Patent Publication Number: US-8996337-B1

Title: Hierarchical position-based dynamics

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     Embodiments of the invention relate generally to physics simulations and more specifically to hierarchical position-based dynamics. 
     2. Description of the Related Art 
     Conventional computer graphics (CG) engines are capable of generating highly realistic images representing real-world or imaginary scenes. These images are typically composed of a plurality of different particles. A collection of such particles can be used to represent different objects in the virtual graphics scene, including rocks, trees, cloth, water, and people, among others. Some CG engines perform physical simulations of the motion of the particles in the virtual graphics scene, thus simulating the motion of the objects represented by those particles. The physical simulations typically apply equations of motion to each particle to predict the position of the particle at a particular time step in the physical simulation. A wide variety of techniques are available for applying equations of motion to particles in the virtual graphics scene. One commonly-used technique is known as the “explicit Euler” technique. 
     In conventional systems, the explicit Euler technique can be used to predict the position of each particle in the virtual graphics scene by integrating a set of forces acting on the particle to generate one or more velocity components for the particle. The velocity components are then integrated to generate a position prediction for the particle. One problem with the explicit Euler technique is that each integral is approximated; thus, the position of the particle may not always be predicted with a high degree of accuracy, especially for larger time steps. Approximation errors can accumulate over multiple time steps, potentially causing the simulation to diverge, which may result in a numerical explosion. 
     One solution to the problems with the explicit Euler technique is to correct the generated position predictions based on a set of constraint equations associated with each particle. A constraint equation restricts the position of a particle and typically takes the form of an equality or of an inequality. For example, an equality constraint equation C 1  that restricts the height H P , of a particle P j  to zero in a virtual graphics scene would be C 1 (P j )=H Pj =0. In another example, an inequality constraint equation C 2  that restricts the distance between particles P j  and P k  to be less than a distance D would be C 2 (P j ,P k )=|P j −P k |−D&lt;0. The constraint equations for each particle can be solved to generate a position correction that, when applied to the position predictions generated by the explicit Euler technique, results in a corrected position prediction. Many techniques for solving constraint equations are known in the art. 
     For example, some physical simulations employ a Gauss-Seidel solver to solve the constraint equations for each particle. However, Gauss-Seidel solvers suffer from certain drawbacks. The main drawback is that each constraint is solved separately from all other constraints. This causes error corrections to propagate slowly through the particles and the solver to converge slowly to the correct solution. If the number of solver iterations is limited by a time budget, as is the case in computer games, the result computed by a Gauss-Seidel may diverge significantly from the true solution. In the case of a cloth simulation, this causes visual stretchiness. 
     There are methods that solve for all constraints simultaneously. Examples are the Conjugate Gradients method and the multi-grid method. For these methods to work, the constraints have to be linearized, which may be time consuming. In addition, the linearization can introduce stability problems because it replaces the true constraint functions by approximations which are only valid in a certain region. 
     As the foregoing illustrates, there remains a need in the art for a more effective way to simulate the motion of particles in a virtual graphics scene. 
     SUMMARY OF THE INVENTION 
     One embodiment of the invention sets forth a computer-implemented method for executing a physics simulation of a plurality of particles in a virtual graphics scene. The method includes the steps of generating a position prediction for a first particle assigned to a first level of a particle hierarchy that includes the first level and a second level, where each particle in the plurality of particles is assigned to either the first level or the second level of the particle hierarchy, and generating a position prediction for a second particle assigned to the second level of the particle hierarchy. The method further includes the steps of generating a corrected position prediction for the first particle based on the position prediction for the first particle and one or more constraint equations associated with the first particle, generating a corrected position prediction for the second particle based on the corrected position prediction for the first particle, and generating an image for storage in a memory and/or display on a display device based on the corrected position prediction for the first particle and the corrected position prediction for the second particle. 
     Advantageously, constraint equations are solved for fewer particles at each timestep in the physics simulation since the second level of the particle hierarchy only includes a subset of the particles in the virtual graphics scene. The corrected position predictions generated for these particles can then be used to generate corrected position predictions for the particles in the first level of the particle hierarchy. By implementing the physics simulation engine, more detailed physics simulations are possible, when compared to prior art techniques. 
     Since fewer constraint equations need to be solved at each timestep, a corrected position prediction can be generated for each particle in the virtual graphics scene and can thus be generated for all of the particles that represent a particular virtual object. The physics simulation engine is thereby capable of generating a physically accurate simulation of the virtual object, allowing physically realistic images to be generated based on that physics simulation. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       So that the manner in which the above recited features of the invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments. 
         FIG. 1  is computer system configured to implement one or more aspects of the invention; 
         FIG. 2  is a conceptual diagram that illustrates a physics simulation engine of  FIG. 1  in greater detail, according to one embodiment of the invention; 
         FIG. 3  sets forth a flowchart of method steps for generating a particle hierarchy, according to one embodiment of the invention; 
         FIG. 4  is a flowchart of method steps for assigning a particle to a first set of particles or to a second set of particles, according to one embodiment of the invention; 
         FIG. 5A  is a flowchart of method steps for collapsing an initial constraint associated with a particle, according to one embodiment of the invention; 
         FIG. 5B  is a conceptual diagram that illustrates an example of collapsing initial constraints associated with a particle, according to one embodiment of the invention; 
         FIG. 6  is conceptual diagram that illustrates the hierarchical solver of FIG.  2  in greater detail, according to one embodiment of the invention; 
         FIG. 7A  sets forth a flowchart of method steps for generating a corrected position prediction for a first particle and a corrected position prediction for a second particle, according to one embodiment of the invention; 
         FIG. 7B  is a conceptual diagram illustrating an example of how a corrected position prediction for particles in a virtual graphics scene may be generated, according to one embodiment of the invention; and 
         FIG. 8  is a flowchart of method steps for updating the particle hierarchy of  FIG. 2 , according to one embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description, numerous specific details are set forth to provide a more thorough understanding of the invention. However, it will be apparent to one of skill in the art that the invention may be practiced without one or more of these specific details. In other instances, well-known features have not been described in order to avoid obscuring the invention. 
     Embodiments of the present invention set forth techniques to create a hierarchy of particle sets before a physics simulation of those particles is initiated. In some embodiments, a particle set 0 contains the entire plurality of the particles. A particle set 1 contains a subset of the particles in the particle set 0. In general, particle set i contains a subset of the particles in particle set i−1. A hierarchy of constraints is also created. A constraint set 0 contains all the constraints. A constraint set 1 contains only constraints for particles in particle set 1. In general, constraint set i only contains constraints for particles in particle set i. Connections between particles in subsequent sets are also created. The particles in particle set 0 have connections to at least two particles in particle set 1. In general, each particle in particle set i has links to at least two particles in particle set i+1. The particles in the last particle set have no links. The hierarchy of particles and constraints can be used to improve the convergence of the solver during the simulation. 
     At each time step, the solver processes all particle sets and constraint sets starting from the highest. In the highest constraint set, the constraint equations belonging to that set are solved using the Gauss-Seidel technique and the positions of the particles in the highest particle set are corrected. The particle connections are then used to propagate the corrections to a lower particle set. A particle in the particle set below the highest particle set has at least two links to particles in the highest particle set (called parents). To determine the correction of this particle, the corrections of its parent particles are averaged. In a second step, the constraint equations in the second to highest constraint set are solved using the Gauss-Seidel technique. The resulting corrections are propagated to the level below. This process is repeated until particle set 0 is reached. 
