JP2000260723A - Method and apparatus for density interpolation at mesh change and machine-readable recording medium having program recorded thereon - Google Patents

Abstract

PROBLEM TO BE SOLVED: To easily calculate the density values of mesh points of a mesh which is changed by storing their sum, and the mesh which expresses density distribution in terms of the density values defined by the positions of the mesh points. SOLUTION: Mesh-changing means 6 makes partial change of a mesh, i.e., executes a step of adding a new point to the mesh, a processing of moving a point of the mesh, and a processing of merging a plurality of points of the mesh into a single point repeatedly, whereby the mesh before the change is corrected to a mesh after the change. Density value calculating means 7 updates the density value of mesh point which was changed. Since the mesh is changed only partially at density value calculation, an operation for calculating overlapping regions between the mesh elements before and after the change or between the control volume regions corresponding to the meshes before and after the change is eliminated.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a technique for expressing a density distribution by a density value at a mesh point position of a discretized mesh. In particular, when a mesh is changed, a density value at a new mesh point position is expressed. The present invention relates to a technique for interpolating while preserving the total amount.

[0002]

2. Description of the Related Art Techniques for mathematically modeling physical phenomena and predicting distributions of various quantities by computer simulation are widely used. In order to perform computer simulation, mesh division for discretizing and treating a space is performed. Then, the entire density distribution is expressed by holding the density value at each mesh point position.
For example, in a semiconductor manufacturing process, the distribution of impurities in a semiconductor that affects the properties of the semiconductor is predicted using a CAD system for the semiconductor manufacturing process. In the semiconductor manufacturing process, since the shape of the analysis region changes, it is necessary to change the mesh as the analysis progresses. Even if the shape does not change, the mesh may be changed so that a fine mesh is arranged in a portion where the spatial variation of the distribution amount is large.

[0003] A method in which a sufficiently fine mesh is prepared in the analysis area in advance and the mesh is not changed may be considered, but the calculation time and memory required for the analysis are almost proportional to the number of mesh points. Inconvenient for two-dimensional and three-dimensional analysis. Therefore, a technique called an adaptive mesh, which changes the mesh to an appropriate one as the analysis proceeds, is used. Due to the change of the mesh, it is necessary to calculate the density value at each mesh point position of the new mesh by interpolation calculation, but with regard to the impurity concentration, it is necessary to perform interpolation so that the total amount of impurities is preserved. Normal.

[0004] An interpolation method for preserving the total amount depends on how to use the discretized mesh to analyze the impurity amount. Here, a case will be described in which a discretization method called a control volume method, which is often used in a CAD system for a semiconductor manufacturing process, is used. A typical processing method in this case is described in, for example,
-97020. An outline of the processing is shown in the flowchart of FIG.

First, the data of the mesh M before the change, the density value of each mesh point, and the data of the mesh Mnew after the change are read (steps 131 and 132 in FIG. 13). Next, the control volume shape information of the meshes M and Mnew is created (step 133). The control volume is an area that is assigned to each mesh point. FIG. 14A shows a two-dimensional discretized mesh 141 and a corresponding boundary 142 of the control volume area. There are several ways to define the control volume area,
The most general method is to define a control volume area of a certain mesh point as a set of points in an analysis area whose distance to the mesh point is shorter than the distance to another mesh point. In that case, in two dimensions, as shown in FIG.
Is a part of the vertical bisector of the surrounding mesh side. The impurity concentration in the control volume region is represented by the concentration value of the mesh point that handles it.
That is, since the density value of the mesh point is discretized as being the average density inside the control volume area, the total impurity amount inside the control volume area (in this example, two-dimensional, the amount per unit thickness) is , Calculated by multiplying the density value of the mesh point by the area of the control volume.

[0006] When the mesh is changed, the control volume area also changes. Modified mesh M
Assume that a region indicated by a solid line 143 in FIG. 14B is given as a control volume region CVnew of a certain mesh point on new. Control volume area CV
In order to obtain the amount of impurities in new, an overlapping area between the control volume area CVnew and the control volume area of the mesh M before change is obtained. Then Figure 14
The four regions R1, R2, R3, and R4 shown in (c) are obtained. region
Since R1 is included in the control volume area covered by a certain mesh point of the mesh M before the change, the value obtained by multiplying the density value at the certain mesh point by the area of the area R1 is the impurity inside the area R1. Amount. Similarly, regarding the regions R2, R3, and R4, the amounts of impurities inside the regions are obtained, and the sum is added to calculate the total amount of impurities inside the control volume region CVnew. The value obtained by dividing this by the area of the control volume area CVnew is defined as the density value at the mesh point that covers the control volume area CVnew. Thus, the changed mesh Mne
The operation of obtaining the density value at each mesh point of w is the operation in step 134 in FIG. According to this procedure, the mesh Mne after the change is maintained without changing the total amount of impurities.
The concentration distribution at w can be determined. Next Step 13
In step 5, the density value of each mesh point of the changed mesh Mnew obtained in step 134 is written out.

