JPH10177591A - Numerical analyzer, its method, and integrated circuit - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a numerical analyzer which can perform numerical analysis in a short time with a simple constitution. SOLUTION: A numerical analyzer is provided with a discretized nabla operator calculating means 8 which calculates a discretized nabla operator which is obtained by discretizing the nabla operator of spatial differentiation based on the element information of each element obtained by dividing a field to be analyzed and a physical quantity calculating means 9 which calculates such a physical quantity that meets the continuous expression of the field to be analyzed by repetitively modifying the physical quantity until the physical quantity can meet the continuous expression by using the calculated discretized nabla operator. Therefore, the numerical analyzer can make numerical analysis in a short time with a simple constitution by reducing the time required for the numerical analysis.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

[Table of Contents] The present invention will be described in the following order.

BACKGROUND OF THE INVENTION Problems to be Solved by the Invention Means for Solving the Problems Embodiments of the Invention (1) First Embodiment (1-1) Principle (1-2) Overall Configuration (1-3) Deltip and Poisson solver tips (1-3-1) Discretized nabla operator (FIG. 5) (1-3-2) Solving Poisson equation (1-3-3) ) Configuration of Delchip (FIGS. 6 to 13) (1-3-4) Configuration of Poisson solver tip (FIGS. 14 to
(FIG. 19) (1-4) Operation and Effect (FIGS. 20 and 21) (2) Second Embodiment (3) Other Embodiments (FIGS. 22 and 23) Effects of the Invention

[0003]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a numerical analysis device, a numerical analysis method, and an integrated circuit, and is suitably applied to, for example, a numerical analysis device that performs a numerical analysis using a finite element method.

[0004]

2. Description of the Related Art Conventionally, in the fields of electronic devices, aerospace, automobiles, etc., various numerical analysis methods are used to analyze heat, fluid, stress, electromagnetic field problems, and the like. The thermo-fluid analysis is used, for example, to analyze the temperature distribution in the electronic device and to examine the installation location of the cooling fan. Electromagnetic fluid analysis is used, for example, to study how to apply electromagnetic waves that can be efficiently transported when transporting mercury using electromagnetic waves in a nuclear reactor. Also, the viscoelastic fluid analysis is used, for example, to study an optimal method of injecting a polymer material when molding using a polymer material. The solid-state stress analysis is used, for example, to study what kind of stress is applied when a force is applied to the housing, and is used to study the strength of the housing.

[0005] Typical examples of such numerical analysis methods include a difference method, a finite volume method, a finite element method and a boundary element method. Although each of these methods has advantages and disadvantages, the finite element method has recently attracted attention because of its high versatility and easy incorporation of boundary conditions. The details of the finite element method are disclosed in documents such as "Takahiko Tanahashi: Computational Fluid Dynamics, 1994, IPC". In any case, when analyzing the field using these methods,
It is a general method to perform calculation using a workstation or a general-purpose computer by giving initial conditions and boundary conditions to each physical quantity, instead of performing it manually.

[0006]

However, in the conventional numerical analysis method, a complicated coefficient matrix must be calculated during the analysis. Therefore, if a workstation or a personal computer is used, an enormous amount of time is required for the numerical analysis. There is an inconvenience. In fact, even if the workstation has a higher computing power than a personal computer, the computing power is about 100 [MFlops], and it takes 10 to 10 hours to analyze about tens of thousands of elements. It is.

As a method for solving this problem, a method of numerical analysis using a supercomputer having a high computing ability (computational ability is equal to or more than a number [GFlops]) can be considered, but a supercomputer generally has a very large cost of about 10 billion yen. It is expensive, requires an installation area as large as a room, and requires a very large amount of power consumption. There is a disadvantage that it cannot be used easily. Such numerical analysis is mainly used for product evaluation, and it is inefficient to spend enormous cost and time in the process of product design.

The present invention has been made in view of the above points, and has as its object to propose a numerical analysis apparatus, a numerical analysis method, and an integrated circuit that can perform numerical analysis in a short time with a simple configuration.

[0009]

According to the present invention, there is provided a numerical analysis apparatus for calculating a physical quantity of a field to be analyzed, in which the element information of each element obtained by dividing the field to be analyzed is provided. An analysis means is provided for calculating a discretized nabla operator obtained by discretizing the spatial derivative nabla operator based on the calculated value, and calculating a physical quantity using the calculated discretized nabla operator. When the discretized nabla operator is calculated in this manner and the physical quantity is obtained using the discretized nabla operator, the time required for the numerical analysis can be reduced because the discretized nabla operator can be calculated by a simple vector operation. Can be reduced. Further, since the discretized nabla operator is calculated, the memory storage of the coefficient matrix as in the related art is not required, so that the configuration of the numerical analysis device can be simplified.

Further, in the present invention, in a numerical analysis apparatus for calculating a physical quantity of a field to be analyzed, a Nabla operator for spatial differentiation is discretely based on element information of each element obtained by dividing the field to be analyzed. By using a discretized nabla operator calculation means for calculating a discretized nabla operator and a discretized nabla operator, the physical quantity is repeatedly corrected so as to satisfy the continuity equation of the field to be analyzed. And a physical quantity calculating means for calculating a physical quantity satisfying the continuous equation. Calculating the discretized nabla operator in this way,
By using the calculated discretized nabla operator to iteratively correct the physical quantity to satisfy the continuous equation, the Poisson equation derived from the field to be analyzed can be easily solved, and the time required for numerical analysis Can be reduced.
The use of the discretized nabla operator eliminates the need for memory of the coefficient matrix as in the related art, so that the configuration of the numerical analysis apparatus can be simplified.

Further, in the present invention, a discretized Nabla operator obtained by discretizing a Nabla operator for spatial differentiation is calculated based on element information of each element obtained by dividing a field to be analyzed. Divide the field to be analyzed and the calculation board that calculates the physical quantity that satisfies the continuity equation by iteratively modifying the physics quantity so as to satisfy the continuity equation of the field to be analyzed by using the modified nabla operator And a computer for supplying element information to the calculation board to calculate the physical quantity and for displaying the physical quantity calculated by the calculation board as a numerical analysis result to constitute a numerical analysis apparatus. I did it. By calculating the discretized nabla operator in this way, and by repeatedly correcting the physical quantity using the discretized nabla operator, by providing a calculation board for calculating a physical quantity that satisfies the continuous equation, The Poisson equation derived from the field to be analyzed can be easily solved, and the time required for numerical analysis can be reduced.

In the present invention, the physical quantity calculated by the computer is displayed in real time by displaying the physical quantity received by the computer from the computer during the iterative calculation. By displaying the physical quantities during the iterative calculation in real time in this way, it is possible to determine how far the analysis is currently progressing in the numerical analysis device and what kind of analysis result is currently obtained. You can easily find out when. Therefore, by looking at this display, it is possible to judge whether the initial conditions and boundary conditions are optimal without waiting for the final result, and if inappropriate, correct those conditions immediately and perform the numerical analysis again. As a result, numerical analysis can be performed efficiently.

Further, in the present invention, element information of each element obtained by dividing the field to be analyzed is input, and based on the element information, a discretized nabla operation is performed by discretizing a navel operator for spatial differentiation. The discretized nabla operator calculation circuit for calculating a child is integrated into an integrated circuit. By thus integrating the discretized nabla operator calculation circuit into an integrated circuit, the numerical analysis device can be further reduced in size, and the configuration of the numerical analysis device can be simplified.

According to the present invention, a discretized nabla operator obtained by discretizing a spatial differential nabla operator is input, and the expression of the continuity of the field to be analyzed is satisfied using the discretized nabla operator. The physical quantity calculation circuit that calculates the physical quantity that satisfies the continuity equation by iteratively correcting the physical quantity in such a manner as described above is integrated. By thus integrating the physical quantity calculation circuit into an integrated circuit, the numerical analysis device can be further reduced in size, and the configuration of the numerical analysis device can be simplified.

[0015]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below in detail with reference to the drawings.

(1) First Embodiment (1-1) Principle In this section, the principle of the present invention will be described first. In a conventional numerical analysis device using a general-purpose computer, the installed CPU (central processing unit) is general-purpose, so that the compiled numerical analysis software becomes redundant. As a result, exchange between registers and access to memory are performed. It took enormous amount of calculation time.
In general, when the finite element method is used for numerical analysis, a large amount of data such as nodes and element information, coefficient matrices, calculation results, and the like must be handled. Therefore, a large memory capacity is required. . By the way, in a device equipped with a small capacity memory such as a general-purpose computer,
The shortage of memory capacity must be compensated for by a hard disk that requires more time to access than the memory. As a result, a general-purpose computer has a problem that the calculation time further increases.

Therefore, the present invention focuses on this point and reduces the time required for numerical analysis by reducing the number of exchanges between registers and memory access, thereby reducing the time required for numerical analysis even in a simple configuration. To realize a numerical analysis device that can perform the analysis. First, in the present invention, a finite element method called GSMAC is used as a numerical analysis technique. This GSM
AC finite element method is a MAC method developed in the field of difference method,
The technique of the SMAC method and the HSMAC method is applied to the finite element method. The feature of the difference method is that the boundary shape of the analysis model needs to be simple, while the shape of the analysis model has arbitraryness. It is expensive and therefore has a wide range of application. The GSMAC finite element method includes heat, fluid,
It can be applied to a wide range of unsteady problems such as stress field and electromagnetic field, and has a feature that numerical analysis can be performed with high accuracy. Furthermore, in the case of the GSMAC finite element method, the Poisson equation is derived when the Navier-Stokes equation is applied to the field to be analyzed. However, the Poisson equation is easily reduced by simultaneously relaxing the field vector and energy. There is a special effect that can be solved. By using the GSMAC finite element method having such features as a numerical analysis technique, the present invention can reduce the time required for numerical analysis.

