JP2000194685A - Simulation method and device - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a simulation method/device which can calculate a correct motion of a simulation object element under a periodic boundary condition where pressure, stress and a couple of forces per volume are constant. SOLUTION: A set pressure, or a set pressure and a couple of forces per a set volume are converted into tensor σex of an external force acting on a cell (S3). Then a numeric solution of an equation of motion of a matrix h showing the cell is found (S4-S10) to show that the time differential d (dh/dt)h-1}/dt of the product (dh/dt)h-1} of the time differential (dh/dt) of the matrix h showing the cell and an inverse matrix h-1 of the matrix h showing the cell are equal to WV(σ-σex)} being a constant multiple of a product of a difference (σ-σex) between the tensor σex of the external force acting on the cell and a tensor σ of a force and of cell volume V. Thus, a satisfactorily large aggregate consisting of the simulation object elements such as molecules and atoms can be simulated.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a simulation method and apparatus, for example, a field of molecular design for optimizing material properties by controlling a molecular structure, and in particular, a cyclic method in which pressure, stress and couple per volume are constant. Under boundary conditions,
The present invention relates to a simulation method and apparatus for simulating a molecular design using a molecular dynamics method.

[0002]

2. Description of the Related Art In order to know the properties of an aggregate of elements to be simulated, such as molecules, atoms, and ions, the force acting between the elements to be simulated is appropriately set.
A simulation method that numerically solves the equation of motion of the element to be simulated, simulates the collective motion, and derives the properties of the aggregate from the collective motion is called a molecular dynamics method. This molecular dynamics method is divided into classical molecular dynamics (usually referred to here as molecular dynamics) and quantum molecular dynamics (usually called first-principles molecular dynamics).

In the classical molecular dynamics method, the elements to be simulated are atoms, molecules, and the like. The elements to be simulated are regarded as classical masses or rigid bodies, and the forces acting between the elements to be simulated are given. However, the force acting between the elements given in advance is empirically determined so that the simulation results are in good agreement with the experimental results.

[0004] The quantum molecular dynamics method is a combination of the molecular dynamics method and a method for determining an electronic state. The element to be simulated is an ion or a nucleus, and the element to be simulated is generally regarded as a classical mass point. further,
Electrons are handled by a method of obtaining an electronic state, and a force acting on an ion or a nucleus which is a simulation target element is obtained from the obtained electronic state.

When performing a simulation of an aggregate that can be regarded as sufficiently large and uniform as compared with the element to be simulated, a parallelepiped which is a basic unit of space is assumed, assuming a periodic boundary condition that the system is periodic. Only the simulation elements inside the cell are calculated.

When imposing constant pressure or stress conditions under periodic boundary conditions, the well-known Parrinello-Rahman method (eg, Phys. Rev. Lett. 45, 11) is used.
96 (1980)), the cell is represented by a matrix, and the position of the simulation target element is expressed as a product of the matrix representing the cell and the relative coordinate vector of the target element. And
Set the kinetic energy of the cell and the potential energy of the cell due to pressure or stress in an appropriate way, set the Lagrangian of the cell and the target element of the simulation,
The equation of motion is derived by solving the Euler-Lagrange equation that Lagrangean satisfies. As a result, a motion equation of a matrix representing a cell and a motion equation of relative coordinates of a target element are obtained. Parrinello-Rahman's method includes the step of finding numerical solutions to these equations of motion.

[0007]

However, the well-known method of Parrinello-Rahman has the following problems. For example, under periodic boundary conditions, there are a plurality of crystallographically equivalent cells. Therefore, there is a problem in that the movements of the obtained simulation target elements should be the same, but the movements will be different if different cells which are crystallographically equivalent are used. Therefore, a problem arises in that the cell cannot be changed to another crystallographically equivalent cell during the simulation.

