Simulator and program

Abstract

To provide a simulator allowed to suppress the lowering in accuracy of simulation even where an object has a curved surface.SOLUTION: A simulator comprises: an input section that inputs a simulation condition including information representative of a shape and position of an object arranged in an analysis space and fluid flowing in the analysis space and initial and boundary conditions; a processing section that defines a plurality of secondary curved-surface elements to reproduce a surface of the object, arranges a plurality of fluid particles representative of the fluid in the analysis space and calculates the shortest distance from respective ones of the plurality of fluid particles to the secondary curved-surface elements and furthermore calculates, on the respective ones of the plurality of fluid particles, a force to undergo from the surface of the object based on a calculation value of the shortest distance and determines a behavior of the plurality of fluid particles based on the force calculated; and an output section that information reflecting the behavior of the plurality of fluid particles is outputted.SELECTED DRAWING: Figure 7

Description

[Technical Field]

[0001] The present invention relates to a simulation device and a program. [Background technology]

[0002] A method of dividing an object into tetrahedral elements (tetra mesh) is used in analyses using molecular dynamics (MD) and renormalization group molecular dynamics (RMD) (Patent Document 1). When an object is divided into a tetra mesh, multiple triangular planar elements appear on the surface of the object. When a fluid comes into contact with an object, fluid particles representing the fluid are subjected to a force from the surface of the object. This force is calculated based on the shortest distance from the fluid particle to the triangular planar element. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] Japanese Patent Publication No. 2020-071692 Summary of the Invention [Problem to be solved by the invention]

[0004] When an object has a curved surface, representing it with multiple triangular planar elements results in the loss of information about the curved surface within each of the planar elements. The misalignment between the actual curved surface and the planar elements that approximate it reduces the accuracy of the simulation.

[0005] An object of the present invention is to provide a simulation device and program that can suppress a decrease in simulation accuracy even when the surface of an object is curved. [Means for solving the problem]

[0006] According to one aspect of the present invention, an input section for inputting simulation conditions; a processing unit that performs a simulation based on the simulation conditions; an output unit that outputs a simulation result obtained by the processing unit; Equipped with the simulation conditions include shapes and positions of objects placed in an analysis space, information representing a fluid flowing in the analysis space, initial conditions, and boundary conditions; The processing unit defining a plurality of quadric surface elements that represent the surface of the object; placing a plurality of fluid particles representing the fluid within the analysis space; calculating the shortest distance from each of the plurality of fluid particles to the quadric surface element; calculating a force acting on each of the plurality of fluid particles from the surface of the object based on the calculated value of the shortest distance; determining behavior of the plurality of fluid particles based on the calculated forces; There is provided a simulation device that outputs information reflecting the behavior of the plurality of fluid particles to the output unit.

[0007] According to another aspect of the present invention, A program that causes a computer to realize a simulation function based on given simulation conditions, The ability to define multiple quadric surface elements that represent the surface of an object in analytical space; a function of placing a plurality of fluid particles representing a fluid within the analysis space; a function of calculating the shortest distance from each of the plurality of fluid particles to the quadric surface element; a function of calculating a force acting on each of the plurality of fluid particles from the surface of the object based on the shortest distance; determining the behavior of the plurality of fluid particles based on the calculated forces; a function of outputting information reflecting the behavior of the plurality of fluid particles to an output unit; A program for realizing this on a computer is provided. [Effects of the Invention]

[0008] By representing the surface of an object using a plurality of quadric surface elements, it is possible to suppress a decrease in simulation accuracy compared to when the surface is represented using planar elements. [Brief explanation of the drawings]

[0009] [Figure 1] FIG. 1 is a schematic diagram showing an example of an analytical model to be simulated. [Figure 2] 2A and 2B are schematic diagrams showing a planar element 15, which is one surface of a tetrahedral element exposed on the surface of the object 10, and the actual surface 10S of the object 10. FIG. [Figure 3] FIG. 3A is a schematic perspective view showing the positional relationship between a fluid particle 20 and one quadric surface element 12, and FIG. 3B is a graph showing an example of the potential generated due to the surface of the object 10. [Figure 4] FIG. 4 is a graph showing an example of the function f(λ). [Figure 5] FIG. 5A is a schematic perspective view of the analytical model, and FIG. 5B is a graph showing the analytical results. [Figure 6] FIG. 6 is a block diagram of a simulation device according to an embodiment. [Figure 7] FIG. 7 is a flowchart showing the procedure of the simulation executed by the processing unit 51 of the simulation device according to this embodiment. [Figure 8] FIG. 8 is a diagram showing an example of an image displayed on the output unit 52. As shown in FIG. [Figure 9] FIG. 9A is a schematic diagram showing a part of an object model of object 10 divided into tetrahedrons, and FIG. 9B is a graph showing the simulation results. DETAILED DESCRIPTION OF THE INVENTION

[0010] A simulation device according to one embodiment will be described with reference to FIGS. 1 to 9B.

