KR20230090208A - Energy-momentum conservation method and apparatus for real-time simulation of deformable objects - Google Patents

Abstract

The present invention discloses an energy-momentum conservation method and apparatus for real-time deformable body simulation. According to the present invention, an energy-momentum conservation device for real-time deformable body simulation, comprising: a processor; And a memory connected to the processor, wherein the memory receives the mass of the deformable body and the position of each vertex of the mesh constituting the deformable body at the current viewpoint, and searches the position of the next viewpoint of each vertex of the mesh. A constraint condition including the total linear momentum, total angular momentum, and total energy of the deformable body and a parameter for interpolating the energy due to the momentum of the deformable body and the total energy are introduced into the objective function for An energy-momentum conservation device is provided that stores program instructions executed by the processor so as to search for an approximate solution of the objective function through

Description

Energy-momentum conservation method and apparatus for real-time simulation of deformable objects

The present invention relates to a method and apparatus for conserving energy-momentum for real-time deformable body simulation.

As a conventional technology of real-time deformable body simulation, the technology of “Projective Dynamics: Fusing Constraint Projection for Fast Simulation” (hereinafter referred to as PD) presented at the international conference SIGGRAPH in 2014 is representative.

In the above paper, the numerical integration problem of the position and velocity of the deformable body is converted into an optimization problem, and the quasi-Newton method is repeatedly performed to find an approximate solution of the optimization problem.

Meanwhile, “FEPR: Fast Energy Projection for Real-Time Simulation of Deformable Objects” presented at SIGGRAPH, an international conference in 2018, is representative of the conventional energy-momentum conservation technique. This preserves the energy-momentum by repeatedly projecting the results (position and velocity) obtained through the random deformable body simulation onto the energy-momentum surface.

Physically, both the total energy and momentum of the deformable body must be conserved, but the above deformable body simulation technique (PD) has a problem in that energy and momentum rapidly decay as the simulation proceeds due to excessive approximation.

Accordingly, when the energy-momentum conservation technique (FEPR) is applied, energy is perfectly conserved, but momentum is not properly conserved. In addition, since this is an additional operation, it is a major factor in slowing down the simulation.

US registered patent 7,610,184

In order to solve the above problems of the prior art, the present invention is to propose a method and apparatus for preserving energy-momentum for real-time deformable body simulation that preserves all energy-momentum in deformable body simulation and significantly reduces the calculation process.

In order to achieve the above object, according to an embodiment of the present invention, an energy-momentum conservation device for real-time deformable body simulation, comprising: a processor; And a memory connected to the processor, wherein the memory receives the mass of the deformable body and the position of each vertex of the mesh constituting the deformable body at the current viewpoint, and searches the position of the next viewpoint of each vertex of the mesh. A constraint condition including the total linear momentum, total angular momentum, and total energy of the deformable body and a parameter for interpolating the energy due to the momentum of the deformable body and the total energy are introduced into the objective function for An energy-momentum conservation device is provided that stores program instructions executed by the processor so as to search for an approximate solution of the objective function through

The constraints include the velocity of the deformable body, and the velocity can be replaced by backward Euler approximation including the position of each vertex and the time step size.

The energy due to the momentum of the deformable body can be defined by the following equation.

[mathematical expression]

는 i번째 메쉬 꼭짓점의 질량, P는 총 선형 운동량, L은 총 각 운동량,

는 관성 텐서의 역수임here,

is the mass of the ith mesh vertex, P is the total linear momentum, L is the total angular momentum,

is the reciprocal of the inertia tensor

The constraint may be defined by the following equation.

[mathematical expression]

는

와

를 보간하기 위한 파라미터임here,

Is

and

is a parameter for interpolating

The objective function may be defined by the following equation.

[mathematical expression]

이고,

은 정규화 가중치임here,

ego,

is the regularization weight

The program instructions may search for the approximate solution through sequential quadratic programming (SQP).

및

인 경우 1에 가까운 값을 가질 수 있다. The above parameters are

and

may have a value close to 1.

According to another aspect of the present invention, as an energy-momentum conservation method for real-time simulation of a deformable body in a device including a processor and a memory, the mass of the deformable body and the position of each vertex of the mesh constituting the deformable body at the current time are received. step; To interpolate the constraint condition including the total linear momentum, total angular momentum, and total energy of the deformable body and the energy by the momentum of the deformable body and the total energy in the objective function for searching the position of the next time point of each vertex of the mesh introducing a parameter for; and searching for an approximate solution of the objective function through the local/global iteration.