[0008] Cleveland (for example, J. et al., Supra) solved this problem by improving the above-mentioned Parrinello-Rahman method.
Chem. Phys. 89, 4897 (1988)). However, the method proposed by Cleveland has a problem that a well-known virial theorem to be established between the set pressure and the actual pressure does not hold, and a correct result cannot be obtained.

Further, in the conventional method for treating the condition of constant stress (Parrinello-Rahman constant stress method), if the reference time is changed in the middle of the simulation, the simulation object obtained is not changed. There is a problem that the movements of the elements are different. This is because, under the condition of constant stress, the potential energy of the cell was set although it was incorrect in principle.

[0010] Further, there is no method for handling the condition of a constant couple per volume, and a simulation cannot be performed under the condition of a constant couple per volume.

The present invention provides pressure, stress and stress by properly setting the kinetic energy of the cell and properly treating the energy of the stress with the couple acting on the cell to derive the equation of motion of the matrix representing the cell. Provided is a simulation method and apparatus for correctly calculating the motion of a simulation target element under a periodic boundary condition in which a couple per volume is constant.

[0012]

In order to achieve the above object, a simulation method according to the present invention provides a sufficiently large aggregate of elements to be simulated, such as molecules and atoms, under a condition that the pressure is constant and a periodic boundary condition. , A step of converting a set pressure into a tensor σex of an external force acting on a cell, and a time differential (dh / d) of a matrix h representing the cell.
t) and the product of the inverse matrix h ⁻¹ of the matrix h representing the cell ｛(dh
/ Dt) h ^-1 } time derivative d {(dh / dt) h ^-1 } /
dt is a constant times ΔW of the product of the difference (σ−σex) between the tensor σex of the external force acting on the cell and the tensor σ of the force and the volume V of the cell.
V (σ-σex)} to obtain a numerical solution of a motion equation of a matrix h representing a cell.

[0013] The simulation method of the present invention provides a sufficiently large aggregate of elements to be simulated, such as molecules and atoms, under the condition that the stress and / or couple per volume are constant and the periodic boundary condition is satisfied. A simulation method for simulating the behavior, wherein a step of converting a set stress and a couple per set volume into a tensor σex of an external force acting on a cell, a time derivative (dh / dt) of a matrix h representing the cell, and The time derivative d {(dh / d) of the product {(dh / dt) h ^-1 } of the inverse matrix h ^-1
t) h ^-1 } / dt is a constant multiple {WV (σ−σex)} of the product of the difference between the tensor σex of the external force acting on the cell and the tensor σ of the force (σ−σex) and the volume V of the cell. Calculating the numerical solution of the equation of motion of the matrix h representing the cell.

[0014] The simulation apparatus of the present invention is a simulation apparatus for simulating the behavior of a sufficiently large aggregate of elements to be simulated such as molecules and atoms under the condition that the pressure is constant and the periodic boundary condition. Means for converting the set pressure into a tensor σex of the external force acting on the cell, and the product ｛(dh / dt) of the time derivative (dh / dt) of the matrix h representing the cell and the inverse matrix h ⁻¹ of the matrix h representing the cell ) H ^-1時間 time derivative d ｛(dh / d
t) h ^-1 } / dt is equal to the constant times {WV (σ-σex)} of the product of the difference (σ−σex) between the tensor σex of the external force acting on the cell and the tensor σ of the force and the volume V of the cell. Means for calculating a numerical solution of a motion equation of a matrix h representing a cell.

The simulation apparatus of the present invention provides a sufficiently large aggregate of elements to be simulated, such as molecules and atoms, under a condition that a couple per pressure and / or volume is constant and a periodic boundary condition. A simulation device for simulating the behavior, comprising means for converting a set pressure and a couple per set volume into a tensor σex of an external force acting on a cell, a time derivative (dh / dt) of a matrix h representing the cell, and The time derivative d {(dh / d) of the product {(dh / dt) h ^-1 } of the inverse matrix h ^-1
t) h ^-1 } / dt is a constant multiple {WV (σ−σex)} of the product of the difference between the tensor σex of the external force acting on the cell and the tensor σ of the force (σ−σex) and the volume V of the cell. Means for calculating a numerical solution of a motion equation of a matrix h representing a cell, which is equal.

