KR101350732B1 - Multi-Resolution Meshless Method for Real-Time Simulation of Deformable Objects - Google Patents

Abstract

According to the present invention, when the geometric information of the object to be simulated is given, the deformation inside the deformation body that does not significantly affect the simulation even when the accuracy of the deformation decreases is calculated as a low resolution node, and the surface requiring accurate calculation is considered in consideration of this. This paper deals with the deformation simulation method using the multi-resolution elementless method, which is calculated using high resolution nodes.

Description

Multi-Resolution Meshless Method for Real-Time Simulation of Deformable Objects

The present invention relates to a method for simulating a deformable body, and more particularly, a method of uniformly generating nodes inside an object when given geometric information of a deformable body to be deformed, and using a node generated by the method. It is about how to simulate the behavior in real time.

More specifically, the deformation inside the deformable body, which does not significantly affect the simulation even if the deformation accuracy is inferior, is calculated as the low resolution node, and the deformation at the surface requiring accurate calculation of the deformation is considered as the high resolution node. The present invention relates to a deformation simulation method using a multi-resolution elementless method.

In order to solve a complex problem, it is easier to approach the solution of the problem by creating an object that is the same as or similar to the object in question and creating a situation that causes the problem. In order to solve such problem experiments or social phenomena, simulation is to repeatedly simulate the situation and grasp the characteristics of the object. Simulation is widely used in the medical field as well as in the social sciences to analyze social phenomena.

Medical simulations refer to realistic situations that can occur in real medical settings and allow trainees to practice and train in this simulation environment. In this simulation environment, there is no risk of harming the actual patient's health, and various cases can be simulated to create a systematic and standardized education curriculum. In addition, the trainee can train as many times as necessary until the skill is fully learned without the burden of dealing with the patient directly. One of the key technologies for implementing such a medical simulation is a technique for simulating the behavior of the deformable body. Simulating the behavior of organs deformed by the trainee's operation during the simulation is one of the most important parts of the medical simulation.

An important aspect of this variant simulation is that the model's response to the user's input must be in real time, and the accuracy of the calculations that are visually reasonable must be guaranteed. In medical simulations, the information delivered to the trainer must be delivered quickly and accurately enough to recognize the deformation and reaction forces of the organs in real time, and the organs must respond accurately to user input. To this end, a model that considers the physical properties of the deformable body, boundary conditions, external forces, etc. is mainly used, and the boundary element method, finite element method, etc. are used as a simulation method.

Non-patent literatures 1-7 are mentioned as such a deformation | transformation simulation method, and patent documents 1, 2, etc. are mentioned. They are an element-based simulation method, which has the advantage of simulating the behavior of the deformable body relatively accurately and stably, but has the disadvantage of generating a uniform element for the object to be deformed. Creating a uniform element for a deformable body having an arbitrary shape requires a lot of computation time and is a difficult task that cannot be automatically generated when the shape of an object is complicated. In addition, when an object is broken or cut during the simulation, there is a problem in that a new element needs to be created, which causes problems in various fields.

In order to improve the problem of the method of using such an element, the elementless method of simulating the behavior of the deformable body without using the element has been developed, and examples thereof include non-patent documents 6 and 7. Since the element-free method does not require elements, it is relatively easy to simulate deformations only with the geometric information of the object to be simulated, but there is a problem that real-time simulation is difficult due to the large amount of computation required to simulate deformations.

1. Republic of Korea Patent Registration No. 1085174 2. Korea Patent Registration No. 1030732

Jeon, Seong-ki et al., The Korean Society for Mechanical Engineering 2004 Spring Conference pp.557-562, 2004. Bro-Nielsen, Morten, Finite Element Modeling in Surgery Simulation, Proceeding of Medicine Meets Virtual Reality 5 (MMVR-5 '97), 1997. 3. DOUG L. JAMES, et al., "Multiresolution Green ''s Function Methods for Interactive Simulation of Large-Scale Elastostatic Objects", ACM Transactions on Graphics, Vol. 22, no. 1, Pages 47-82, January 2003. M. Muller, et al., "Point Based Animation of Elastic, Plastic and Melting Objects", Eurographics / ACM SIGGRAPH Symposium on Computer Animation, 2004. 5. Markus Becker, et al., "Corotated SPH for deformable solids", Eurographics Workshop on Natural Phenomena, 2009. 6. FRIES T.-P., et al., "Classification and Overview of Meshfree Methods" Tech. rep., TU Brunswick, Germany Nr. March, 2003. 7. Lim, Y.J., et al., "On the use of meshfree methods and a geometry based surgical cutting in multimodal medical simulations." HAPTICS, pp. 295-301. IEEE Computer Society, 2004

