..alpha.m) is measured as: ##EQU41## The velocity used in generating the friction coefficient for the time step t then is determined as:
U.sup..alpha. (t)=(1-w.sub.r)U.sup..alpha. (t-1)+w.sub.r U.sup..alpha.m (t).
Next, the pressure gradient at the facet is derived from the pressure gradient in the fluid adjacent to the facet. In particular, the pressure gradient at the facet is determined as the pressure gradient in voxels that affect the facet. Initially, p, the pressure, is determined for each voxel x (step 1315) as:
p.sup.xm =.rho..sup.xm T.sup.xm,
where T.sup.x, E.sup.x, U.sup.x and .rho..sup.x are, respectively, the temperature, energy, velocity, and density of the voxel x, and are determined as: ##EQU42## The pressure used in generating the friction coefficient for the time step t then is determined as:
p.sup.x (t)=(1-w.sub.r)p.sup.x (t-1)+w.sub.r p.sup.xm (t).
Next, the pressure gradient .gradient.p is determined for each voxel (step 1320). For a voxel at a location (x, y, z) that is unaffected by a surface or by a transition between voxels of different sizes, the pressure gradient may be determined as: ##EQU43## For a voxel which has no neighbor in a particular direction (because of its proximity to a surface), or whose neighbor in a particular direction is of a different voxel size, the pressure gradient is determined as: ##EQU44## where
e.sub.x',y',z' is 1 if a voxel exists at x',y',z' and is 0 otherwise,
d.sub.x',y',x' is the distance from the center of the voxel x',y',z' to the voxel x,y,z.
This is 1 if x',y',z' is the same size as x,y,z, or 1.5 if larger by a factor of 2, or 0.75 if smaller by a factor of 2.
The pressure gradient .gradient.p.sup..alpha. at a facet is then determined as a weighted average of the pressure gradients of voxels affected by the facet (step 1325): ##EQU45##
The tangential velocity U.sub.t.alpha. is then determined by subtracting the normal component of the facet velocity from the facet velocity (step 1330):
U.sub.t.sup..alpha. =U-(U.multidot.n.sub..alpha.)n.sub..alpha..
The pressure gradient .gradient.p.sub.t.sup..alpha. in the tangential direction, is determined (step 1335):
.gradient.p.sub.t.sup..alpha. =.gradient.p.sup..alpha. .multidot.U.sub.t.sup..alpha..
Finally, the friction coefficient is generated (step 1340) using the pressure gradient .gradient.p.sub.t.sup.60 in the expression provided above for the friction coefficient.
Other embodiments are within the scope of the following claims.

Number	Name	Date
4729098	Cline et al.	Mar 1988
4809202	Wolfram	Feb 1989
4969116	Wada et al.	Nov 1990
4989166	Akasaka et al.	Jan 1991
5038302	Kaufman	Aug 1991
5255212	Kondoh et al.	Oct 1993
5361385	Bakalash	Nov 1994
5377129	Molvig et al.	Dec 1994
5408638	Sagawa et al.	Apr 1995
5442733	Kaufman et al.	Aug 1995
5537641	da Vitoria Lobo et al.	Jul 1996
5548694	Gibson	Aug 1996
5594671	Chen et al.	Jan 1997
5606517	Traub et al.	Feb 1997

Number	Date	Country
WO 9201993	Feb 1992	WOX
9721195	Jun 1997	WOX

Computer simulation of physical processes

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Entry
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