In addition to the control volume method, there is a finite element method as a generally widely used discretization method. In the case of the finite element method, it is common to assume that the density distribution changes linearly in the mesh element. In a typical finite element method using simplex (triangle in 2D, tetrahedron in 3D) for mesh elements, a method of interpolating so that the total amount is preserved before and after the mesh change, for example, Procedures are conceivable. First,
Overlap areas between the mesh elements before and after the change are determined, and the amount in each overlap area is determined assuming that the density distribution in the mesh before the change is a linear change. Then, the amount in each mesh element after the change can be known. The density value at each mesh point after the change is obtained by adding the volume of the mesh element after the change including the mesh point at the vertex and the volume of the mesh element after the change including the mesh point at the vertex. Obtained by dividing by the amount.

[0008]

A problem with the prior art is that it is necessary to find an overlapping area between mesh elements before and after the change or between control volume areas corresponding to the mesh before and after the change. The processing is simple when the analysis space is one-dimensional, but in two-dimensional or three-dimensional space, it is easy to find the overlap between regions stably due to the need to consider various special overlapping methods. Not. Therefore, it is difficult to apply the conventional interpolation method to a high-dimensional mesh.

[Object of the Invention] The object of the present invention is to
Under the condition that the total amount is saved, the density value at the mesh point after the change without performing the work of finding the overlapping area between the mesh elements before and after the change or the control volume area corresponding to the mesh before and after the change Is to be calculated.

[0010]

According to the density interpolating method at the time of changing the mesh according to the present invention, when the original mesh expressing the density distribution is changed by the density value at each mesh point position, the new mesh after the change is changed. The density value at the mesh point position is
A density interpolating method at the time of mesh change in which interpolation is performed while preserving the total amount, a process of adding one mesh point starting from the original mesh, a process of merging a plurality of mesh points into one mesh point, or A mesh changing step of changing the original mesh to the new mesh by repeatedly performing a process of moving a mesh point to be deleted to a mesh point to be added; a process of adding one mesh point in the mesh changing step; Each time a process of merging a plurality of mesh points into one mesh point or a process of moving a mesh point to be deleted to a mesh point to be added is performed, the added mesh point, the mesh point after merging, or The density value at the mesh point of The total amount of before and after contains a density value calculation step of determining to be the same.

In the density interpolation method for changing the mesh of the present invention, the density value at the corrected mesh point is determined while performing one of the operations of adding, moving, or merging one mesh point. .

In determining the density value, it is assumed that only the density value at one mesh point is treated as an unknown, and the density value at the other points is unchanged. The mesh elements to be changed are only a part of the elements, and the total amount before the mesh change in the area where the element groups are merged can be obtained from the control volume shape corresponding to the element shape before the mesh change. On the other hand, the total amount of the mesh after the change in the same region can be easily described by an equation in which the density value at one mesh point is an unknown value because the control volume shape corresponding to the element shape after the mesh change is known. . Therefore, it is possible to know a density value for which the total amount is preserved only by determining the unknown value so that both are equal, and it is not necessary to find an overlapping area between mesh elements or between control volume areas.

As a device suitable for carrying out the above-described density interpolating method at the time of changing the mesh, the density interpolating device at the time of changing the mesh according to the present invention uses the original density distribution expressed by the density value at each mesh point position. When the mesh changes,
A density interpolating apparatus at the time of mesh change for interpolating the density value at the mesh point position of the new mesh after the change while preserving the total amount thereof, the processing of adding one mesh point starting from the original mesh. A mesh changing means for changing the original mesh to the new mesh by repeatedly performing a process of merging a plurality of mesh points into one mesh point or a process of moving a mesh point to be deleted to a mesh point to be added. (6 in FIG. 1), processing for adding one mesh point in the mesh changing means (6 in FIG. 1), processing for merging a plurality of mesh points into one mesh point, or adding mesh points to be deleted. Every time the process of moving to a mesh point is performed, the density value at the added mesh point or the merged mesh point is As unchanged density value other than Mesh points, and a density value calculating means (7 in FIG. 1) to determine as the total amount before and after the mesh update the same.

[0014]

Embodiments of the present invention will now be described in detail with reference to the drawings.