The present invention was developed to solve the problems of the prior art as described above, and an object of the present invention is to provide a method for generating a deformable body using the elementless method and a simulation method using the same.

In particular, it is an object of the present invention to provide a method capable of simulating the behavior of a deformable body in real time using the elementless method.

In more detail, the present invention provides a method for uniformly generating nodes inside a deformation body given the geometric information of the deformation object to be simulated, and the deformation inside the deformation body that does not significantly affect the simulation even if the deformation accuracy is low. It is an object of the present invention to provide a deformation simulation method using a multi-resolution elementless method that calculates a node with a high resolution node.

The method for generating nodes of a deformable body for simulation according to the present invention for solving the above problem comprises the steps of: generating a bounding box surrounding the deformable body from the geometric shape information of a given deformable body; Dividing any one of three reference sides of the generated bounding box in x, y, and z directions at a resolution specified by a user; Dividing the remaining sides based on the resolution obtained by dividing the reference sides; Creating a node at a center point of each divided region; And removing nodes outside the deformed body among the generated nodes.

The preferred reference side is the shortest of the three sides.

According to another aspect, a simulation method of a deformation body using the multiresolution elementless method is characterized by simulating deformation of a deformation body from displacement in at least one low resolution node array having a relatively low resolution and at least one high resolution node array having a relatively high resolution. It is done.

A method for simulating a deformable body using such a multi-resolution elementless method, the method of generating a node comprising: generating a bounding box surrounding a deformable body from geometric shape information of a given deformable body; Dividing any one of three sides of the generated bounding box in the x, y, and z directions by a user-specified resolution; Dividing the remaining sides based on the resolution obtained by dividing the reference sides; Creating a node at a center point of each divided region; And removing nodes that are outside of the deformable body among the generated nodes.

In the process of calculating the displacement of the node, the low resolution node is used as it is, and the high resolution node is divided into an active node near the surface of the deformable body and a passive node inside the deformable object. It is preferable to calculate the displacement of the high resolution passive node by using and to calculate the displacement of the active node of the high resolution node.

에 의해 구해지고, 고해상도 절점 중 패시브 절점에서의 변위 U _p 는

에 의해 구해지며, 고해상도 절점 중 액티브 절점에서의 변위 U _h 는

에 의해 구해질 수 있다. Displacement U _s at low resolution node is

Determined by, the displacement U _p at the passive node

Obtained by, the displacement U _h of the active node

Can be obtained by

In addition, in the step of dividing the reference side to a specified resolution, it is preferable to select the side that is the shortest among the three sides.

The present invention can reduce the time required to generate the element by not using the element, and there is an effect that can be easily simulated even if the deformed body is broken or cut during the simulation.

Deformation inside the deformable body, which does not significantly affect the simulation even when the accuracy of deformation is poor, can be calculated with low resolution nodes and high resolution nodes on surfaces that require accurate deformation calculations to reduce the simulation time. Even if a fracture occurs in the deformable body during the simulation, there is an effect that can easily enable a new simulation.

1 is a perspective view of a step of forming a bounding box in the node generation process of the deformable body for simulation according to the present invention
2 is a node generation process diagram of the deformable body for simulation according to the present invention;
3 is a position diagram of nodes used in the simulation method according to the present invention;
Figure 4 is an illustration of a multiresolution elemental deformation model in the simulation method according to the present invention

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings. In the drawings of the present invention, the sizes and dimensions of the structures are enlarged or reduced from the actual size in order to clarify the present invention, and the known structures are omitted so as to reveal the characteristic features, and the present invention is not limited to the drawings . DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings. In the following description, well-known functions or constructions are not described in detail to avoid obscuring the subject matter of the present invention.