In the present invention, a discretized nabla operator obtained by a vector operation is used as a nabla operator (that is, an operator of spatial differentiation) used when solving a Poisson equation. Thus, in the present invention,
The time required to calculate the nabla operator can be reduced. In addition, since the time for calculating the Nabla operator can be reduced, the Nabla operator may be calculated each time, and the memory capacity can be reduced because there is no need for storage. In addition, since it is not necessary to access a hard disk unlike the related art, the time required for numerical analysis can be further reduced. By using the discretized nabla operator in this way, the present invention can reduce the time required for numerical analysis as a whole and simplify the configuration of the numerical analysis device.

(1-2) Overall Configuration In FIG. 1, reference numeral 1 denotes a numerical analysis apparatus to which the present invention is applied as a whole. A personal computer or a work station is used to manage the numerical analysis operation of the entire numerical analysis apparatus 1. Is provided. The host computer 2 includes a main body 2A in which various electric circuits such as a CPU (Central Processing Unit), a memory, and a hard disk are stored, and a monitor 2B for displaying various information such as analysis results.

The main body 2A of the host computer 2 is connected to a keyboard (not shown) for inputting initial conditions and boundary conditions, which are parameters for numerical analysis, and a predetermined transmission path (for example, PCI). (Peripheral
An expansion unit 5 is connected via a Component Interconnect (bus) 3. A plurality of calculation-dedicated boards 4 are mounted in the extension unit 5 as means for analyzing physical quantities, so that the host computer 2 and the plurality of calculation-dedicated boards 4 are connected via the transmission path 3. ing. Here, a plurality of dedicated calculation boards 4 are connected to the host computer 2 in order to perform parallel calculation using the plurality of dedicated calculation boards 4. One substrate 4 may be used.

As shown in FIG. 2, an interface circuit 6, a memory 7, and a chip 8
And a Poisson solver chip 9 are implemented. The interface circuit 6 receives the node and element information S1 sent from the host computer 2 via the transmission line 3, and stores them in the memory 7 sequentially. Deruchitsupu 8 is a discrete nabla operator calculating means for calculating a discrete nabla operator ∇ _a and element volume V _e will be described later, each time it is necessary to discretize nabla operator ∇ _a later Poisson solver multichip 9, It reads the nodes and elements information S1 from the memory 7, and calculates a discrete nabla operator ∇ _a and element volume V _e on the basis of the nodes and elements information S1 read.

The Poisson solver 9 is a physical quantity (specifically, a vector and energy of a field to be analyzed, which satisfies a continuous equation for each time step, as will be described later. And energy H), and the discretized nabla operator ∇ _a and element volume V _e calculated by Delchip 8
Is used to calculate the physical quantity of the element. The calculated physical quantity is sent to the host computer 2 as calculation result information S2 via the interface circuit 6 and the transmission path 3. That is, in the numerical analysis device 1, a part for solving the Poisson equation which requires the longest time among the analysis times (that is, a part for calculating a physical quantity) and a part for calculating a coefficient matrix required for the solution method (that is, a discretized nabla operator ∇ _a part for calculating a) is obtained by the calculation dedicated substrate 4.

Here, the numerical analysis device 1 having such a configuration operates based on numerical analysis software built in the host computer 2.
This software can be roughly divided into three types: software called a preprocessor, software called an analysis solver, and software called a postprocessor.
It is constituted by one.

The preprocessor is software for creating an analysis model, and divides a field to be analyzed into each element and sets initial conditions and boundary conditions. The post processor is software for displaying an analysis result on the monitor 2B using the obtained physical quantity or for evaluating the analysis result. On the other hand, the analysis solver is software for obtaining a physical quantity for each element. In this embodiment, the physical solver is obtained by using the GSMAC finite element method. Although this analysis solver basically operates on the host computer 2, a part that closes most of the calculation time (that is, a part that calculates a physical quantity by solving a Poisson equation, and a discretized nabla operator ∇ _a required in the solution method) Is calculated using the dedicated calculation board 4 by activating the dedicated calculation board 4.

Here, a numerical analysis procedure in the numerical analysis apparatus 1 will be described using a flowchart shown in FIG. In the numerical analysis apparatus 1, in step SP2 entered from step SP1, the preprocessor is started to create an analysis model (that is, to divide an analysis target field into elements), and to calculate on each calculation dedicated board 4. Allocate elements. Incidentally, in the numerical analysis device 1, since a plurality of calculation-dedicated substrates 4 are provided to parallelize the calculation processing, such an allocation processing is performed. As an allocation method, adjacent elements are allocated to different calculation-dedicated substrates 4 so that adjacent elements are not allocated to the same calculation-dedicated substrate 4.

In the next step SP3, the analysis solver is started, and the node and element information S1 are supplied to each dedicated calculation board 4 based on the allocation performed by the preprocessor, and the respective dedicated calculation board 4 is used. To determine the physical quantity of each element. Incidentally, the calculation of the physical quantity, using the discrete nabla operator ∇ _a and element volume V _e calculated in Deruchitsupu 8, obtained by solving the Poisson equation Poisson solver multichip 9.

In the next step SP4, it is determined whether or not the physical quantities have been calculated for all the elements. If the calculation has not been completed for all the elements, the procedure returns to step SP2 to allocate the remaining elements. The physical quantity is calculated again by the analysis solver, and when the calculation is completed for all the elements, the process proceeds to the next step SP5. In step SP5, the post processor is started, and the analysis result is displayed on the monitor 2B using the physical quantities of the respective elements obtained by the analysis solver. When this is completed, the process moves to the next step SP6, and the numerical analysis processing ends.

Next, an analysis solver which is the most important part in the numerical analysis procedure will be described with reference to a flowchart shown in FIG. In the analysis solver, step SP
In step SP11 entered from step 10, first, the node and element information S1 are loaded from the host computer 2 into the memory 7 of the dedicated calculation board 4. In the next step SP 12, Deruchitsupu 8 reads the nodes and elements information S1 is stored in the memory 7, obtaining the discretization nabla operator ∇ _a and element volume V _e of the element based on the nodes and elements information S1 . Then, in the next step SP13, the delchip 8 calculates the discretized nabla operator ∇ _a
And the element volume V _e to the Poisson solver chip 9.

[0029] At next step SP14, the physical quantity of elements Poisson solver multichip 9 satisfies the continuity equation using discretization nabla operator ∇ _a and element volume V _e (i.e. ∇ · v = 0) (i.e. Find the field vector v and energy H). Specifically, the Poisson equation is solved by repeatedly correcting the physical quantity so as to satisfy the continuity equation, and thereby the physical quantity satisfying the continuity equation is obtained. In the next step SP15, the Poisson solver 9 converts the obtained physical quantity into an interface circuit 6.
And to the host computer 2 via the transmission path 3. When this ends, the process moves to the next step SP16 and ends. Incidentally, the analysis solver shown here is performed in parallel on each calculation-dedicated board 4.

(1-3) Deltip and Poisson solvers (1-3-1) Discretized nabla operator This section describes the discretized nabla operator. The discretized nabla operator ∇ _a is _a discretized version of the spatial derivative nabla operator ∇, and the elemental mean of the gradient of the shape function is given by

(Equation 20) Is defined by Where a is the local node number of the element,
N _a is a shape function. Ω _e is an element area S _e in the case of two dimensions, and is an element volume V _e in the case of three dimensions.
Incidentally, all the integral calculations here are in the physical space (x, y,
z), but not in the computation space (ξ, η, ζ). That is, the calculation is performed after being mapped from the physical space to the calculation space.

Here, when the three-dimensional discretized nabla operator ∇ _a is redefined by the field integration method, the following equation is obtained.

(Equation 21) It becomes as shown in. In this case, if the hexahedron shown the shapes of the elements in FIG. 5, the shape function N _a are each eight nodal (coordinates (-1, -1, -1), (+ 1, -1, -1),
(-1, + 1, -1), (-1, -1, + 1), (+ 1, -1 ,, + 1),
As local coordinates consisting of (-1, + 1, + 1), (+ 1, + 1, -1), (+ 1, + 1, + 1),

(Equation 22) It becomes as shown in. Where a = 1 to 8 and l _a , ξ _a , η _a , ζ _a , ξ _a η _a , η _a ζ _a , ζ _a ξ
_a and ξ _a η _a ζ _a are linearly independent basis vectors.

[0032] Incidentally the calculation is performed is mapped from the physical space to the calculation space as described above, in which one combines the physical space and computation space, the Jacobian matrix [J _ij] and the inverse matrix of the Jacobian [J _ij ] ^-1 . The Jacobian inverse matrix [J _ij ] ^-1 is given by the following equation, where the cofactor matrix is [A _ij ].

(Equation 23) Can be described as follows. Where J is Jacobian and its definition is

(Equation 24) become that way. Each component of the cofactor matrix [A _ij ] is expressed by the following equation.

(Equation 25) Is defined by

Further, the element average of each component of the cofactor matrix is calculated by using the average value of the increment (Δr) of the position vector in the ξ, η, and ζ directions as follows:

(Equation 26) It is represented as By the way, the average value of the vector Δr in each direction is given by

[Equation 27]

[Equation 28]

(Equation 29) It is represented as

Here, when the components of the discretized nabla operator by the volume integral method shown in the equation (21) are described in the calculation space, the following equation is obtained.

