SIMULATION APPARATUS, SIMULATION METHOD, AND COMPUTER READABLE MEDIUM STORING PROGRAM

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First Claim
1. A simulation apparatus comprising:
 an input unit that acquires a value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of an interaction potential between particles determined according to a material to be simulated and an initial condition of a particle disposition; and
a processing unit that analyzes behavior of the particles by a molecular dynamics method based on the initial condition acquired by the input unit, using the interaction potential defined by the values of the parameters acquired by the input unit.
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Abstract
A value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of an interaction potential between particles determined according to a material to be simulated and an initial condition of a particle disposition are input to an input unit. A processing unit analyzes behavior of the particles by a molecular dynamics method using the interaction potential defined by the values of the parameters input to the input unit and based on the initial condition input to the input unit.
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7 Claims
 1. A simulation apparatus comprising:
an input unit that acquires a value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of an interaction potential between particles determined according to a material to be simulated and an initial condition of a particle disposition; and a processing unit that analyzes behavior of the particles by a molecular dynamics method based on the initial condition acquired by the input unit, using the interaction potential defined by the values of the parameters acquired by the input unit.  View Dependent Claims (2, 3)
 4. A simulation method comprising:
analyzing behavior of particles by a molecular dynamics method using an interaction potential defined by a value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of the interaction potential between the particles determined according to a material to be simulated, as a simulation condition.  View Dependent Claims (5, 6)
 7. A computer readable medium storing a program that causes a computer to execute a process, the process comprising:
a function of analyzing behavior of particles by a molecular dynamics method using an interaction potential defined by a value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of the interaction potential between the particles determined according to a material to be simulated, as a simulation condition.
1 Specification
The content of Japanese Patent Application No. 2018204194, on the basis of which priority benefits are claimed in an accompanying application data sheet, is in its entirety incorporated herein by reference.
Certain embodiment of the present invention relates to a simulation apparatus, a simulation method, and a computer readable medium storing a program.
There is a known simulation method of analyzing deformation when external force is applied to a material having a random shape by a molecular dynamics method or a renormalization group molecular dynamics method (for example, Patent Document 1). Hereinafter, in the present specification, the molecular dynamics method and the renormalization group molecular dynamics method are collectively and simply referred to as a molecular dynamics method.
In the simulation method based on the molecular dynamics method, it is possible to analyze an elastic region using a model in which a material to be simulated is represented by an aggregate of a plurality of particles and the particles are coupled by a spring. A spring constant of the spring that couples the particles is determined such that a Young'"'"'s modulus of the material is reproduced regardless of a particle disposition based on a Voronoi area derived from tetra mesh information with positions of the plurality of particles as nodes.
Japanese Unexamined Patent Publication No. 200937334
According to an embodiment of the present invention, there is provided a simulation apparatus including: an input unit that acquires a value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of an interaction potential between particles determined according to a material to be simulated and an initial condition of a particle disposition; and a processing unit that analyzes behavior of the particles by a molecular dynamics method based on the initial condition acquired by the input unit, using the interaction potential defined by the values of the parameters acquired by the input unit.
According to another aspect of the invention, there is provided a simulation method including: analyzing behavior of particles by a molecular dynamics method using an interaction potential defined by a value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of the interaction potential between the particles determined according to a material to be simulated, as a simulation condition.
According to yet another aspect of the invention, there is provided a computer readable medium storing a program that causes a computer to execute a process. The process includes a function of analyzing behavior of particles by a molecular dynamics method using an interaction potential defined by a value of a parameter defining a linear term and a value of a parameter defining a nonlinear term of the interaction potential between the particles determined according to a material to be simulated, as a simulation condition.
In the analysis of the linear elastic material, the spring constant of the spring that couples the particles can be determined by a known method. In the simulation method in the related art of determining the spring constant by this method, it is impossible to perform a simulation such as deformation of the nonlinear elastic material.
There is a need for providing a simulation apparatus, a simulation method, and a computer readable medium storing a program capable of performing a simulation such as deformation of a nonlinear elastic material.