     This method highly improves the convergence rate of the Gauss-Seidel constraint solver. This is because error corrections can propagate much faster through a system of a large number of particles and constraints. The reason is that constraints in higher levels directly connect particles that are connected by a long series of constraints in the original particle system. The improved convergence rate results in more accurate particle positions especially when the number of solver iteration is limited due to time restrictions as present in computer games. The visual result is that cloth and soft bodies look less stretchy. The specific techniques that implement this functionality are described in detail below. 
       FIG. 1  is a block diagram that illustrates a computer system  100  configured to implement one or more aspects of the invention. As shown, the computer system  100  includes a system memory  102 , a central processing unit (CPU)  104 , a graphics processing unit (GPU)  106 , a physics processing unit (PPU)  108 , one or more input/output (I/O) devices  110 , a network connection  112 , a GPU memory  114 , and a PPU memory  116 . The CPU  104 , the GPU  106 , the PPU  108  and the I/O devices  110  are each associated with one or more drivers  124  stored in the system memory  102 . The drivers  124  are software programs that may be executed by the various processing units of the computer system  100 , including the CPU  104 , the GPU  106 , the PPU  108 , and/or the I/O devices  110  to translate program instructions into machine code. For example, the PPU  108  may execute the driver  124  associated with the PPU  108  to translate program instructions into machine code native to the PPU  108 . 
     The CPU  104  is the primary processor in the computer system  100  and is configured to execute a software application  118 , a rendering engine  120 , a physics simulation engine  122 , and one or more of the drivers  124  stored in the system memory  102 . The software application  120 , the rendering engine  122 , and the physics simulation engine  122  cooperate to generate physically accurate computer graphics, as described in greater detail below in  FIGS. 2-8 . When executing these software programs, the CPU  104  may read data from or write data to the system memory  102 . In one embodiment, the CPU  104  is coupled to the GPU  106  and to the PPU  108 . 
     The GPU  106  and the PPU  108  are co-processors that supplement the processing capabilities of the CPU  104 . The GPU  106  increases the graphics processing capabilities of the computer system  100 . In one embodiment, the computer system  100  includes multiple GPUs that operate in concert or independently to perform graphics processing operations. The GPU  106  is coupled to the GPU memory  114  and to the system memory  102 . The GPU  106  executes software programs stored in the GPU memory  114  or the system memory  102 . The GPU  106  reads data from and/or writes data to the GPU memory  114  and/or to the system memory  102  when executing software programs. The software programs executed by the GPU  106  configure various hardware components (not shown) within the GPU  106  to perform different graphics processing tasks. The GPU  106  is accessible by software programs executing on the CPU  104  and/or the PPU  108 . For example, the rendering engine  120  may access the GPU  106  to perform graphics processing operations. 
     The PPU  108  is a specialized processing unit that performs physics calculations to increase the physics processing capabilities of the computer system  100 . The PPU  108  may be, for example, a PhysX™ chip. The PPU  108  may be integrated on the same chip as the CPU  104  and/or the GPU  106  or, alternatively, may be located on an add-in card coupled to the computer system  100 . In one embodiment, the computer system  100  includes multiple PPUs that operate in concert or independently to perform physics processing operations. The PPU  108  is coupled to the PPU memory  116  and to the system memory  102 . The PPU  108  executes software programs stored in the PPU memory  116  and/or the system memory  102 . The PPU  108  reads data from and/or writes data to the PPU memory  116  and/or the system memory  102  when executing software programs. The software programs executed by the PPU  108  may configure various hardware components (not shown) within the PPU  108  to perform different physics processing tasks. The PPU  108  is accessible by software programs executing on the CPU  104  and/or the GPU  106 . For example, the physics simulation engine  122  may access the PPU  108  to perform physics processing operations. 
     In one embodiment, the GPU  106  and the PPU  108  are integrated onto a single chip that provides both graphics processing and physics processing functionality. In another embodiment, the CPU  104 , the GPU  106  and the PPU  108  are all integrated onto a single chip that performs general processing operations, graphics processing operations, and physics processing operations for the computer system  100 . Various other architectural configurations are also within the scope of embodiments of the present invention. 
     The CPU  104 , the GPU  106 , and the PPU  108  are each coupled to the I/O devices  110 . The I/O devices  110  include input devices, such as a keyboard, a mouse, a video game controller, a microphone, a touchpad, a scanner, a stylus, a CD-ROM drive, and a DVD drive, among others. The I/O devices  110  also include output devices, such as a cathode-ray tube (CRT) monitor, a liquid crystal display (LCD) monitor, a printer, and a speaker, among others. The CPU  104 , the GPU  106  and the PPU  108  receive data from the I/O devices  110  and transmit data to the I/O devices  110 . Data received from the I/O devices  110  may be stored in the system memory  102 , the GPU memory  114 , and/or the PPU memory  116 . 
     The CPU  104 , the GPU  106 , and the PPU  108  may access a network via the network connection  112 . The network connection  112  may be an Ethernet connection, a cable connection, a wireless connection, or a telephone connection and may provide a connection to any type of network, including the World Wide Web, the Internet, a local area network (LAN), a wide area network (WAN), an intranet, a cellular network, or any other technically feasible type of network. When the network connection  112  is implemented as an Ethernet connection, the computer system  100  includes an Ethernet controller. The CPU  104 , the GPU  106 , and the PPU  108  may download data from or upload data to remote computing systems and/or external memory units via the network connection  112 . The data downloaded via the network connection  112  may be stored in the system memory  102 , the GPU memory  114 , and/or the PPU memory  116 . 
     As described, the system memory  102  includes the software application  118 , the rendering engine  120 , and the physics simulation engine  122 . The software application  118  may be any technically feasible software application that generates a virtual graphics scene, including a video game, a computer-aided design (CAD) application, or an animation application, among others. The virtual graphics scene may include one or more virtual objects. For example, the software application  118  could be a video game that generates a virtual graphics scene representing a particular level in the video game. The virtual graphics scene could include a virtual object, such as a character controlled by a user via the I/O devices  110 . The software application  118  generates an image based on the virtual graphics scene using the rendering engine  120  and the physics simulation engine  122  and then outputs the image to the I/O devices  110 , such as a display device. 
     The software application  118  accesses the rendering engine  120  to generate pixels that make up the image. The pixels may then be output to a display device. The rendering engine  120  may implement a variety of rendering approaches, including ray tracing, ray casting, radiosity, and/or rasterization. The rendering engine  120  may be implemented as a software rendering engine or as a hardware rendering engine. The rendering engine  120  may also be implemented as a combination of software and hardware. The rendering engine  120  may offload graphics processing tasks onto the GPU  106  to increase processing efficiency. 
     The software application  118  also accesses the physics simulation engine  122  to implement a physics simulation of the virtual objects in the virtual graphics scene. For a particular virtual object in the virtual graphics scene, the physics simulation engine  122  may simulate the motion of each particle associated with that virtual object. For example, the virtual graphics scene could include a curtain hanging in a doorway, and the physics simulation engine  122  could simulate the motion of the curtain by simulating the motion of each particle associated with the curtain. Then, when the software application  118  generates the image, the curtain may appear physically accurate and physically realistic. 
     At a current time step in the physics simulation, each particle occupies a current position in the virtual graphics scene. For example, if the virtual graphics scene is a three-dimensional (3D) space represented in Cartesian coordinates, then the current position of a particle could be represented by X, Y, and Z coordinates. At the current time step, the physics simulation engine  122  simulates the motion of the particle in the virtual graphics scene to predict the position of the particle at a subsequent time step in the physics simulation. 
     The physics simulation engine  122  simulates the motion of the particle by integrating one or more external forces associated with the particle at each time step to generate a velocity prediction for the particle. The external forces could be, for example, a gravitational force, a magnetic force, or a drag force. Each external force includes force components that act in one or more different directions. If the virtual graphics scene is represented in Cartesian coordinates, as described above, then the force components Fx, Fy, and Fz would act in X, Y, and Z directions, respectively. The physics simulation engine  122  integrates each force component associated with the particle to generate a corresponding velocity component for the particle. Returning to the above example, the physics simulation engine  122  could integrate the Fx, Fy, and Fz force components to generate Vx, Vy, and Vz velocity components. The velocity components represent the velocity prediction for the particle. 