According to the present invention, it is possible to simulate deformation of a deformable body by solving the problem of the element generation process and the breakage of the deformable body that occur in the element-based simulation method by using the element. As a result, the method can be divided into a method for generating a node necessary for calculating the deformation of the deformation in the simulation and a method for simulating the deformation of the deformation from the generated node. First, a method for generating the node is described.

In the method for generating a node of a deformable body for simulation according to the present invention, a step of forming a bounding box for forming a frame for recognizing the deformable body, dividing the deformable body to a desired resolution, and forming one of the points inside the divided region as a node And removing the nodes outside the deformable body.

The bounding box can be set in the region of information surrounding the deformable body from geometrical shape information of a given deformable body, as shown in FIG.

The dividing of the deformable body may be performed by dividing one reference side of three sides in the x, y, and z directions of the generated bounding box to a resolution specified by a user, and then dividing the remaining sides based on the resolution obtained by dividing the reference side. It can be made by the method.

In the step of dividing into the desired resolution, the reference side is preferably set to the reference side, which is the shortest of the three sides (x side on the drawing). If you specify the resolution by setting the longest side of the bounding box as the reference side, you will not get enough resolution when dividing the shorter side, so you will not get enough resolution. It is preferable to divide.

The method of setting the node inside the divided region is sufficient for any one point inside the region, but if the nodes set in the divided region are different from each other, the distance between neighboring nodes is not constant and accurate simulation cannot be achieved. It is preferable to set the position of the node to a constant point of each divided area, and to set the location of the node more easily, the center point of the divided area is set to the node. (See Figure 2 (d))

After the node is set, the node located at the part out of the deformable body is removed as shown in FIG.

The method of simulating deformation of a deformable body according to the invention is made by a combination of information obtained from nodes of different resolutions.

That is, the deformation of the deformable body is simulated from displacements in at least one low resolution node array having a relatively low resolution and at least one high resolution node array having a relatively high resolution.

Since the method for creating the node has already been described above, a detailed description thereof will be omitted.

When an external force is applied outside the deformable body, the displacement at each node can be obtained by solving the differential equation

(1)

(One)

질량행렬을 나타내고, n은 변형체 내부의 절점의 수를 나타낸다. C 는

댐핑행렬, K 는

강성행렬을 나타낸다. 그리고 u 는 각 절점에서의 변위를 나타내는

변위벡터이며, f 는 각 절점에서의 힘을 나타내는

힘벡터이다. 변형체의 시뮬레이션에서는 식 (1)을 풀어서 매 스텝마다 변위를 업데이트함으로써 연속적으로 변형되는 모습을 구현하게 된다. In Equation (1) above, M is

The mass matrix is shown, and n is the number of nodes inside the deformable body. C is

Damping matrix, K is

It shows the stiffness matrix. And u represents the displacement at each node

Displacement vector, f is the force at each node

Power vector. In the simulation of the deformable body, equation (1) is solved and the displacement is updated at every step to realize the continuous deforming mode.

When nodes are created inside the deformable body in the same manner as described above, nodes of uniform distribution are created not only on the surface of the deformable body but also inside the deformable body. Since the real-time simulation of the deformable body only concerns the deformation at the deformable surface, only the nodes of the densely distributed distribution are effective near the deformable surface, and the debris inside the deformable body do not need dense nodes. Therefore, in the present invention, in order to improve the increase in the calculation amount of the simulation as the number of nodes is increased, a deformation-free simulation method based on the elementless method using multiresolution nodes which reduces the calculation amount of the simulation is developed.

In the elementless method-based deformation simulation method of the present invention, the deformation calculation inside the deformation that does not require accurate calculation is performed quickly using low resolution nodes, and the calculation near the surface requiring accurate calculation is accurate using high resolution nodes. . By calculating the strain in this way, the overall calculation can be reduced while maintaining the accuracy of the strain near the surface.

In order to distinguish between the part requiring accurate calculation and the part requiring rough calculation, an area at a distance from the surface of the deformable body was defined as the surface layer. The surface layer to be defined may vary depending on the type of deformation.