[Explanation of Configuration] As shown in FIG. 1, the embodiment of the present invention comprises a data processing device 1 operated by program control, a storage device 2 for holding data, and a recording medium 3. You.

In the initial state, the storage device 2 stores data on the mesh before the change (coordinate values of mesh points, connection information between mesh points, connection information between mesh elements, density values). Each time the density value is stored in the data area 11 by the density value calculating means 7, the data area 11
Data area 9 in which the contents of
And adding or moving one mesh point or merging operation to one mesh point with respect to the mesh indicated by the data stored in the data area 10 and the data stored in the data area 9. A data area 11 for holding mesh data in a state where
A change plan data area 12 for holding target points, operation order, and the like for addition, movement, and merging operations of mesh points is included.

The data processing apparatus 1 includes a mesh sequential change planning means 5 for operating a mesh point of the original mesh one by one to plan a change procedure for sequentially forming a changed mesh; A mesh changing means for manipulating mesh points one by one in accordance with a change procedure planned by the change planning means; and a density value at the changed mesh point every time one mesh point is changed by the mesh changing means. , A data input / output unit 8 for inputting / outputting data to / from the storage device 2, and a control for creating a density distribution of the mesh after the change by appropriately using these units 5 to 8. Unit 4 is included.

The outline of the functions of these means 4 to 8 is as follows.

The mesh sequential change planning means 5 performs
Data relating to the subsequent mesh is read out from the data areas 9 and 10 via the data input / output means 8, and a plan for sequentially changing the mesh M before the change to the mesh Mnew after the change is created. Procedures and the like are stored in the area 12 as necessary.

The mesh changing means 6 adds one point of the mesh points, moves one point, and merges them into one point, and stores the corrected mesh data in the data area 11 via the data input / output means 8.

Based on the contents of the data areas 9 and 11, the density value calculating means 7 calculates the density values of the mesh points that have undergone the addition, movement and merging processing so that the total amount is stored. save.

A data input / output unit 8 for inputting / outputting data to / from the storage device 2 refers to data stored in the storage device 2 and stores the data.

The control unit 4 includes a mesh sequential change planning unit 5
, A modification of the mesh using the mesh changing means 6 according to the plan, a calculation of the density value of the changed mesh point using the density value calculating means 7, Processing such as updating the data in the area 9 is executed in an appropriate order.

The recording medium 3 includes a disk, a semiconductor memory,
It is another recording medium in which a program for causing the data processing device 1 including a computer to function as a density interpolating device when changing the mesh is recorded. The program recorded on the recording medium 3 is the data processing device 1
By controlling the operation of the data processing device 1, the mesh sequential change planning means 5, the mesh changing means 6, the density value calculating means 7, and the data input / output means 8 are realized on the data processing device 1.

[Description of Operation] FIG. 3 shows an outline of a processing procedure performed by the control unit 4 of FIG. With reference to this flowchart, the overall operation of the present embodiment will be described in detail.

First, the mesh data and density information of the mesh M before the change are read from the data area 9 (step 31 in FIG. 3). Subsequently, the mesh data relating to the changed mesh Mnew is read from the data area 10 (step 3).
2). The mesh Mnew is obtained by appropriately adjusting the number of mesh points and the position according to a change in the density distribution or the like. The task performed in the present invention is to obtain a density value for each mesh point of the changed mesh Mnew.

The iterative process composed of the following steps 33, 34, 35 and 37 is a portion where the mesh M is gradually modified so that the original mesh M matches the changed mesh Mnew. However, in one modification, adding one point of mesh point, moving one point of mesh point,
Alternatively, any process of merging several mesh points into one point is performed.

According to the change plan data generated by the mesh sequential change planning means 5 in FIG. 1, the mesh is formed by using any one of the methods (adding one point, moving one point, or merging into one point) in the mesh changing means 6. Step 33 in FIG. 3 takes charge of the work of correcting and storing the mesh data of the corrected mesh Mtmp in the data area 11. As for the change plan data, the change plan from the first mesh M to Mnew may be created at once, or may be partially created as appropriate in the repetitive processing.

In the next step 34, the density value at the corrected mesh point is determined by using the density value calculating means 7 so that the total amount is stored, and this density value is stored in the data area 11. At this time, since it is possible to determine the value by a simple calculation assuming that there is only one mesh point whose density value is undetermined, it is not necessary to detect an overlapping area between mesh elements or between control volumes. The details will be described later.

