Priority is claimed to Japanese Patent Application Nos. 2016-017588, filed Feb. 2, 2016, and 2015-173647, filed Sep. 3, 2015 the entire content of each of which is incorporated herein by reference.
Technical Field
Certain embodiments of the present invention relate to a simulation method, a simulation apparatus, and a simulation program capable of simulating a heat conduction phenomenon of a simulation target.
Description of Related Art
A simulation method based on a molecular dynamics method (MD method) is known as a method for researching phenomena of materials science in general by using computers on the basis of classical mechanics, quantum mechanics, or the like. A renormalization group molecular dynamics method (RMD method) which is a technique developed from the MD method in order to handle a macro scale system has been proposed. With respect to a heat conduction phenomenon of a simulation target, in the MD method or the RMD method, normally, only heat conduction based on lattice vibration (phonons) can be handled. For this reason, if a heat conduction phenomenon of metal in which free electrons greatly contribute to the heat conduction phenomenon is simulated using the MD method or the RMD method, in many cases, results far from an actual phenomenon may be obtained.
In the related art, a simulation method capable of correctly handling a heat conduction phenomenon using the MD method or the RMD method has been proposed. In such a simulation method, temperature parameters are given to particles. By solving a heat conduction equation based on a temperature of each particle at a certain time point, a temperature of each particle at the next time point is calculated. By solving the heat conduction equation, it is possible to perform a simulation in which specific heat and a thermal conductivity of an actual simulation target are considered.
According to an embodiment of the present invention, there is provided a method of performing a coupled simulation of a structural-elastic phenomenon and a heat conduction phenomenon of a simulation target including a plurality of particles, including: performing numerical calculation of a motion equation capable of being transformed into an equation of the same form as that of a heat conduction equation with respect to a term of a spatial temperature distribution and a term of a derivative of temperature with respect to time to perform a simulation of the heat conduction phenomenon of the simulation target.
According to another embodiment of the present invention, there is provided a simulation apparatus that performs a coupled simulation of a structural-elastic phenomenon and a heat conduction phenomenon of a simulation target including a plurality of particles, including: a storage unit that stores a motion equation capable of being transformed into an equation of the same form as that of a heat conduction equation with respect to a term of a spatial temperature distribution and a term of a derivative of temperature; a central processing unit that performs numerical calculation of the motion equation stored in the storage unit to calculate a temperature distribution of the simulation target; and an output unit that outputs a result of the numerical calculation performed by the central processing unit.
According to still another embodiment of the present invention, there is provided a simulation program that causes a computer to execute a coupled simulation of a structural-elastic phenomenon and a heat conduction phenomenon of a simulation target including a plurality of particles, in which the simulation program realizes a function of performing numerical calculation of a motion equation capable of being transformed into an equation of the same form as that of a heat conduction equation with respect to a term of a spatial temperature distribution and a term of a derivative of temperature with respect to time to perform a simulation of the heat conduction phenomenon of the simulation target.
Generally, temperature change due to a heat conduction phenomenon is smoother than change in particle positions due to a structural phenomenon or an elastic phenomenon (hereinafter, referred to as a “structural-elastic phenomenon”). A time pitch when a heat conduction phenomenon is simulated is set to be longer than a time pitch when a structural-elastic phenomenon is simulated. However, in a coupled simulation for handling both of a heat conduction phenomenon and a structural-elastic phenomenon, the heat conduction phenomenon may also be simulated with a short time pitch for simulating the structural-elastic phenomenon.
In analysis in a temperature field, generally, a temperature distribution in a steady state is attracting interest. If the heat conduction phenomenon is simulated with a short time pitch until the temperature distribution reaches a steady state, the calculation time is lengthened.
It is desirable to provide a simulation method capable of preventing a calculation time for handling a heat conduction phenomenon from being lengthened. Further, it is desirable to provide a simulation apparatus capable of preventing the calculation time from being lengthened. Furthermore, it is desirable to provide a simulation program capable of preventing the calculation time from being lengthened.
In a simulation of the heat conduction phenomenon, temperature parameters are given to respective particles, and calculation of numerical values of temperatures is performed with the same time pitch Δt as that in the dynamic analysis. When the calculation of the numerical values of the temperatures is performed, by solving a motion equation equivalent to a heat conduction equation instated of solving the heat conduction equation, analysis of a temperature field is performed. Here, the “equivalent” means that the motion equation can be transformed into an equation of the same form of the heat conduction equation with respect to a term of a spatial temperature distribution and a term of a derivative of temperature with respect to time of the thermal conduction equation. If the position of a mass point in the motion equation corresponds to “temperature”, a gradient of a temporal change of a temperature (first-order derivative of the temperature) corresponds to a speed in the motion equation. In this specification, the gradient of the temporal change of the temperature is referred to as a “rate of temperature change”.