[Equation 30] It is represented as Using element averaging here, this equation becomes

(Equation 31) It can be transformed as follows. The element average of the derivative of the shape function is

(Equation 32) And the mean of the elements of the cofactor matrix <A _{ij> e}
Is obtained from the definition formula of the cofactor matrix, so the approximate expression of the discretized nabla operator ∇ _a by the volume integral method is

[Equation 33] become that way. Further, this equation uses the average value in each direction of the vector Δr shown in equations (27) to (29), and

(Equation 34) It can be transformed into At this time, the element volume V _e is given by

(Equation 35) It is represented as

[0035] The use of this discrete nabla operator ∇ _a,
The gradient, divergence and rotation are given by

[Equation 36]

(37)

(38) It is represented as

[0036] If to determine the discrete nabla operator ∇ _a as an operator for spatial differentiation required the slave connexion every field analysis, calculated since performed by a simple vector operations as shown in Equation (34) Time can be significantly reduced. By the way, when the coefficient matrix was calculated using the Gauss-Legendre numerical integration formula as in the past, it was necessary to store the coefficient matrix once calculated in the memory because the calculation took a long time. If such a discretized nabla operator ∇ _a is used, the calculation does not take much time unlike the conventional case, and the calculation may be performed each time. By using the Supporting connexion discrete nabla operator ∇ _a, becomes a memory of enormous capacity as in the prior art unnecessary, it is possible to sufficiently cope with even less memory generic computer. It should be noted that the Delchip 8 includes nodes and element information S stored in the memory 7.
1 is read out and based on the resulting node coordinates (34)
And (35) determine the discrete nabla operator ∇ _a and element volume V _e by calculating the equation, it is the main task.

(1-3-2) Solution of Poisson Equation In this section, the solution of the Poisson equation will be described. Here, it is assumed that the field to be analyzed is a velocity field, and the velocity vector v and energy H are obtained as physical quantities.
First, there are Navier-Stokes equations and continuity equations as motion equations governing the field to be analyzed. This equation is:

[Equation 39] It is described as follows. Applying the incompressible condition to this equation of motion gives

(Equation 40) The Poisson equation for the modified speed potential φ as shown in FIG. The Poisson equation is calculated using the following GSMAC finite element method algorithm to calculate the velocity v
And the total energy H is simultaneously relaxed.

First, an initial value at time t = 0 is given. Next, a predicted value of the speed is obtained. Next, a corrected speed potential is introduced, and the speed v and the energy H (Bernoulli function) are calculated by the following equation.

[Equation 41] Solve by the simultaneous relaxation method while satisfying. That is, the iterator k
= 0,

(Equation 42) Is set as the initial value, and the following equation

[Equation 43]

[Equation 44]

[Equation 45]

[Equation 46]

[Equation 47] Is performed. As a result, for example, assuming that ε = 0.001,

[Equation 48] Is satisfied, v ^{n + 1} = v ^{k + 1} , H ^{n + 1} = H ^{k + 1} , and the process proceeds to the next time step.

The Bernoulli function H is represented by the following equation.

[Equation 49] It is represented by ∇ is the Nabla operator for spatial differentiation,
Next formula

[Equation 50] It is represented by

As described above, the Poisson equation can be solved by simultaneously correcting the velocity v and the energy (ie, Bernoulli function) H. In addition, Poisson Solvachip 9
The, the use of discrete nabla operator ∇ _a supplied from Deruchitsupu 8 as nabla operator ∇, described above using elements volume V _e supplied from the Deruchitsupu 8 (42) - (47) The main task is to find the corrected velocity vector v and energy H by performing the iterative operation shown in the equation.

(1-3-3) Configuration of Delchip In this section, a specific configuration of the delchip 8 will be described. However, the notations shown in the figure indicate the contents as shown in FIG. 6, respectively. First, the delchip 8 is constituted by the blocks shown in FIGS. The coordinates of the eight local nodes read from the memory 7 (the local node coordinates are stored in the memory 7 as node and element information S1).
Are stored for each of the x, y, and z components and input to the 入 力 direction vector calculation unit 10, the η direction vector calculation unit 11, and the ζ direction vector calculation unit 12.

The ξ direction vector calculation unit 10 shown in FIG. 7 performs the calculation of the above equation (27), and calculates the average value of the vector Δr in the ξ direction. The x-coordinate (elm [ie] [1] .x) of the first node is input to the adder 10B after the sign is inverted by the sign inverter 10A, and
The x-coordinate (elm [ie] [2] .x) of the node is directly used as the adder 10B.
Is input to Accordingly, the x component of the vector (Δr) ₂₁ is calculated in the adder 10B. Also, the third node x
The coordinates (elm [ie] [3] .x) are directly input to the adder 10B, and the x-coordinate (elm [ie] [4] .x) of the fourth node is sign-inverted via the sign inverter 10C. Adder 10B
, The x component of the vector (Δr) ₃₄ is calculated in the adder 10B. The x coordinate (elm [ie] [5] .x) of the fifth node is input to the adder 10B after sign inversion through the sign inverter 10D, and the x coordinate (elm [ie ] [6] .x) is the adder 10
By being input to B, the adder 10B calculates the x component of the vector (Δr) ₆₅ . The x-coordinate (elm [ie] [7] .x) of the seventh node is directly input to the adder 10B, and the x-coordinate (elm [ie] [8] .x) of the eighth node is sign-inverted. Adder 1 after sign inversion via 10E
By being input to 0B, the adder 10B calculates the x component of the vector (Δr) ₇₈ . Adder 10
B represents the calculated vectors (Δr) ₂₁ , (Δr) ₃₄ ,
The x components of (Δr) ₆₅ and (Δr) ₇₈ are further added, and the resultant addition value is output to the multiplier 10F. The multiplier 10F multiplies the added value by 0.25 (that is, 1/4). Thus, the x component (dr [1] [1]) of the average value of the vector Δr in the ξ direction is calculated.

Similarly, the y-coordinates (elm [i
e] [1] .y to elm [ie] [8] .y) are direct or sign inverter 10
By being input to the adder 10H via G, 10I, 10J, and 10K, the vectors (Δr) ₂₁ , (Δr) ₃₄ , (Δr) _65, and (Δr) in the adder 10H.
The y component of ₇₈ is calculated, and the sum of them is calculated. The added value is input to the multiplier 10L, where
By multiplying by 0.25, the y component (dr [1] [2]) of the average value of the vector Δr in the ξ direction is calculated. Similarly, the z coordinate of the first to eighth nodes (elm [ie] [1] .z to elm [i
e] [8] .z) are direct or sign inverters 10M, 10P, 10
By being input to the adder 10N via Q and 10R, the vector (Δr) ₂₁ ,
The z components of (Δr) ₃₄ , (Δr) ₆₅ and (Δr) ₇₈ are calculated, and the sum of them is calculated. The added value is input to the multiplier 10S, where it is multiplied by 0.25, thereby calculating the z component (dr [1] [3]) of the average value of the vector Δr in the ξ direction.

The η direction vector calculator 11 shown in FIG.
Performs the calculation of the above equation (28), and calculates the average value of the vector Δr in the η direction. The η-direction vector calculation unit 11 is composed of a multiplier, an adder, and a sign inverter, like the ξ-direction vector calculation unit 10.
X coordinates of the first to eighth nodes (elm [ie] [1] .x to elm [ie] [8].
x), the x component (dr [2] [1]) of the average value of the vector Δr in the η direction is calculated, and the y coordinate (elm) of the first to eighth nodes is calculated.
Using [ie] [1] .y to elm [ie] [8] .y), the y component (dr [2] [2]) of the average value of the vector Δr in the η direction is calculated.
Using the z coordinate of the eighth node (elm [ie] [1] .z to elm [ie] [8] .z), the z component (dr [2] of the average value in the η direction of the vector Δr
[3]) is calculated.

The ζ direction vector calculator 12 shown in FIG.
Performs the calculation of the above equation (29), and calculates the average value of the vector Δr in the ζ direction. The ζ direction vector calculation unit 12 is also composed of a multiplier, an adder, and a sign inverter, like the ξ direction vector calculation unit 10.
X coordinates of the first to eighth nodes (elm [ie] [1] .x to elm [ie] [8].
x) is used to calculate the x component (dr [3] [1]) of the average value in the ζ direction of the vector Δr, and the y coordinate (elm) of the first to eighth nodes is calculated.
Using [ie] [1] .y to elm [ie] [8] .y), the y component (dr [3] [2]) of the average value of the vector Δr in the ζ direction is calculated.
Using the z coordinate of the eighth node (elm [ie] [1] .z to elm [ie] [8] .z), the z component (dr [3] of the average value in the η direction of the vector Δr
[3]) is calculated.

The average values (dr [1] [1] to dr [3] [3]) of the vector Δr thus obtained in the η, η, and ζ directions are, as shown in FIG. The data is input to the volume calculator 13. The element volume calculation unit 13 performs the calculation of the above equation (35), and calculates the average value of the vector Δr in the ξ, η, and ζ directions (dr [1] [1] to dr [3] [3] ) Is used to calculate the element volume V _e . First, the y component (dr [1] [2]) in the ξ direction and the z component (dr [2] [3]) in the η direction are input to the multiplier 13A and multiplied, and the multiplication result is input to the adder 13B. Is done. The z component (dr [1] [3]) in the ξ direction and the y component (dr [2] [2]) in the η direction are input to the multiplier 13C and multiplied.
The result of the multiplication is input to the adder 13B after the sign is inverted by the sign inverter 13D. The adder 13B adds each of the input multiplication results to obtain the x of the outer product.
The component is calculated and output to the multiplier 13E. The x component (dr [3] [1]) in the ζ direction is input to the multiplier 13E, and the multiplier 13E multiplies the x component of the outer product by the x component in the ζ direction to obtain the outer product and the inner product. The x component is calculated and output to the adder 13F.