According to another aspect of the present invention, a computer readable recording medium on which a program for performing the above method is recorded is provided.

According to the present invention, there is an advantage in that the amount of calculation can be reduced while more accurately preserving the energy-momentum.

In addition, according to the present invention, since the optimization problem of Projective Dynamics is converted into a constraint-optimization problem for energy-momentum, there is an advantage of accurately conserving energy and momentum in deformable body simulation.

1 is a diagram showing the configuration of an energy-momentum conservation device for real-time deformable body simulation according to a preferred embodiment of the present invention.
2 is a diagram showing deformable body simulation results of a technique according to the present embodiment and a conventional technique.
3 is a diagram showing total energy, elastic energy, and angular momentum over time.
4 is a diagram showing an energy graph according to the solver iteration (number of iterations) of the technique according to the present embodiment and the conventional technique.

Since the present invention can make various changes and have various embodiments, specific embodiments are illustrated in the drawings and described in detail.

However, this is not intended to limit the present invention to specific embodiments, and should be understood to include all modifications, equivalents, and substitutes included in the spirit and scope of the present invention.

The present invention uses the objective function derived from the backward Euler numerical integration method to find the position and velocity, which are the results of simulation of the deformable body of the next frame ( n +1th), using information at the current time of the deformable body. do.

As described above, although the conventional PD solver based on backward Euler is stable, it does not conserve momentum and total energy and has a problem in that excessive damping occurs. In this embodiment, the energy-momentum constraint is introduced to Solve the problem.

1 is a diagram showing the configuration of an energy-momentum conservation device for real-time deformable body simulation according to a preferred embodiment of the present invention.

Referring to FIG. 1 , the device according to the present embodiment may include a processor 100 and a memory 102 .

The processor 100 may include a central processing unit (CPU) capable of executing a computer program or other virtual machines.

Memory 102 may include a non-volatile storage device such as a non-removable hard drive or a removable storage device. The removable storage device may include a compact flash unit, a USB memory stick, and the like. Memory 102 may also include volatile memory, such as various random access memories.

The memory 102 stores program instructions for energy-momentum conservation for real-time deformable body simulation.

The program instructions according to the present embodiment receive the mass of the deformable body and the position of each vertex of the mesh constituting the deformable body at the current viewpoint, and convert the deformable body to an objective function for searching the position of the next viewpoint of each vertex of the mesh. A constraint condition including the total linear momentum, total angular momentum, and total energy of and parameters for interpolating the energy by the momentum of the deformable body and the total energy are introduced, and an approximate solution of the objective function is obtained through the local/global iteration. explore

Constraints according to this embodiment include the velocity of the deformable body, and the velocity is replaced by backward Euler approximation including the position of each vertex and the time step size. In addition, energy by the momentum of the deformable body is separately defined so that energy-momentum can be conserved in the deformable body simulation.

Hereinafter, the backward Euler numerical integration method will be first described, and then the energy-momentum conservation method for real-time deformable body simulation according to the present embodiment will be described in detail.

와 속도

의 집합으로 설명될 수 있으며, 변형체의 상태는 물리 법칙에 의해 시간이 지남에 따라 이산 시간 샘플

에서 점진적인 변화를 나타낸다. The state of a mesh (transform) composed of m vertices is the vertex position in 3D space.

with speed

can be described as a set of discrete time samples over time by the laws of physics

represents a gradual change in

의 상태가 주어지면, 오일러 수칙 적분은 다음과 같이

의 위치 및 속도를 포함하는 다음 상태를 근사적으로 구할 수 있다.

Given the state of , the Euler's Law integral is

The following state including the position and velocity of can be approximated.

는 외부 힘,

는 내부 힘으로서

이고, 여기서

는 j번째 유한 요소의 위치 에너지를 나타낸다. where h is the time step size, M is the mass matrix,

is the external force,

is the internal force

and here

represents the potential energy of the j-th finite element.

Equation 1 is expressed as follows.

이다. here,

am.

를 찾는 방법은 다음의 목적함수

를 최소화하는

를 찾는 최적화 문제로 변환될 수 있다. In Equation 2, the new state

The way to find is the following objective function

to minimize

can be transformed into an optimization problem to find

대신 x를 사용한다. In order to simplify the notation below

use x instead.

