WO2007092037A2 - Method of simulating deformable object using geometrically motivated model - Google Patents
Method of simulating deformable object using geometrically motivated model Download PDFInfo
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- WO2007092037A2 WO2007092037A2 PCT/US2006/028162 US2006028162W WO2007092037A2 WO 2007092037 A2 WO2007092037 A2 WO 2007092037A2 US 2006028162 W US2006028162 W US 2006028162W WO 2007092037 A2 WO2007092037 A2 WO 2007092037A2
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
- G06T17/20—Finite element generation, e.g. wire-frame surface description, tesselation
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F11/00—Error detection; Error correction; Monitoring
- G06F11/22—Detection or location of defective computer hardware by testing during standby operation or during idle time, e.g. start-up testing
- G06F11/26—Functional testing
- G06F11/261—Functional testing by simulating additional hardware, e.g. fault simulation
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F13/00—Interconnection of, or transfer of information or other signals between, memories, input/output devices or central processing units
- G06F13/10—Program control for peripheral devices
- G06F13/105—Program control for peripheral devices where the programme performs an input/output emulation function
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2111/00—Details relating to CAD techniques
- G06F2111/10—Numerical modelling
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
Definitions
- Embodiments of the present invention relate generally to methods of simulating deformable objects. More particularly, embodiments of the invention relate to methods of simulating deformable objects using a geometrically motivated underlying model.
- One common technique used to simulate deformable objects first creates a virtual model of an object (e.g., a mesh or point cloud) and then applies simulated physical forces such as tension, friction, gravity, pressure, etc., to the discrete points of the virtual model.
- virtual models have been used to represent a wide variety of materials under different conditions. For example, researchers have developed virtual models for clothing, plastic, rubber, and so on. In addition, researchers have also developed virtual models to simulate complex, unique behaviors of these objects, such as fracturing and melting.
- Some of the more common approaches to simulating deformable objects involve finite difference methods, mass-spring systems, boundary element methods, finite element methods, and implicit surfaces and mesh-free particle systems.
- the term "stability" here refers to a process's tendency to respond in a reasonable way to minor deviations in its inputs.
- implicit numerical integration techniques produce accurate simulations as various input parameters are varied, such as the mass of simulated points, the timestep of the integration, and so on.
- explicit numerical integration techniques may produce simulations where the overall energy of a system erroneously increases by simply varying the timestep of the integration, the mass of simulated points, or the stiffness of a simulated object. As a result, explicit numerical integration techniques can produce highly unrealistic simulation results.
- FIG. 1 is a diagram illustrating one way in which instability arises in deformable object simulations using explicit numerical integration.
- Fig. 1 is a diagram illustrating one way in which instability arises in deformable object simulations using explicit numerical integration.
- a simple one dimensional deformable object is modeled as a mass-spring system.
- the mass-spring system relies on Newton's second law of motion.
- Newton's second law of motion the acceleration of an object produced by a net force is directly proportional to the magnitude of the net force in the direction of the net force, and inversely proportional to the mass of the object.
- deformable objects at least part of the net force at any point is created by displacement of the point from an equilibrium position. The displacement of a point from its equilibrium position creates potential energy, i.e., "deformation energy", causing the point to be pulled toward the equilibrium position.
- a mass-spring system 100 comprises a spring 101 with a resting length / ⁇ , and two point masses 102 and 103 both having mass "m".
- Point mass 102 is fixed at an origin and point mass 103 is located atx(t) at an initial time "f".
- the location of point mass 103 is updated by a modified Euler integration scheme after a timestep "/?". According to the modified Euler integration scheme, the velocity "v" and position "x" of point mass 103 at time “t+h” are computed using the following equations (1) and (2):
- Equation (1) uses an explicit Euler step and equation (2) uses an implicit Euler step.
- Equations (1) and (2) can be represented as a system matrix "Af' multiplied by a
- system matrix "E” represents a discrete system
- the spectral radius of system matrix "E” i.e., the maximum magnitude of eigenvalues e ⁇ and ej
- the magnitude of eigenvalue e ⁇ converges to 1 with eo ⁇ ⁇ 1 for h 2 k — > ⁇ .