     The physics simulation engine  122  then integrates the velocity components to generate a position prediction for the particle at a subsequent time step. Returning again to the above example, the physics simulation engine  122  could integrate the Vx, Vy, and Vz velocity components to generate X, Y, and Z coordinates representing the position prediction for the particle at the subsequent time step. Those skilled in the art will recognize that force components, velocity components, and position coordinates can be represented in a wide variety of different coordinate systems including, for example, cylindrical or spherical coordinate systems, among others, all of which fall within the scope of the present invention. In one embodiment, the physics simulation engine  122  implements the explicit Euler numerical integration algorithm when integrating the force components and the velocity components. Those skilled in the art will recognize that a wide variety of numerical integration techniques may be used to integrate the force and velocity components of each particle, all of which fall within the scope of the present invention. 
     Depending on the technique used to integrate the force and velocity components of the particle, the physics simulation engine  122  may generate a position prediction for the particle with varying degrees of accuracy. The physics simulation engine  122  is configured to improve the accuracy of the position prediction by generating a corrected position prediction based on the position prediction. The physics simulation engine  122  may generate the corrected position prediction for the particle by enforcing one or more constraint equations, or “constraints,” associated with the particle. 
     As described in greater detail below in  FIGS. 2-3  and  5 - 7 , a constraint may be an “initial” constraint that is initially associated with the particle, a “hierarchical” constraint that is generated based on an initial constraint, or a “collision” constraint that is generated by the physics simulation engine  122  dynamically during the physics simulation when two or more particles collide. The physics simulation engine  122  enforces a constraint, either an initial constraint, hierarchical constraint, or collision constraint, by modifying the position prediction for the particle until the constraint is satisfied, thus generating a corrected position prediction. For example, if a constraint C 1  restricts the height of the particle to be greater than 10, and the position prediction for the particle specifies that the height of the particle would be equal to 9 at a subsequent time step, then the physics simulation engine  122  could modify the position prediction for the particle until the height is greater than 10, thus generating a corrected position prediction for the particle that satisfies the constraint. In situations where a constraint is associated with two or more particles, the physics simulation engine  122  modifies the position predictions for the two or more particles to generate corrected position predictions for the two or more particles that satisfy the constraint associated with those particles. 
     The physics simulation engine  122  may also generate a corrected position prediction for certain particles in the virtual graphics scene based on the corrected position predictions for one or more other particles in the virtual graphics scene, as described in greater detail in FIGS.  2  and  6 - 7 . 
     Once the physics simulation engine  122  generates a corrected position prediction for each particle in the virtual graphics scene, the physics simulation engine  122  generates a corrected velocity prediction for the particle based on the current position of the particle, the corrected position prediction for the particle, and the size of the time step. The physics simulation engine then sets the current position of the particle to the corrected position prediction corresponding to that particle, sets the current velocity of the particle to the corrected velocity prediction for the particle, and advances to the subsequent time step in the physics simulation. 
     When the physics simulation engine  122  sets the current position of each particle to the corrected position prediction corresponding to the particle, the physics simulation engine  122  also transmits the current position of each particle to the software application  118 . The software application  118  may then implement the rendering engine  120  to render an image representing the virtual graphics scene. The image may then be displayed on a display device included in the I/O devices  110 . 
       FIG. 2  is a conceptual diagram that illustrates the physics simulation engine  122  in greater detail, according to one embodiment of the invention. As shown, the physics simulation engine  122  includes a hierarchy generator  202 , a particle hierarchy  204 , a hierarchical solver  206 , and a particle database  208 . The particle database  208  includes, for each particle in the virtual graphics scene, an initial position and an initial velocity for the particle (e.g., the position and velocity of the particle before the physics simulation engine  122  executes the physics simulation), the external forces acting on the particle, a current velocity and a current position of the particle (e.g., the velocity and position of the particle at a current time step), and a set of initial constraints associated with the particle. 
     The particle hierarchy  204  is a data structure that is used by the hierarchical solver  206  to enforce the constraints associated with each particle in the virtual graphics scene. The particle hierarchy  204  includes at least a first level and a second level. The hierarchy generator  202  is configured to generate the particle hierarchy  204  prior to the physics simulation engine  122  executing the physics simulation. When generating the particle hierarchy  204 , the hierarchy generator  202  assigns each particle to the first level or to the second level of the particle hierarchy  204 , as described in greater detail below in  FIGS. 3-4 . In one embodiment, particles having a “fine” level of detail are assigned to the first level of the particle hierarchy  204 , while particles having a “coarse” level of detail are assigned to the second level of the particle hierarchy  204 . In alternative embodiments, the particle hierarchy  204  includes any number of different levels to which particles are assigned having different levels of detail. 
     A first particle assigned to the first level of the particle hierarchy  204  is referred to herein as a “child” particle of a second particle assigned to the second level of the particle hierarchy  204  when the first particle and the second particle are both associated with at least one common constraint. The second particle is referred to herein as a “parent” particle of the first particle. Each child particle is associated with at least two parent particles, and each parent particle is associated with at least one child particle. When generating the particle hierarchy  204 , the hierarchy generator  202  generates a “correction weight” between each child particle and each parent particle of the child particle. A correction weight is a value, such as an integer value or a decimal value, used by the hierarchical solver  206  to generate corrected position predictions for each child particle based on the corrected position predictions for the parent particles of the child particle, as described herein. 
     When the physics simulation engine  122  executes the physics simulation, at the current time step in the physics simulation, the hierarchical solver  206  generates a velocity prediction and a position prediction for each particle in the virtual graphics scene, including each parent particle and each child particle. The hierarchical solver  206  then generates a corrected position prediction for each parent particle by enforcing the initial constraints, hierarchical constraints, and collision constraints associated with the parent particle. 
     For each child particle, the hierarchical solver  206  identifies the parent particles of the child particle. The hierarchical solver  206  then generates a corrected position prediction for the child particle based on the difference between the position prediction and the corrected position prediction generated for each parent particle of the child particle and based on the correction weights between those parent particles and the child particle. In this fashion, the physics simulation engine  122  generates a corrected position prediction for each particle in the virtual graphics scene. 
     The hierarchical solver  206  then generates the corrected velocity prediction, as described in  FIG. 1 . The hierarchical solver  206  accesses the particle database  208  and updates the current velocity and current position of each particle to reflect the corrected position prediction and corrected velocity prediction for that particle, respectively. The physics simulation engine  122  may then proceed to a subsequent time step in the physics simulation. 
       FIG. 3  sets forth a flowchart of method steps for generating the particle hierarchy  204 , according to one embodiment of the invention. Persons skilled in the art will understand that, although the method  300  is described in conjunction with the systems of  FIGS. 1-2 , any system configured to perform the method steps, in any order, is within the scope of the present invention. In one embodiment, the method  300  described herein is implemented by the hierarchy generator  202  prior to the physics simulation engine  122  executing the physics the physics simulation. 
     As shown, the method  300  begins at step  302 , where the hierarchy generator  202  receives a plurality of particles and a plurality of constraints from the particle database  208 . Each of the particles in the plurality of particles occupies an initial position in the virtual graphics scene and has an initial velocity. Each particle is also associated with a set of external forces acting on the particle and a set of initial constraints. 
     At step  304 , the hierarchy generator  202  includes each particle within the plurality of particles in either a first set of particles or a second set of particles. The first set of particles includes child particles that correspond to parent particles included in the second set of particles. A technique for including each particle in either the first set of particles or the second set of particles is described in greater detail below in  FIG. 4 . According to the technique described in  FIG. 4 , each particle in the virtual graphics scene is assigned to either the first level or the second level of the particle hierarchy  204  based on one or more constraints associated with the particle. 
     At step  306 , the hierarchy generator  202  selects a first particle included in the first set of particles. At step  308 , the hierarchy generator  202  generates a correction weight between the first particle and each parent particle of the first particle. Each correction weight is generated according to Equation 1 
     