[0011] FIG. 1 is a schematic diagram showing an example of an analytical model to be simulated. An elastic object 10 and a plurality of fluid particles 20 representing a fluid are arranged in an analytical space. The object 10 is divided into a plurality of tetrahedral elements, and object particles 11 are arranged at each vertex of the tetrahedral elements. Each surface of the tetrahedral elements exposed on the surface of the object 10 is approximated by a quadric surface element 12. As an example, the object 10 is tubular, and a plurality of fluid particles 20 are arranged in the internal space of the tube. In the simulation, analysis is performed on the flow of fluid in the internal space of the object 10, and the stress and deformation acting on the elastic object 10.

[0012] Next, a method for approximating the surface of object 10 with multiple quadric surface elements 12 will be described with reference to FIGS. 2A and 2B. FIGS. 2A and 2B are schematic diagrams showing a planar element 15, which is one surface of a tetrahedral element exposed on the surface of object 10, and the actual surface 10S of object 10. Although the planar element 15 is triangular and the surface 10S of object 10 is curved, FIGS. 2A and 2B reduce the dimensions by representing the planar element 15 as a straight line and the surface 10S as a curve. Object particles 11 are placed on both ends of the planar element 15. Note that FIGS. 2A and 2B exaggerate the difference between the planar element 15 and the actual surface 10S.

[0013] As shown in FIG. 2A, the vertices (both ends in FIG. 2A) of the planar elements 15 are located on the surface 10S of the object 10. A plurality of points 16 are generated on the surface 10S. As shown in FIG. 2B, the curved surface (curve in FIG. 2B) on which the plurality of points 16 are arranged is approximated with quadric surface elements 12 (quadratic curves in FIG. 2B). This approximation can be performed by applying, for example, the least squares method.

[0014] FIG. 3A is a schematic perspective view showing the positional relationship between a fluid particle 20 and one quadric surface element 12. The fluid particle 20 is placed near the quadric surface element 12. FIG. 3B is a graph showing an example of the potential generated due to the surface of the object 10. The horizontal axis represents the distance r from the surface of the object 10, and the vertical axis represents the potential U. As the distance r approaches 0 from a positive value, the potential U increases, and becomes infinite at the distance r=0. Furthermore, as the distance r increases, the potential U asymptotically approaches 0. In order to calculate the interaction between the fluid particle 20 and the surface of the object 10, the shortest distance r between the fluid particle 20 and the quadric surface element 12 is used. min (Figure 3A) needs to be calculated.

[0015] Next, the shortest distance r between the fluid particle 20 (FIG. 3A) and the quadric surface element 12 (FIG. 3A) is calculated. min A method for calculating the quadric surface element 12 in the xyz orthogonal coordinate system can be expressed by the following equation.

number

[0016] The length of a line segment having both ends of an arbitrary point P(x, y, z) and a point Q(s, t, u) constrained on the quadric surface element 12 is expressed by the following formula.

number

[0017] The position of the point P(x, y, z) where the length of the line segment PQ is the shortest is found using the Lagrange multiplier method. First, the following function S(x, y, z) is defined.

number

number

[0018] Partially differentiate the function L(x,y,z,λ) with respect to the variables x,y,z,λ, and find λ when the result is 0. In other words, solve the following simultaneous equations to find the value of the Lagrange multiplier λ that gives the extreme value.

number

[0019] From the four equations (5a) to (5d), we obtain the following quintic equation f(λ) with the Lagrange multiplier λ as a variable.

number

[0020] A quintic equation cannot be solved algebraically. Next, a method for finding a solution to equation (6) will be described with reference to FIG. 4. FIG. 4 is a graph showing an example of a function f(λ). First, a quartic equation that sets the first derivative f'(λ) of the function f(λ) to 0 is solved to find the extreme values ​​of the function f(λ). As a method for finding the solution to a quartic equation, for example, the Ferrari method can be used. FIG. 4 shows four extreme values ​​E1, E2, E3, and E4, in order from the smallest value of the variable λ.

[0021] Next, find the inflection points of the function f(λ) by solving a cubic equation that sets the second derivative f''(λ) of the function f(λ) to 0. Figure 4 shows the three inflection points F1, F2, and F3, in order from the smallest variable λ.

