SPH-based hyperelastic simulation method and device, and computer program stored in a computer-readable recording medium for carrying out the method

Abstract

To provide an SPH-based hyperelastic simulation method and apparatus.SOLUTION: An SPH-based hyperelastic simulation apparatus includes a processor, and a memory connected to the processor. The memory stores program instructions executed by the processor to define states of each of m particles in a discrete time as a set of positions and velocities, for an SPH-based deformable body composed of m particles, approximate hyperelastic energy of each of the m particles using a rest-pose volume, a material parameter, a vectorized deformation gradient, and a projection of the vectorized deformation gradient in order to optimize an objective function for solving new states of each of the m particles, search for an initial approximation of a Hessian matrix using the approximated hyperelastic energy, and simulate the SPH-based deformable body based on the searched initial approximation.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to physics-based animation techniques, and more particularly to a method and apparatus for simulating hyperelastically deformable bodies based on SPH. [Background technology]

[0002] A representative example of conventional technology for deformable body simulation based on SPH (Smoothed Particle Hydrodynamics) is the technology described in the paper "Fast Corotated Elastic SPH Solids with Implicit Zero-Energy Mode Control" (hereinafter referred to as "existing technique"), published in the international journal PACMCGIT (Proceedings of the ACM on Computer Graphics and Interactive Techniques).

[0003] The technique applies a backward Euler method to approximate the next position and velocity of a deformable body of particles, but does so by solving two linear systems in sequence.

[0004] Here, the solutions of the two linear systems are the Cholesky factorization and the preconditioned conjugate gradient, respectively.

[0005] Existing techniques are optimized to represent the simplest corotated model among hyperelastic models, and therefore cannot simulate the movement of deformable bodies on an SPH basis using more sophisticated models such as the neo-Hookean or St. Venant-Kirchoff. Furthermore, if a high elastic modulus or a large time interval is used, the simulation results may become unstable. [Prior art documents] [Patent documents]

[0006] [Non-Patent Document 1] Fast Corotated Elastic SPH Solids with Implicit Zero-Energy Mode Control Paper Summary of the Invention [Problem to be solved by the invention]

[0007] The present invention provides a method and apparatus for SPH-based hyperelastic simulation that can improve accuracy while reducing complexity. [Means for solving the problem]

[0008] In order to solve the above-mentioned technical problems, according to one embodiment of the present invention, there is provided an SPH (Smoothed Particle Hydrodynamics)-based hyperelastic simulation device, comprising a processor and a memory coupled to the processor, wherein the memory stores program instructions executed by the processor to: define the states of each of the m particles at discrete time as a set of positions and velocities for an SPH-based deformable body composed of m particles; approximate the hyperelastic energy of each of the m particles using a rest-pose volume, material parameters, vectorized deformation gradients, and a projection of the vectorized deformation gradients in order to optimize an objective function for solving a new state for each of the m particles; find an initial approximation of a Hessian matrix using the approximated hyperelastic energy; and simulate the SPH-based deformable body based on the found initial approximation.

[0009] The deformation gradient is defined by the following mathematical formula:

number

[0010] The superelastic energy is defined by the following formula:

number

[0011] The new state is expressed by the following equation:

number

[0012] The internal force is evaluated with a negative slope, and the new state is expressed as:

number

[0013] The optimization problem for the objective function is reformulated as follows:

number

[0014] The program instructions may optimize the objective function through an iterative approach using the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm.

[0015] According to another aspect of the present invention, there is provided a method for simulating SPH-based hyperelasticity in an apparatus having a processor and a memory, the method including the steps of: generating an SPH-based deformable body consisting of m particles; defining, for the SPH-based deformable body consisting of m particles, the states of each of the m particles at discrete time as a set of positions and velocities; approximating the hyperelastic energy of each of the m particles using a rest-pose volume, material parameters, vectorized deformation gradients, and a projection of the vectorized deformation gradients in order to optimize an objective function for solving the new states of each of the m particles; searching for an initial approximation of a Hessian matrix using the approximated hyperelastic energy; and simulating the SPH-based deformable body based on the searched initial approximation.

[0016] According to yet another aspect of the present invention, there is provided a computer program stored on a computer-readable recording medium for carrying out the above method. [Effects of the Invention]

[0017] According to the present invention, it is possible to simulate the motion of deformable bodies on SPH substrates by applying more sophisticated models such as the neo-Hookean and St. Venant-Kirchoff.

[0018] Furthermore, according to the present invention, the existing technique is extended to an optimization problem, which is solved using L-BFGS (Limited-Memory BFGS), and stable simulation results can be generated even when using a higher elastic modulus or a larger time step than the existing technique. [Brief explanation of the drawings]

[0019] [Figure 1] 1 is a diagram illustrating the configuration of an SPH-based hyperelasticity simulation apparatus according to a preferred embodiment of the present invention. [Figure 2] 1 is a diagram illustrating an L-BFGS algorithm based on A0 according to an embodiment of the present invention. [Figure 3] 10 shows the results of a solver tested on various elastic models. [Figure 4] 10 is a diagram showing a simulation result according to the present embodiment and a simulation result according to the existing technique; [Figure 5] 10 is a diagram showing a result of applying the L-BFGS algorithm according to the present embodiment and a simulation result of the existing Newton method. DETAILED DESCRIPTION OF THE INVENTION

[0020] Although the present invention can be modified in various ways and can have various embodiments, specific embodiments are illustrated in the drawings and described in detail in the detailed description, but it should be understood that this is not intended to limit the present invention to the specific embodiments, and that the present invention includes all modifications, equivalents, and alternatives that fall within the spirit and technical scope of the present invention.