       
         
           
             
               
                 
                   
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     According to Equation 1, a correction weight denoted w jk  is generated for a particle P j  and a particle P k  based on the initial distance between particles P j  and P k , denoted d P1P2 , and based on the maximum possible distance between particles P j  and P k , denoted max(d jk ). max(d jk ) is based on a distance constraint associated with particles P j  and P k  and defines the maximum allowable distance between those particles. In addition, the increment ε is added to the denominator of Equation 1 to ensure that the denominator of Equation 1 does not equal zero. In one embodiment, the increment   is a decimal value. Equation 1 may be applied to generate a correction weight between the first particle and each of the parent particles of the selected particle. Once the hierarchy generator  202  has generated the correction weights for the first particle, the hierarchy generator  202  normalizes each of these correction weights by dividing the correction weight by the sum of the correction weights associated with the first particle. 
     At step  310 , the hierarchy generator  202  “collapses” initial constraints associated with the first particle. As referred to herein, “collapsing” constraints associated with the first particle includes generating new hierarchical constraints between certain neighbor particles of the first particle, as described in greater detail below in  FIG. 5A  and illustrated by example in  FIG. 5B . As also described in  FIGS. 5A and 5B , the new hierarchical constraints are generated based on initial constraints between the first particle and each of the certain neighbor particles. 
     At step  311 , the hierarchy generator  202  determines whether additional particles in the first set of particles may be selected and processed according to steps  306 ,  308 , and  310 . If additional particles may be selected and processed, then the method  300  returns to step  306  and proceeds as described above. Otherwise, the method  300  proceeds to step  312 . 
     At step  312 , the hierarchy generator  202  generates the particle hierarchy  204 . The particle hierarchy  204  has a first level and a second level. At step  314 , the hierarchy generator  202  assigns the particles included in the first set of particles to the first level of the particle hierarchy  204 . At step  316 , the hierarchy generator assigns the particles included in the second set of particles to the second level of the particle hierarchy  204 . The hierarchy generator  202  also includes the hierarchical constraints generated for each particle in the particle hierarchy  204 . 
     Those skilled in the art will recognize that although the method  300  describes a technique for generating the particle hierarchy  204  with a first level and a second level, the method  300  may be implemented by the hierarchy generator  202  to generate the particle hierarchy  204  having any number of levels. For example, the hierarchy generator  202  could generate the particle hierarchy  204  with three levels by implementing steps  302 ,  304 ,  306 ,  308 , and  310  of the method  300  with the plurality of particles in a first pass to generate the first and second sets of particles. The hierarchy generator  202  could then implement steps  304 ,  306 ,  308 , and  310  of the method  300  with the particles included in the second set of particles to generate a third set of particles. The hierarchy generator  202  could then assign the particles included in the first, second, and third sets of particles to the first, second, and third levels of the particle hierarchy  204 , respectively. 
       FIG. 4  is a flowchart of method steps for including a particle in either the first set of particles or the second set of particles, according to one embodiment of the invention. Persons skilled in the art will understand that, although the method  400  is described in conjunction with the systems of  FIGS. 1-2 , any system configured to perform the method steps, in any order, is within the scope of the present invention. In one embodiment, the method  400  is implemented in order to perform step  304  of the method  300 , as previously described in  FIG. 3 . 
     As shown, the method  400  begins at step  402 , where the hierarchy generator  202  includes the plurality of particles in the second set of particles. Initially, all of the particles in the plurality of particles are included in the second set of particles. However, through implementing steps  404 ,  406 ,  408 ,  410 ,  412 , and  414  of the method  400  repeatedly, as further described herein, the hierarchy generator  202  transfers a portion of the particles initially included in the second set of particles to the second set of particles. 
     At step  404 , the hierarchy generator  202  selects a particle in the second set of particles. At step  406 , the hierarchy generator  202  identifies in the second set of particles a first neighborhood of particles associated with the selected particle. As referred to herein, a “neighborhood” of particles associated with a particular particle includes any particle associated with a constraint with which the particular particle is also associated. For example, if a distance constraint C 1  restricts the distance between two particles P j  and P k  to be less than a particular value, then the neighborhood of particle P j  includes the particle P k  and the neighborhood of the particle P k  includes the particle P j . The particles P j  and P k  are referred to herein as “sharing” the constraint C 1 . Additionally, any two or more particles associated with a same constraint, such as the particles P j  and P k  described in the above example, are referred to herein as “neighbor” particles to one another. The first neighborhood of particles therefore includes each neighbor particle of the selected particle that is included in the second set of particles, where each neighbor particle shares a constraint with the selected particle. 
     At step  408 , the hierarchy generator  202  determines that the first neighborhood of particles includes at least M particles, where M is an integer value. In one embodiment, M is equal to 2. If the first neighborhood includes at least M particles, then the method  400  proceeds to step  410 . 
     At step  410 , the hierarchy generator  202  identifies in the first set of particles a second neighborhood of particles associated with the selected particle. Initially, when the first set of particles does not include any particles, the second neighborhood of particles does not include any particles. However, after subsequent repetitions of steps  404 ,  406 ,  408 ,  410 ,  412 , and  414 , as previously described, the first set of particles may include a number of particles and, thus, the first neighborhood of particles may include a portion of those particles. 
     At step  412 , the hierarchy generator  202  determines that each particle in the second neighborhood of particles has at least N neighbors in the second set of particles, where N is an integer value. In one embodiment, N is equal to M. In a situation where the second neighborhood of particles does not include any particles, the method  400  proceeds to step  414 . If each particle in the second neighborhood of particles has at least N neighbors in the second set of particles, then the method  400  also proceeds to step  414 . 
     At step  414 , the hierarchy generator  202  transfers the selected particle from the second set of particles to the first set of particles. At step  416 , the hierarchy generator  202  determines that additional particles may be selected and processed according to steps  404 ,  406 ,  408 ,  410 ,  412 , and  414 . If additional particles are to be selected and processed, then the method  400  returns to step  404  and proceeds as described above. Otherwise, the method  400  terminates, thereby allowing the method  300  to advance from step  304  to step  306 . 
       FIG. 5A  is a flowchart of method steps for collapsing initial constraints associated with the first particle, according to one embodiment of the invention. Persons skilled in the art will understand that, although the method  500  is described in conjunction with the systems of  FIGS. 1-2 , any system configured to perform the method steps, in any order, is within the scope of the present invention. In one embodiment, the method  500  is implemented to perform step  310  of the method  300 , as previously described in  FIG. 3 . 