In step 35, it is checked whether or not the modified mesh Mtmp matches the modified mesh Mnew to be finally obtained.
The density information of p may be output as the density information of the mesh Mnew (step 36). If they do not match, it is necessary to continue the correction.
The process returns to step 33 with tmp as the mesh M before the change. The processing of step 37 can be realized by, for example, copying the contents of the data area 11 to the data area 9. Further, the reference destination of the data of the mesh M is changed to the data area 11 relating to the mesh Mtmp, the data area 9 relating to the original mesh M is released, and the mesh M is newly added.
It can also be realized by data processing such as securing a data area for tmp. As still another implementation method, since the modification to the mesh that is performed in one iteration is partial, the mesh M before modification is replaced with the mesh M before modification instead of the data area 11 that holds the entire modified mesh Mtmp. There is also a method of generating data (block 21 in FIG. 2) of difference information with the subsequent mesh Mtmp. Since some mesh elements are recreated by processing such as addition or movement of mesh points, information on mesh elements to be corrected is also held in addition to information on mesh points to be corrected. Using the difference information, the data of the mesh M can be updated to the data after the correction.

Now, the method of calculating the density value in step 34 in FIG. 3 will be described in detail by taking the case where one of the mesh points is moved.

FIG. 4 shows the mesh point p0 shown in FIG.
Shows the state of change of the control volume areas 42 and 43 corresponding to the mesh elements when the element moves to the mesh point q shown in FIG. It is assumed that the mesh point q after the movement is inside one of the mesh elements around the mesh point p0 before the movement. If not, there must be a mesh element before modification including the point q after the movement, so either one of the vertices should be processed as if it has moved, or the mesh point adding and merging operations must be performed. Should be processed by a combination of

Considering the merged area 41 of the six mesh elements formed around the point q after the movement, the mesh points p1 to p6 other than the mesh point q have the same mesh positions before the correction. Is a point. Therefore, it can be said that it is an appropriate measure to change only the density value of the mesh point q without changing the density values at these mesh points p1 to p6. In addition, the total amount of the inside of the merged area 41 after the modification shown in FIG.
Since it is equal to the total amount inside the merged area 41 before the modification shown in FIG. 4A, the total amount inside the merged area 41 after the modification can be obtained by calculating the total amount inside the merged area 41 before the modification. That is, if the area of the control volume area of each mesh point p0 to p6 and the density value of each mesh point p0 to p6 are multiplied by the corresponding one, and the results of each multiplication are added, the total amount inside the merged area 41 after correction is obtained. (Since it is a two-dimensional example, the amount per unit thickness) can be calculated.

Next, in FIG. 4 (b), the merged area 41 is set as a processing target area, and the density value at the mesh point q may be obtained so that the total amount inside the merged area 41 does not change. That is, since the area of the control volume area of each of the mesh points p1, p2, p3, p4, p5, and p6 within the merged area 41 can be calculated, the corresponding mesh points p1, p2, p3, p4, p5 ,
By subtracting the amount obtained by multiplying by the density value at p6 from the total amount inside the merged area 41 obtained using the mesh before correction, the amount that should be inside the control volume at the mesh point q is obtained. This may be divided by the control volume area of the mesh point q.

In order for the total amount to be unchanged by this method, it is prerequisite that there is no change in the mesh element or the control volume area outside the merging area 41, but strictly speaking, the merging is performed as shown below. There are cases where the control volume area outside the area 41 is affected.

First, FIG. 4A shows a case where the control volume area of the mesh point p0 is outside the merging area 41. In this case, the same processing as described above may be performed with a merged area obtained by adding a mesh element overlapping the control volume area to the merged area 41 as a processing target area. The presence or absence of the overlap can be detected by examining the position of the circumcenter of each mesh element around the mesh point p0, so that an operation for finding the overlap area is not required.

In addition, in FIG. 4B, a case where the control volume area of the mesh point q is outside the merging area 41 can be considered. For example, when the mesh point q is closer to the mesh side connecting the mesh points p2 and p3, the control volume area of the mesh point q protrudes outside the merged area 41. If a mesh element exists in the protruding portion, the same processing as described above may be performed with a merged area obtained by adding the mesh element to the merged area 41 as a processing target area.

The most troublesome case is, for example, the mesh points q, p
This is the case where the outer center of the mesh element composed of p2 and p3 is outside the merged area 41, and the mesh side connecting the mesh points p2 and p3 is part of the boundary of the analysis area.
This case can be dealt with by changing the control volume area in the mesh element formed by the mesh points q, p2, and p3. Normally, the connection between the middle point of the mesh side and the outer center is the control volume boundary.However, for example, when the control volume is changed to the connection between the middle point of the mesh side and the center of gravity, the control volume is
You can avoid getting out of the office. A flag is set on a mesh element employing such a method, and the calculation method is switched when calculating the control volume area in the element. If the final mesh Mnew is a mesh suitable for the control volume method, there is no such mesh element in Mnew, so such a mesh element created when modifying the mesh will be changed again. And eventually should not remain. In this way, it is possible to prevent the control volume area from being changed outside the merging area 41.