와 2차 거리 측정을 위해 사용하는 아래와 같은 잠재적 함수

를 선택하는데 있다. One of the novelties of the PD technique is the auxiliary projection variable

and the latent function used to measure the second order distance:

is to choose

는 양의 가중치이고,

는 이산 미분 연산자에 대한 희소 행렬이다.

is a positive weight,

is a sparse matrix for the discrete differential operator.

를 수학식 4로 대체하면 다음과 같은 증강된 목적함수

를 얻을 수 있다. of Equation 3

Replacing with Equation 4, the following augmented objective function

can be obtained.

The first term of Equation 5 acts as an inertia that moves x toward y, and the second term acts as an elasticity that penalizes the deformation of x.

(초기화)이고,

및

는 로컬 및 글로벌 단계에서 최소화된다. The PD technique uses an iterative shift optimization technique, where:

(initialization),

and

is minimized in the local and global steps.

는 고정되고,

는 풀어진다. at the local level

is fixed,

is released

는 요소 유형(예를 들어, 사면체 또는 스프링)에 따라 정의되는 구속조건 집합체(constraint manifold)이다. here,

is a constraint manifold defined according to the element type (e.g. tetrahedron or spring).

가 고정되고 수학식 5의 목적함수가 최소화되도록

가 풀어진다. at the global level

is fixed and the objective function of Equation 5 is minimized

is released

This can be expressed as a system of linear equations:

이다. here,

am.

은 Symmetric positive-definite이고, 로컬/글로벌 반복 동안 일정하게 유지된다. system matrix

is symmetric positive-definite and remains constant during local/global iterations.

에 대한 최적화 문제로 다시 나타낼 수 있다. Since the PD technique is a quasi-Newton method that minimizes Equation 5 using an approximate Hessian matrix, a single iteration of the local/global step in the PD technique is as follows

It can be re-expressed as an optimization problem for

이다. here,

am.

는 찾는 것은 다음 선행 시스템을 푸는 것과 같다.

is equivalent to solving the following preceding system.

계산에는 로컬 단계가 포함되며, 수학식 9의 해로

를 업데이트하는 것은 글로벌 단계와 같다.

The calculation includes a local step, as a solution of Equation 9.

Updating the is the same as the global step.

), 총 각 운동량(total angular momentum,

) 및 총 에너지(

)의 3가지의 중요한 물리량을 고려한다. In this embodiment, the energy-momentum conservation condition (constraint condition) is introduced to solve the excessive damping of the PD technique, and the total linear momentum,

), total angular momentum,

) and total energy (

) considers three important physical quantities.

In the mesh configuration, the above physical quantities are defined as follows.

,

및

는 각각 i번째 꼭짓점의 질량, 속도 및 위치를 나타낸다. here,

,

and

represent the mass, velocity and position of the ith vertex, respectively.

로 v를 대체하면 운동량과 에너지는 다음과 같이 정의된다. According to this embodiment, inverse Euler approximation to reduce the complexity of Equation 10

Substituting v with , the momentum and energy are defined as:

가 에너지와 운동량을 보존하도록 최적화 문제의 해를 제한하는 것을 목표로 한다. In this embodiment, the new state

We aim to constrain the solution of the optimization problem so that is conserves energy and momentum.

,

및

를

에서 보존해야 할 물리량이라 하고,

인 닫힌 시스템에서

및

이다.

,

and

cast

is the physical quantity to be conserved in

In a closed system where

and

am.

However, in general, each physical quantity should be updated with an approximate amount of energy transferred when considering an external force such as gravity.

However, since Equation 12 does not consider the dependence between energy and momentum, a situation in which energy and momentum are incompatible may occur.

및

와 같은 상황을 가정하면,

는 0이 아닌 운동 에너지를 반환하지만, 0이어야 하는

과 양립하지 못한다. 즉, 모든 구속조건을 만족하는 해가 존재하지 않는 문제가 있을 수 있다. for example,

and

Assuming a situation like

returns non-zero kinetic energy, but

is incompatible with That is, there may be a problem in which there is no solution that satisfies all constraints.

To this end, according to this embodiment, energy by momentum is defined as follows.