     As shown, the method  500  begins at step  502 , where the hierarchy generator  202  identifies, in the second set of particles, a first neighborhood of particles associated with the first particle. The first neighborhood of particles includes particles in the second set of particles that share a constraint with the first particle. Accordingly, each of the particles in the first neighborhood of particles is a parent particle of the first particle. 
     At step  504 , the hierarchy generator  202  determines an average position of the particles included in the first neighborhood of particles. The hierarchy generator  202  determines the average position by averaging corresponding position coordinates of each particle in the first neighborhood of particles. For example, the hierarchy generator could average an X position coordinate of each particle in the first neighborhood to generate an average X position coordinate, and so forth. 
     At step  506 , the hierarchy generator  202  identifies, in the first neighborhood of particles, a second particle having a position closest to the average position. The hierarchy generator  202  identifies the second particle by determining the distance from each particle in the first neighborhood of particles to the average position and then selecting the particle having the least distance to the average position. 
     At step  508 , for each neighbor particle of the first particle, excluding the second particle, that is not a neighbor of the second particle, the hierarchy generator  202  generates a hierarchical constraint between the neighbor particle and the second particle. 
     The hierarchical constraint is based on one or more initial constraints between the neighbor particle and the first particle and on one or more initial constraints between the first particle and the second particle. For example, if the first particle shared a first distance constraint with the neighbor particle and also shared a second distance constraint with the second particle, then the hierarchical constraint would be a third distance constraint based on the first and second distance constraints. In this example, the hierarchical constraint could also be based on the initial distance between the neighbor particle and the second particle. Those skilled in the art will recognize that the method  500  may be implemented for any particle included in the first set of particles. 
     Once the hierarchy generator  202  generates a hierarchical constraint for the first particle, the method  500  terminates, thereby allowing the method  300  to advance from step  310  to step  312 . 
       FIG. 5B  is a conceptual diagram that illustrates an example of collapsing initial constraints associated with a particle, according to one embodiment of the invention. As shown,  FIG. 5B  includes a particle diagram  550 . The particle diagram  550  includes parent particles  510 ,  512 ,  516 , child particle  516 , initial constraints  518 ,  520 ,  522 , hierarchical constraints  524 ,  526 , and average position  522   
     Child particle  516  corresponds to the first particle described in  FIG. 5 . Parent particle  510  and child particle  516  are both associated with initial constraint  518 , parent particle  512  and child particle  516  are both associated with initial constraint  522 , and parent particle  514  and child particle  516  are both associated with initial constraint  520 . Accordingly, parent particles  510 ,  512 , and  514  are each neighbors of child particle  516  and are thus included in a neighborhood associated with child particle  516 . The neighborhood associated with the child particle  516  corresponds to the first neighborhood of particles generated at step  502  of the method  500 . 
     Position  522  represents an average of the positions of each of the parent particles  510 ,  512 , and  514 . Position  522  corresponds to the average position generated at step  504  of the method  500 . Parent particle  512  occupies a position closest to the position  522 , and so parent particle  512  is selected as the second particle, as described in step  506  of the method  500 . Parent particle  510  and  514  do not share a constraint with parent particle  512 , and so the hierarchy generator  202  generates hierarchical constraints  524  and  526  between parent particle  510  and parent particle  512 , and between parent particle  514  and parent particle  512 , respectively. Generating the hierarchical constraints  524  and  526  in this fashion corresponds to step  510  of the method  500 . 
     Once the hierarchy generator  202  has generated the particle hierarchy  204 , the physics simulation engine  122  may execute the physics simulation. As previously described, the physics simulation engine  122  implements the hierarchical solver  206  to execute the physics simulation. 
       FIG. 6  is conceptual diagram that illustrates the hierarchical solver  206  of  FIG. 2  in greater detail, according to one embodiment of the invention. As shown, the hierarchical solver  206  includes a prediction engine  602 , position predictions  604 , a correction engine  606 , corrected positions  604 . 
     The prediction engine  602  is configured to generate the position predictions  604  based on the current position, current velocity, and set of external forces associated with each particle included in the particle database  208  at each time step in the physics simulation. The position predictions  604  includes a position prediction and a velocity prediction for each particle in the virtual graphics scene. For each particle in the virtual graphics scene, at a current time step in the physics simulation, the prediction engine  602  retrieves the set of external forces acting on the particle, the current velocity of the particle, and the current position of the particle from the particle database  208 . The prediction engine  602  integrates the set of external forces acting on the particle to generate a velocity prediction for the particle. The prediction engine  602  then integrates the velocity prediction to generate a position prediction for the particle. In one embodiment, the prediction engine  602  implements the explicit Euler technique to integrate the set of external forces and to integrate the velocity prediction. In alternative embodiments, another numerical integration technique may be used to generate the position prediction. The prediction engine  602  generates a predicted position for each particle in the virtual graphics scene to generate the position predictions  604 . 
     At the current time step, the correction engine  606  retrieves the position predictions  604 , the particle hierarchy  204 , and the initial constraints associated with each particle from the particle database  208 . The correction engine  206  then generates zero or more collision constraints for each particle in the virtual graphics scene. The correction engine  206  generates a collision constraint for a particle when the position prediction generated for the particle is identical to the position prediction for at least one other particle. A collision constraint associated with two particles takes the form C coll (P j , P k )=|P I −P k |&gt;0, and P j  and P k  represent the respective positions of the two particles. A collision constraint may be associated with any number of different particles having the same position prediction. 
     Once the correction engine  606  generates the collision constraints for each particle, the correction engine  606  enforces each of the initial constraints, hierarchical constraints, and collision constraints associated with the parent particles (i.e., the particles assigned to the second level of the particle hierarchy). The correction engine  606  enforces each constraint by modifying the position predictions corresponding to the parent particles associated with the constraint. The correction engine  606  enforces each constraint separately by implementing a non-linear Gauss-Seidel algorithm. 
     The non-linear Gauss-Seidel algorithm approximates a particular constraint C associated with one or more parent particles according to Equation 2:
 