[0021] The terms used in this specification are merely used to describe particular embodiments and are not intended to limit the present invention. The singular expressions include the plural expressions unless the context clearly dictates otherwise. In this specification, the terms "comprise" or "have" are intended to specify the presence of features, numbers, steps, operations, components, parts, or combinations thereof described in the specification, and should be understood not to preclude the possibility of the presence or addition of one or more other features, numbers, steps, operations, components, parts, or combinations thereof.

[0022] Furthermore, it goes without saying that the components of the embodiments described with reference to each drawing are not limited to being applied only to that embodiment, but may be embodied to be included in other embodiments within the scope that maintains the technical idea of ​​the present invention, and that even if separate description is omitted, multiple embodiments may be embodied again in a single integrated embodiment.

[0023] In addition, when describing the present invention with reference to the accompanying drawings, the same or related reference numerals are used for the same components regardless of the reference numerals, and redundant description thereof will be omitted. When describing the present invention, if it is determined that a detailed description of the related known technology would unnecessarily obscure the gist of the present invention, the detailed description thereof will be omitted.

[0024] FIG. 1 is a diagram showing the configuration of an SPH-based hyperelastic simulation device according to a preferred embodiment of the present invention. As shown in FIG. 1, the SPH-based hyperelastic simulation device includes a processor 100 and a memory 102.

[0025] The processor 100 includes a CPU (central processing unit) and a GPU (graphics processing unit) that execute computer programs, as well as a virtual machine. Memory 102 includes volatile storage devices such as fixed hard drives and removable storage devices, including CompactFlash units, USB memory sticks, etc. Memory 102 may also include volatile memory, such as various types of random access memory.

[0026] The memory 102 according to this embodiment stores program commands for simulating an SPH-based deformable body composed of m particles. These program instructions are stored on a computer-readable recording medium.

[0027] The program commands according to this embodiment define the state of each of the m particles at discrete time as a set of positions and velocities for an SPH-based deformable body composed of m particles, approximate the hyperelastic energy of each of the m particles using a rest-pose volume, material parameters, vectorized deformation gradients, and a projection of the vectorized deformation gradients in order to optimize an objective function for solving a new state of each of the m particles, search for an initial approximation of a Hessian matrix using the approximated hyperelastic energy, and simulate the SPH-based deformable body based on the searched initial approximation.

[0028] According to this embodiment, the objective function is optimized by an iterative approach using the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm.

[0029] The SPH-based hyperelastic simulation process according to this embodiment will now be described in detail. The SPH-based hyperelastic simulation according to this embodiment is based on deformation gradient and zero-energy mode suppression.

[0030] These will be explained in detail below. Elastic energy models (such as corotated, Neo-Hookean, and St. Venant-Kirchoff) are widely used for elastically deforming bodies.

[0031] JPEG0007749242000009.jpg14149

[0032] The deformation gradient for the existing SPH foundation deformation body is expressed as follows:

number

[0033]

number

number

number

[0034] Next we present the implicit penalty forces that suppress the zero energy modes previously proposed.

number

number

[0035] The optimization process is described in detail below. Given the state JPEG0007749242000019.jpg25140tn, the implicit Euler scheme is n+1 Approximate the next state of

number

number

[0036] To solve for the new state xn+1 in Equation 8, it is reformulated as an optimization problem to find xn+1 that minimizes the following objective function g(x):

number

number

[0037] Despite the fast convergence properties of Newton's method, the Hessian matrix per solver iteration This is less efficient because JPEG0007749242000026.jpg11128 must be calculated and factorized.

[0038] In this embodiment, the L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) algorithm is used. JPEG0007749242000027.jpg17143

[0039] Below we present an effective approximation of the matrix for this embodiment and demonstrate the efficiency of the optimization-based elastic SPH solid solver through experiments.

[0040] In the following, an initial approximation of the Hessian matrix is ​​described. The initial approximation of the Hessian, denoted by A0, has a significant effect on the convergence speed of L-BFGS. JPEG0007749242000028.jpg53143

number

[0041]

number

number

number

[0042] FIG. 2 is a diagram illustrating the L-BFGS algorithm based on A0 according to the present embodiment. Figure 3 shows the results of the solver tested on various elastic models.

[0043] Referring to FIG. 3, it can be seen that the model according to this embodiment better represents the torsion of the elastic beam than the corotated model on the left.

