JP5546339B2 - Analysis method and analysis apparatus - Google Patents

Description

  The present invention relates to an analysis method and an analysis apparatus for analyzing a system.

  An implicit method using the finite element method is widely used to know in what state a system to which a constant force or torque is input converges to a balanced position. However, the convergence is extremely poor particularly when a large number of contacts are involved, and it is not uncommon for the calculation to fail especially when the object is accompanied by a rigid body motion.

Seiichi Koshizuka, “Development of an implicit particle calculation model for elastic bodies”, Proceedings of the Lecture on Computational Engineering, 1999, 4, p. 37-40

  Therefore, in recent years, analysis by the explicit method of the particle method (for example, see Non-Patent Document 1) has begun to be performed, but vibration remains due to the energy conservation law as it is.

  The present invention has been made in view of these problems, and an object thereof is to provide a technique for efficiently dissipating energy and obtaining a steady state of a system quickly in an explicit solution.

  One embodiment of the present invention relates to an analysis method. This analysis method is an analysis method for analyzing a system by numerically calculating a motion equation of a particle for a system including a large number of particles, and is a non-conservative force acting on the particles and dissipating the energy of the system. The step of acquiring the non-conservative force obtained on the premise, the step of numerically calculating the motion equation of the particle including the acquired non-conservative force, and whether or not the system has reached a steady state based on the calculation result Determining.

  According to this aspect, the energy of the system can be efficiently dissipated by considering the non-conservative force obtained on the premise of dissipating the energy of the system.

  It should be noted that any combination of the above-described constituent elements, or those obtained by replacing the constituent elements and expressions of the present invention with each other between apparatuses, methods, systems, computer programs, recording media storing computer programs, and the like are also included in the present invention. It is effective as an embodiment of

  According to the present invention, energy can be efficiently dissipated in an explicit solution.

It is a flowchart for demonstrating the analysis process which concerns on embodiment. It is a block diagram which shows the function and structure of an analyzer. FIGS. 3A and 3B are explanatory diagrams illustrating an application example of the analysis method according to the embodiment to mechanism analysis. It is a graph which shows attenuation | damping of a kinetic energy by the case where various energy dissipation processes are taken, and the case where it is not so.

  In the analysis by the explicit method using the particle method, vibration remains, and it is necessary to remove energy by some method in order to obtain a steady state. At this time, for example, a method of zeroing the kinetic energy at certain intervals may be considered, but depending on the problem, it may not be appropriate.

  In general finite element methods, the implicit method is the mainstream, and the explicit method is often used mainly for unsteady dynamic analysis. Recently, a method called a static explicit method has also been proposed, which is a method of developing time while searching for a balance position of forces. From these facts, it can be said that the finite element method has no idea of dissipating energy.

  The particle method is solved mainly by the explicit method, but there are no ideas to dissipate energy because the purpose of analysis is mostly dynamic such as fracture and friction. There is also an example in which the bending position of the cantilever beam is analyzed (for example, see Non-Patent Document 1), but in this case, an implicit algorithm is used.

  In DEM (Distinct Element Method), the damping coefficient at the time of collision is determined empirically, but this simulates the coefficient of restitution and is not used for the purpose of obtaining a steady state.

  The present invention will be described below based on preferred embodiments with reference to the drawings. The same or equivalent components, members, and processes shown in the drawings are denoted by the same reference numerals, and repeated descriptions are omitted as appropriate.

  In the analysis method according to the embodiment, in the explicit method of the particle method, the steady state of the system is obtained in consideration of the energy dissipation of the system to be analyzed. Thereby, the steady state of the system can be obtained more quickly.

FIG. 1 is a flowchart for explaining an analysis process according to the embodiment. The process shown in FIG. 1 is executed by the analysis apparatus 10 (see FIG. 2) according to the embodiment. First, the analysis apparatus 10 generates an analysis model (S102). The analysis apparatus 10 may create an analysis model based on input information from the user, or may acquire a model by reading an analysis model that has already been created. This analysis model is a particle model suitable for the molecular dynamics method, and is configured as a particle system S composed of N particles. The particles may be atoms or molecules.
In the particle system S, each of the N particles is initially arranged, and an initial velocity is given to each of the N particles.