The z component (dr [1] [3]) in the ξ direction and the x component (dr [2] [1]) in the η direction are input to a multiplier 13G and multiplied. 13H. The x component in the ξ direction (dr [1] [1]) and the z component in the η direction (dr
[2] [3]) are input to the multiplier 13I and multiplied. The result of the multiplication is input to the adder 13H after the sign is inverted by the sign inverter 13J. The adder 13H calculates the y component of the outer product by adding the input multiplication results, and outputs this to the multiplier 13K. Multiplier 13K
Is input with the y component of the ζ direction (dr [3] [2]), and the multiplier 13K multiplies the y component of the outer product by the y component of the ζ direction to obtain the y components of the outer product and the inner product. The calculated value is output to the adder 13F.

The x component (dr [1] [1]) in the ξ direction and the y component (dr [2] [2]) in the η direction are input to the multiplier 13L and multiplied. 13M. The y component in the ξ direction (dr [1] [2]) and the x component in the η direction (dr
[2] [1]) are input to the multiplier 13N and multiplied. The result of the multiplication is input to the adder 13M after the sign is inverted by the sign inverter 13P. The adder 13M calculates the z-component of the outer product by adding the input multiplication results, and outputs this to the multiplier 13Q. Multiplier 13Q
, The z component of the ζ direction (dr [3] [3]) is input, and the multiplier 13Q multiplies the z component of the outer product by the z component of the を direction to obtain the z components of the outer product and the inner product. The calculated value is output to the adder 13F.

The adder 13F calculates the element volume (volume [ie]) by adding the x, y, and z components of the input outer product and inner product, respectively. This element volume (volume [i
e]) is input to a reciprocal unit 14 for obtaining a reciprocal in a subtraction format, where a reciprocal operation is performed to generate a reciprocal (rvolume [ie]) of the element volume. The element volume (volume [ie]) and the reciprocal of the element volume (rv)
olume [ie]) is supplied to the Poisson solver chip 9.

The average values (dr [1] [1] to dr [3] [3]) of the vector Δr in the ξ, η, and ζ directions are, as shown in FIGS. It is also input to the x direction cross product addition unit 15, the y direction cross product addition unit 16, and the z direction cross product addition unit 17 that constitute the calculation unit. These x direction cross product adder 1
5, y direction cross product addition unit 16 and z direction cross product addition unit 17
Is

(Equation 51) The x, y, and z components of the variable c shown in (1) are calculated for each node. Incidentally, the variable c is obtained by multiplying the discretized nabla operator ∇ _a by the element volume V _e, as can be seen by comparing with the equation (34). First, the x direction cross product adder 15
, The y component (dr [2] [2]) in the η direction and the z component (dr [3] [3]) in the ζ direction are input to the multiplier 15A and multiplied, and the result of the multiplication is a sign inverter. After the sign is inverted by 15, it is input to the adder 15 </ b> C. The z component (dr [2] [3]) in the η direction and the y component (dr [3] [2]) in the ζ direction are input to the multiplier 15D and multiplied, and the multiplication result is input to the adder 15C. Is done.

The y component (dr [3] [2]) in the ζ direction and the z component (dr [1] [3]) in the ξ direction are input to the multiplier 15E and multiplied. After the sign is inverted by 15F, it is input to the adder 15C. Z in the ζ direction
The component (dr [3] [3]) and the y component in the ξ direction (dr [1] [2]) are input to the multiplier 15G and multiplied, and the multiplication result is input to the adder 15C. The y component (dr [1] [2]) in the ξ direction and the z component (dr [2] [3]) in the η direction are input to the multiplier 15H and multiplied. The multiplication result is output by the sign inverter 15I. After the sign is inverted, it is input to the adder 15C. The z component in the （direction (dr [1] [3]) and the y component in the η direction (dr
[2] [2]) are input to the multiplier 15J and multiplied, and the multiplication result is input to the adder 15C. The adder 15C calculates an outer product addition by adding the input multiplication results. Thus, the x component (cx [ie] [1]) of the variable c at the first node is obtained by multiplying the outer product addition by 0.25 by the multiplier 15K.

Similarly, the x-direction cross product adder 15 performs the arithmetic processing of the equation (51) using dr [1] [2] to dr [3] [3], thereby obtaining the second to the second. X of variable c at 8 nodes
The components (cx [ie] [2] to cx [ie] [8]) are obtained. Note that cx
The circuit for obtaining [ie] [2] to cx [ie] [8] has almost the same configuration as the circuit for obtaining cx [ie] [1], but depending on the nodes to be obtained, (ξ _a , η _a , ζ _a ), The positions of the sign inverters are slightly different.

As shown in FIG. 11, the y-direction cross product adder 1
6 has substantially the same configuration as that of the x-direction cross product addition unit 15, and performs the arithmetic processing of Expression (51) using the input dr [1] [1] to dr [3] [3], thereby obtaining The y components (cy [ie] [1] to cx [ie] [8]) of the variable c at the first to eighth nodes are obtained. Similarly,
As shown in FIG. 12, the z-direction cross product addition unit 17 has substantially the same configuration as the x-direction cross product addition unit 15, and the input dr [1]
By performing the arithmetic processing of equation (51) using [1] to dr [3] [2], the z component (cz) of the variable c at the first to eighth nodes is obtained.
[ie] [1] ～ cz [ie] [8]).

The x, y, and z components (cx [ie] [1] to cx [ie] of the variable c at the first to eighth nodes thus obtained.
[8], cy [ie] [1]-cy [ie] [8], cz [ie] [1]-cz [ie]
[8]) is input to the multiplication circuit 18 as shown in FIG. 13, and is multiplied by the reciprocal value (rvolume) of the element volume. Thereby, the x, y, z components (del x [ie] [1] to del x [ie] of the discretized nabla operator ∇ _a regarding the first to eighth nodes
[8], del y [ie] [1] ~ dely [ie] [8], del z [ie] [1] ~ del
z [ie] [8]) is discretized nabla operator ∇ _a is determined as shown in equation ((34 regarding i.e. first to eighth node) determined). The discretized nabla operator ∇ _{a at} each node thus obtained is supplied to the Poisson solver 9.

(1-3-4) Structure of Poisson Solver Chip In this section, a specific structure of the Poisson solver chip 9 will be described. However, also in this case, the notation shown in the figure indicates the contents as shown in FIG. In the Poisson solver 9, the velocity vector v is calculated as the field vector, and the pressure P is calculated as the energy H.

[0056] First Deruchitsupu of 8 discrete nabla operator ∇ _a supplied from the x, y, z components (del x [ie] [1 ] ~del
x [ie] [8], del y [ie] [1] to del y [ie] [8], del z [ie] [1]
To del z [ie] [8]) are input to the first inner product calculation unit 19 as shown in FIG. Further, the first inner product calculation unit 19 supplies the x, y, z components (u [elm [i
e] [1] .n]-u [elm [ie] [8] .n], v [elm [ie] [1] .n]-v [e
lm [ie] [8] .n], w [elm [ie] [1] .n] to w [elm [ie] [8] .n]
) Is also entered. The first inner product calculation unit 19 calculates the above (4
3) The variable D ^k shown in the equation is obtained. However,
Here, in order to simplify the calculation, equation (43) is not used, and the following equation is used.

(Equation 52) The variable D ^k is calculated using the relational expression shown in (1). Note that this equation (52) is

(Equation 53)

(Equation 54)

[Equation 55] Are derived from the three relational expressions shown in FIG. First
In the inner product calculation unit 19 uses the multiplier 19A~19I multiplies each component with each other discrete nabla operator ∇ _a and the velocity vector v, using the adder 19J The multiplication result between each of its components And add. Thereby, the variable D ^k (= d [ie]) based on the equation (52) is calculated.

The variable D ^k is input to the multiplier 20 as shown in FIG. 15, where the simultaneous relaxation coefficient −λ ⁻¹ (=
rum [ie]) (the simultaneous relaxation coefficient -λ ^-1 is specified by the host computer 2). As a result, the correction speed potential φ ^k (= fg [ie]) corresponding to the above equation (45) is calculated. The modified speed potential φ ^k is input to the subsequent multiplier 21 where it is multiplied by the reciprocal value dt ^-1 (= rdt) of the time step. The multiplication result φ ^k / dt (= dp [ie]) is input to the adder 22, where it is added to the pressure P (= p [ie]) obtained last time. As a result, the pressure P is repeatedly corrected, and as described above (47)
The next expression corresponding to the expression

[Equation 56] Is calculated as shown in FIG.

[0058] Further Deruchitsupu 8 discrete nabla operator ∇ _a of x supplied from, y, z components (del x [ie] [1 ] ~del
x [ie] [8], del y [ie] [1] to del y [ie] [8], del z [ie] [1]
~ Del z [ie] [8]) is also input to the first speed correction unit 23 as shown in FIG. The speed correction unit 23 corrects the speed vector v.

[Equation 57] Iteratively corrects the velocity vector v using the formula shown in

[0059] In the first speed correcting section 23, discrete nabla operator ∇ _a of x, y, z components (del x [ie] [1 ]
~ Del x [ie] [8], del y [ie] [1] to del y [ie] [8], del z
[ie] [1] to del z [ie] [8]) are multipliers 23A to 23A, respectively.
3I, where φ ^k / dt (= dp [i
e]). The results of the multiplication are respectively added to adders 23
J to 23S, where each component of the velocity vector v (uu [elm [ie] [1] .n] to uu [elm [ie] [8] .n], vv [elm [ie]
[1] .n] ~ vv [elm [ie] [8] .n], ww [elm [ie] [1] .n] ~ ww [elm
[ie] [8] .n]). Thereby, each component of the corrected speed vector v shown in the equation (57) is calculated.