The calculation method of the density value has been described in detail in the case of the mesh point moving process. However, in the case of the mesh point addition and merging process, the density value can be obtained by essentially the same calculation method. .

FIG. 5 shows the mesh shown in FIG.
FIG. 6 shows how the control element areas 52 and 53 corresponding to the mesh elements change when the mesh point q shown in FIG. 6 is added. FIG. 6 shows the mesh point p0 shown in FIG. The state at the time of merging with q is shown. Merged area of mesh elements generated around point q
The density value of the added mesh point or the merged mesh point q is calculated to save the total amount inside (the area 51 shown in FIG. 5 or the area 61 shown in FIG. 6). As described in the move process, mesh elements are added to the merged area or the control volume is temporarily changed so that the control volume area does not change outside the merged area of the mesh element to be processed. Will be performed as appropriate.

FIG. 6 shows a case where the mesh point q after the merging process exists as the original mesh point. However, the original mesh may be located at a position where no mesh point exists. In FIG. 6, two points are merged into one point, but three or more points may be merged into one point. Of course, in these cases, too, the control volume area
It is necessary to set the processing target area so as to include all places where the changes have been made to 62 and 63.

As described above, the addition of one point of a mesh point, the movement of one point, and the operation of merging into one point are performed without obtaining an overlapping area between mesh elements or control volume elements, and a density value in which a total amount is stored. Can be calculated. If the mesh is finally changed to the desired mesh by repeating this correction operation, the density distribution at the mesh points of the new mesh can be obtained while keeping the total amount.

This density interpolation method can also be applied to the finite element method which assumes that the density value changes linearly within the mesh element. FIG. 7 shows that the two-dimensional finite element method using a triangular mesh element is used.
An example of how the mesh element changes when the mesh point p0 shown in FIG. 9A moves to the mesh point q shown in FIG. In this case, consider the region 71 where the mesh element has changed. Regarding the mesh points p1, p2, p3, p4, and p5, since the point positions are the same, it is reasonable to consider that the density value at each mesh point does not change. Then, since there is no change in the total amount outside the region 71,
The density value at the mesh point q in FIG. 7B may be determined so that the amount inside the area 71 does not change.

Assuming that the density value changes linearly within the triangular mesh element, the total amount QT per unit thickness inside a certain element T is given by the following equation (1).

QT = (sum of density values at three vertices of T) ÷ 3 × (area of T) (1)

Since the density values of all mesh points in FIG. 7 (a) before the mesh change are known, the quantity QT within each mesh element obtained from this equation (1) is used as the quantity Qtotal inside the area 71. Is easily obtained by adding Also, in FIG. 7B after the mesh is changed, Qtotal can be written in the form of the following equation (2) by adding the amounts inside the mesh elements T1, T2, T3, T4, and T5.

Qtotal = 1/3 ｛(density value at point p1) ・ (sum of areas of T1 and T2) + (density value at point p2) ・ (sum of areas of T2 and T3) + (point p3 (Density value at point) · (sum of areas of T3 and T4) + (density value at point p4) · (sum of areas of T4 and T5) + (density value at point p5) · (area of T5 and T1) + (Concentration value at point q) · (sum of areas of T1, T2, T3, T4, T5)｝ (2)

In the equation (2), all the values other than the density value at the mesh point q are known values, so that the density value at the mesh point q can be easily obtained. Similarly, it is possible to calculate the density value while keeping the total amount for the operation of adding or merging mesh points.

Next, the operation of this embodiment will be described with reference to an example in which the impurity concentration is interpolated in the simulation of the impurity thermal diffusion process in the semiconductor manufacturing process.
FIG. 8 is a flowchart of the heat diffusion simulation. In step 81, the impurity distribution in the initial state (state at time t = 0) is set. That is, the analysis space is divided into meshes, and the density value at each mesh point is set.
At step 82, the time t is updated at a certain time interval Δt, and at step 83, the impurity distribution after the elapse of Δt is obtained by solving the diffusion problem. Tend is the analysis end time, and the updating of the time is restricted in step 82 so as not to exceed this time.