After defining the surface layer, displacements in the surface layer are calculated using high resolution nodes, and displacements in areas other than the surface layer are calculated using low resolution nodes. Thus, in order to calculate the displacement in each area by dividing the deformable body into two areas, as shown in FIG. First, nodes belonging to the surface layer region were defined as active nodes and nodes belonging to regions other than the surface layer as passive nodes.

The displacement at the active node is accurately calculated by solving the equation of Eq. (1) that reflects the physical properties of the actual deformation, and the displacement at the passive node is quickly calculated by approximating the displacement calculated at the low resolution node.

The displacement U _s of an object at low resolution nodes can be obtained by the differential equation below.

(2)

(2)

행렬이다. N _s 은 저해상도 절점의 수, f _s 는 저해상도 절점에서의 힘을 나타내는

벡터이다. 위의 식 (2)를 계산하여 구한 저해상도 절점에서의 변위 U _s 는 고해상도 절점 중 패시브 절점에서의 변위를 계산할 때 사용된다.Where M _s is the mass matrix of the low resolution nodes, C _s is the damping matrix of the low resolution nodes, and K _s is the stiffness matrix of the low resolution nodes,

It is a matrix. N _s is the number of low resolution nodes, f _s is the force at the low resolution nodes

It is a vector. The displacement U _s at the low resolution node obtained by calculating Equation (2) above is used to calculate the displacement at the passive node among the high resolution nodes.

Next, the displacement at the passive node among the high resolution nodes is calculated. The displacement at the passive node is calculated by approximating the displacement of the low resolution node. That is, since the displacement at the low resolution node is calculated, the displacement at the passive node can be quickly calculated by interpolating the displacement of the low resolution node around each passive node.

The displacement at the passive node can be calculated using Equation (3) below.

(3)

(3)

백터, U _p 는 패시브 절점에서의 변위를 나타내는

벡터이다. 여기서 n _s , n _p 는 각각 저해상도 절점과 패시브 절점의 수를 나타낸다. T는

행렬이고, 저해상도 절점의 변위로부터 패시브 절점의 변위를 계산하는 맵핑행렬을 나타낸다.In Equation (3) above, U _s represents the displacement at the low resolution node.

Vector, U _p represents the displacement at the passive node

It is a vector. Where n _s and n _p represent the number of low resolution nodes and passive nodes, respectively. T is

It is a matrix and represents a mapping matrix for calculating the displacement of the passive node from the displacement of the low resolution node.

The next step is to calculate the displacement at the active node among the high resolution nodes. The differential equation for calculating the displacement at the full high resolution node is

(4)

(4)

질량행렬, C _h 는 고해상도 절점의 댐핑을 나타내는

뎀핑행렬, K _h 는 고해상도 절점의 강성을 나타내는

강성행렬로 n _h 는 전체 고해상도 절점의 수를 나타내고, f _h 는 고해상도 절점에서의 힘을 나타내는

힘 벡터이다. 앞에서 고해상도 절점을 패시브 절점과 액티브 절점으로 구분하였는데 식 (4) 역시 절점의 종류에 따라서 아래의 수식 (5)와 같이 분리될 수 있다.In equation (4) above, M _h represents the mass of the high resolution node.

Mass matrix, C _h , represents damping of high resolution nodes

The damping matrix, K _h , represents the stiffness of the high resolution node

In the stiffness matrix, n _h represents the total number of high resolution nodes and f _h represents the force at the high resolution nodes.

Force vector. The high resolution nodes are divided into passive nodes and active nodes. Equation (4) may be separated according to the type of nodes as shown in Equation (5) below.

(5)

(5)

벡터, f _a 는 액티브 절점에서의 힘 벡터를 나타내는

벡터이다. 여기서 n _a 는 액티브 절점의 수를 나타낸다. 액티브 절점에서의 변위는 식 (5) 전체를 풀어서 계산하는 것이 아니라 액티브 절점에 해당하는 부분만 독립적으로 풀어서 계산할 수 있다. 패시브 절점에서의 변위 (u _p )를 계산하였으므로, 액티브 절점에서의 변위 u _a 는 식 (5)에서 액티브 절점에 해당하는 부분만 풀어서 계산할 수 있다. Equation (5) is expressed by dividing equation (4) into parts corresponding to active nodes and passive nodes. In equation (5) u _a represents the displacement at the active node

Vector, f _a represents the force vector at the active node

It is a vector. Where n _a represents the number of active nodes. The displacement at the active node can be calculated by solving only the part corresponding to the active node independently, rather than solving the entire equation (5). Since the displacement u _p at the passive node is calculated, the displacement u _a at the active node can be calculated by solving only the portion corresponding to the active node in equation (5).