  The analysis apparatus 10 acquires the preserving force that acts on each particle of the particle system S (S104). The analysis device 10 may acquire a preserving force acting on the particles based on input information from the user. Alternatively, the analysis device 10 calculates and obtains a preservative force acting on the particles based on a potential energy function (hereinafter, also referred to as a potential function or simply a potential as appropriate) representing an interaction between particles in the particle system S. May be.

  The analysis apparatus 10 acquires a non-conservative force that is a non-conservative force acting on the particles and is obtained on the premise that the energy of the particle system S is dissipated (S106). In step S106 for obtaining the non-conservative force, the analysis apparatus 10 attenuates the dissipative force obtained on the assumption that the internal vibration of the particle system S is attenuated and the translational motion of the particle system S among the non-conservative forces acting on the particles. And the viscous force obtained on the premise of performing (S108, S110). When it is assumed that the particles i and j in the particle system S are coupled with the spring constant k, the dissipative force exerted on one of the particles i and j on the other is calculated from the following Equation 1, The viscous force acting on the particle i is calculated from Equation 2 below.

Where m _i is the mass of particle i, m _j is the mass of particle j, v _i is the velocity of particle i, v _j is the velocity of particle j, t is time, v _r is the relative velocity of particles i and j,
Is the relative velocity of the particles i and j at (n-1 / 2) dt, where n is an arbitrary integer, and M is the reduced mass in the relative motion of the particles i and j.

However,
Is the velocity of the particle i at (n-1 / 2) dt, where n is an arbitrary integer, and τ is a time constant determined by the magnitude of the viscous resistance.
As understood from Equations 1 and 2, the dissipative force and the viscous force are forces depending on the velocity of the particles.

The analysis device 10 adds the dissipative force, the viscous force, and the storage force for each particle, and obtains a total force (hereinafter referred to as a combined force) acting on the particle (S112).
The analysis apparatus 10 calculates at least one of the position and velocity of the particle by numerically solving the obtained equation of motion of the particle including the total force. The analysis apparatus 10 moves the particles based on the calculation result (S114).

  The analysis apparatus 10 determines whether or not the particle system S has reached a steady state based on the calculation result of at least one of the particle position and velocity (S116). For example, when the total kinetic energy of the particle system S is smaller than a predetermined lower limit value, the analysis apparatus 10 determines that the particle system S has reached a steady state. When it is determined that the particle system S has not reached the steady state (N in S116), the analysis apparatus 10 returns the process to step S104. When it is determined that the particle system S has reached the steady state (Y in S116), the analysis apparatus 10 ends the molecular dynamics calculation process and provides the physical quantity of the particle system S in the steady state to the user.

  FIG. 2 is a block diagram illustrating the function and configuration of the analysis apparatus 10. Each block shown here can be realized by hardware such as a computer (CPU) (central processing unit) and other elements and mechanical devices, and software can be realized by a computer program or the like. Here, The functional block realized by those cooperation is drawn. Therefore, it is understood by those skilled in the art who have touched this specification that these functional blocks can be realized in various forms by a combination of hardware and software.

  The analysis device 10 includes a calculation unit 12, a storage unit 24, an input unit 26, and an output unit 28. The calculation unit 12 includes a model generation unit 14, a conserving force acquisition unit 16, a non-conserving force acquisition unit 18, a summation unit 20, a molecular dynamics calculation unit 22, and a steady state determination unit 30.

The model generation unit 14 sets initial positions and initial velocities of N particles of the particle system S based on information input from the user via the input unit 26 or information already stored in the storage unit 24.
The storage force acquisition unit 16 acquires the storage force acting on each particle of the particle system S. The storage force acquisition unit 16 calculates a gradient of potential representing the interaction between particles in the particle system S, and acquires the storage force according to the calculated gradient.

  The non-preserving force acquisition unit 18 calculates and acquires the dissipative force based on Equation 1. The non-preserving force acquisition unit 18 calculates and acquires a viscous force based on Equation 2. Hereinafter, the derivation process of Equation 1 and Equation 2 will be described.