In step 84, a new mesh suitable for the spatial change of the impurity concentration calculated in step 83 is generated. A more specific description will be given using a one-dimensional example shown in FIG. FIG. 10A shows the impurity distribution in the initial state. The position of the step shown on the horizontal axis is the position of the mesh point, and the step width becomes smaller as the density change degree increases. By the diffusion calculation, the density distribution changes as shown in FIG. Due to the diffusion phenomenon, the change in density is generally slower, and the tail of the distribution in the right direction is widened. According to this change, an operation of changing to a mesh as shown by the horizontal axis in FIG. 8C is performed in step 84 of FIG.

Steps 85 to 88 are portions for obtaining the density value in the new mesh by interpolation. If a new density value is determined, it is determined in step 89 whether or not the end time has been reached. If not, the flow returns to step 82 to continue the diffusion calculation.

In the part of the interpolation calculation of the impurity concentration, first,
In step 85, the mesh point positions of the original mesh and the new mesh are compared, and points to be added and points to be deleted are detected. All points to be added first are added while giving density values one by one (step 86), and then points to be deleted are deleted one by one (step 86).
87, 88). The deletion operation is realized by a merging operation to adjacent mesh points. In this example, first in step 87,
For each point to be deleted, check the closest point among the mesh points used for the new mesh,
Step 88 merges to that point. At this time, the merging operation cannot be performed unless the point to be deleted and the merging destination point are adjacent (connected by a mesh side), so the processing is performed from the point in such a state. The merging destination points are not limited to the points used in the new mesh, and it is of course possible to merge the adjacent mesh points in an arbitrary order. In this embodiment, all the operations of adding mesh points are performed first. Although it is possible to perform the adding operation and the merging operation in an arbitrary order, for example, if all the merging operations are performed first in a case where the positions of most mesh points are changed, it is extremely rough. The mesh represents the impurity concentration distribution temporarily. Even if mesh points are added from such a state, an accurate density distribution cannot be reconstructed. Therefore, it is usually more appropriate to prioritize the adding operation over the merging operation. The procedure for updating the impurity concentration value in each of the adding operation and the merging operation is as described above.

In this example, no mesh point moving operation is used. The mesh point moving operation can be performed by a combination of the mesh point adding operation and the mesh point merging operation. Therefore, even if the mesh changing means 6 in FIG. Can be realized. FIG. 9 shows a flowchart of an embodiment using a moving operation. The difference from the previous example is
That is, the moving operation is included in step 96. In this step 96, an object which can be moved from a point to be deleted to a point to be added is detected. In particular,
If there is a mesh point to be deleted among the vertices of the mesh element before modification including the point to be added, it means that the mesh element can be moved. The procedure for updating the impurity concentration value during the moving operation is as described above.

While the connection relation of mesh points always changes in the addition or merging operation, there is a case where it is not necessary to change the connection relation of mesh points in the moving operation depending on the position of the moved point. In this case, there is no need to regenerate the mesh elements, nor to modify the mesh point and the connection information of the elements. Since it is sufficient to update the coordinate data of the mesh points and the data of the density value, the correction work becomes extremely simple. In addition, if there is no move operation, it is possible to perform two corrections of addition and merging with one operation, so processing can be performed faster when move operation is possible Can be expected. Accordingly, as shown in the flowchart of FIG. 9, the processing can be performed most efficiently by first performing the moving operation in step 96, and then sequentially performing the adding operation and the merging operation in steps 97 and 98.

When the conventional interpolation method is applied to a mesh that has been discretized by the control volume method and a fine mesh distribution is obtained from a coarse mesh distribution, a coarse stepwise distribution results. There is. FIG. 11 shows an example of a one-dimensional distribution. Figure (a)
Represents the initial distribution state. The density values at the mesh points p0, p1, p2, p3, p4, and p5 at the positions along the horizontal axis are indicated by black circles. The area of each rectangular area depicted in FIG. 7A corresponds to the total amount (per unit area) of the corresponding mesh point in the control volume area. Therefore, in the interpolation method that preserves the total amount, the sum of the areas of the rectangular regions does not change. FIG. 3B shows mesh points p1, p2,
Delete all p3 and p4, and use the conventional interpolation method to
5 shows a state in which the concentration at the position of p5 is determined. From this state, if the mesh points p1, p2, p3, and p4 are added again to determine the density distribution by the conventional interpolation method, the state shown in FIG.
The coarse step-like distribution change shown in (b) is reflected as it is, which is completely different from the original distribution state.