[0044] FIG. 4 is a diagram showing a simulation result according to the present embodiment and a simulation result according to the existing technique. As shown in Figure 4(a), when L-BFGS is run once, the simulation is somewhat unstable. However, as shown in Figure 4(b), when the number of L-BFGS runs is increased to two, the simulation quickly stabilizes. As shown in Figure 4(c), when a simulation is performed using the existing technique, it is unstable. In order to stabilize the simulation using the existing technique, as shown in Figure 4(d), the size of h must be reduced, which increases the complexity.

[0045] FIG. 5 shows the results of applying the L-BFGS algorithm according to this embodiment and the simulation results of the existing Newton method. From this, it can be seen that there is not much difference between the results of applying the L-BFGS algorithm according to this embodiment and the results of the existing Newton method.

[0046] The above-described embodiments of the present invention have been disclosed for illustrative purposes, and those skilled in the art will recognize that various modifications, changes, and additions may be made within the spirit and scope of the present invention, and such modifications, changes, and additions should be considered to fall within the scope of the following claims. [Explanation of symbols]

[0047] 100 processors 102 memory

Claims

1. A hyperelasticity simulation device based on SPH (Smoothed Particle Hydrodynamics), a processor; a memory coupled to the processor; The memory includes: For an SPH-based deformable object consisting of m particles, the state of each of the m particles at discrete time is defined as a position and velocity set, approximating a hyperelastic energy of each of the m particles using a rest-pose volume, material parameters, vectorized deformation gradients, and a projection of the vectorized deformation gradients to optimize an objective function for solving a new state of each of the m particles; Using the approximated hyperelastic energy, find an initial approximation of the Hessian matrix; simulating the SPH-based variant based on the searched initial approximation; storing program instructions to be executed by said processor; The program instruction optimizing the objective function by an iterative approach using the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm; The Hessian matrix is [Equation 15] and by searching the initial approximation 【number】 is changed to a constant approximation, where X is the particle position, h is the time step size, M is the mass matrix, and H is the zero-energy mode Hessian. Superelastic energy per particle 【number】 is approximated as follows: [0011] Here, V ｉ is the rest-pose volume, W ｉ is composed of material parameters such as Poisson's ratio and Young's modulus, and vec(·) is a 3x3 matrix converted into a 9x1 vector, Z ｉ teeth 【number】 of, [0016] and projected onto a manifold that satisfies the above. 【number】 is expressed in matrix-vector product form as follows: [0012] The constant approximation is defined as follows: [0013] Hessian matrix A ０ The initial approximation of is completed in B as shown in the following formula: [0014] An SPH-based hyperelastic simulation device characterized by:

2. 2. The SPH-based hyperelastic simulation device of claim 1, wherein the deformation gradient is defined by the following mathematical formula: [Equation 1] 【number】

3. 3. The SPH-based hyperelastic simulation device of claim 2, wherein the hyperelastic energy is defined by the following mathematical formula: [Equation 4] Here, Ψ represents an elastic energy density function, which varies depending on the type of elastic material of the deformable body.

4. 4. The SPH-based hyperelastic simulation device of claim 3, wherein the new state is expressed by the following equation: [Equation 7] 【number】

5. 5. The SPH-based hyperelastic simulation device according to claim 4, wherein the internal force is evaluated with a negative gradient, and the new state is expressed by the following equation: [Equation 8]

6. 6. The SPH-based hyperelastic simulation device according to claim 5, wherein the optimization problem of the objective function is reformulated by the following equation: [Equation 9]

7. A method for simulating hyperelasticity based on SPH (Smoothed Particle Hydrodynamics) in an apparatus including a processor and a memory, comprising: generating an SPH-based variant consisting of m particles; For an SPH-based deformation body composed of the m particles, a step of defining the states of each of the m particles at discrete time as a set of positions and velocities; approximating the hyperelastic energy of each of the m particles using a rest-pose volume, material parameters, vectorized deformation gradients, and a projection of the vectorized deformation gradients to optimize an objective function for solving a new state for each of the m particles; using the approximated hyperelastic energy to find an initial approximation of the Hessian matrix; and simulating a variant of the SPH-based model based on the initial approximation found; The objective function is optimized by an iterative approach using the L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) algorithm; The Hessian matrix is [Equation 15] and by searching the initial approximation 【number】 is changed to a constant approximation, where X is the particle position, h is the time step size, M is the mass matrix, and H is the zero-energy mode Hessian. Superelastic energy per particle 【number】 is approximated as follows: [0011] Here, V ｉ is the rest-pose volume, W ｉ is composed of material parameters such as Poisson's ratio and Young's modulus, and vec(·) is a 3x3 matrix converted into a 9x1 vector, Z ｉ teeth 【number】 of, [0016] and projected onto a manifold that satisfies the above. 【number】 is expressed in matrix-vector product form as follows: [0012] The constant approximation is defined as follows: [0013] Hessian matrix A ０ The initial approximation of is completed in B as shown in the following formula: [0014] An SPH-based hyperelastic simulation method.

8. A computer program stored on a computer-readable recording medium for performing the method of claim 7.

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