As shown in FIG. 17, the velocity vector v
Is also repeatedly corrected in the second speed correction unit 24. This second speed correction unit 24 is given by the following equation:

[Equation 58] Modify the velocity vector v using the formula shown in. In the second speed correction unit 24, multipliers 24A to 24A
The time step (dt), the concentrated mass (rfm), and the components (uu [in], vv [i]
n] and ww [in]). The multiplication result is input to the adders 24D to 24F, where it is added to each component (u [in], v [in], w [in]) of the speed vector v at the time n. Thereby, each component of the corrected speed vector v shown in the equation (58) is calculated.

[0061] Further, as shown in FIG. 18, x discretization nabla operator ∇ _a supplied from Deruchitsupu 8, y, z components (del x [ie] [1 ] ~del x [ie] [8], del y [ie] [1]-del y
[ie] [8], del z [ie] [1] to del z [ie] [8]) are also input to the second inner product calculation unit 25. The second inner product calculation unit 2
5 includes x, y, and z components (u [elm [ie]) of the velocity vector v.
[1] .n]-u [elm [ie] [8] .n], v [elm [ie] [1] .n]-v [elm
[ie] [8] .n], w [elm [ie] [1] .n]-w [elm [ie] [8] .n]
Is also entered. The second inner product calculation unit 25 calculates (5
2) Calculate the variable D ^k using the equation. First, in the second inner product calculating section 25, a multiplier 25A~25I multiplies each component with each other discrete nabla operator ∇ _a and velocity vector v using adder 25J The multiplication result between each of its components
Add using Thereby, the variable D ^k (= d [ie]) based on the equation (52) is calculated.

As shown in FIG. 19, this variable D ^k is input to the following absolute value circuit 26, where it is converted into a variable e by converting it to an absolute value. This variable e is input to the following maximum value detection circuit 27, where the maximum value (divm
ax), and a new maximum value (divmax) is detected. This maximum value is input to the following magnitude judgment unit 28, where it is compared with the convergence judgment reference value (eps1). As a result, if the maximum value (divmax) is still larger than the convergence criterion value (eps1), the iterative correction of the velocity vector v and the pressure P is continued,
If it is smaller than the convergence criterion value (eps1), the process proceeds to the next time step. The Poisson solver chip 9 is provided with a memory for storing data therein, and temporarily stores various data generated in the calculation process in the iterative correction in the memory, and stores the data as necessary. Is read. For example, the above-mentioned maximum value (divmax) is stored in this memory, and is read out at any time when the maximum value is detected. In addition, this Poisson Solvachip 9
The components of the velocity vector v and the pressure P, which have been repeatedly corrected in the above, are sent to the host computer 2 via the transmission path 3 as described above.

(1-4) Operation and Effect In the above configuration, the numerical analysis device 1 first supplies the node and element information S1 from the host computer 2 to the calculation-dedicated board 4 when performing numerical analysis. The node and element information S1 are sent to the memory 7 of the dedicated calculation board 4 via the transmission path 3 and stored in the memory 7. Delchip 8
Is a Poisson solver 9 with the discretized Nabla operator ∇ _a
Each time the node is required, the node and element information S1
Is read, and a discretized nabla operator ∇ _a and an element volume V _e are calculated based on the node and the element information S1. The Poisson solver 9 uses the discretized nabla operator ∇ _a and the element volume V _e calculated in the del tip 8 to obtain a physical quantity (ie, ∇ · v = 0) at each time step so as to satisfy a continuous equation (ie, ∇ · v = 0). The field vector v and the energy H) are iteratively corrected, thereby calculating a physical quantity satisfying the continuous equation. This physical quantity is sent to the host computer 2 and processed as a numerical analysis result.

As described above, in the numerical analysis apparatus 1, the part for solving the Poisson equation, which takes the longest time in the numerical analysis process, is calculated by the Poisson solver 9 of the calculation-dedicated substrate 4, so that the host computer 2 In this case, the number of exchanges between registers and memory access can be reduced, and the time required for numerical analysis can be greatly reduced as compared with the related art. Further, in the numerical analysis apparatus 1 calculates a discrete nabla operator ∇ _a in Deruchitsupu 8, in solving Poisson equation was to use the discrete nabla operator ∇ _a as nabla operator ∇. In this way, since the discretized nabla operator ∇a can be obtained by a simple operation such as _a vector operation, the time for calculating the nabla operator ∇ can be greatly reduced, and the entire time required for the numerical analysis is also reduced. It can be significantly reduced. Further, in the numerical analysis apparatus 1, by using that it is possible to calculate a short time discretization nabla operator ∇ _a was to ask every time it is necessary to the discretized nabla operator ∇ _a, as in the prior art Therefore, the capacity of the memory itself can be reduced, and the numerical analysis device itself can be simplified.

[0065] Also In the numerical analysis device 1, together with an integrated circuit portion for obtaining the discretization nabla operator ∇ _a, generates a calculated only substrate 4 parts to solve the Poisson equation and an integrated circuit, the computation-substrate 4, numerical analysis can be realized with a simple configuration using a personal computer or a work station, which is a general-purpose computer. Therefore, in the case of the numerical analysis device 1, the installation area and power consumption required for the numerical analysis device can be reduced as compared with the case where a supercomputer is used as in the related art. Incidentally, the discretized nabla operator ∇ _a and the solution of the Poisson equation can be used for various numerical analyses.
_The Delchip 8 for calculating a and the Poisson solver 9 for solving the Poisson equation have high versatility and have an effect when they can be applied to various devices.

According to the above configuration, the discretized nabla operator ∇ _a is calculated by the delchip 8 and the Poisson solver 9 is used to solve the Poisson equation using the discretized nabla operator ∇ _a. By obtaining the vector of the field to be analyzed, it is possible to easily realize the numerical analysis apparatus 1 that can perform numerical analysis in a short time with a simple configuration.

The numerical analysis apparatus 1 displays an analysis result by using the calculated physical quantity. FIGS. 20 and 21 show display examples at this time. FIG. 20 shows an analysis of the flow of air in the video camera when the air holes are provided in the video camera. FIG. 21 shows an analysis of the temperature distribution in the video camera. By using the numerical analysis apparatus 1 in this manner, for example, the flow and temperature distribution of air in the electronic device can be easily analyzed, and the user can easily be notified of the flow and temperature distribution of air. . Actual displays in the numerical analysis apparatus 1 are shown in Reference Drawings 1 and 2. Incidentally, reference drawing 1 is a display when analyzing the flow of air in the video camera, and reference drawing 2 is a display when analyzing the temperature distribution in the video camera.

(2) Second Embodiment The numerical analysis apparatus according to the second embodiment basically includes a first
It has almost the same configuration as the numerical analysis apparatus 1 shown in the embodiment, and differs from the numerical analysis apparatus 1 only in the method of displaying the physical quantities (that is, field vectors and energies) of the respective elements calculated by the calculation-dedicated substrate 4. is there.

When a numerical analysis is mainly performed by a workstation or a personal computer as in the prior art, the computer must concentrate on calculating the coefficient matrix and solving the Poisson equation. There was not enough room to process. However, as described in the first embodiment, computation-substrate 4 is provided with a Poisson solver multichip 9 solving Deruchitsupu 8 and Poisson's equation for calculating the discrete nabla operator ∇ _a,
Approximately 90% or more of the numerical analysis is performed by the dedicated calculation board 4
In this case, the load on the host computer 2 is almost eliminated, and processing with a large load such as displaying a moving image can be processed at a sufficient speed.

Therefore, in the second embodiment, when the calculation is performed by the Delchip 8 or the Poisson solver 9, the physical quantity in the middle of the calculation is monitored in real time by utilizing the fact that the host computer 2 has extra processing. 2B
(In this case, real-time means
This means a drawing speed that enables animation display in almost real time.) By doing so, the operator can determine whether the initial conditions in the numerical analysis, such as the element size, are optimal or not, without waiting for the final result, by looking at the real-time display. it can.
Therefore, when the initial conditions are inappropriate, the initial conditions can be quickly corrected and the numerical analysis can be performed again, so that the numerical analysis can be performed efficiently. Further, not only can the numerical analysis be efficiently performed as described above, but also an abnormal part during the calculation which does not appear in the final result can be found, which is also effective when developing software for a numerical analysis apparatus.

Here, the numerical analysis apparatus according to the second embodiment will be specifically described below with reference to the drawings. FIG.
In FIG. 22, in which the same reference numerals are given to the corresponding parts,
Numeral 30 designates a numerical analysis apparatus according to the second embodiment as a whole, comprising a configuration of a dedicated calculation board 31 and a host computer 2.
Except for the display method, the configuration is almost the same as that of the numerical analysis device 1 of the first embodiment.

As shown in FIG. 23, the structure of the Poisson solver chip 32 of the calculation-dedicated board 31 is different from that of the first embodiment. This Poisson solve tip 3
2 is a physical quantity that satisfies the equation of continuity (that is, a vector and energy of a field, for example, a velocity vector v and energy H in the case of a velocity field) as in the first embodiment.
A physical quantity calculation means for calculating a, calculates a physical quantity that satisfies the continuity equation by repeating modify the physical quantity by using the discrete nabla operator ∇ _a and element volume V _e calculated in Deruchitsupu 8. However, this Poisson solve chip 32
Outputs not only the physical quantities converged to satisfy the continuous equation as in the first embodiment, but also outputs the physical quantities before the convergence, that is, during the iterative calculation. That is, in other words, the Poisson solver 32 always outputs the calculated physical quantity regardless of whether or not the convergence has occurred. At that time, the Poisson solver multichip 32 also outputs with the physical quantity convergence determination flag epsilon _a indicating whether or not data that the physical quantity is converged to be outputted. The physical quantity (v and H) and the convergence judging flag epsilon _a is sent to a first embodiment similarly to calculation result information S2 to the host computer 2.