- the magnitude of ⁇ / is only smaller than one where timestep "h” is less than 2 1— . Where timestep "h” is greater than 2 J— , the system is unstable.
- Fig. IB shows the result of performing an integration step starting with v(t) — 0. h 2 k
- point mass 103 overshoots equilibrium position IQ, by a distance greater than the distance between x(t) and / ⁇ .
- x(t + h) -I 0 > x(t) - I 0 .
- the potential energy of system 100 increases after timestep "h”. Since system 100 had zero kinetic energy at time "f ⁇ the overall (i.e., kinetic plus potential) energy of the system is erroneously increased after timestep " ⁇ ".
- a method of modeling a deformable object comprises modeling deformable elasticity for the object by pulling a deformed shape towards a defined goal shape.
- a method of simulating a deformable object comprises defining positions and velocities for a plurality of points in a deformed shape, and updating the positions and velocities according to the positions of points in a goal shape.
- a method of describing object deformation in a simulation comprises defining elastic forces associated with the object deformation in proportion to distances between points in a deformed shape and points in a goal shape, resolving the object deformation toward an equilibrium using an explicit integration scheme, and resolving the explicit integration scheme by matching on a point by point basis an original shape and the deformed shape and then pulling points corresponding to the deformed shape towards a corresponding points in the goal shape.
- Figs. IA and IB are diagrams of a conventional one dimensional mass-spring system used to simulate a deformable object
- Fig. 2 is a diagram illustrating a simulation of a two dimensional deformable object according to one embodiment of the present invention
- Fig. 3 is a flowchart illustrating a method of simulating a deformable object according to an embodiment of the present invention.
- Embodiments of the present invention provide various methods of modeling and simulating deformable objects. These methods can be readily applied to a wide range of computer-related applications such as scientific visualization, computer graphics, computer animated films, and video games, to name but a few.
- Selected embodiments of the invention are particularly well-suited for applications requiring a high level of computational efficiency. For instance, some embodiments of the invention are capable of performing real-time simulations on deformable objects with complicated geometries and/or material properties using only a small portion of a computer's data processing bandwidth.
- Video games are one application that requires a high level of computational efficiency.
- state-of-the-art video games tend to incorporate a variety of realistic effects, such as characters and objects that interact with their environment in real time as if governed by the laws of physics.
- Such characters and objects are often formed by deformable objects including, for example, clothing, bendable or stretchable materials, and so on.
- a deformable object simulation according to embodiments of the invention is typically performed by running a software application on a computational platform including at least one microprocessor and memory.
- the term "run” here describes any process in which a hardware resource associated with a computational platform performs an operation under the direction (directly or indirectly) of a software resource.
- the software application receives geometric data and physics data defining a configuration of the deformable object and any external forces acting on the object.
- the "configuration" of a deformable object is broadly defined as a description of all physical attributes of the object itself, including, for example, the locations and masses of discrete elements constituting the object (e.g., points, surfaces, etc.), any connectivity and movement of those discrete elements, and so forth.
- the software application simulates the deformable object by updating the object's configuration based on internal forces of the object such as elastic tension, the movement of discrete elements comprising the object, and any external forces acting on the object such as gravity, pressure, or friction, or impact forces from collisions with other objects.
- the computational platform typically comprises one or more central processing units (CPUs) and one or more memories.
- the one or more memories store the software application and loads it into the one or more CPUs for execution.
- a deformable object simulation may also be performed with more than one software application.
- the simulation could be performed by two software applications running in two execution threads on a single CPU, on two different processor cores, or two different CPUs.
- one of the applications generally comprises a main application defining the geometric and physics data
- the other application generally comprises a physics application running in parallel with the main application and updating the geometric and physics data.
- deformable object refers broadly to any collection of data capable of representing an object comprising elements that can be arranged with varying spatial relationships.
- a deformable object could comprise a mesh, a surface, or a set of points.
- a deformable object generally further comprises parameters related to the way the object tends to deform.