Further, the storage medium of the present invention stores a program for simulating the behavior of a sufficiently large aggregate of elements to be simulated such as molecules and atoms under the condition that the pressure is constant and the periodic boundary condition. A storage medium for storing in a readable manner,
The program includes at least a module for converting a set pressure into a tensor σex of an external force acting on a cell, a time derivative (dh / dt) of a matrix h representing a cell, and an inverse matrix h ⁻¹ of a matrix h representing the cell. The time derivative d ｛(dh / dt) h ^-1 ｝ / dt of the product ｛(dh / dt) h ^-1が is given by the difference (σ−σex) between the tensor σex of the external force acting on the cell and the tensor σ of the force. A module for calculating a numerical solution of the equation of motion of the matrix h representing the cell, which is equal to a constant multiple {WV (σ−σex)} of the product of the volume V of the cell.

Further, the storage medium of the present invention provides a sufficiently large aggregate of elements to be simulated, such as molecules and atoms, under the condition that the stress and / or couple per volume are constant or one of them is constant and the periodic boundary condition is satisfied. A storage medium storing a program for simulating behavior in a computer-readable manner, wherein the program converts at least a set stress and a couple per set volume into a tensor σex of an external force acting on a cell. And the time derivative of the matrix h representing the cell (dh / dt) and its inverse matrix h ^-1 , the product of ({dh / dt) h ^-1 }, the time derivative d {(dh / dt) h ^-1 } / dt Is equal to the constant multiple {WV (σ−σex)} of the product of the difference (σ−σex) between the tensor σex of the external force acting on the cell and the tensor σ of the force and the volume V of the cell. Characterized in that it comprises a module for obtaining the numerical solution of h equations of motion.

Here, there are further provided calculation condition parameters such as pressure, time interval and convergence parameter for performing a simulation, and initial condition parameters such as a relative coordinate position of an atom, a time derivative thereof, a matrix representing a cell and a time derivative thereof. .

[0019]

DESCRIPTION OF THE PREFERRED EMBODIMENTS <Concept of Simulation Method of the Present Invention> The reason why a correct simulation result can be obtained by the first method of the present invention will be described below.

The virial theorem is a well-known theorem known in statistical mechanics, which is a set of constant pressures.
The theorem is that the average of the actual pressure is equal to the set pressure. However, the traditional Parrinello-Rahman method and Clevela
With the method proposed by nd, the virial theorem does not hold. Pa
The method of rrinello-Rahman and the method proposed by Cleveland first formulate the kinetic energy and potential energy of the simulation target element and the cell, and then derive the equation of motion between the matrix representing the cell and the relative coordinates of the target element. , The kinetic energy formula of the cell is Parrinello-Rah
The man method differs from the one proposed by Cleveland.
As a result, the equation of motion of the relative coordinates of the target element is common between the two methods, but the equation of motion of the matrix representing the cell is different. That is, an appropriate equation for the kinetic energy of the cell has not been provided.

The virial theorem is a theorem established by a set of constant pressures well known in statistical mechanics. In statistical mechanics, the pressure is constant. There are other sets of constant pressures that are different. In this set, a theorem corresponding to the Virial theorem that the average of the product of the actual pressure and the volume is equal to the product of the average of the volume and the set pressure is established.

The following shows that the present invention can simulate a collective motion in which the theorem corresponding to the virial theorem is established.

The set is subject to a periodic boundary condition, and its basic vector is defined as v (a), v (b), v (c).
(That is, when there is a simulation element in v (r), j, k, l are arbitrary integers, and v (r) + jv
(It is assumed that the same simulation element always exists in (a) + kv (b) + lv (c)). Note that the vector x is represented by v
Expressed by (x). At this time, a matrix representing a cell corresponding to the boundary condition is defined as h = (v (a), v (b), v (c)).