Next, a simulation apparatus and a simulation method according to an embodiment will be described with reference to
The processing unit 21 performs a simulation by a molecular dynamics method based on the input simulation condition and outputs a simulation result to the output unit 22. The simulation result includes information representing behavior of particles when a simulation target is represented by an aggregate of a plurality of particles. The processing unit 21 includes, for example, a computer, and a program for causing the computer to execute a simulation by the molecular dynamics method is stored in the storage unit 23. The output unit 22 includes a communication apparatus, a removable media writing apparatus, a display, and the like.
In this embodiment, in order to simulate deformation of a nonlinear elastic material, a nonlinear term is introduced into an interaction potential between the particles used in the molecular dynamics method. Hereinafter, a method of introducing the nonlinear term of the interaction potential will be described.
An interaction potential φ(r_{ij}) between a particle i and a particle j is represented by the following equation.
[Formula 1]
ϕ(r_{ij})=½k_{ij}(r_{ij}−r_{ij0})^{2} (1)
Here, k_{ij }is a spring constant of a spring that couples the particles i and j, r_{ij }is a distance between the particles i and j, and r_{ij0 }is a distance between the particles i and j when the spring has a natural length.
Next, a method of determining the spring constant k_{ij }will be described with reference to
In this embodiment, a nonlinear term is added to the interaction potential of equation (1), and the interaction potential is defined by the following equation (2).
[Formula 2]
ϕ(r_{ij})=½k_{ij}(r_{ij}−r_{ij0})^{2}+a_{ij3}(r_{ij}−r_{ij0})^{3}+a_{ij4}(r_{ij}−r_{ij0})^{4} (2)
In Equation (2), thirdorder and fourthorder terms are added as the nonlinear term, but higherorder terms may be added. In this embodiment, up to the fourthorder term is considered as the nonlinear term of the interaction potential.
Coefficients a_{ij3 }and a_{ij4 }of the thirdorder term and the fourthorder term in equation (2) are defined by the following equation (3).
In a case where a material to be handled is the linear elastic material, it is possible to determine the spring constant k_{ij }from the Young'"'"'s modulus of the material as described with reference to
Further, it is assumed that the deformation when the nonlinear elastic material is compressed is substantially linear and nonlinearity appears when a tensile stress is applied to the nonlinear elastic material. When a compressive strain is generated in the nonlinear elastic material, that is, when r_{ij}≤r_{ij0}, the interaction potential can be represented by the following equation (4).
[Formula 4]
ϕ(r_{ij})=½Ak_{ij}(r_{ij}−r_{ij0})^{2} (4)
When an extending strain is generated in the nonlinear elastic material, that is, when r_{ij}>r_{ij0}, the interaction potential can be represented by the following equation (5).
When A=1 and a_{3}=a_{4}=0, the interaction potential of equation (5) has the same shape as the interaction potential when the linear elastic material is handled.
A simulation is performed to obtain the deformation of the material when a tensile test is performed using the interaction potentials of equations (4) and (5). Hereinafter, a simulation result will be described.
In a case where the simulation is performed using the interaction potential shown in equation (5), an obtained relationship between the strain and the stress is required to be the same even though the simulation model has a different dimension. Hereinafter, results of obtaining the relationships between the stress and the strain of simulation models having different dimensions will be described.
In a case where the simulation is performed using the interaction potential shown in equation (5), parameters A, a_{3}, and a_{4 }are required to be determined so as to reproduce physical properties of the material to be simulated. However, it is unclear how the parameters of these nonlinear terms affect the relationship between the stress and the strain. Therefore, it is impossible to determine the values of these parameters directly from the physical properties of the material. It is possible to find the values of these parameters, for example, by repeating trial and error. Hereinafter, an example of a procedure for finding optimum values of these parameters will be described. Here, the “optimum value” does not mean an optimum value among all combinations of the values but means a preferable value with which the simulation can be performed with sufficiently high accuracy.
First, initial values of the parameters A, a_{3}, and a_{4 }are set (step SA1). The initial value may be determined from an empirical rule based on, for example, the Young'"'"'s modulus in a linear deformation region of the nonlinear elastic material to be simulated. Based on the set values of the parameters A, a_{3}, and a_{4}, the relationship between the stress and the strain is obtained by the molecular dynamics method using the interaction potential of equation (5) (step SA2). This relationship is referred to as second stressstrain relationship data.