The host computer 2 thins out the physical quantities received in real time at predetermined intervals, performs predetermined signal processing on the thinned physical quantities, and
The physical quantity is displayed on the monitor 2B as an analysis result. At that time, in the host computer 2, the physical quantity based on the convergence determination flag epsilon _a, it is determined whether or not converge,
If the physical quantities of all the elements have converged, graphic data indicating the final result is displayed on the monitor 2B; otherwise, graphic data indicating that analysis is in progress is displayed on the monitor 2B. .

Here, the internal configuration of the host computer 2 will be described with reference to FIG. However, in FIG. 24, only the circuit blocks relating to the display in the host computer 2 are shown. C of host computer 2
The PU 33 receives the calculation result information S2 from the calculation-dedicated board 4 as described above, and extracts a physical quantity from the calculation result information S2 at every predetermined timing (for example, 100 [n
s] is extracted every 10 [ms]).
Then, the CPU 33 performs signal processing for display on the physical quantity, and outputs the resulting graphic data to the video chip 34. The CPU33 is computed based on the convergence determination flag epsilon _a information in S2, it generates graphic data showing an intermediate final results or analysis, and outputs this to the Bideochitsupu 34.

The video chip 34 includes a video buffer 34A, a graphic controller 34B, a latch circuit 34C, an attribute controller 34D, a video DAC (Digital Analog Converter) 34E, and a CRT.
(Cathode-Ray Tube) A controller 34F and a sequencer 34G are provided. Graphic data provided from the CPU 33 is sequentially stored in the video buffer 34A, and the video buffer 34A is provided at a predetermined timing.
, And outputs the result to the monitor 2B, so that graphic data indicating the result of the numerical analysis is displayed on the monitor 2B.

At this time, in the video chip 34, C
Upon receiving a control signal from the PU 33, the graphics controller 34B controls writing to the video buffer 34A and reading from the video buffer 34A. Attribute controller 34D
The selected display color is controlled by the video DAC 34E, and the selected display color is instructed to the monitor 2B by the video DAC 34E. The screen display size is controlled by setting the reading range of the video buffer 34A by the CRT controller 34F, and the display timing is controlled by the sequencer 34G. A latch circuit 34C interposed between the graphic controller 34B and the video buffer 34A is a register for mediating data set in a graphic register (a type of memory) inside the graphic controller 34B to the video buffer 34A. It is.

In this way, the host computer 2 supplies the graphic data generated by the CPU 33 to the monitor 2B via the video buffer 34A, so that the graphic data indicating the numerical analysis result is displayed on the monitor 2B. It has been done. Here, the signal processing performed by the CPU 33 when generating graphic data will be described. Since the physical quantity supplied from the calculation-dedicated board 31 is three-dimensional data, when displaying the physical quantity on a monitor screen that is a two-dimensional plane, the CPU 33 must convert the three-dimensional data into two-dimensional data. That is, this is the most important signal processing performed by the CPU 33.

The two-dimensional conversion processing will be specifically described below. First, it is assumed that the velocity vector v and the pressure P are output as the energy H from the Poisson solver 32, and the velocity vector v is represented by a coordinate value (X
It is assumed that vector data from (1, Y1, Z1) to coordinate values (X2, Y2, Z2) is output, and scalar data at coordinate values (X1, Y1, Z1) is output as pressure P. In this case, the two-dimensional conversion processing includes:
X which is a monitor plane by performing parallel projection on vector data
What is necessary is just to perform the process which projects on a Y plane. In the parallel projection on the XY plane, a vector indicating the direction of projection is represented by (X
p, Yp, Zp), the following equation

[Equation 59] Is obtained by multiplying the coordinate values of the starting point and the ending point of the vector data by the transformation matrix T represented by. That is, in this case, first, the coordinate values (X1, Y1, Z
1) is multiplied by a transformation matrix T to obtain start point coordinates (X1 ′, Y1 ′) on the XY plane, and then the end point coordinate values (X2, Y2, Z2) are multiplied by the transformation matrix T to obtain XY. End point coordinates (X2 ', Y
2 '). A vector connecting the start point coordinates (X1 ', Y1') and the end point coordinates (X2 ', Y2') is the vector data after conversion. After performing the conversion process using the conversion matrix T, the CPU 33 generates, for example, bit map data of an arrow indicating the converted vector data,
This is stored in the video buffer of the video chip 34 as graphic data.

The scalar data indicating the pressure P is displayed by the CPU 33 by color-coding a fixed area near the starting point coordinates (X1 ', Y1') or an arrow indicating vector data. In this case, if a table of, for example, scalar amounts and corresponding display colors (a color that changes stepwise from red to blue, for example) is prepared in advance, the CPU 33 refers to the table. However, the colors can be easily classified according to the scalar amount. Further, by performing such a display, the operator can easily visually identify the difference in size, which is convenient. By the way, when a certain area is painted, the CPU 33 performs an interpolation process between the adjacent area and the adjacent area, whereby the color is continuously changed so that the pressure distribution is displayed more easily. . Note that the CPU 33 does not perform the above-described conversion processing for all vector data every time, but only for vector data that needs to be displayed and changed vector data (for example, vector data that changes every time during calculation). This reduces the load on the display processing.

Here, the numerical analysis device 30 having such a configuration is similar to that of the first embodiment, except that the host computer 2
Numerical analysis operation is performed based on numerical analysis software built in. This software can be broadly divided into three types: software called a preprocessor, software called an analysis solver, and software called a postprocessor.

The preprocessor is software for creating an analysis model, and divides a field to be analyzed into elements and sets initial conditions and the like. The post processor is software for displaying the obtained physical quantity as an analysis result on the monitor 2B or for evaluating the analysis result. On the other hand, the analysis solver is software for obtaining a physical quantity for each element. In this embodiment, the physical solver is obtained by using the GSMAC finite element method. Although this analysis solver basically operates on the host computer 2, a part that closes most of the calculation time (that is, a part that calculates a physical quantity by solving a Poisson equation, and a discretized nabla operator ∇ _a required in the solution method) Is calculated by the calculation-dedicated board 32 by activating the calculation-dedicated board 32.

Here, the numerical analysis procedure of the numerical analysis device 30 according to the second embodiment will be described with reference to the flowchart shown in FIG. In the numerical analysis device 30, first, in step SP21 entered from step SP20, the preprocessor is started, an analysis model is created (that is, the field to be analyzed is divided into each element), and calculation is performed by each calculation-dedicated board 31. Allocate elements. By the way,
In this numerical analysis device 30, since a plurality of calculation-dedicated boards 31 are provided to parallelize the calculation processing, such allocation processing is performed. As an allocation method, adjacent elements are allocated to different calculation-dedicated boards 31,
Adjacent elements are not allocated to the same dedicated calculation board 4.

In the next step SP22, the analysis solver is started, and based on the allocation performed by the preprocessor, the node and element information S1 are stored in each dedicated calculation board 31.
Is supplied, and the physical quantity of each element is obtained by using the calculation dedicated substrate 31. Incidentally, the calculation of the physical quantity is based on the discretized nabla operator ∇ _a and the element volume V _e calculated in Delchip 8.
By solving the Poisson equation with the Poisson solver 32.

In the next step SP23, the post processor is started up, and the physical quantity output from the dedicated calculation board 31 is displayed on the monitor 2B as an analysis result, and
Graphic data indicating that analysis is in progress is displayed on the monitor 2B. At next step SP24, based on the convergence determination flag epsilon _a, determines whether the physical quantity is a value has converged, if the converged value step SP2
Returning to step 2, the calculation is continued, and if the value converges, the process proceeds to the next step SP25. At step SP25, it is determined whether or not the physical quantities have been calculated for all the elements. If the calculation has not been completed for all the elements, the procedure returns to step SP21, the remaining elements are allocated, and the physical solver is again used by the analysis solver. Is calculated, and when the calculation is completed for all the elements, the process proceeds to the next step SP26. Determining whether the physical quantity is obtained is calculated for all the elements, the convergence determination flag epsilon _a for all elements is performed based on whether or not indicate convergence. At step SP26, graphic data indicating that the currently displayed analysis result is the final result is displayed, and the process moves to the next step SP27 and ends.

In the above configuration, also in the case of the numerical analysis device 30 of the second embodiment, when performing numerical analysis, first, the host computer 2 supplies the node and element information S1 to the calculation-dedicated board 31. The node and the element information S1 are sent to the memory 7 of the calculation-dedicated board 31 via the transmission path 3 and stored in the memory 7. Each time the Poisson solver chip 32 needs the discretized nabla operator ∇ _a , the delchip 8 reads the node and element information S1 from the memory 7, and based on the node and element information S1, the discretized nabla operator ∇ _a and The element volume V _e is calculated. The Poisson solver 32 uses the discretized nabla operator よ う_a and the element volume V _e calculated in Delchip 8 to iteratively modify the physical quantity so as to satisfy the continuity equation, thereby converting the continuity equation. Calculate the physical quantity that satisfies. Then, the Poisson solver 32 sends the physical quantity to the host computer 2 via the transmission path 3. At that time, Poisson Solvatip 3
No. 2 outputs not only the physical quantity converged so as to satisfy the continuous expression, but also outputs the physical quantity before convergence, that is, the physical quantity during the iterative calculation in real time. The Poisson solver multichip 32, the output physical quantity is output together also physical quantity convergence determination flag epsilon _a indicating whether or not the convergence data.