Here, the kinetic energy of the cell is

[0025]

(Equation 1)

And the potential energy of the cell is

[0027]

(Equation 2)

(Where Pex is the set pressure and V is the volume of the cell). If we derive the equation of motion of the cell,

[0029]

(Equation 3)

## EQU1 ## Here, TrA is the sum of the diagonal terms of matrix A, dot h is the time derivative of h, and h ⁻¹ is h
And I is a 3 × 3 unit matrix. Also, W
Is an arbitrary value that can be determined by the user, and is selected for effective simulation. In addition, it is easily understood that this equation of motion does not depend on how to take a cell unlike the method of Parrinell-Rahman.

The external force tensor σex acting on the cell
Is defined as the sum of the set stress tensor and the set couple tensor per volume. Although the specific definition of the couple per volume tensor is not described here, when the set couple per volume is 0, the set couple tensor per volume is 0. Therefore, in this case, the set couple per volume is 0,

[0032]

(Equation 4)

Is as follows. By substituting equation (4) for equation (3), the equation solved in the present invention is obtained.

When the time average of the equation (3) is taken, the left side becomes 0, so that the time average of the right side is 0. By the way, the trace of the force tensor σ is minus its definition or three times the actual pressure,

[0035]

(Equation 5)

Therefore, it can be seen that the theorem corresponding to the virial theorem holds for the motion of the simulation target element obtained by the present invention.

Next, the reason why a correct simulation result can be obtained by the second method of the present invention will be described.

The conventional stress-constant method, Parrinello-R
Ahman's constant stress method assumed that the energy that the set stress exerted on the cell was the potential energy. That is, when the cell is moved quasi-statically, it is assumed that the energy of the cell is a function of only the matrix h representing the cell.
However, the energy that the set stress exerts on the cell is actually
It turned out that it depends not on the function of only the matrix h representing the cell but on how the cell is deformed. In other words, when the cell is deformed and the value of h changes from t1 to t2 during the period from t1 to t2, the energy applied to the cell by the set stress is not a function of only h1 and h2. It depends on how it was transformed. Therefore, the traditional Parrin
ello-Rahman's method was giving incorrect results.

As a result of a detailed study, the energy exerted on the cell by the set stress and the set couple when the cell is slightly changed is
It turned out to be as follows.

[0040]

(Equation 6)

Here, det represents a determinant, and h ^−1t is a transposed matrix of an inverse matrix of h. However, the tensor σex of the external force acting on the cell is the sum of the tensor of the set stress and the couple tensor per set volume. The couple tensor is a 3 × 3 antisymmetric tensor corresponding to the vector when the couple is represented as a vector, and can be naturally defined similarly to the stress tensor. That is, the couple is represented by a three-dimensional axial vector. Since a three-dimensional axial vector can be generally converted into a 3 × 3 antisymmetric tensor, the couple can be expressed as a 3 × 3 antisymmetric tensor. Dividing this by the cell volume gives the couple tensor per volume. That is, the couple N per volume is

[0042]

(Equation 7)

Then, the couple tensor per volume is

[0044]

(Equation 8)

## EQU5 ## Therefore, when the set stress is the tensor Tij and the couple vector per set volume is N = (N1, N2, N3), the external force tensor σex is

[0046]

(Equation 9)

Is as follows.

As in the case of the constant pressure method of the first method, the kinetic energy of the cell is defined as the equation (1), and the equation of motion is derived.

[0049]

(Equation 10)

Is as follows.

By further providing a method for obtaining the solution of this equation, the motion of the element to be simulated under a constant stress and a couple per fixed volume can be correctly calculated.

<Structural Example of Simulation Apparatus According to this Embodiment> A molecular dynamics simulation apparatus according to an embodiment of the present invention will be described below with reference to the drawings.