(Dissipative force, Formula 1)
The dissipative force is derived on the assumption that the internal vibration of the particle system S is attenuated. Consider a case where particles i and j of a particle system S are coupled with a spring constant k. The dissipation equation is given by Equation 3 below.
However, γ (t) is a coefficient indicating the strength of dissipation.

Here, to define the relative velocity v _r and center-of-gravity velocity v _g according to Equation 4 below, divided into motion and relative motion of the center of gravity.

At this time, the dissipation equation of Equation 3 is
It becomes. The relative motion solution is
It becomes.

From Equations 5 and 6, it can be seen that even during the dissipation process, the momentum of the center of gravity is preserved and the energy of the relative motion is dissipated. Hereinafter, numerical integration is performed on these.
When Equation 6 is discretized,
It becomes. However,
Is the relative velocity of particles i and j at (n + 1/2) dt, ^where n is an arbitrary integer, and γ ^{n−1 / 2} is the value of γ at (n−1 / 2) dt, where n is an arbitrary integer. .
Summarizing Equation 8,
It becomes.

From Equation 3, the discretized dissipation equation for particle i is
Therefore, substituting Equation 9 into Equation 10,
Is obtained.

Here, in order to overdampen the particle system S without vibrating,
And it is sufficient. By substituting equation 12 into equation 11, the equation for v _{i in} equation 1 is obtained. The equation for v _{j in} Equation 1 can be obtained in the same manner.

  Since this discretization uses the value of the previous step as the relative velocity value, it is partially implicit (only past information is used), except when the spring constant k is extremely large. It is difficult to break down.

(Viscous force, equation 2)
The viscous force is derived on the assumption that the translational motion of the particle system S is attenuated. This translational attenuation is expressed as attenuation by viscous resistance.
An equation relating to the viscous resistance acting on the particle i inside the particle system S (hereinafter referred to as a viscosity equation) is given by the following Equation 13.
However, γ (t) is a coefficient indicating the strength of viscosity.
By solving Equation 13,
Get.

From Equation 14, the time constant τ is
Given in.
The value of τ determines the magnitude of viscous resistance. From the experience of the present inventor as a person skilled in the art, τ is suitably about 1000 dt to 10000 dt.

By the same procedure as Equation 8, Equation 9, and Equation 10,
Is obtained. Therefore, Expression 2 is obtained by substituting Expression 15 into Expression 16. By using Formula 1 and Formula 2, it is possible to dissipate vibration and translation energy efficiently.

The summing unit 20 stores a conserving force FC _ij according to a potential gradient representing an interaction between the particle i and other particles j (j = 1, 2,..., N (j ≠ i)). 16 and add up for j. For the particle i, the storage power thus added is referred to as the total storage power ΣFC _i .
The summation unit 20 acquires the dissipating force FD _ij exerted from other particles (j = 1, 2,..., N (j ≠ i)) from the non-conservative force acquisition unit 18 and adds up the j for the particle i. . For the particle i, the combined dissipative force is referred to as a combined dissipative force ΣFD _i .
The summation unit 20 acquires the viscous force FV _i from the non-preserving force acquisition unit 18 for the particle i.
The summation unit 20 adds the total storage force ΣFC _i , the total dissipation force ΣFD _i, and the viscous force FV _i for the particle i to obtain a total force FT _i (= ΣFC _i + ΣFD _i + FV _i ) acting on the particle i. .

The molecular dynamics calculation unit 22 numerically integrates the equation of motion of the particle i including the total force FT _i obtained by the summation unit 20. Note that the Verlet method is often used for numerical integration. Regarding the Verlet method, for example, J. et al. M.M. Thijssen, “Computational Physics”, Cambridge University Press (1999), p. 175.

  The molecular dynamics calculation unit 22 updates the position and velocity of the particle i with the new position and velocity of the particle i obtained by numerically integrating the equation of motion. Note that it is not essential to update both the position and the speed, and only one of them may be updated.

  The steady state determination unit 30 determines that the particle system S has reached a steady state when the total kinetic energy of the particle system S is smaller than the lower limit value. When it is determined that the particle system S has not reached the steady state, the steady state determination unit 30 returns the processing to the storage force acquisition unit 16 and uses the position and velocity of each particle updated by the molecular dynamics calculation unit 22. The new molecular dynamics calculation is started. When the steady state determination unit 30 determines that the particle system S has reached the steady state, the molecular dynamics calculation is terminated.