The host computer 2 extracts the physical quantity output in real time at a predetermined timing, and sequentially displays the extracted physical quantity on the monitor 2B as an analysis result in a graphic manner. At that time, if it is determined to be the middle by connexion analysis convergence determination flag epsilon _a, displays the graphic data indicating the way analysis monitor 2B, numerical and physical quantity of all elements Convergence Analysis If it is determined that is completed, graphic data indicating the final result is displayed.

Thus, the Poisson solver tip 32
The host computer 2 outputs the physical quantities during the iterative calculation, and the host computer 2 sequentially displays the physical quantities during the iterative calculation so that the physical quantities calculated by the Poisson solver are displayed on the monitor 2B in real time. As a result, a numerical analysis result that is constantly changing toward the final result is displayed on the monitor 2B. By displaying the physical quantity under analysis in real time in this way, the operator can know how far the analysis is currently progressing in the numerical analysis device 30 and what kind of analysis result is currently obtained. You can easily know when you are. Therefore, by looking at this display, it is possible to judge whether the initial conditions and boundary conditions are optimal without waiting for the final result, and if inappropriate, correct those conditions immediately and perform the numerical analysis again. As a result, numerical analysis can be performed efficiently.

According to the above configuration, the physical quantity to be analyzed is displayed on the monitor 2B in real time by outputting the physical quantity during the iterative calculation from the Poisson solver chip 32, so that the operator can perform the calculation. Numerical analysis device 3 that can easily check the numerical analysis situation that changes from moment to moment and thus has improved usability
0 can be realized.

(3) Other Embodiments In the above embodiment, the discretized nabla operator ∇ _a
Although the description has been given of the case where both the part for calculating the equation and the part for solving the Poisson equation are chipped, the present invention is not limited to this, and only one may be chipped and the other may be calculated by the host computer 2. This is because a sufficient effect can be obtained by this alone in terms of performing a numerical analysis in a simple configuration and in a short time. However, the present invention is not limited to this, and the two calculation portions may be integrated into one chip, and further, the memory 7 and the interface circuit 6 may be integrated into one chip. In this way, the configuration of the calculation-dedicated board 4 can be further simplified.

In the above embodiment, the case where the shape of the element is a hexahedron was described. However, the present invention is not limited to this, and the shape of the element may be a tetrahedron as shown in FIGS. 26 and 27. It may be a pentahedron. Incidentally, the case of four tetrahedral elements and 5 tetrahedral elements may be deformed (34) and (35) defining equation discretization nabla operator ∇ _a and element volume V _e as shown in formula more simple expression. This will be described below.

First, in the case of a tetrahedral element, the shape function N _a
Is

[Equation 60] It is represented as Here, a = 1 to 4, and l _a , ξ _a , η _a , and ζ _a are linearly independent basis vectors. Therefore, the average values of the vector Δr in the ξ, η, and ζ directions are respectively expressed by the following equations.

[Equation 61]

(Equation 62)

[Equation 63] It is represented as Therefore, if the average value of this vector Δr is used, the element volume V _e is given by the following equation.

[Equation 64] Where the discretized nabla operator ∇ _a is

[Equation 65] It is represented as

In the case of a pentahedral element, the shape function N _a
Is

[Equation 66] It is represented as Here, a = 1 to 6, and l _a , ξ
_a , η _a , l _a ζ _a , ξ _a ζ _a , η _a ζ _a are linearly independent basis vectors. Therefore, ξ, η,
The average value in the ζ direction is

[Equation 67]

[Equation 68]

[Equation 69] It is represented as Therefore, if the average value of this vector Δr is used, the element volume V _e is given by the following equation.

[Equation 70] Where the discretized nabla operator ∇ _a is

[Equation 71] It is represented as Incidentally, since equations (34) and (35) are definition equations for a three-dimensional space, they include tetrahedral elements and pentahedral elements. Even if equations (34) and (35) are used, tetrahedral elements and It is possible to obtain the discretized nabla operator ∇ _a and the element volume V _e of the face element.

In the above embodiment, the field to be analyzed is described as a velocity field. However, the present invention is not limited to this.
The field to be analyzed may be another field such as a temperature field, a stress field, an electromagnetic field, or the like. By the way, in that case,
v may be a temperature vector, a stress vector, or a magnetic flux density vector or a current density vector.

In the above embodiment, the GSMAC
The case where the discretized nabla operator ∇ _a is applied to the numerical analysis by the finite element method has been described. However, the present invention is not limited to this.
_a may be applied. Because Using discrete nabla operator ∇ _a, because with may reduce the calculation time of at least coefficient matrix may lower the capacity of the memory.

In the above-described embodiment, the case has been described where the Poisson solvers 9 and 32 have a memory therein, and the data generated in the calculation process is temporarily stored in the memory. The present invention is not limited to this, and such data may be stored in a free area of the memory 7. For example, in the case of the second embodiment, FIG.
As shown in (1), if the Poisson solver chip 41 can access the free area of the memory 7 on the dedicated calculation board 40, and the data generated in the calculation process is temporarily stored using the free area of the memory 7, good. With this configuration, the configuration of the Poisson solver can be further simplified because the memory is eliminated.

In the above-described second embodiment, the CPU
Although the case where the parallel projection is performed as the two-dimensional conversion processing in 33 has been described, the present invention is not limited to this, and the perspective projection may be performed based on an arbitrary viewpoint. By the way, in the case of perspective projection, the transformation matrix is slightly different, but basically the transformation process may be performed by multiplying the transformation matrix by the coordinate value as in the parallel projection.

[0097]

As described above, according to the present invention, based on the element information of each element obtained by dividing the field to be analyzed,
The time required for numerical analysis can be reduced by calculating the discretized nabla operator by discretizing the spatial differential nabla operator and calculating the physical quantity using the calculated discretized nabla operator. Thus, a numerical analysis device having a simple configuration and capable of performing numerical analysis in a short time can be realized. Further, according to the present invention, based on element information of each element obtained by dividing a field to be analyzed, a discretized Nabla operator obtained by discretizing a Nabla operator of spatial differentiation is calculated, and the calculated discretized The time required for numerical analysis by calculating the physical quantity that satisfies the continuity equation by repeatedly correcting the physical quantity so as to satisfy the continuity equation of the field to be analyzed using the Nabla operator Thus, it is possible to realize a numerical analysis apparatus that can perform numerical analysis in a short time with a simple configuration.

[Brief description of the drawings]

FIG. 1 is a block diagram showing an overall configuration of a numerical analysis device according to an embodiment of the present invention.

FIG. 2 is a block diagram showing a configuration of a dedicated calculation board.

FIG. 3 is a flowchart showing the entire procedure of numerical analysis.

FIG. 4 is a flowchart showing a processing procedure of an analysis solver.

FIG. 5 is a schematic diagram illustrating a hexahedral element.

FIG. 6 is a table showing notation contents used in block diagrams of Delchip and Poisson solvers.

FIG. 7 is a block diagram showing a configuration of a delchip.

FIG. 8 is a block diagram showing a configuration of a delchip.

FIG. 9 is a block diagram showing a configuration of a delchip.

FIG. 10 is a block diagram showing a configuration of a delchip.

FIG. 11 is a block diagram showing a configuration of a delchip.

FIG. 12 is a block diagram showing a configuration of a delchip.

FIG. 13 is a block diagram showing a configuration of a delchip.

FIG. 14 is a block diagram showing a configuration of a Poisson solver chip.

FIG. 15 is a block diagram showing a configuration of a Poisson solver chip.

FIG. 16 is a block diagram showing a configuration of a Poisson solver chip.

FIG. 17 is a block diagram showing a configuration of a Poisson solver chip.

FIG. 18 is a block diagram showing a configuration of a Poisson solver chip.

FIG. 19 is a block diagram showing a configuration of a Poisson solver chip.

FIG. 20 is a schematic diagram showing the flow of air in the video camera analyzed by the numerical analysis device.

FIG. 21 is a schematic diagram illustrating a temperature distribution in the video camera analyzed by the numerical analysis device.

FIG. 22 is a block diagram showing an overall configuration of a numerical analysis device according to a second embodiment.

FIG. 23 is a block diagram showing a configuration of a calculation-dedicated board according to the second embodiment.

FIG. 24 is a block diagram for describing an internal configuration of a host computer.

FIG. 25 is a flowchart for explaining a numerical analysis procedure according to the second embodiment.

FIG. 26 is a schematic diagram for describing a tetrahedral element.

FIG. 27 is a schematic diagram used for describing a pentahedron element.

FIG. 28 is a block diagram for explaining a Poisson solver chip according to another embodiment.

[Explanation of symbols]

1, 30 numerical analysis device, 2 host computer, 2A main body, 2B monitor, 3 transmission line,
4, 31, 40 ... Calculation dedicated board, 5 ... Expansion unit, 6 ... Interface circuit, 7 ... Memory, 8 ...
... Delchip, 9, 32, 41 ... Poisson Solvatip.