[0022] Next, compare two adjacent extreme values ​​of the variable λ. If the signs of the two extreme values ​​are opposite, there is a solution between them. In the example shown in Figure 4, there is a solution S2 between extreme values ​​E1 and E2, and there is a solution S3 between extreme values ​​E2 and E3. There is no solution between extreme values ​​E3 and E4.

[0023] Next, we will explain how to determine whether or not there is a solution for the range of the variable λ outside the λ that gives the first extreme value E1 and the λ that gives the fourth extreme value E4. The sign of the first extreme value E1 is compared with the sign of the first derivative f'(λ) in the range outside the λ that gives that extreme value E1. If the signs of both are the same, a solution exists outside the λ that gives the first extreme value E1. If the signs of both are different, no solution exists outside the λ that gives the first extreme value E1. The same applies to the range outside the λ that gives the fourth extreme value E4. In the example shown in Figure 4, a solution S1 exists outside the λ that gives the first extreme value E1, but no solution exists outside the λ that gives the fourth extreme value E4.

[0024] If a solution exists between two adjacent extreme values ​​of the variable λ, the solution is found using Newton's method. When applying Newton's method, it is advisable to use the λ that gives the inflection point as the initial value. For example, in the example shown in Figure 4, when finding a solution between extreme values ​​E1 and E2, the λ that gives the inflection point F1 is used as the initial value. When finding a solution outside the λ that gives the first extreme value E1 and the fourth extreme value E4, a value of λ that is slightly outside the λ that gives the extreme value is used as the initial value for Newton's method.

[0025] For each of the obtained solutions, the coordinates (x, y, z) of point P are found using equations (5a) to (5c). Then, the length of line segment PQ is calculated using equation (2). The shortest length is adopted as the solution for the shortest distance from point Q to the quadric surface element 12.

[0026] Next, referring to FIGS. 5A and 5B, the shortest distance r from the fluid particle 20 to the quadric surface element 12 (FIG. 3A) is min The results of an analysis to evaluate the suitability of the above-mentioned method for determining σ are described below.

[0027] Fig. 5A is a schematic perspective view of the analytical model. A rectangular parallelepiped container 25 is placed in the analytical space 30. A spherical object 10 and a plurality of fluid particles 20 are contained in the container 25. Fig. 5A shows only the lower half of the container 25 in the height direction. The position of the object 10 is fixed.

[0028] The Lennard-Jones potential was used as the interaction potential between the fluid particles 20. Furthermore, instead of the potential shape shown in Fig. 3B, the interaction potential between the fluid particles 20 and the surface of the object 10 was one in which the potential U increases linearly as the distance r decreases within a range where the distance r is smaller than a certain value. The potential energy of the fluid particle 20 due to the surface of the object 10, the potential energy between the fluid particles 20, and the kinetic energy of the fluid particle 20 were calculated.

[0029] 5B is a graph showing the analysis results. The horizontal axis represents time in units of seconds, and the vertical axis represents energy in arbitrary units. The solid lines SF, Fp, Fk, and T in the graph represent the potential energy of the fluid particles 20 due to the surface of the object 10, the potential energy between the fluid particles 20, the kinetic energy of the fluid particles 20, and the total energy of the fluid particles 20, respectively.

[0030] Because object 10 is fixed, the temperature of fluid particle 20 is not transferred to object 10, and at equilibrium, the temperature of the fluid remains constant. As time passes, the system reaches equilibrium, and the total energy of fluid particle 20 remains constant, as shown in Figure 5B.

[0031] If the potential due to the surface of the object 10 is not spatially continuous, the equilibrium state of the system cannot be maintained. In this evaluation result, as shown in FIG. 5B, the equilibrium state of the system is maintained. This evaluation result shows that the shortest distance r between the fluid particle 20 and the quadric surface element 12 that approximates the surface of the object 10 is min It was confirmed that the continuity of the potential obtained using this calculation method is maintained.

[0032] 6 is a block diagram of a simulation device according to this embodiment. The simulation device according to this embodiment includes an input unit 50, a processing unit 51, an output unit 52, and a storage unit 53. Simulation conditions are input from the input unit 50. Furthermore, various instructions (commands) and the like are input to the input unit 50 by an operator. The input unit 50 is composed of, for example, a communication device, a removable media reader, a keyboard, and the like.

[0033] The processing unit 51 performs a simulation based on the input simulation conditions and outputs the processing results to the output unit 52. The processing unit 51 includes, for example, a computer, and a program for causing the computer to execute each procedure of the simulation is stored in the storage unit 53. The output unit 52 includes a communication device, a removable media writing device, a display, etc.

[0034] FIG. 7 is a flowchart showing the procedure of the simulation executed by the processing unit 51 of the simulation device according to this embodiment.

[0035] First, the processing unit 51 acquires the simulation conditions input to the input unit 50 (step SA1). The simulation conditions include information defining an object to be placed in the analysis space, information defining a fluid to be placed in the analysis space, information defining the potential that the surface of the object exerts on fluid particles (for example, the potential shown in FIG. 3B), boundary conditions of the analysis space, and initial and end conditions of the simulation. The information defining the object includes, for example, the shape of the object and elasticity information of the object. The elasticity information of the object includes, for example, information defining the interaction potential between object particles. The information defining the fluid includes information defining the interaction potential between fluid particles.

[0036] For example, the Lennard-Jones potential is used as the interaction potential acting between fluid particles 20 (FIG. 1). The interaction potential acting between matter particles 11 (FIG. 1) is defined by, for example, the spring constant when two mutually adjacent matter particles 11 are connected by a spring.

[0037] When the processing unit 51 acquires the simulation conditions, it places a plurality of fluid particles 20 representing a fluid in the analysis space and assigns an initial velocity value to each of the fluid particles 20 (step SA2). Furthermore, the processing unit 51 represents an object 10 (FIG. 1) placed in the analysis space with a plurality of object particles 11 (FIG. 1), and defines a plurality of quadric surface elements 12 (FIGS. 1 and 3) representing the surface of the object 10 (step SA3).

[0038] Next, the processing unit 51 calculates the force acting on each of the plurality of fluid particles 20 based on the interaction potential between the fluid particles (step SA4). Furthermore, the processing unit 51 calculates the force acting on each of the plurality of matter particles 11 based on the interaction potential between the matter particles (step SA5).

[0039] Next, the processing unit 51 calculates, for each of the plurality of fluid particles 20, the potential that the fluid particle 20 receives from the surface of the object 10 and the shortest distance r from the fluid particle 20 to the quadric surface element 12. min Based on the calculated values ​​in FIG. 3A, the force acting on the fluid particle 20 is calculated (step SA6). The quadric surface element 12 receives a force due to a reaction from the fluid particle 20.

[0040] Next, the processing unit 51 distributes the force that the quadric surface element 12 receives from the fluid particle 20 to the object particle 11 arranged at the vertex of the quadric surface element 12, and calculates the force acting on the object particle 11 (step SA7).

[0041] Next, the processing unit 51 solves the equation of motion for each of the fluid particles 20 based on the forces acting on the fluid particles 20, and updates the positions and velocities of the fluid particles 20 (step SA8). Furthermore, the processing unit 51 solves the equation of motion for each of the object particles 11 based on the forces acting on the object particles 11, and updates the positions and velocities of the object particles 11, and redefines the quadric surface elements 12 of the object 10 (step SA9).

[0042] The processing unit 51 repeats the procedures from step SA4 to step SA9 until the analysis termination condition is satisfied (step SA10). When the analysis termination condition is satisfied, the processing unit 51 outputs the simulation result to the output unit 52 (FIG. 6) (step SA11) and terminates the analysis process. The output unit 52 is, for example, an image display device, and displays the simulation result as an image.

[0043] 8 is a diagram showing an example of an image displayed on the output unit 52. The processing unit 51 displays an image of the stress distribution within the object 10 based on the force acting on the object particle 11. For example, a perspective view of the object 10 is displayed in shades of gray, and the shades of gray represent the magnitude of the stress.

[0044] Furthermore, the state of the fluid flow may be displayed as an image in addition to the distribution of stress applied to the object 10. For example, the output unit 52 may display information including the positions of the plurality of fluid particles 20 and the velocity vectors of the fluid particles 20.

[0045] Next, the results of an actual simulation performed using the simulation device according to the above embodiment will be described with reference to Figures 9A and 9B. In this simulation, a flow analysis of a polymer fluid filled in a cylindrical object 10 (Figure 1) was performed. The distribution of the flow velocity of the polymer fluid inside the cylindrical pipe can be theoretically calculated.

[0046] FIG. 9A is a schematic diagram showing a portion of an object model of object 10 divided into tetrahedrons. Object 10 has an inner diameter of 50 mm and is divided into tetrahedrons with dimensions of approximately 10 mm. The inner surface 10S of object 10 is cylindrical. Because a cylindrical surface is a quadric surface, quadric surface elements 12 approximating surface 10S coincide with the cylindrical surface 10S. When surface 10S is divided into triangular planar elements 15, an error D occurs between the approximation and the actual surface 10S. The maximum value of this error D is approximately 1 mm.

[0047] In the simulation, the diameter of the fluid particles 20 was set to 2 μm, and the number of fluid particles was set to 30,000. A periodic boundary condition was applied to the central axis direction of the cylindrical shape. A volume force in the central axis direction was applied to the fluid.

[0048] Figure 9B is a graph showing the simulation results. The horizontal axis represents the distance from the center of the cylinder in units of mm, and the vertical axis represents the flow velocity in units of m / s. The open circles in the graph shown in Figure 9B represent the analysis results when the surface of the object 10 is approximated by quadric surface elements 12 (Figure 1) using the method of the above embodiment. The solid circles represent the analysis results when the surface of the object 10 is approximated by triangular plane elements. The dashed line represents the theoretical solution.

[0049] When the surface of object 10 is approximated with quadric surface elements 12, analysis results closer to the theoretical solution are obtained than when the surface is approximated with triangular plane elements. When the surface of object 10 is approximated with triangular plane elements, the deviation from the theoretical solution is large, particularly in the range close to the surface of object 10. This is because when the surface of object 10 is approximated with triangular plane elements, an error D (FIG. 9A) of approximately 1 mm maximum occurs between the actual surface 10S and the plane elements 15. In the method according to the above embodiment, this error is small, and high analysis accuracy is obtained.

[0050] Next, the excellent effects of the above embodiment will be explained. In the above embodiment, the surface of the object 10 is approximated by quadric surface elements 12 (FIG. 1), which allows for higher approximation accuracy of the object surface compared to approximation by triangular plane elements. As a result, it is possible to improve the accuracy of analysis.

[0051] Furthermore, as shown in FIG. 8, the distribution of stress applied to the object 10 is displayed as an image, so that the user can easily recognize the distribution of stress.

[0052] Next, a modification of the above embodiment will be described. In the above embodiment, an example in which the object 10 is cylindrical has been described, but the above embodiment can also perform highly accurate simulations for objects of other shapes. In the above embodiment, an example in which the object 10 is an elastic body has been described, but even if the object 10 is a rigid body, analysis using the above embodiment can be performed.

[0053] The above-described embodiments are merely examples, and the present invention is not limited to the above-described embodiments. For example, it will be obvious to those skilled in the art that various modifications, improvements, combinations, etc. are possible. [Explanation of symbols]

[0054] 10 objects 10S Object surface 11 Object particles 12 Quadratic surface elements 15 Planar elements 16 Points on the surface of an object 20 fluid particles 25 Container 30 Analysis space 50 Input section 51 Processing section 52 Output section 53 Storage section

Claims

1. an input section for inputting simulation conditions; a processing unit that performs a simulation based on the simulation conditions; an output unit that outputs a simulation result obtained by the processing unit; Equipped with the simulation conditions include shapes and positions of objects placed in an analysis space, information representing a fluid flowing in the analysis space, initial conditions, and boundary conditions; The processing unit defining a plurality of quadric surface elements that represent the surface of the object; placing a plurality of fluid particles representing the fluid within the analysis space; calculating the shortest distance from each of the plurality of fluid particles to the quadric surface element; calculating a force acting on each of the plurality of fluid particles from the surface of the object based on the calculated value of the shortest distance; determining behavior of the plurality of fluid particles based on the calculated forces; A simulation device that outputs information that reflects the behavior of the plurality of fluid particles to the output unit.

2. 2. The simulation device according to claim 1, wherein the Lagrange multiplier method is used in calculating the shortest distance.

3. the simulation conditions include elasticity information of the object; The processing unit disposing a plurality of object particles that reproduce the object; calculating a force acting on each of the quadric surface elements from the plurality of fluid particles; calculating a force acting on each of the plurality of object particles based on the force acting on the quadric surface element; determining a stress or strain generated in the object based on the forces acting on the plurality of object particles; 3. The simulation device according to claim 1, wherein information representing stress or strain occurring in the object is output to the output unit.

4. 3. The simulation device according to claim 1, wherein the calculation of the shortest distance involves solving a quintic equation related to Lagrange multipliers, and calculating the shortest distance based on the solution.

5. A program that causes a computer to realize a simulation function based on given simulation conditions, The ability to define multiple quadric surface elements that represent the surface of an object in analytical space; a function of placing a plurality of fluid particles representing a fluid within the analysis space; a function of calculating the shortest distance from each of the plurality of fluid particles to the quadric surface element; a function of calculating a force acting on each of the plurality of fluid particles from the surface of the object based on the shortest distance; determining the behavior of the plurality of fluid particles based on the calculated forces; a function of outputting information reflecting the behavior of the plurality of fluid particles to an output unit; A program that makes the computer realize this.

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