In analysis of the temperature field, by setting a time constant of a temperature change to be variable, a sloe of the temperature change becomes steeper than a gradient of an actual temperature change. Thus, it is possible to reduce a time until a calculation result with respect to the temperature reaches a steady state. Here, the “steady state” means that the rate of temperature change at a position of each particle is zero.
If the gradient of the temperature change is made steep in the simulation, a temperature obtained in the simulation is overshot, so that a vibration waveform may appear. In this embodiment, by applying a method for relaxing the vibration in a short time, the reduction of the time until the temperature field reaches the steady state is achieved. As the method for relaxing the vibration in a short time, for example, a method called a fast inertial relaxation engine (FIRE) may be applied. The FIRE method is disclosed in ErikBitzek, Pekka Koskinen, Franz Gahler, Michael Moseler, and Peter Gumbsch, “Structural Relaxation Made Simple”, Physical Review Letters, 97, 170201 (2006).
Referring to
In step 11, dynamic analysis of each particle is executed for only one time pitch. For example, by numerically solving a motion equation of each particle, the position and speed of the particle after a time pitch Δt are calculated. In step 12, a calorific value is calculated based on the result of the dynamic analysis. The calorific value P is calculated using the following expression in a contact portion against a different member.
P=μ|F·vr|
Here, μ represents a frictional coefficient, F represents a force applied to the contact portion, and vr represents a relative speed.
In step 13, by solving the motion equation equivalent to the heat conduction equation using the calorific value of each particle, a temperature of each particle after the time pitch Δt is calculated.
In a case where the temperature field does not reach the steady state, in step 132, a parameter (attenuation coefficient) for regulating an attenuation term of the motion equation equivalent to the heat conduction equation is set to be smaller than an original value, or is set to be zero. Until the temperature field reaches the steady state, the motion equation is solved by setting the attenuation coefficient of the motion equation to be smaller than the original value or to be zero. Thus, it is possible to make the temperature change steeper than an original change.
In a case where the temperature field reaches the steady state, in step 133, the attenuation coefficient of the motion equation equivalent to the heat conduction equation becomes equal to the original value. The original value of the attenuation coefficient is determined based on a product of a density of a simulation target and a specific heat thereof, as described in detail later.
After step 132 or step 133, in step 134, numerical calculation of the motion equation equivalent to the heat conduction equation is executed for one time pitch. Here, a value set in step 132 or step 133 is used as the attenuation coefficient of the motion equation. After step 132, the procedure returns to the flowchart of
In step 14 of
In step 15, for example, if the rate of temperature change of each particle becomes equal to or lower than a reference value, it is determined that the temperature field reaches the steady state. In a case where it is determined that the temperature field reaches the steady state, the procedure returns to step 11, and the dynamic analysis is executed for the next one time pitch. In a case where it is determined that the temperature field does not reach the steady state, in step 16, it is determined whether a condition for setting the rate of temperature change to zero is satisfied. This determination is performed for each time pitch of the numerical calculation of the motion equation equivalent to the heat conduction equation.
In step 16, in a case where the determination condition is not satisfied, the procedure returns to step 11, and the dynamic analysis is executed for one time pitch. In step 16, in a case where it is determined that the determination condition is satisfied, in step 17, the rate of temperature change of each particle is set to be zero. Then, the procedure returns to step 11, and then, the dynamic analysis is executed for one time pitch. The technique of step 16 and step 17 is referred to as FIRE.
Then, the FIRE will be described with reference to
A broken line b in
In order to reduce the number of steps of numerical calculation until the temperature field enters the steady state, it is preferable that the attenuation coefficient is set to be small. If the attenuation coefficient is small, even if the same external force is applied, change in a calculation value of a temperature becomes large. Here, since the influence of an elastic term relatively becomes great, vibration occurs in the calculation value of the rate of temperature change.
If the motion equation equivalent to the heat conduction equation is solved by setting the attenuation coefficient to be small, as indicated by the broken line b, the change in the calculation result of the temperature obtained by the numerical calculation becomes steep. Further, the calculation value of the temperature comes close to the temperature in the steady state while being vibrated.