In the simulation method of the deformable body using the multi-resolution elementless method according to the present invention by the above process, the deformation inside the object that does not significantly affect the simulation even if the accuracy of the deformation is inferior, it is necessary to calculate the low-definition node and calculate the accurate deformation. On the surface, we can calculate the high resolution nodes and reduce the simulation time.

Claims (11)

Generating a bounding box surrounding the deformable body from the geometric shape information of the given deformable body;
Dividing the bounding box at regular intervals;
Creating a node at a point in each divided region; And
Node generation method for the deformation simulation, characterized in that it comprises the step of removing the nodes outside of the deformation of the generated nodes The method of claim 1,
Splitting the bounding box
Deformation simulation, characterized in that any one of the three sides in the x, y, z direction of the generated bounding box is divided at a resolution specified by the user, and then the remaining sides are divided based on the resolution at which the reference side is divided. Node creation method 3. The method of claim 2,
In the step of dividing the reference side to a specified resolution, the reference side is the node generation method for the deformation simulation, characterized in that the shortest side of the three sides The method of claim 1,
The node is a node generation method for the deformation simulation, characterized in that the center point in each divided region In the simulation method of the deformed body,
The interior of the variant is calculated as an array of at least one low resolution node that is relatively low resolution,
Deformation method of the deformable body using the multi-resolution elementless method, characterized in that the surface of the deformable body simulates the deformation of the deformable body from the displacement calculated with at least one high-resolution node arrangement. The method of claim 5,
How to create a node
Generating a bounding box surrounding the deformable body from the geometric shape information of the given deformable body;
Dividing any one of three sides of the generated bounding box in x, y, and z directions by a user-specified resolution;
Dividing the remaining sides based on the resolution obtained by dividing the reference sides;
Creating a node at a center point of each divided region; And
Simulation method of the deformable body using the multi-resolution elementless method characterized in that it comprises the step of removing the nodes that are outside of the deformity among the generated nodes The method of claim 5,
The low resolution node is used as it is,
The high resolution node is divided into an active node near the surface of the deformable body and a passive node inside the deformable body,
Using the deformation at the low resolution node to calculate the displacement of the high resolution passive node,
Simulation method of the deformable body using the multi-resolution elementless method characterized in that the displacement of the active node of the high resolution node The method of claim 7, wherein
The displacement u _s at the low resolution node is

Simulation method of the deformable body using the multi-resolution elementless method characterized in that obtained by
Where M _s is the mass matrix of the low resolution node, C _h Is the damping matrix of the low resolution nodes, K _h is the stiffness matrix of the low resolution nodes, and

Where n _s is the number of low resolution nodes, f _s is the force vector at the low resolution nodes) The method of claim 7, wherein
The displacement u _p at the passive node among the high resolution nodes is

Simulation method of the deformable body using the multi-resolution elementless method characterized in that obtained by
(Where u _s is the displacement at low resolution nodes, u _p is the displacement at passive nodes, u _s is

Vector u _p The

Vector, n _s is the number of low resolution nodes, n _p is the number of passive nodes, T is

Mapping matrix) The method of claim 7, wherein
The displacement u _h at the active node among the high resolution nodes is a differential equation

Simulation method of the deformable body using the multi-resolution elementless method characterized in that
Where M _h is the mass matrix of the high resolution node, C _h is the damping matrix of the high resolution node, and K _h is the stiffness matrix of the high resolution node,

Where n _h is the total number of high resolution nodes and f _h is the force vector at the high resolution nodes) The method according to claim 6,
In the resolution division step, the reference side is the shortest of the three sides, and the simulation method of the deformable body using the multi-resolution elementless method

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