  The storage unit 24 stores a molecular dynamics calculation program, a particle model generation program, and the like for performing the analysis method according to the embodiment. The storage unit 24 stores various data necessary for executing these programs. The calculation unit 12 reads necessary programs and data from the storage unit 24 and executes processing.

The input unit 26 includes an input device such as a keyboard and a mouse for receiving user input related to processing executed by the calculation unit 12. The input unit 26 may be configured to receive input from a recording medium such as a flexible disk, CD, or DVD.
The output unit 28 is an output device including a display device such as a display, a printer, and the like. The user confirms various physical quantities such as stress of the particle system S that has reached a steady state via the output unit 28. The output unit 28 may be configured to output the result to a recording medium.

  FIGS. 3A and 3B are explanatory diagrams illustrating an application example of the analysis method according to the embodiment to mechanism analysis. FIG. 3A is an explanatory diagram showing an initial state of the mechanism 50 that rotates by applying torque and the fixed rigid pin 60. The mechanism 50 includes a shaft portion 52 and a tooth portion 54. In this application example, torque is applied to the shaft portion 52 as shown in FIG. 3A, and at the same time, the entire mechanism 50 is moved toward the fixed rigid body pin 60. When the mechanism 50 and the fixed rigid pin 60 are sufficiently close, the tooth portion 54 of the mechanism 50 and the fixed rigid pin 60 come into contact with each other. In this application example, the steady state, that is, the balance position after contact is determined.

  At this time, if energy is not dissipated, all energy is stored, and the mechanism 50 continues to repeatedly collide with the fixed rigid pin 60. To avoid this, consider the case where the kinetic energy is forced to zero periodically.

  In this case, since the mechanism 50 needs to move the distance until it hits the fixed rigid body pin 60, once the kinetic energy is reduced to zero, it must be restarted from the acceleration, which is inefficient. In addition, depending on the model, the timing of energy dissipation and the period of vibration may resonate and vibration may increase.

Therefore, as in the analysis method according to the present embodiment, non-conservative force obtained on the premise of dissipating system energy is introduced. In this case, when the dissipating force is introduced as the non-preserving force, the mechanism 50 quickly hits the fixed rigid body pin 60, so that the time required for the simulation can be shortened. Moreover, when the time to reach a steady state can be sufficiently reduced by introducing the dissipative force, it is preferable because the calculation load can be reduced as compared with the case where both the viscous force and the dissipative force are considered.
However, in the case of dissipative force, after the mechanism 50 hits the fixed rigid body pin 60, the collision with the fixed rigid body pin 60 may be repeated several times. At that time, it is considered that the collision interval gradually becomes shorter and eventually reaches the balanced position. However, in order to obtain the balanced position, it is necessary to wait for the collision to end.

When a viscous force is introduced as a non-preserving force, the mechanism 50 quickly hits the fixed rigid body pin 60, so that the time required for the simulation can be shortened. Further, when the time required to reach a steady state can be sufficiently reduced by introducing the viscous force, it is preferable because the calculation load can be reduced as compared with the case where both the viscous force and the dissipating force are considered.
However, after the collision with the fixed rigid body pin 60, it is expected that the vibration generated inside due to the impact is not easily dissipated. If the viscous resistance seems to be large enough to dissipate the internal vibration immediately, there is a possibility that the time until the collision with the fixed rigid pin 60 cannot be ignored as in the case where the kinetic energy is periodically made zero.

  When both dissipative force and viscous force are introduced, vibration generated inside due to the impact of the collision can be effectively damped while suppressing the repetition of the collision. Can be obtained.

  FIG. 3B is an explanatory diagram showing the Mises stress distribution when the mechanism 50 and the fixed rigid pin 60 are in a balanced position. In this application example, the initial state of FIG. 3A can be given to the analysis method according to the embodiment to quickly obtain the steady state of FIG. 3B, and the time required for simulation can be shortened.