The calculation value of the temperature when the FIRE is applied is indicated by a solid line c. At a time point t1, it is assumed that the condition of step 16 (
An example of the condition of step 16 (
Parameters which serve as a presumption condition for executing a simulation is input from the input unit 22. The parameters include an inter-particle potential, an external field, a density, a specific heat, a heat conduction coefficient, and the like, for example. The initial state of the particle determined in step 10 (
A control program, a simulation program, and various parameters that define a motion equation numerically calculated in a simulation, and the like are stored in the storage unit 21. The simulation program realizes a function for performing the simulation shown in
Then, in step 13 (
A Lagrangian L of a classical scalar field ϕ may be expressed as the following expression as disclosed in Advanced Quantum Mechanics (Addison Wesley, 1967), for example.
Here, σs[ϕ(x)] represents a constraint condition. λs represents an undetermined constant of the Lagrangian. δ represents the δ Dirac function. An exponent 3 attached to δ and an exponent 3 attached to dx mean three-dimensions. Σ means summing up with respect to x. When x is a value other than xs, a value of the δ function is zero. In Expression (1), ρ(x), κ, and Q(x) do not mean specific physical quantities.
As the constraint condition σs[ϕ(x)], the Dirichlet boundary condition or the Neumman boundary condition may be used. The Dirichlet boundary condition may be expressed as the following expression.
[Expression 2]
σs[ϕ(x)]=ϕs−ϕs0 (2)
Expression (2) means that ϕ(x) is a constant when x=xs.
The Neumman boundary condition may be expressed as the following expression.
[Expression 3]
σs[ϕ(x)]=κ∇sϕs−qs (3)
Expression (3) means that a gradient of ϕ(x) is a constant when x=xs. ϕ(xs) is expressed as ϕ(s). ∇s represents a gradient at the position of x=xs. ϕs0 is a constant based on the constraint condition.
Expression (1) for defining the Lagrangian of a continuum is discretized on an one-dimensional superlattice, and a discretized scalar field is referred to as ϕi. The Lagrangian L of the discretized scalar field ϕi is expressed as the following expression.
Here, ΔVi represents a linear volume element. ∇i±1 represents a difference operator. The difference operator is defined as the following expression.
Here, Δ represents an inter-particle distance.
If the scalar field ϕi is considered as particle attributes, for example, a speed, a temperature, and the like, ϕi satisfies the following Lagrange's motion equation.
Here, γi represents an attenuation coefficient. From Expression (4) and Expression (6), the following motion equation with respect to an i-th particle is obtained.
Here, the first term on the right side in Expression (7) represents an elastic term, the second term thereof represents an attenuation term, the third term thereof represents an external force, and the fourth term thereof represents a constraint condition. j=±1 below Σ of the first term on the right side means that a sum with respect to particles on opposite sides of the i-th particle is used. s below Σ of the fourth term on the right side corresponds to the number of constraint conditions.
In a steady state where (d2/dt2)ϕi becomes zero, Expression (7) may be transformed as follows.
Since the first term on the right side in Expression (8) represents difference approximation of κ∇2ϕ(x), Expression (8) results in a diffusion equation. Expression (8) is obtained under that assumption that the left side in Expression (7) is zero. Accordingly, if ρi on the left side in Expression (7) is sufficiently small, it can be understood that a solution of Expression (6) comes close to a solution of the diffusion equation (8). That is, by solving Expression (6) using a numerical solution of an ordinary differential equation, it is possible to approximately solve the diffusion equation of Expression (8). As the numerical solution of the ordinary differential equation, the Verlet scheme, the Leap-frog scheme, the Gear method, or the like may be used.
The heat conduction equation may be expressed as the following expression.
Here, ρ represents density, Cv represents specific heat, T represents temperature, t represents time, K represents heat conductivity, and Q represents a calorific value per unit volume.
If ϕi, γi, κi, and Qi in Expression (8) are respectively considered as the temperature T, a product of the density ρ and the specific heat Cv, the heat conductivity K, and the calorific value Q, it can be understood that Expression (8) is the same form as that of the heat conduction equation of Expression (9). Specifically, a term of a spatial distribution of ϕi in Expression (8) and a term of a derivative of ϕi with respect to time may be associated with a term of a spatial temperature distribution and a term of a derivative of temperature with respect to time in Expression (9), respectively. Thus, under the condition that ρi is sufficiently small, it can be said that the motion equation of Expression (7) is equivalent to the heat conduction equation of Expression (9).