 C ( P+ΔP )≈ C ( P )+∇ P   *C ( P )+Δ P= 0  Equation 2
 
     In Equation 2, P represents a vector that includes the position predictions for each parent particle associated with the constraint C. ΔP represents a correction that may be applied to the position predictions P so that the position predictions P satisfy the constraint C. ∇P represents the derivative of C with respect to each element of P. According to the non-linear Gauss-Seidel algorithm, ΔP is restricted to be in the direction of ∇P. This restriction conserves linear and angular momentum and also prevents the constraint C from being underdetermined. ΔP can then be solved to yield Equation 4:
 
Δ P=λ*∇   P   *C ( P )  Equation 3
 
     In equation 3, λ is a Lagrange multiplier that is determined using conventional techniques. Based on Equation 3, the correction ΔPj can be found for a particular particle having a position prediction Pj according to Equations 4 and 5:
 
Δ P   j   =−s*w   j *∇ P   *C ( P )  Equation 4
 
     In equation 4, wj is equal to 1/mj, where mj is the mass of the particle, and the quantity s is provided by Equation 5: 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       C 
                       ⁡ 
                       
                         ( 
                         P 
                         ) 
                       
                     
                     
                       
                         ∑ 
                         k 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           w 
                           k 
                         
                         * 
                         
                           
                              
                             
                               
                                 ∇ 
                                 Pk 
                               
                               ⁢ 
                               
                                 * 
                                 
                                   C 
                                   ⁡ 
                                   
                                     ( 
                                     P 
                                     ) 
                                   
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     In Equation 5, the value k represents all k particles associated with the constraint C. The correction engine  606  may thus generate a correction ΔPk for each parent particle associated with the constraint C by implementing the non-linear Gauss-Seidel algorithm described herein. The correction engine  606  may then update the position prediction associated with the particle to reflect the correction ΔPk associated with the particle. 
     The correction engine  606  implements the non-linear Gauss-Seidel algorithm described above to enforce each of the constraints associated with the parent particles in the particle hierarchy  204 . For each constraint and for each parent particle associated with that constraint, the correction engine  606  generates a different correction for the parent particle and then updates the position prediction corresponding to that parent particle. Accordingly, the position prediction for the parent particle may be updated multiple times depending on the number of constraints with which the parent particle is associated. 
     In one embodiment, the correction engine  606  enforces the constraints associated with some or all of the parent particles in the particle hierarchy  204  in multiple passes. Since the correction engine  606  enforces each constraint separately, a correction is occasionally applied to one or more position predictions causing those position predictions to no longer satisfy a constraint that was previously satisfied. In this situation, the correction engine  606  implements the non-linear Gauss-Seidel algorithm to enforce the constraint that was previously satisfied. In this fashion, the correction engine  606  implements the non-linear Gauss-Seidel algorithm to enforce any constraint that is not satisfied until all of the constraints associated with the parent particles are satisfied. 
     The correction engine  606  generates the corrected position predictions  608  for the parent particles in the particle hierarchy  204  by enforcing each constraint equation associated with those parent particles at each timestep. The correction engine  606  then generates a corrected position prediction for each child particle in the particle hierarchy  204  based on the corrected position predictions generated for the parent particles corresponding to those child particles and based on the correction weights, as further described herein. 
     For each child particle, the correction engine  204  identifies the parent particles of the child particle. The correction engine  602  then generates a corrected position prediction P corr  for the child particle according to Equation 6: 
     