In a case where a calculation fails during the calculation of step SA2 (step SA3), the calculation ends, the values of parameters A, a_{3}, and a_{4 }are updated (step SA6), and the process of step SA2 is repeated. Here, the case where the calculation fails means, for example, a case where a calculation for dividing by zero is performed. A maximum value, a minimum value, and a pitch width at the time of the updating of the values of the parameters A, a_{3}, and a_{4 }are determined in advance.
In a case where the second stressstrain relationship data is obtained with no failure in the calculation of step SA2, the first stressstrain relationship data representing the measured relationship between the stress and the strain is compared with the second stressstrain relationship data obtained by the calculation. From the comparison, an error of the second stressstrain relationship data with respect to the first stressstrain relationship data is obtained (step SA4).
As the error of the second stressstrain relationship data with respect to the first stressstrain relationship data, for example, it is preferable to adopt a determination coefficient (R_{2 }value) with the function f (ε) representing the first stressstrain relationship data obtained from the measured value as a regression equation and the second stressstrain relationship data obtained from the calculation result as a sample value. Hereinafter, this determination coefficient is referred to as “the determination coefficient of the stressstrain relationship”.
Further, second Young'"'"'s modulusstrain relationship data is obtained from the second stressstrain relationship data obtained by the calculation. The second Young'"'"'s modulusstrain relationship data can be represented, for example, by a ratio of a stress increment to a strain ε increment at two adjacent points of the discrete second stressstrain relationship data. In step SA4, an error of the second Young'"'"'s modulusstrain relationship data with respect to the first Young'"'"'s modulusstrain relationship data is obtained together with the error of the second stressstrain relationship data with respect to the first stressstrain relationship data. As this error, it is preferable to adopt a determination coefficient (R_{2 }value) with a function df(ε)/dε (
The processes from step SA2 (
Hereinafter, a method of determining the optimum values of the parameters A, a_{3}, and a_{4 }will be described with reference to
Values of the parameters A, a_{3}, and a_{4 }when the determination coefficient of the Young'"'"'s modulusstrain relationship and the determination coefficient of the stressstrain relationship are the largest are adopted as the optimum values. In the example shown in
As shown in
The processing unit 21 acquires the value of the parameter A defining the linear term, the value of the spring constant k_{ij}, and the values of the parameters a_{3 }and a_{4 }defining the nonlinear terms of the interaction potential between the particles determined according to the material to be simulated, and the simulation condition such as the initial condition of the particle disposition (step SB1). The parameter A and the spring constant k_{ij }may be handled as one linear parameter as A·k_{ij}. The particle is disposed based on the acquired simulation condition to generate a simulation model (step SB2).
The behavior of the particles is analyzed by the molecular dynamics method using the interaction potentials of equations (4) and (5) based on the input values of the parameter A, the spring constant k_{ij}, and the parameters a_{3 }and a_{4 }(step SB3). After the analysis, the analysis result is output to the output unit 22 (step SB4). For example, the output unit 22 may display a change in the position of the particle or a change in the shape of the tetra mesh as a graphic in a time series.
Next, an excellent effect of the embodiment will be described. According to this embodiment, it is possible to perform a mechanism analysis in consideration of the nonlinear region of the nonlinear elastic material. Using the parameter optimization method shown in
Next, a modification example of the above embodiment will be described. In the above embodiment, the optimum values of the parameters A, a_{3}, and a_{4 }are determined using the determination coefficient of the Young'"'"'s modulusstrain relationship and the determination coefficient of the stressstrain relationship in step SA7 of
As shown in
In the above embodiment, up to the fourthorder nonlinear term is considered as the interaction potential of equation (5), but a higherorder nonlinear term of the fifthorder or more may be considered. In this case, the number of nonlinear parameters increases.
The above embodiment is an example, and the present invention is not limited to the above embodiment. For example, it is apparent to those skilled in the art that various changes, improvements, combinations, and the like can be made.
It should be understood that the invention is not limited to the abovedescribed embodiment, but may be modified into various forms on the basis of the spirit of the invention. Additionally, the modifications are included in the scope of the invention.