Claims (34)

[Claims] 1. A numerical analysis apparatus for calculating a physical quantity of a field to be analyzed, wherein a Nabla operator for spatial differentiation is discretized based on element information of each element obtained by dividing the field to be analyzed. A numerical analysis apparatus comprising: an analysis unit that calculates a discretized nabla operator and calculates the physical quantity using the calculated discretized nabla operator. 2. The analysis means according to claim 1, wherein, when said element has a hexahedral shape, said physical space is (x,
y, z), the calculation space is (ξ, η, ζ), the position vector in the physical space is r (= (x, y, z)), and the base vectors in the calculation space are ξ _a , η _a , ζ _a , Assuming that the volume of the element is V _e and the discretized nabla operator is ∇ _a , The numerical analysis device according to claim 1, wherein the discretized nabla operator is calculated by performing a vector operation shown in (1). 3. The analysis means, together with the discretized nabla operator, comprises: Numerical Analysis device according to claim 2, characterized in that to calculate the element volume V _e shown. 4. The analysis means, when the shape of the element is a tetrahedron, converts the physical space to (x,
y, z), the calculation space is (ξ, η, ζ), the position vector in the physical space is r (= (x, y, z)), and the base vectors in the calculation space are ξ _a , η _a , ζ _a , Assuming that the volume of the element is V _e and the discretized nabla operator is ∇ _a , The numerical analysis device according to claim 1, wherein the discretized nabla operator is calculated by performing a vector operation shown in (1). 5. The analyzing means, together with the discretized nabla operator, comprises: Numerical analysis apparatus according to claim 4, characterized in that to calculate the element volume V _e shown. 6. When the shape of the element is a pentahedron, the analyzing means converts the physical space to (x,
y, z), the calculation space is (ξ, η, ζ), the position vector in the physical space is r (= (x, y, z)), and the base vectors in the calculation space are ξ _a , η _a , ζ _a , Assuming that the volume of the element is V _e and the discretized nabla operator is ∇ _a , The numerical analysis device according to claim 1, wherein the discretized nabla operator is calculated by performing a vector operation shown in (1). 7. The analyzing means, together with the discretized nabla operator, comprises: Numerical Analysis device according to claim 6, characterized in that to calculate the element volume V _e shown. 8. The numerical analysis apparatus according to claim 1, wherein the physical quantity is a vector of the field to be analyzed. 9. A numerical analysis apparatus for calculating a physical quantity of a field to be analyzed, wherein a Nabla operator for spatial differentiation is discretized based on element information of each element obtained by dividing the field to be analyzed. By using a discretized nabla operator calculating means for calculating a discretized nabla operator, and using the discretized nabla operator, by iteratively correcting the physical quantity so as to satisfy the continuity equation of the field to be analyzed A numerical value calculating unit that calculates a physical amount that satisfies the continuity equation. 10. The numerical analysis apparatus according to claim 9, wherein said physical quantity calculating means calculates a vector and energy of said field to be analyzed as said physical quantity. 11. The continuous expression is obtained by replacing the Nabla operator with ∇,
Assuming that the vector of the field is v, The numerical value analysis apparatus according to claim 10, wherein the physical quantity calculation means repeatedly corrects the field vector and the energy using the discretized nabla operator as the nabla operator ∇. . 12. The discretized nabla operator calculating means, when the shape of the element is a hexahedron, converts the physical space into (x,
y, z), the calculation space is (ξ, η, ζ), the position vector in the physical space is r (= (x, y, z)), and the base vectors in the calculation space are ξ _a , η _a , ζ _a , Assuming that the volume of the element is V _e and the discretized nabla operator is ∇ _a , 10. The numerical analysis apparatus according to claim 9, wherein the discretized nabla operator is calculated by performing a vector operation shown in (1). 13. The discretized nabla operator calculating means, together with the discretized nabla operator, comprises: 13. The numerical analysis device according to claim 12, wherein the element volume V _e shown in (1) is calculated. 14. The discretized nabla operator calculating means, when the shape of the element is a tetrahedron, converts the physical space to (x,
y, z), the calculation space is (ξ, η, ζ), the position vector in the physical space is r (= (x, y, z)), and the base vectors in the calculation space are ξ _a , η _a , ζ _a , Assuming that the volume of the element is V _e and the discretized nabla operator is ∇ _a , 10. The numerical analysis apparatus according to claim 9, wherein the discretized nabla operator is calculated by performing a vector operation shown in (1). 15. The discretized nabla operator calculating means, together with the discretized nabla operator, comprises: Numerical Analysis device according to claim 14, characterized in that to calculate the element volume V _e shown. 16. The discretized nabla operator calculating means converts the physical space to (x,
y, z), the calculation space is (ξ, η, ζ), the position vector in the physical space is r (= (x, y, z)), and the base vectors in the calculation space are ξ _a , η _a , ζ _a , Assuming that the volume of the element is V _e and the discretized nabla operator is ∇ _a , 10. The numerical analysis apparatus according to claim 9, wherein the discretized nabla operator is calculated by performing a vector operation shown in (1). 17. The discretized nabla operator calculating means, together with the discretized nabla operator, comprises: Numerical Analysis device according to claim 16, characterized in that to calculate the element volume V _e shown. 18. A numerical analysis apparatus for calculating a physical quantity of a field to be analyzed, wherein a Nabla operator for spatial differentiation is discretized based on element information of each element obtained by dividing the field to be analyzed. Calculate a discretized nabla operator, and use the discretized nabla operator to repeatedly modify the physical quantity so as to satisfy the equation of the continuity of the field to be analyzed, thereby obtaining a physical quantity that satisfies the equation of the continuity. Calculating the physical quantity by supplying the element information to the calculation board and dividing the field of the analysis object into the fields to be analyzed, and calculating the physical quantity, and calculating the physical quantity by the calculation board. A computer for displaying the physical quantity as a numerical analysis result. 19. The calculation board outputs a physical quantity in the middle of the iterative calculation, and the computer displays the physical quantity in the middle of the iterative calculation received from the calculation board to display the physical quantity calculated by the calculation board in real time. 19. The numerical analysis device according to claim 18, wherein the numerical value is displayed as: 20. The calculation board outputs a convergence determination flag indicating whether or not the physical quantity has converged so as to satisfy a continuous equation, together with the physical quantity, and the computer outputs the convergence determination flag based on the convergence determination flag. 20. The numerical analysis apparatus according to claim 19, wherein it is determined whether or not the physical quantity has converged, and if the physical quantity has converged, a result indicating a final result is displayed. 21. The calculation board according to claim 18, wherein the calculation board calculates the vector and the energy of the field satisfying the continuous equation by iteratively correcting the vector and the energy of the field to be analyzed. Numerical analysis equipment. 22. A numerical analysis method for calculating a physical quantity of a field to be analyzed, wherein a Nabla operator for spatial differentiation is discretized based on element information of each element obtained by dividing the field to be analyzed. A numerical analysis method comprising: calculating a discretized nabla operator; and calculating the physical quantity using the calculated discretized nabla operator. 23. When the shape of the element is a hexahedron, the physical space is (x, y, z), the calculation space is (ξ, η, ζ),
The position vector in the physical space is r (= (x, y, z)), the base vectors in the calculation space are ξ _a , η _a , ζ _a , the volume of the element is V _e , and the discretized nabla operator is ∇ _{For a} , the following equation 23. The numerical analysis method according to claim 22, wherein the discretized nabla operator is calculated by performing a vector operation shown in (1). 24. The following equation, together with the discretized nabla operator: 24. The numerical analysis method according to claim 23, wherein the element volume V _e shown in FIG. 25. The numerical analysis method according to claim 22, wherein the physical quantity is a vector of the field to be analyzed. 26. A numerical analysis method for calculating a physical quantity of a field to be analyzed, wherein a Nabla operator for spatial differentiation is discretized based on element information of each element obtained by dividing the field to be analyzed. Calculate a discretized nabla operator, and use the discretized nabla operator to repeatedly correct the physical quantity so as to satisfy the continuity equation of the field to be analyzed, thereby obtaining a physical quantity satisfying the continuity equation. A numerical analysis method characterized by calculating 27. The method according to claim 27, wherein the physical quantity is the vector and energy of the field to be analyzed, and the vector and energy of the field satisfying the continuity equation are calculated by iteratively correcting the vector and energy of the field. 27. The numerical analysis method according to claim 26. 28. The continuous equation is represented by the following equation, where ナ is the Nabla operator, and v is the vector of the field. The numerical analysis method according to claim 27, wherein the field vector and the energy are iteratively corrected using the discretized nabla operator as the nabla operator ∇. 29. An integrated circuit used in a numerical analysis device for analyzing a physical quantity of a field to be analyzed, wherein element information of each element obtained by dividing the field to be analyzed is input, and based on the element information. An integrated circuit comprising a discretized nabla operator calculating circuit for calculating a discretized nabla operator by discretizing a spatial differential nabla operator. 30. The discretized nabla operator calculating circuit, when the shape of the element is a hexahedron, converts the physical space to (x,
y, z), the calculation space is (ξ, η, ζ), the position vector in the physical space is r (= (x, y, z)), and the base vectors in the calculation space are ξ _a , η _a , ζ _a , Assuming that the volume of the element is V _e and the discretized nabla operator is ∇ _a , 30. The integrated circuit according to claim 29, wherein the discretized nabla operator is calculated by performing a vector operation shown in the following. 31. The discretized nabla operator calculation circuit, together with the discretized nabla operator, comprises the following equation: 31. The integrated circuit according to claim 30, wherein the element volume V _e shown in the following is calculated. 32. An integrated circuit used in a numerical analysis apparatus for calculating a physical quantity of a field to be analyzed, wherein a discretized nabla operator obtained by discretizing a spatial differential nabla operator is input, and A physical quantity calculation circuit that calculates a physical quantity that satisfies the continuity equation by iteratively modifying the physical quantity so as to satisfy the continuity equation of the field to be analyzed using a child. circuit. 33. The integrated circuit according to claim 32, wherein the physical quantity calculation circuit calculates a vector and an energy of the field to be analyzed as the physical quantity. 34. The continuous expression is obtained by replacing the Nabla operator with ∇,
Let v be the vector of the field, 34. The integrated circuit according to claim 33, wherein the physical quantity calculation circuit iteratively corrects the field vector and energy using the discretized nabla operator as the nabla operator ∇.