FIG. 12 shows a change in distribution when a similar operation is performed using the interpolation method of the present invention. FIG. 7A shows the initial state, and FIG. 7B shows the distribution state after the mesh point p1 is merged with the mesh point p2. FIG. 3C shows the state after the mesh point p2 is further merged with the mesh point p3, and FIG.
Is a state after the mesh point p3 is merged with the mesh point p4 and the mesh point p4 is merged with the mesh point p5. In this state, a rough step-like distribution similar to FIG. 11 (b) is obtained. Next, FIG. 12E shows a state in which the mesh point p1 has been added again. Hereinafter, when the mesh points p2, p3, and p4 are sequentially added, a distribution state obtained by linearly interpolating the distribution of FIG. 12D is obtained as shown in FIG. 12F. As described above, when the interpolation method of the present invention is used, a distribution that changes linearly is obtained. In a numerical simulation, a distribution that changes discontinuously artificially may be used, but in most cases, it is appropriate to assume that the distribution changes continuously. Therefore, the interpolation method of the present invention is more suitable in many cases.

[0057]

A first effect of the present invention is that the density value can be easily calculated when the density value is interpolated, particularly in the interpolation in a high dimension. Development is easier. The reason why the density value calculation is simplified is that it is not necessary to detect overlap between mesh elements or between control volume regions because density values are interpolated while changing the mesh one by one mesh point. .

The second effect of the present invention is that when the distribution of fine meshes is obtained by interpolation based on the distribution of coarse meshes on the premise of the control volume method, a coarse step-like shape caused by the original coarse meshes is obtained. , But a distribution that changes linearly is obtained. The reason that this effect can be obtained is that the density value at that point position is calculated by interpolation while adding mesh points one by one.

[Brief description of the drawings]

FIG. 1 is a block diagram illustrating an example of an embodiment of the present invention.

FIG. 2 is a diagram showing another configuration example of the storage device 2.

FIG. 3 is a flowchart showing an outline of the operation of the exemplary embodiment of the present invention.

FIG. 4 is a diagram for explaining a mesh point moving operation when a control volume method is assumed as a discretization method.

FIG. 5 is a diagram for explaining an operation of adding a mesh point when a control volume method is assumed as a discretization method.

FIG. 6 is a diagram for explaining a merging operation of mesh points when a control volume method is assumed as a discretization method.

FIG. 7 shows a case where a finite element method is assumed as a discretization method.
FIG. 9 is a diagram for explaining a mesh point moving operation.

FIG. 8 is a flowchart showing a first specific example of the operation of the exemplary embodiment of the present invention.

FIG. 9 is a flowchart illustrating a second specific example of the operation of the exemplary embodiment of the present invention.

FIG. 10 is a diagram showing a typical change in concentration distribution and a change in mesh point position in an impurity diffusion calculation in a one-dimensional example.

FIG. 11 is a diagram illustrating a change in a distribution state calculated according to a change from a fine mesh to a coarse mesh and vice versa, assuming a control volume method in a conventional interpolation method.

FIG. 12 is a diagram showing a change in a distribution state calculated according to a change from a fine mesh to a coarse mesh and vice versa, assuming a control volume method in the interpolation method of the present invention.

FIG. 13 is a flowchart for explaining a conventional technique.

FIG. 14 is a diagram for explaining a state in which control volume areas overlap in the related art.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 ... Data processing apparatus 2 ... Storage device 3 ... Recording medium 4 ... Control part 5 ... Mesh sequential change plan means 6 ... Mesh change means 7 ... Density value calculation means 8 ... Data input / output means 9 ... Data about mesh M before change Area 10: Data area of mesh Mnew after change 11: Data area of mesh Mtmp corrected by one point 12: Change plan data area 21: Difference information data area of M and Mtmp 41: Around mesh point q after movement Merged area of formed mesh elements 42 Boundary line of control volume area corresponding to mesh before moving operation 43 Boundary line of control volume area corresponding to mesh after moving operation 51 Around added mesh point q The merged area of the mesh elements generated at the time 52 ... The boundary of the control volume area corresponding to the mesh before the addition operation 53 Boundary line of the control volume area corresponding to the mesh after the addition operation 61 ... Merged area of mesh elements generated around the mesh point q after the merging 62 ... Boundary line of the control volume area corresponding to the mesh before the merging operation 63 … The boundary line of the control volume area corresponding to the mesh after the merging operation 71… the area where the mesh element has changed by the moving operation 141… the two-dimensional discretized mesh 142 before the change… the control volume area corresponding to the mesh 141 Boundary line 143… Control volume area of a certain mesh point of the changed mesh

Claims (10)