       
         
           
             
               
                 
                   
                     P 
                     corr 
                   
                   = 
                   
                     
                       P 
                       j 
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             P 
                             ⁡ 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           w 
                           jk 
                         
                         * 
                         
                           ( 
                           
                             
                               P 
                               k 
                             
                             - 
                             
                               Q 
                               k 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     In Equation 6, P(j) is a set of indices that identifies the parent particles of the child particle. P j  is the position prediction for the child particle. w jk  is a correction weight between the child particle and a parent particle of the child particle. P k  represents the position prediction for the parent particle generated by the prediction engine  602 , and Q k  represents the corrected position prediction for the parent particle generated by the correction engine  606 . The correction engine  602  implements Equation 6 to generate corrected position predictions for each child particle in the particle hierarchy  204 . 
     Once the correction engine  606  has generated a corrected position prediction for each particle in the particle hierarchy  204 , the correction engine  606  generates a corrected velocity prediction based on the difference between the current position of the particle, the corrected position prediction generated for the particle, and the size of the time step. The correction engine  606  then accesses the particle database  208  and updates the current position of the particle to reflect the corrected position prediction, and updates the current velocity to reflect the corrected velocity prediction. The physics simulation engine  122  may then proceed to a subsequent time step in the physics simulation. 
       FIG. 7  sets forth a flowchart of method steps for generating a corrected position prediction for a first particle and a corrected position prediction for a second particle, according to one embodiment of the invention. Persons skilled in the art will understand that, although the method  700  is described in conjunction with the systems of  FIGS. 1-2  and  6 , any system configured to perform the method steps, in any order, is within the scope of the present invention. 
     As shown, the method  700  begins at step  702 , where the hierarchical solver  206  receives the plurality of particles in the virtual graphics scene, the initial constraints associated with those particles, and the particle hierarchy  204 . At the current time step, each particle occupies a current position in the virtual graphics scene and has a current velocity. Each particle is associated with a set of external forces acting on the particle and has a particular mass value. 
     At step  704 , the prediction engine  602  within the hierarchical solver  206  generates a position prediction for each particle in the virtual graphics scene. The prediction engine  602  integrates the set of external forces acting on the particle to generate a velocity prediction for the particle. The prediction engine  602  then integrates the velocity prediction to generate a position prediction for the particle. In one embodiment, the prediction engine  602  implements the explicit Euler numerical integration technique to generate the position prediction for the particle. In an alternative embodiment, the correction engine  602  implements a different numerical integration technique. The prediction engine  602  stores the position predictions for each particle in position predictions  604 . 
     At step  706 , the correction engine  606  generates zero or more collision constraints for each particle in the virtual graphics scene. The correction engine  606  generates a collision constraint for a particle when the position prediction generated for the particle is identical to the position prediction for at least one other particle. A collision constraint associated with two particles takes the form C coll (P j , P k )=|P j −P k |&gt;0, and P j  and P k  represent the respective positions of the two particles. A collision constraint may be associated with any number of different particles having the same position prediction. 
     At step  708 , the correction engine  606  generates a corrected position prediction for each parent particle in the particle hierarchy  204  by enforcing one or more constraints associated with those parent particles. The constraints may include initial constraints, hierarchical constraints and/or collision constraints. The correction engine  606  enforces each constraint separately by modifying the position predictions for each parent particle associated with the constraint using the non-linear Gauss-Seidel algorithm described in  FIG. 6 . 
     At step  710 , the correction engine  606  selects a child particle in the particle hierarchy  204 . As previously described, the child particle is associated with at least two parent particles in the particle hierarchy  204 . 
     At step  712 , the correction engine  606  generates a corrected position prediction for the child particle based on the current position of the child particle, the current positions of the at least two parent particles, the corrected position predictions generated for the at least two parent particles, and the correction weights between the child particle and the at least two parent particles. Those skilled in the art will recognize that step  712  may be repeated for each child particle in the particle hierarchy. 
     At step  714 , the correction engine  606  generates a corrected velocity prediction for each particle in the virtual graphics scene. The correction engine  606  generates the corrected velocity prediction based on the difference between the current position of the particle, the corrected position prediction generated for the particle, and the size of the time step. 
     At step  716 , the correction engine  606  sets the current velocity of each particle in the virtual graphics scene to the corrected velocity prediction corresponding to the particle. At step  718 , the correction engine  606  sets the current position of each particle in the virtual graphics scene to the corrected position prediction corresponding to the particle. The correction engine  606  sets the current velocity and the current position of each particle to the corrected velocity prediction and the corrected position prediction corresponding to the particle, respectively, by accessing the particle database  208 . 
     At step  720 , the hierarchical solver advances to a subsequent time step in the physics simulation. 
     In embodiments where the particle hierarchy  204  includes greater than two levels, the correction engine  606  may implement steps  708 ,  710 , and  712  repeatedly to generate corrected position predictions for each level of the particle hierarchy  204 . 
     Once the correction engine  606  updates the current velocity and current position of each particle in the virtual graphics scene, the physics simulation engine  122  transmits the current position of each particle in the virtual graphics scene to the software application  118 . The software application  118  may then implement the rendering engine  120  to render an image of the virtual graphics scene for output to a display device. 
       FIG. 7B  is a conceptual diagram illustrating an example of how a corrected position prediction for particles in a virtual graphics scene may be generated, according to one embodiment of the invention. As shown,  FIG. 7B  includes a particle diagram  750 . The particle diagram  750  included parent particles  722 ,  724 ,  726 , child particle  728 , corrected position predictions  730 ,  732 ,  734 ,  736 , and correction weights  738 ,  740 ,  742 . 
     Each of parent particles  722 ,  724 , and  726  is a parent particle of the child particle  728 . Parent particle  722  and child particle  728  are both associated with correction weight  738 , parent particle  724  and child particle  728  are both associated with correction weight  740 , and parent particle  726  and child particle  728  are both associated with correction weight  742 . 
     The location of each of the parent particles  722 ,  724 ,  726 , and child particle  728  corresponds to the position prediction generated for that particle, as described in conjunction with step  704  of the method  700 . Corrected position predictions  730 ,  732 , and  734  correspond to the corrected position predictions generated for parent particles  722 ,  724 , and  726 , respectively, as described in conjunction with  708  of the method  700 . 