FIG. 1 is a block diagram functionally showing the molecular dynamics simulation processing in the present embodiment.

The present apparatus simulates the behavior of atoms under a constant pressure condition under a periodic boundary condition by a molecular dynamics method. The molecular dynamics simulation processing according to the present embodiment is shown in FIG. As described above, under the control of the control unit 11, the external force tensor calculation unit 12, the calculation condition parameter 17 and the initial condition parameter 18 are used.
The calculation is performed through the force tensor calculator 13, the cell matrix calculator 14, the force calculator 15, and the relative coordinate calculator 16, and is output to the analysis result data 19.

The control section 11 controls each calculation section. The external force tensor calculation unit 12 converts the set pressure into an external force tensor according to the above equation (4). Cell matrix calculator 14
Updates the matrix representing the cells using the external force tensor and the force tensor. The force calculation unit 15 calculates the force acting on the atom that is the element to be simulated by using an appropriate method. The relative coordinate calculation unit 16 solves the equation of motion of the relative coordinates of the simulation target element, and updates the relative coordinates of the simulation target element and its time derivative.

FIG. 2 is a diagram showing a configuration of the simulation apparatus according to the present embodiment. This apparatus is composed of, for example, a general-purpose computer, and includes a computer main body 21 composed of a CPU 22 and a RAM 23,
T24, a keyboard 25 and a mouse 26 as input devices,
An external storage device 27 and a printer 28 as an output device are connected by a bus 29.

The CPU 22 controls the entire apparatus, accesses the RAM 23 according to input instructions from the keyboard 25 and the mouse 26, and executes a molecular dynamics simulation process by activating a control program stored in the RAM 23. The RAM 23 has an external storage device 27
A program load area 231 for storing various control programs (PA to PE shown in FIG. 1) installed from the external storage device, and for storing various parameters (DA and DB shown in FIG. 1) also installed from the external storage device 27. And an analysis result storage area 233 for storing analysis results (DC shown in FIG. 1). When the simulation apparatus is configured as a dedicated machine, a ROM for storing fixed programs and parameters may be further provided.

The external storage device 27 is used to store various data necessary for the molecular dynamics simulation processing in the present device. The stored data in this case include various control programs (external force tensor calculation program PA, force tensor calculation program PB, cell matrix calculation program PC, force calculation program PD, relative coordinate calculation program PE), as well as calculation condition parameters DA, There are initial condition parameter DB and analysis result data DC. In this example, these data are installed from the external storage device 27 to the RAM 23 during the molecular dynamics simulation.

The keyboard 25 and the mouse 26 are for inputting data and giving instructions. The CRT 24 is for displaying data, and is used here for displaying simulation results. The printer 28 is for printing data, and is used here for printing out the simulation result.

<Example of Operation of Simulation Apparatus of the Present Embodiment> Next, an example of operation of the simulation apparatus of the present embodiment will be described. FIG. 3 is a flowchart showing the operation of the molecular dynamics simulation process.

When simulating the behavior of a certain molecule, first, calculation conditions parameters (DA) such as pressure, time interval, and convergence parameter to be simulated are input (step S1), and the relative coordinate position of the atom and its time derivative are calculated. , An initial condition parameter (DB) such as a matrix representing a cell and its time derivative (step S)
2). The input of these condition parameters is performed through a keyboard 25 and a mouse 26 shown in FIG.
There is a method in which these parameters are stored in the external storage device 27 in advance, and are appropriately read from the external storage device 27 and input.

The condition parameters thus input are given to the CPU 22. Thereby, the CPU 22
Are the various programs (P
A to PE), the following molecular dynamics simulation processing is executed. The operation of the molecular dynamics simulation process will be described with reference to FIG.