는 관성 텐서의 역수를 나타내며,

를 사용하여 근사화된다.

represents the reciprocal of the inertia tensor,

is approximated using

Then the constraints are defined as:

는

와

를 보간하고, 이에 따라 본 실시예에 따른 최적화 문제는 다음과 같이 정의된다. here,

Is

and

interpolates , and accordingly, the optimization problem according to the present embodiment is defined as follows.

는 수학식 5에서 정의된 것이고,

은 정규화 가중치이다. here,

Is defined in Equation 5,

is the normalization weight.

는 상기한 바와 같이

와

를 보간하는 최적의

를 찾기 위해 사용된다. According to this embodiment, an approximate solution is found through SQP (Sequential Quadratic Programming), an iterative technique, and the

As mentioned above

and

Optimal to interpolate

is used to find

는 최적화를 통해 구해지고,

및

이 서로 양립할 수 있을 때,

는 0이 되고 K는 버려진다. 그렇지 않으면

는 에너지가 운동량과 양립할 수 있음을 보장한다.

is obtained through optimization,

and

When these are compatible with each other,

becomes 0 and K is discarded. Otherwise

guarantees that energy is compatible with momentum.

및

에서

는 1에 가깝게 된다. above mentioned family

and

at

becomes close to 1.

2 is a diagram showing simulation results of a deformable body of a technique according to the present embodiment and a conventional technique, and FIG. 3 is a graph showing total energy, elastic energy, and angular momentum over time.

2 and 3 show simulations of elasticity and rotational motion of the deformable body, and as shown in FIGS. 2 and 3, it can be confirmed that the technique according to the present embodiment preserves energy-momentum well compared to the prior art. .

4 is a diagram showing an energy graph according to the solver iteration (number of iterations) of the technique according to the present embodiment and the conventional technique.

It can be seen that the technique according to the present embodiment uses a significantly smaller number of iterations than the prior art.

The embodiments of the present invention described above have been disclosed for illustrative purposes, and those skilled in the art having ordinary knowledge of the present invention will be able to make various modifications, changes, and additions within the spirit and scope of the present invention, and such modifications, changes, and additions will be considered to fall within the scope of the following claims.

Claims (10)

As an energy-momentum conservation device for real-time deformable body simulation,
processor; and
Including a memory coupled to the processor,
the memory,
Receiving the mass of the deformable body and the position of each vertex of the mesh constituting the deformable body at the current time point,
To interpolate the constraint condition including the total linear momentum, total angular momentum, and total energy of the deformable body and the energy by the momentum of the deformable body and the total energy in the objective function for searching the position of the next time point of each vertex of the mesh Introduce parameters for
To search for an approximate solution of the objective function through the local / global iteration,
An energy-momentum conservation device for storing program instructions executed by the processor. According to claim 1,
The constraint condition includes the velocity of the deformable body, and the velocity is replaced by an inverse Euler approximation including a position of each vertex and a time step size. According to claim 1,
An energy-momentum conservation device in which the energy by the momentum of the deformable body is defined by the following equation.
[mathematical expression]

here,

is the mass of the ith mesh vertex, P is the total linear momentum, L is the total angular momentum,

is the reciprocal of the inertia tensor According to claim 3,
The constraint is an energy-momentum conservation device defined by the following equation.
[mathematical expression]

here,

Is

and

is a parameter for interpolating According to claim 4,
The objective function is an energy-momentum conservation device defined by the following equation.
[mathematical expression]

here,

ego,

is the regularization weight According to claim 5,
The program commands search for the approximate solution through SQP (Sequential Quadratic Programming). According to claim 6,
The above parameters are

and

An energy-momentum conservation device with a value close to 1 for . An energy-momentum conservation method for real-time deformation body simulation in a device including a processor and a memory,
receiving a mass of the deformable body and a position of each vertex of a mesh constituting the deformable body at a current viewpoint;
To interpolate the constraint condition including the total linear momentum, total angular momentum, and total energy of the deformable body and the energy by the momentum of the deformable body and the total energy in the objective function for searching the position of the next time point of each vertex of the mesh introducing a parameter for; and
An energy-momentum conservation method comprising the step of searching for an approximate solution of the objective function through the local / global iteration. According to claim 8,
The constraint condition includes the velocity of the deformable body, and the velocity is replaced by an inverse Euler approximation including a position of each vertex and a time step size. A computer-readable recording medium on which a program for performing the method according to claim 8 is recorded.

Images