- the deformable object may comprise a goal state for each of its elements, and a stiffness parameter specifying how easily each element approaches its goal state.
- Fig. 2 is a diagram illustrating a method of modeling and simulating a deformable object according to one embodiment of the invention.
- the method of Fig. 2 is described in relation to a two-dimensional (2D) object.
- the method can be readily applied to objects expressed in higher (e.g., 3D) or lower (e.g., ID) dimensions.
- a deformable object 200 is modeled by an "actual shape” 202 corresponding to a deformed state of deformable object 200, and a “goal shape” 203 corresponding to a non-deformed state of deformable object 200.
- Actual shape 202 comprises four "actual points” xj, x?, x-3, and x*
- goal shape 203 comprises four "goal points” gi, g 2 , g 3 , and g 4 corresponding to respective actual points xj, x?, X 3 , and X 4 .
- Goal shape 203 is defined by matching an "original shape” 201 comprising four "original points” X 1 0 , x° , x° , and x° to corresponding actual points xi, x?, X 3 , and X 4 in actual shape 202.
- goal shape 203 is a matched version of original shape 201.
- match here refers to a process of transformation applied to original shape 201 such that goal shape 203 will approximate actual shape 202.
- such a transformation can comprise, for example, a linear transformation, a higher order (e.g., quadratic) transformation, or some combination thereof.
- deformable elasticity in deformable object 200 is then modeled by pulling actual points xj, X 2 , X 3 , and X 4 toward corresponding goal points gj, g 2 , g 3 , and g 4 , as indicated by arrows in Fig. 2.
- the notations x ; ° , X 1 , and g are used to refer to points and also to locations of the points. For example, the location of an actual point x, is denoted as simply X 1 .
- One way to match original shape 201 to actual shape 202 is by rotating and translating original shape 201.
- the amount of rotation and translation can be determined by minimizing some distance between points in goal shape 203 and corresponding points in actual shape 202.
- the translation and rotation could be chosen to minimize a weighted least squares distance between corresponding points in goal shape 203 and actual shape 202.
- Correspondences between points in original shape 201 and actual shape 202 are generally defined a priori, for example, when original shape 201 and actual shape 202 are first defined.
- corresponding points are labeled with the like subscripts.
- original point x° corresponds to actual and goal points xj and gj.
- the configuration of goal shape 203 can be computed by finding a rotation matrix R and translation vectors to and t such that the following equation (5) is minimized: ⁇ w ⁇ R ⁇ x « -t o ) + t-x,) 2 (5).
- w represents a mathematical weight associated with original and actual points x° and X 1 .
- the weight assigned to each point is the point's mass.
- weights W 1 are substituted by masses "777," throughout.
- Equation (9) defines a first matrix A qq , and a second matrix A pq .
- First matrix A qq is symmetric, and therefore it contains a scaling component, but no rotational component.
- second matrix ⁇ contains both a symmetric component and a rotational component.
- the rotational component of second matrix ⁇ is rotation matrix R and the symmetric component of second matrix A pq is a symmetric matrix S.
- ⁇ represents a "stiffness" of deformable object 200.
- deformable object 200 is rigid, deformable shape 202 tends to approach the configuration of goal shape 203 very quickly.
- ⁇ ⁇ 1 deformable shape 202 still approaches the configuration of goal shape 203, however, it does so more slowly.
- equation (13) the term represents a system matrix similar to h ⁇ - a system matrix "E” in equation (3). However, unlike the eigenvalues of system matrix "E”,
- the magnitudes of the eigenvalues of the term are always one (1), regardless of the respective values of ⁇ and timestep h. IN particular, the eigenvalues of the term
- FIG. 3 is a flowchart illustrating a method of simulating a deformable object according to an embodiment of the invention. In the description that follows, exemplary method steps are denoted by parentheses (XXX).
- the method comprises defining an actual shape corresponding to a deformed state of the deformable object (301), defining a goal shape corresponding to a non-deformed state of the deformable object (302), and updating the location of each point in the actual shape by pulling the point toward a corresponding point in the goal shape (303).