The control unit 11 sequentially performs calculations necessary for the molecular dynamics simulation processing through the calculation units (12 to 16), and based on the calculation results, the relative coordinates of the atoms given as the initial conditions and the relative coordinates of the atoms. The time derivative, the matrix representing the cell and the time derivative are updated. in this case,
First, the control unit 11 calculates the external force tensor calculation unit 12
The pressure is converted into an external force tensor σex according to the equation (4) (step S3). If the stress and / or the couple per volume are constant instead of the constant pressure condition, the control unit 11 causes the external force calculation unit 12 to set the stress and the set force in accordance with the equation (9). The set couple per volume is converted into an external force tensor σex.

Next, the control section 11 calculates a force tensor by the force tensor calculation section 13 (Step S).
4). In this embodiment, normally, according to the equation (11) used in classical molecular dynamics in which each atom is treated as a mass point,
Calculate the force tensor.

[0065]

[Equation 11]

Where the position of the i-th atom is v (ri), and v (si) is

[0067]

(Equation 12)

The relative coordinates are as follows.

The formula for calculating the force tensor differs depending on whether the element to be simulated is a mass point or a rigid body, and whether it is classical molecular dynamics or quantum molecular dynamics.

The equation (11) is well known as an equation that defines the stress of an atom in an atom simulation under a constant pressure condition under a periodic boundary condition. However, strictly speaking, the left side is not a stress but a force tensor as referred to in the present invention.

Next, the control section 11 solves the equation of motion (10), which is the equation of motion of the matrix representing the cell, by the cell matrix calculation section 14 and updates the matrix representing the cell based on the above calculation result (step S5). . This equation of motion is a second-order ordinary differential equation, and is solved using a frog jump method, a predictor-corrector method, or the like, which is a usual method for solving the second-order ordinary differential equation. Note that the dot h or the like at the previous time is appropriately stored according to the solution method.

Next, the control section 11 calculates the force acting on the element to be simulated by the force calculation section 15 (step S6).

It should be noted that molecular dynamics include molecular dynamics in a narrow sense (often called classical molecular dynamics for distinguishing from first-principles molecular dynamics described below) and first-principles molecular dynamics (often classical molecular dynamics). Dynamics, which is called quantum molecular dynamics). In classical molecular dynamics, in order to determine the force acting on the element of interest to be simulated, the coordinates of the other (often multiple) elements to be simulated when the element of interest to be simulated is taken as the origin of the coordinates, A function having a value of a force exerted on the target simulation target element by another simulation target element is given. This function is empirically determined so that the simulation reflects the properties of the actual material. In quantum molecular dynamics, the electronic state is determined by a method such as density functional theory, which determines the electronic state corresponding to the position of the element to be simulated, such as an ion or a nucleus. Is calculated.

The present embodiment relates to a classical molecular dynamics simulation. Therefore, the force calculation unit 15
The force acting on the atom is calculated from the coordinates of the electrons using the function of the force given in advance. However, the invention is not limited to classical molecular dynamics, but is equally applicable to quantum molecular dynamics.

Next, the control unit 11 calculates and updates the relative coordinates of the simulation target element and its time derivative by the relative coordinate calculation unit 16 (step S7).

In this embodiment, the well-known equation of motion of an atom

[0077]

(Equation 13)

Is solved. Where the mass of the i-th atom is m
The force acting on the i, i-th atom is fi. This equation of motion is a second-order ordinary differential equation, and is solved using a frog jump method, a predictor-corrector method, or the like, which is a usual method for solving the second-order ordinary differential equation. Note that dv of the previous time is appropriately determined according to the solution method.
(S) / dt and the like are stored. In some methods of molecular dynamics, the element to be simulated is treated as a rigid body, but the present invention can be applied in exactly the same way.

Next, the control section 11 performs output processing such as the relative position of the element to be simulated, the time derivative thereof, the matrix representing the cell, etc. obtained by the above calculations (step S8).

Thereafter, the same processing as described above is repeated while updating the time by the time step Δt until the end condition is satisfied (steps S9 and S10). The termination condition is, for example, a convergence condition of a physical property value calculated from an analysis time or a collective motion of a simulation target element, and is given as a calculation condition in advance.