  FIG. 4 is a graph showing the attenuation of kinetic energy with and without various energy dissipation processes. 4 indicates the kinetic energy of the mechanism 50 in arbitrary units, and the horizontal axis indicates time in arbitrary units. The data indicated by reference numeral 70 in FIG. 4 corresponds to the case where neither the dissipative force nor the viscous force is taken in the application example shown in FIG. The data indicated by reference numeral 72 in FIG. 4 corresponds to the case where the dissipative force is incorporated in the application example shown in FIG. The data indicated by reference numeral 74 in FIG. 4 corresponds to the case where the dissipation force and the viscous force are taken in the application example shown in FIG. As is clear from FIG. 4, according to the analysis method according to the present embodiment, by introducing the dissipative force and the viscous force into the molecular dynamics calculation, the energy of the system is preferably dissipated to obtain a steady state faster. Can do.

  In the above-described embodiment, examples of the storage device are a hard disk and a memory. Based on the description in this specification, each unit is realized by a CPU (not shown), an installed application program module, a system program module, a memory that temporarily stores the contents of data read from the hard disk, and the like. It is understood by those skilled in the art who have touched this specification that they can do this.

  The configuration and operation of the analysis method and the analysis apparatus 10 according to the embodiment have been described above. These embodiments are exemplifications, and it is understood by those skilled in the art that various modifications can be made to each component and combination of processes, and such modifications are within the scope of the present invention. .

  3 and 4, the case where the analysis method according to the present embodiment is applied to the system including the mechanism 50 and the fixed rigid pin 60 has been described. However, the present invention is not limited to this. For example, for contact analysis of a gear when a load is applied. Can also be applied. Especially for structure / mechanism analysis where there are many contact points such as gears, it is difficult to obtain a convergent solution if there are many contact points, but the analysis method according to this embodiment is an explicit solution. This is preferable because a solution is always found.

  In the embodiment, the case where a dissipative force or a viscous force is employed as the non-preserving force has been described. However, the present invention is not limited to this, and for example, a frictional force may be employed.

  DESCRIPTION OF SYMBOLS 10 Analysis apparatus, 12 Computation part, 14 Model production | generation part, 16 Conservation force acquisition part, 18 Non-conservation force acquisition part, 20 Summation part, 22 Molecular dynamics calculation part, 24 Storage part, 26 Input part, 28 Output part, 30 Steady state determination unit.

Claims (5)

An analysis method for analyzing a system by numerically calculating a motion equation of a particle for a system including a large number of particles using a molecular dynamics method ,
Obtaining a non-conservative force acting on the particles and obtained on the premise of dissipating the energy of the system;
Numerically calculating the equation of motion of the particle including the acquired non-conservative force;
Based on the calculation results, seen including a determining whether the system has reached a steady state,
The method for obtaining the non-conserving force includes the step of obtaining the non-conserving force depending on the velocity of the particles . Step of acquiring the nonconservative force to claim 1, characterized in that it comprises a step of obtaining a nonconservative force obtained on the assumption damping vibrations of the inside of the system to a non-conserved force acting on the particle Analysis method described. Step of acquiring the nonconservative force, according to claim 1 or 2, characterized in that it comprises a step of obtaining a nonconservative force obtained on the assumption that attenuates the translational movement of the system a nonconservative force acting on the particle Analysis method described in 1. An analysis device for analyzing a system by numerically calculating a motion equation of a particle for a system including a large number of particles using a molecular dynamics method ,
A non-conserving force acquisition unit that acquires a non-conserving force obtained on the premise of dissipating energy of the system, which is a non-conserving force acting on particles
A calculation unit that numerically calculates a motion equation of a particle including the non-conservative force acquired by the non-conservative force acquisition unit;
A steady state determination unit that determines whether or not the system has reached a steady state based on a calculation result in the calculation unit ;
The non-conserving force acquisition unit acquires a non-conserving force depending on the velocity of particles . A computer program for causing a computer to perform a function of analyzing a system by numerically calculating a motion equation of a particle using a molecular dynamics method for a system including a large number of particles,
A non-conservative force acting on the particles and a non-conservative force obtained on the premise of dissipating the energy of the system;
A function for numerically calculating the equation of motion of the particle including the acquired non-conservative force;
A function of determining whether or not the system has reached a steady state based on a calculation result ;
The computer program is characterized in that the acquiring function acquires a non-conservative force depending on the velocity of particles .