- Positions of actual points in the actual shape are generally defined by events in a software application, such as initialization and subsequent updates of the actual shape's configuration. Meanwhile, positions of goal points in the goal shape can be computed in a variety of different ways.
- the positions of goal points gi in the goal shape are computed by defining an original shape comprising original points and rotating and translating the original shape to match the actual shape using a translation vector and rotation matrix, e.g., as described by equation (10).
- the translation vector is generally computed from the respective centers of mass of the original and actual shapes, as described by equations (6) and (7), and the rotation matrix is generally computed by polar decomposition of a linear transformation matrix computed according to equation (9).
- center of mass x° n ⁇ in equation (10) and relative locations q t in equations (8) and (9) are computed before any timesteps of the simulation are executed.
- second matrix A pq is a 3 x
- Second matrix A pq is decomposed into rotation matrix "i?" and symmetric matrix
- S ⁇ ' i.e., UA p T q A pq
- S ⁇ ' is computed by diagonalizing the symmetric matrix A ⁇ q A pq using 5-10 Jacobi rotations, where the computational complexity of each Jacobi rotation is constant.
- the positions of goal points g,- are computed by a linear transformation using rotation matrix "R” and linear transformation matrix "A” according to the following equation (14):
- equation (14) the term ⁇ is a control parameter used to control the locations of goal points gi.
- Linear transformation matrix "A” is divided by lJdet(A) to ensure that the volume of the goal shape relative to the original shape is conserved by equation (14).
- Linear transformation matrix "A” is generally constructed by forming first matrix A qq before any simulation timesteps are executed and then forming second matrix A pq with each timestep.
- equation (14) can generate goal points g r - closer the locations of actual points x / . Accordingly, equation (14) is generally better at simulating more highly deformed, and/or less rigid objects.
- goal points gi Another way to compute goal points gi is by performing a quadratic transformation on the original points.
- Quadratic transformation matrix A [AQM] ⁇ a R 3x9 preferably minimizes the equation ⁇ w, (Aq ⁇ -pi) 2 , and is computed by the following equation (16): .
- a symmetric matrix A qq e R 9x9 and vector q t in equation (16) can be computed before a simulation begins.
- the control parameter/? can be used with quadratic transformation matrix A to further control the positions of the goal points.
- Still another way to compute goal points g is by dividing actual points x, into overlapping clusters and then computing a separate transformation matrix for each cluster.
- actual points x, represented by a volumetric mesh can be divided into clusters where each cluster comprises points adjacent to a common element of the volumetric mesh
- a mesh can be regularly divided into overlapping cubical regions, and then a cluster can be formed by all points in each cubical region.
- Equation (11) becomes the following equation (17):
- the method described in relation to Fig. 3 can also be used to simulate plastic deformable objects.
- a plastic deformable object is simulated by representing a deformation state SP for the object.
- the deformation state S p is initialized with the identity matrix "/" and then updated with each timestep of the simulation.
- Deformation state SP is updated based on symmetric matrix S derived by the polar decomposition of second matrix A pq of equation (9).
- Symmetric matrix S represents a deformation of the original shape in an unrotated reference frame. Accordingly, where an amount of deformation (i.e., a distance) JS -/
- timestep "/?” and the parameter c creep are used to control the plasticity of the deformable object.
- the plasticity can be bound - / exceeds a threshold value c max .
- state matrix is set by the following equation
- state matrix SP is divided by ydet(S p ) every time it gets updated. Note that each time S p is updated, first and second matrices A qq and A pq must also be updated.
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DE112006003728T DE112006003728T5 (en) | 2006-02-03 | 2006-07-21 | A method of simulating a deformable object using a geometrically motivated model |
JP2008553222A JP2009529161A (en) | 2006-02-03 | 2006-07-21 | A method for simulating deformable objects using geometry-based models |
GB0813908A GB2449377B (en) | 2006-02-03 | 2006-07-21 | Method of simulating deformable object using geometrically motivated model |
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US11/346,299 US7650266B2 (en) | 2005-05-09 | 2006-02-03 | Method of simulating deformable object using geometrically motivated model |
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