     The corrected position prediction  736  is generated for the child particle  728  based on the position predictions generated for the parent particles  722 ,  724 , and  726 , the corrected position predictions  730 ,  732 , and  734  corresponding to those particles, and the corrections weights  738 ,  740 , and  742 . Generating the corrected position prediction  736  in this fashion is described in conjunction with step  712  of the method  700 . 
     In addition to generating corrected position predictions for each particle in the virtual graphics scene via the prediction engine  602  and the correction engine  606 , the hierarchical solver  206  may also update the particle hierarchy  204  dynamically at a particular time step in the physics simulation under certain circumstances. When the distance between two particles that share a distance constraint exceeds a threshold value, the hierarchical solver  206  may update the particle hierarchy  204 . 
       FIG. 8  is a flowchart of method steps for updating the particle hierarchy  204 , according to one embodiment of the invention. Persons skilled in the art will understand that, although the method  800  is described in conjunction with the systems of  FIGS. 1-2  and  6 , any system configured to perform the method steps, in any order, is within the scope of the present invention. The hierarchical solver  206  implements the method  800  when the distance between a first particle and second particle exceeds a threshold value, and the first particle and the second particle share a distance constraint. 
     As shown, the method  800  begins at step  802 , where the hierarchical solver identifies a first neighborhood of particles associated with the first particle. This first neighborhood of particles includes any particles that share a constraint with the first particle. 
     At step  804 , the hierarchical solver inserts a “split plane” through the first particle. The split plane separates the particles included in the first neighborhood into two groups that reside on opposing sides of the split plane. The split plane is inserted such that a line connecting the first particle to the second particle is perpendicular to the split plane. 
     At step  806 , the hierarchical solver  206  removes all constraints between the first particle and the particles included in the first neighborhood of particles. At step  808 , the hierarchical solver  206  removes all constraints between the parent particles of the first particle. The constraints removed in steps  806  and  808  may include initial constraints, hierarchical constraints and/or collision constraints. 
     At step  810 , the hierarchy generator  206  generates a cloned particle by cloning the first particle. The cloned particle occupies a position on a first side of the split plane. In one embodiment, the cloned particle has the same mass value as the first particle. 
     At step  812 , the hierarchical solver  206  generates one or more hierarchical constraints between the cloned particle and any particles included in the first neighborhood of particles residing on the first side of the split plane. The hierarchical constraints generated for the cloned particle are distance constraints that restrict the position of the cloned particle relative to the particles included in the first neighborhood of particles residing on the first side of the split plane. 
     At step  814 , the hierarchical solver  206  generates one or more hierarchical constraints between the first particle and any particles included in the first neighborhood of particles residing on the second side of the split plane. The hierarchical constraints generated for the first particle are distance constraints that restrict the position of the first particle relative to the particles included in the first neighborhood of particles residing on the second side of the split plane. 
     By implementing the method  800 , the hierarchical solver  206  updates the particle hierarchy  204  in situations where particles that share a distance constraint have a distance from one another that exceeds a threshold value. This feature of the hierarchical solver  206  allows the physics simulation  122  to simulate virtual objects that may break apart or tear. For example, if the particles in the virtual graphics scene represent a piece of cloth, and each particle in the piece of cloth shares a distance constraint with an adjacent particle, then the cloth may be torn by removing the distance constraints shared between particles on either side of the tear and inserting cloned particles along the edges of the tear, as described. 
     In sum, the physics simulation engine executes a physics simulation to simulate the motion of virtual objects in a virtual graphics scene by simulating the motion of particles that compose the virtual objects. At each timestep in the physics simulation, the physics simulation engine generates a position prediction for each particle in the virtual graphics scene. The physics simulation engine then generates a corrected position prediction for the particle by enforcing constraints associated with the particle using a particle hierarchy. Finally, the physics simulation engine updates the position of the particle in the virtual graphics scene to reflect the corrected position prediction. 
     The particle hierarchy includes at least a first level and a second level. Prior to the physics simulation engine executing the physics simulation, a hierarchy generator generates the particle hierarchy by assigning each particle in the virtual graphics scene to either the first level or the second level of the particle hierarchy. The hierarchy generator generates correction weights between the particles in the first level of the particle hierarchy and the particles in the second level of the particle hierarchy. The hierarchy generator also generates hierarchical constraints for the particles in the second level of the particle hierarchy by collapsing initial constraints associated with particles in the first level of the particle hierarchy. 
     When the physics simulation engine executes the physics simulation, a hierarchical solver generates position predictions for each particle in the virtual graphics scene. The hierarchical solver then generates corrected position predictions for the particles in the second level of the hierarchy by implementing a non-linear Gauss-Seidel algorithm. The hierarchical solver also generates corrected position predictions for the particles in the first level of the particle hierarchy based on the corrected position predictions generated for the particles in the first level of the particle hierarchy. 
     Advantageously, constraint equations are solved for fewer particles at each timestep in the physics simulation since the second level of the particle hierarchy only includes a subset of the particles in the virtual graphics scene. The corrected position predictions generated for these particles can then be used to generate corrected position predictions for the particles in the first level of the particle hierarchy. By implementing the physics simulation engine, more detailed physics simulations are possible, when compared to prior art techniques. 
     Since fewer constraint equations need to be solved at each timestep, a corrected position prediction can be generated for each particle in the virtual graphics scene and can thus be generated for all of the particles that represent a particular virtual object. The physics simulation engine is thereby capable of generating a physically accurate simulation of the virtual object, allowing physically realistic images to be generated based on that physics simulation. 
     While the forgoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof. For example, aspects of the present invention may be implemented in hardware or software or in a combination of hardware and software. One embodiment of the invention may be implemented as a program product for use with a computer system. The program(s) of the program product define functions of the embodiments (including the methods described herein) and can be contained on a variety of computer-readable storage media. Illustrative computer-readable storage media include, but are not limited to: (i) non-writable storage media (e.g., read-only memory devices within a computer such as CD-ROM disks readable by a CD-ROM drive, flash memory, ROM chips or any type of solid-state non-volatile semiconductor memory) on which information is permanently stored; and (ii) writable storage media (e.g., floppy disks within a diskette drive or hard-disk drive or any type of solid-state random-access semiconductor memory) on which alterable information is stored. Such computer-readable storage media, when carrying computer-readable instructions that direct the functions of the present invention, are embodiments of the present invention. 
     Therefore, the scope of the present invention is determined by the claims that follow.