[Claims] When the original mesh expressing the density distribution is changed by the density value at each mesh point position, the density value at the mesh point position of the new mesh after the change is interpolated while preserving the total amount. A density interpolating method at the time of changing the mesh, wherein the process of adding one mesh point starting from the original mesh or the process of merging a plurality of mesh points into one mesh point is repeatedly performed. A mesh change step of changing the mesh to the new mesh, and a process of adding one mesh point or a process of merging a plurality of mesh points into one mesh point is performed in the mesh change step. The density value at a mesh point or a mesh point after merging is used as the density value at other Concentration interpolation method when the mesh changes, wherein a total amount of before and after the Interview update and a density value calculation step of determining to be the same. 2. In the mesh changing step,
2. The density interpolation method according to claim 1, wherein the process of adding one mesh point is performed prior to the process of merging a plurality of mesh points into one mesh point. 3. In the mesh changing step,
In order to change the original mesh to the new mesh, a process of moving a mesh point to be deleted to a mesh point to be added is also performed. In the density value calculation step, the mesh point to be deleted should be added. When the process of moving to the mesh point is performed, the density value at the moved mesh point is determined so that the density value other than the mesh point remains unchanged, and the total amount before and after the mesh update is the same. 2. The density interpolation method according to claim 1, wherein the mesh is changed. 4. In the mesh changing step,
4. The density interpolation method according to claim 3, wherein the process of moving the mesh point to be deleted to the mesh point to be added is given the highest priority, and the process of adding one mesh point is given the highest priority next. 5. When the original mesh expressing the density distribution is changed by the density value at each mesh point position, the density value at the mesh point position of the new mesh after the change is interpolated while preserving the total amount. A density interpolating apparatus for changing a mesh to be performed, wherein a process of adding one mesh point starting from the original mesh or a process of merging a plurality of mesh points into one mesh point is repeatedly performed. Mesh changing means for changing the mesh to the new mesh, and each time the processing for adding one mesh point or the processing for merging a plurality of mesh points into one mesh point is performed in the mesh changing means, Updates the density value at the mesh point or the merged mesh point, leaving the density values other than the mesh point unchanged Concentration interpolator when the mesh changes, wherein a total amount of later and a density value calculating means for determining to be the same. 6. The apparatus according to claim 5, wherein said mesh changing means performs a process of adding one mesh point prior to a process of merging a plurality of mesh points into one mesh point. Density interpolation device when changing the mesh described. 7. The apparatus according to claim 7, wherein said mesh changing means also performs a process of moving a mesh point to be deleted to a mesh point to be added in order to change said original mesh to said new mesh, The calculating means, when a process of moving the mesh point to be deleted to a mesh point to be added is performed,
6. The method according to claim 5, wherein a density value at the mesh point after the movement is determined such that the total value before and after the mesh update is the same, with the density values other than the mesh point unchanged. Density interpolator. 8. The mesh changing means has a configuration in which a process of moving a mesh point to be deleted to a mesh point to be added is given the highest priority, and a process of adding one mesh point is given priority next. Claim 7
Density interpolation device when changing the mesh described. 9. When the original mesh expressing the density distribution is changed by the density value at each mesh point position, the density value at the mesh point position of the new mesh after the change is interpolated while preserving the total amount. A machine-readable recording medium on which a program for realizing a density interpolating device at the time of mesh change is recorded by a computer, wherein the computer is configured to add one mesh point starting from the original mesh or a plurality of processes. Mesh changing means for changing the original mesh to the new mesh by repeatedly performing a process of merging the mesh points into one mesh point; processing for adding one mesh point in the mesh changing means or a plurality of mesh points Each time the process of merging into one mesh point is performed, the added mesh A machine that records a program for functioning as a density value calculating means for determining the density value at a point or a mesh point after merging so that the density value before and after the mesh update is the same, with the density value at other mesh points unchanged. A readable recording medium. 10. When the original mesh expressing the density distribution is changed by the density value at each mesh point position, the density value at the mesh point position of the new mesh after the change is interpolated while preserving the total amount. A machine-readable recording medium on which a program for realizing a density interpolating device at the time of changing a mesh by a computer is recorded, wherein the computer is configured to add one mesh point starting from the original mesh. By repeatedly performing the process of merging the mesh points into one mesh point or the process of moving the mesh point to be deleted to the mesh point to be added,
Mesh changing means for changing the original mesh to the new mesh; processing for adding one mesh point in the mesh changing means; processing for merging a plurality of mesh points into one mesh point; or adding a mesh point to be deleted. Every time the process of moving to the power mesh point is performed, the density value at the added mesh point or the mesh point after merging, the density value other than the mesh point is unchanged, and the total amount before and after the mesh update is the same. A machine-readable recording medium on which a program for functioning as density value calculation means is determined.