In this way, the collective motion of the simulation target element is sequentially simulated at each time interval Δt. The simulation results are shown in FIG.
Displayed through 24. If necessary, the simulation result is printed by the printer 28,
Alternatively, it can be stored in the external storage device 27. When the simulation result is stored in the external storage device 27, the simulation result can be read out and displayed or printed at any time, and furthermore, the simulation processing can be continued from the simulation result.

The present invention may be applied to a system composed of a plurality of devices (for example, a host computer, an interface device, a reader, a printer, etc.) or may be applied to an apparatus composed of one device.

An object of the present invention is to provide a storage medium storing a program code of software for realizing the functions of the above-described embodiments to a system or an apparatus, and to provide a computer (or CPU) of the system or apparatus.
Or MPU) reads and executes the program code stored in the storage medium.

In this case, the program code itself read from the storage medium implements the functions of the above-described embodiment, and the storage medium storing the program code constitutes the present invention. Examples of a storage medium for supplying the program code include a floppy disk, hard disk, optical disk, magneto-optical disk, and C
A D-ROM, a CD-R, a magnetic tape, a nonvolatile memory card, a ROM, and the like can be used.

When the computer executes the readout program code, not only the functions of the above-described embodiment are realized, but also the OS (Operating System) running on the computer based on the instruction of the program code. ) May perform some or all of the actual processing, and the processing may realize the functions of the above-described embodiments.

Further, after the program code read from the storage medium is written into a memory provided in a function expansion board inserted into the computer or a function expansion unit connected to the computer, based on the instruction of the program code, It goes without saying that the CPU included in the function expansion board or the function expansion unit performs part or all of the actual processing, and the processing realizes the functions of the above-described embodiments.

When the present invention is applied to the storage medium, the storage medium stores program codes corresponding to the above-described flowcharts. In brief, however, the program load area 231 shown in FIG. Will be stored in the storage medium. That is, the program code of at least the external force tensor calculation program module, the force tensor calculation program module, the cell matrix calculation program module, the force calculation program module, and the relative coordinate calculation program module may be stored in the storage medium.

[0088]

As described above, according to the present invention, the kinetic energy of a matrix representing a cell can be obtained by appropriately setting the kinetic energy of the cell and properly treating the energy of the stress with a couple acting on the cell. By providing the guidance, it is possible to provide a simulation method and apparatus that correctly calculates the motion of the simulation target element under a periodic boundary condition in which pressure, stress, and couple per volume are constant.

[0089]

[Brief description of the drawings]

FIG. 1 is a diagram showing a configuration example of a simulation device according to the present embodiment.

FIG. 2 is a diagram illustrating an example of a hardware configuration of a simulation device according to the present embodiment.

FIG. 3 is a flowchart illustrating an operation example of the simulation device according to the present embodiment;

Claims (7)