Hereinbefore, a case where the classical scalar field ϕ is discretized on the one-dimensional superlattice has been described. In order to expand the above-mentioned theory to a three-dimensional complicated shape, a finite volume method may be applied to Expression (7). If the finite volume method is applied to Expression (7), the following expression is obtained.
Here, Σ of the first term on the right side in Expression (10) means that summing-up is performed with respect to all particles which are in contact with an i-th particle. ΔSij represents an area of a contact surface between the Voronoi polyhedrons i and j. mi is expressed as the following expression.
[Expression 11]
mi=ρiΔVi (11)
mi is a parameter corresponding to a mass in the motion equation, and thus, mi is referred to as a virtual mass.
The virtual mass mi should satisfy the following lower limit condition from a stability condition of numerical integration.
With respect to a time pitch dt, an optimal value may be determined based on an inter-particle distance (mesh size) and a sound speed when the structural-elastic phenomenon is analyzed. The lower limit value of the virtual mass mi is determined from the time pitch dt determined based on the structural-elastic phenomenon and Expression (12). Further, as described above, under the condition that ρi is sufficiently small, the solution of Expression (6) comes close to the solution of the diffusion equation (8). Thus, it is preferable that ρi is sufficiently smaller than a true density ρ of a simulation target. That is, it is preferable that the following expression is satisfied.
[Expression 13]
ρΔVi>>mi (13)
By replacing γi in Expression (10) with a product of a density ρi and a specific heat Cvi and replacing ϕi with a temperature Ti, the following expression is obtained. Cvi represents a specific heat at the position of the i-th particle.
By solving Expression (14) which is the same form as that of the motion equation, it is possible to perform a simulation of a temperature field. Since ϕi in Expression (10) is replaced with Ti in Expression (14), the dimension of mi in Expression (14) is different from the dimension of mi in Expressions (10) and (11). It can be understood that the dimension of the virtual mass mi is the same as the dimension of the product of the density ρi, the specific heat Cv, the volume element ΔVi, and the time pitch dt from a term including ΔVi on the right side in Expression (14). That is, the dimension of the virtual mass mi becomes [J/K] [s].
From Expression (12), a satisfactory condition of the virtual mass mi is given as the following expression.
Furthermore, a satisfactory condition of the virtual mass mi is empirically given as the following expression from a result of various numerical calculations.
[Expression 16]
ρiCviΔViτ>>mi
10dt<τ<100dt (16)
From Expression (15) and Expression (16), it is recommended that mi be set as expressed in the following expression, as an example.
[Expression 17]
mi=10ρCvΔVidt (17)
As described above, by solving the motion equation (14) equivalent to the heat conduction equation (9), it is possible to perform analysis of a temperature field.
In step 13 (
Then, in step 132 (
A rod-shaped model shown in
Heat energy jA per unit area and per unit time flowing from the central region 30 to one intermediate region 31 is expressed as the following expression.
When a temperature of the end portion region 33 is represented as T and a heat transfer coefficient between the end portion region 33 and the hot bath is represented as h, heat energy jB per unit area and per unit time flowing from the end portion region 33 to the hot bath is expressed as the following expression.
[Expression 19]
jB=h(T−T0) (19)
The heat energy per unit time flowing from the central region 30 to the intermediate region 31 is the same as the heat energy per unit time flowing from the end portion region 33 to the hot bath. Thus, the following expression is established.
[Expression 20]
jASi=jBSj (20)
The following expression is derived from Expression (18) to Expression (20).
From Expression (21), it can be understood that a surface temperature in a steady state is determined only a calorific value from a heat source and a surface area of a simulation target. The surface temperature in the steady state does not depend on a density and a specific heat. Accordingly, even if the motion equation (14) is solved in a case where the attenuation coefficient ρiCvi in Expression (14) is set to an arbitrary value or zero, it is possible to calculate the surface temperature in the steady state.
If the attenuation coefficient ρiCvi is set to be small, as indicated by the broken line b in
If the attenuation coefficient ρiCvi is set to be zero, the vibration is not attenuated and is continuously present. In this embodiment, the vibration of the calculation values of the temperatures is converged in a short time by applying FIRE thereto, as indicated by the solid line c in
During a period of time until a temperature field reaches a steady state, that is, a period of time when an attenuation coefficient is set to a value different from an original value to execute step 13, a substantial meaning of a time pitch when solving a motion equation equivalent to a heat conduction equation is different from the time pitch applied to dynamic analysis in step 11 (
In order to check effects of the simulation method according to this embodiment, a simulation of a temperature field of an object was performed. Hereinafter, the simulation result will be described with reference to
Heat energy flows from the right end of the object 41 to the object 41. Heat is transferred from the object 41 to the object 42 through a contact surface between the right end of the object 41 and the left end of the object 42. Heat energy flows out from the left end of the object 42. Heat energy per unit time flowing into the object 41 and heat energy per unit time flowing out from the left end of the object 42 are equal to each other. Thus, heat balance between the object 41 and the object 42 is maintained as zero. A temperature field of the objects 41 and 42 finally reaches a steady state. That is, the rate of temperature change becomes zero.
It can be understood that during a period of time taken until a temperature of the left end of the object 42 reaches a temperature of the steady state, a rate of temperature change calculated by the simulation according to this embodiment is larger than a rate of temperature change calculated using the finite element method. Thus, it is possible to cause the heat conduction phenomenon to reach the steady state with a small number of time pitches. This is an effect obtained by setting the attenuation coefficient of the motion equation equivalent to the heat conduction equation in step 132 (
If the attenuation coefficient of the motion equation is set to zero, a vibration waveform normally appears in a simulation result. However, a vibration waveform does not appear in the simulation result illustrated in
In the example illustrated in
If the calculation value of the temperature reaches about 240 K, it is determined in step 131 (
As illustrated in
A preferable time pitch when a temperature field is simulated is about 10000 times a preferable time pitch when a structural-elastic phenomenon is simulated. Accordingly, in a method of executing analysis of the temperature field and analysis of the structural-elastic phenomenon in combination with the same time pitch, the number of time pitches in numerical calculation until the temperature field reaches the steady state becomes massive. Thus, in reality, it is not possible to perform a simulation until the temperature field reaches to the steady state.
In the above-described embodiment, since it is possible to arbitrarily set the attenuation coefficient of the motion equation equivalent to the heat conduction equation, it is possible to make a time constant of the temperature change close to a time constant of the structural-elastic phenomenon. Thus, by coupling the analysis of the heat conduction phenomenon and the analysis of the structural-elastic phenomenon, it is possible to perform the simulation until the heat conduction phenomenon reaches the steady state.
Then, before describing another embodiment, a simulation method based on a reference example will be described with reference to
First, it is determined in step 1351 whether each particle in a temperature field satisfies an acceleration algorithm application condition, for each particle.
Hereinafter, the acceleration algorithm application condition will be described. A determination parameter Pi is defined as follows.
Fi is a product of a second-order derivative of a temperature Ti of an article i and a virtual mass mi. This Fi corresponds to a force for changing a temperature of the particle i. When Fi is positive, it is considered that a force in a direction where a temperature increases acts on the particle i, and when Fi is negative, it is considered that a force in a direction where the temperature decreases acts on the particle i. The sign of Fi is the same as the sign of an acceleration of change in the temperature Ti. Vi corresponds to a rate of temperature change of the particle i.
When the determination parameter Pi is positive, the acceleration algorithm application condition is satisfied, and when the determination parameter Pi is negative or zero, it is determined that the acceleration algorithm application condition is not satisfied.
Then, a physical meaning of the determination parameter Pi will be described. The fact that the determination parameter Pi is positive means that the sign of the acceleration of change in the temperature Ti and the sign of the rate of change in the temperature Ti are the same. That is, in a case where the determination parameter Pi is positive, when the temperature Ti increases, a force for further increasing a rising rate of the temperature Ti acts on the particle i. Contrarily, when the temperature Ti decreases, a force for further increasing a lowering rate of the temperature Ti acts on the particle i. This state may be considered as a state where the temperature Ti does not reach the steady state and is changing toward the steady state.
The fact that the determination parameter Pi is negative means that a force for decreasing the rising rate of the temperature Ti or a force for decreasing the lowering rate of the temperature Ti acts on the particle i. Accordingly, when the determination parameter Pi is negative, it is considered that the temperature Ti of the particle i is close to the steady state.
When the determination parameter Pi is positive, it is determined that the acceleration algorithm application condition is satisfied, and when the determination parameter Pi is negative, it is determined that the acceleration algorithm application condition is not satisfied. In a case where the determination parameter Pi is zero, it may be determined that the acceleration algorithm application condition is satisfied or is not satisfied. In this embodiment, when the determination parameter Pi is zero, it is determined that the acceleration algorithm application condition is not satisfied.
In a case where it is determined in step 1351 that the acceleration algorithm application condition is satisfied, that is, in a case where the sign of the rate of change in the temperature Ti and the sign of the acceleration of change in the temperature Ti are the same, in step 1352, the attenuation coefficient of the motion equation equivalent to the heat conduction equation is set to be smaller than a value at a current time point.
Specifically, ρiCviΔVi which is the attenuation coefficient (coefficient of Ti dot) of the motion equation (Expression 14) equivalent to the heat conduction equation is set to be smaller than a value at a current time point. For example, the attenuation coefficient is set to be 0.5 times a value at a current time point. By setting the attenuation coefficient to be small, it is possible to make the temperature change steeper.
In a case where it is determined in step 1351 that the acceleration algorithm application condition is not satisfied, that is, in a case where the sign of the rate of change in the temperature Ti and the sign of the acceleration of change in the temperature Ti are different from each other, in step 1353, ρiCviΔVi which is the attenuation coefficient (coefficient of Ti dot) of the motion equation (Expression 14) equivalent to the heat conduction equation is restored to an original value.
After steps 1352 and 1353, in step 1354, the motion equation (Expression 14) is executed for one time pitch using the attenuation coefficient after correction. Then, in step 14, it is determined whether a simulation termination condition is satisfied. The determination condition may be set to be the same as that in step 14 illustrated in
Then, a result obtained by actually performing a simulation using the simulation method illustrated in
When a contact force of a particle i and a particle j is represented as fij, a relative speed of the particle i and the particle j is represented as vij, and a frictional coefficient between the particle i and the particle j is represented as μij, a calorific value Qi per unit volume of the particle i is expressed as the following expression.
When comparing
Hereinafter, a cause of the vibration of the overall temperature vibrates even after the temperature field reaches the steady state will be described. In the simulation target illustrated in
Next, an embodiment capable of removing vibration of the overall temperature will be described with reference to
In the embodiment illustrated in
As a determination condition relating to whether the attenuation coefficient is to be fixed, an elapsed time from the start of a simulation or the number of time pitches may be used. In this case, if the elapsed time or the number of time pitches exceeds a determination upper limit value, the attenuation coefficient may be fixed to an original value. At a time point determined that the temperature field reaches the steady state, the attenuation coefficient may be fixed to the original value.
Since the simulation is performed in a similar way to the reference example shown in
Next, still another embodiment will be described with reference to
If it is determined in step 1351 that the acceleration algorithm application condition is satisfied, even after the attenuation coefficient is restored to the original value in step 1353, the attenuation coefficient becomes smaller than the original value again. On the other hand, if the virtual mass mi is once increased, even in a case where it is determined that the acceleration algorithm application condition is satisfied in step 1351, the virtual mass mi does not return to the original value. That is, the virtual mass mi maintains the increased state.
Hereinafter, the process of step 1355 will be described in detail. In step 1355, the virtual mass mi is set to be 1.05 times a value at a current time point, for example. Here, in a case where the virtual mass mi after increase exceeds a predetermined upper limit value mimax, the virtual mass mi is set to be equal to the upper limit value mimax. Accordingly, there is no case where the virtual mass mi exceeds the upper limit value mimax.
Times until a temperature field reaches a steady state are approximately the same in the reference example illustrated in
In the embodiment illustrated in
Since the virtual mass mi and the attenuation coefficient return to the original values after the overall temperature is converged on a constant value without vibration, the same termination state as in
Hereinbefore, the present invention has been described with reference to the embodiments, but the present invention is not limited thereto. It is obvious to those skilled in the art that various modifications, replacements, combinations and the like may be made, for example.
It should be understood that the invention is not limited to the above-described 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.
Number | Date | Country | Kind |
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2015-173647 | Sep 2015 | JP | national |
2016-017588 | Feb 2016 | JP | national |
Number | Name | Date | Kind |
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6915245 | Hinton | Jul 2005 | B1 |
Number | Date | Country |
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2899655 | Jul 2015 | EP |
5441422 | Mar 2014 | JP |
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Number | Date | Country | |
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20170068759 A1 | Mar 2017 | US |