[Claims] 1. An object to be simulated such as a molecule or atom.
The condition that the pressure of a sufficiently large aggregate of
Simulation of behavior under periodic boundary conditions
This is a simulation method that converts the set pressure into the tensor σex of the external force acting on the cell.
The time derivative (dh / dt) of a matrix h representing a cell and the cell
Inverse matrix h of matrix h ^-1｛(Dh / dt) h^-1｝
Time derivative d ｛(dh / dt) h^-1｝ / Dt is added to the cell
The difference between the external force tensor σex and the force tensor σ (σ−
σex) and the constant times the product of the cell volume V ΔWV (σ−σe)
x) Equation of motion of the matrix h representing the cell, equal to｝
Obtaining a numerical solution of
Method. 2. A simulation method for simulating the behavior of a sufficiently large aggregate composed of elements to be simulated, such as molecules and atoms, under the condition that stress and / or couple per volume are constant and one of them is constant and periodic boundary conditions are satisfied. A step of converting a set stress and a couple per set volume into a tensor σex of an external force acting on the cell ^{; and} a product of a time derivative (dh / dt) of a matrix h representing the cell and its inverse matrix h ⁻¹ The time derivative of {(dh / dt) h ^-1 } d ｛(d
h / dt) h ^-1 } / dt is a constant multiple of the product of the difference (σ-σex) between the tensor σex of the external force acting on the cell and the tensor σ of the force and the volume V of the cell, ｛WV (σ-σex) Obtaining a numerical solution of a motion equation of a matrix h representing a cell, which is equal to｝. 3. An object to be simulated such as a molecule or an atom.
The condition that the pressure of a sufficiently large aggregate of
Simulation of behavior under periodic boundary conditions
A simulation device that converts the set pressure into a tensor σex of the external force acting on the cell.
And the time derivative (dh / dt) of a matrix h representing a cell and the cell
Inverse matrix h of matrix h ^-1｛(Dh / dt) h^-1｝
Time derivative d ｛(dh / dt) h^-1｝ / Dt is added to the cell
The difference between the external force tensor σex and the force tensor σ (σ−
σex) and the constant times the product of the cell volume V ΔWV (σ−σe)
x) Equation of motion of the matrix h representing the cell, equal to｝
Means for obtaining a numerical solution of
Device. 4. A simulation apparatus for simulating the behavior of a sufficiently large aggregate of elements to be simulated, such as molecules and atoms, under the condition that the pressure and / or the couple per volume are constant and one of the constants is a periodic boundary condition. Means for converting a set pressure and a couple per set volume into a tensor σex of an external force acting on the cell ^; a product of a time derivative (dh / dt) of a matrix h representing the cell and its inverse matrix h ⁻¹ The time derivative of {(dh / dt) h ^-1 } d ｛(d
h / dt) h ^-1 } / dt is a constant multiple of the product of the difference (σ-σex) between the tensor σex of the external force acting on the cell and the tensor σ of the force and the volume V of the cell, ｛WV (σ-σex) Means for calculating a numerical solution of a motion equation of a matrix h representing a cell, which is equal to｝. 5. A storage medium for storing, in a computer-readable manner, a program for simulating the behavior of a sufficiently large aggregate of elements to be simulated, such as molecules and atoms, under constant pressure and periodic boundary conditions. A module for converting the set pressure into at least a tensor σex of an external force acting on a cell, a time derivative (dh / dt) of a matrix h representing a cell, and an inverse matrix h of a matrix h representing the cell. Product with ^-1 {(dh / dt) h ^-1 }
Difference time derivative d {(dh / dt) h -1} / dt is the tensor tensor σex and power of the external force acting on the cell σ of (.sigma.
σex) and the constant times the product of the cell volume V ΔWV (σ−σe)
x) a module for calculating a numerical solution of a motion equation of a matrix h representing a cell, which is equal to｝. 6. A program for simulating the behavior of a sufficiently large aggregate of elements to be simulated, such as molecules and atoms, under the condition that stress and / or couple per volume are constant and one of them is constant and periodic boundary conditions are satisfied. A module for converting at least a set stress and a couple per set volume into a tensor σex of an external force acting on the cell, and a matrix h representing the cell. The product of (dt / dt) and its inverse matrix h ^-1 {d (dt / dt) h ^-1 }
h / dt) h ^-1 } / dt is a constant multiple of the product of the difference (σ-σex) between the tensor σex of the external force acting on the cell and the tensor σ of the force and the volume V of the cell, ｛WV (σ-σex) A module for calculating a numerical solution of a motion equation of a matrix h representing a cell, which is equal to｝. 7. It further comprises calculation condition parameters such as pressure, time step, convergence parameter and the like for performing a simulation, and initial condition parameters such as relative coordinate position of an atom, its time derivative, a matrix representing a cell and its time derivative. 7. The storage medium according to claim 5, wherein: