Tetrahedral mesh generation and recording medium storing...

Details Tetrahedral mesh generation and recording medium storing... Tetrahedral mesh generation and recording medium storing...

1998-09-21

2001-06-26

Vo, Cliff N. (Department: 2772)

Computer graphics processing and selective visual display system

Computer graphics processing

Three-dimension

C345S419000

Reexamination Certificate

active

06252601

ABSTRACT:

FIELD OF THE INVENTION
This invention relates to a system and method for generating a three-dimensional discretized mesh and, more particularly, to a system and method for generating a three-dimensional discretized mesh used to solve partial differential equation numerically by computer.
BACKGROUND OF THE INVENTION
Description of the Related Art
The following known literature relating to the above-mentioned technology will be referred to below:
(1) Japanese Patent Kokai Publication No. JP-A-7-121579
(2) Japanese Patent Kokai Publication No. JP-A-8-329284
(3) M. S. Mock, “Tetrahedral elements and the Scharfetter-Gummel Method,” Proc. NASECODE IV, June, 1985, pp. 36-47
(4) P. Fleischmann and S. Selberherr, “A New Approach to Fully Understand Unstructured Three-dimensional Delaunay Mesh Generation with Improved Element Quality,” Proc. SISPAD, September, 1996, pp. 129-130
Solving partial differential equations numerically involves generating a discretized mesh which tesselates an analytic domain into minute domains, deriving simultaneous equations which approximate the original partial differential equations and then solving the simultaneous equations. Methods of deriving the simultaneous equations include the finite-element method and the finite-difference method, with the particular method that is used depending upon the nature of the problem to be solved. For example, in the case of a device simulator which computes the electrical characteristic of a semiconductor device through use of a computer, extensive use is made of a method referred to as the control volume method or box integration method derived as a finite-element method. The method defines a minute domain allotted to each discretized grid point and uses a trial function that has a value of 1 within a minute domain and a value of 0 elsewhere.
A method of generating a three-dimensional discretized mesh used in discretized approximation of an equation is set forth in say of Japanese Patent Kokai Publication No. JP-A-7-121579, which deals with the finite-element method. Japanese Patent Kokai Publication No. JP-A-7-121579 proposes a finite-element mesh generation method that includes forming intermediate mesh by partitioning a three-dimensional model to be analyzed into the finite elements of a plurality of tetrahedrons and the finite elements of a pentahedron and/or hexahedron, subdividing each finite element of this intermediate mesh by the finite elements of a hexahedron and forming the entire analytical model as a hexehedral mesh.
When numerical analysis is performed using the control volume method, however, the numerical analysis is rendered unstable if the discretized mesh has not undergone tesselation referred to as Delaunay partitioning. Ordinarily, therefore, it is not possible to apply a mesh generation method for the finite-element method that does not take the nature of such a mesh into account.
Examples of a mesh generation method for a case where a three-dimensional control volume method is used are described in “Tetrahedral elements and the Scharfetter-Gummel Method,” Proc. NASECODE IV, pp. 36-47, June 1985 by S. Mock and in the Japanese Patent Kokai Publication No. JP-A-8-329284. These methods involve repeating an operation which includes first creating a Delaunay mesh that includes an analytic domain and then adding a single grid point to this mesh to locally correct the mesh. The correction is carried out in such a manner that the Delaunay-partitioned nature of the mesh will not be lost.
In order to perform the tetrahedral mesh revision by this method, it is necessary to delete mesh elements together with their connecting information and set the connecting information again correctly. Developing a program for executing such processing correctly requires great care and is not easy.
A method of obtaining a Delaunay-partitioned mesh by the advancing front method is disclosed as a method through which the tetrahedral mesh correction operation can be avoided. For example, see “A New Approach to Fully Understand Unstructured Three-dimensional Delaunay Mesh Generation with Improved Element Quality,” Proc. SISPAD, pp. 129-130, September, 1996.
An overview of the conventional advancing front method will be described with reference to
FIGS. 8 and 9
.
FIG. 8
is a block diagram illustrating the relationship between processing content and control data in the conventional advancing front method, and
FIG. 9
is a flowchart illustrating the processing procedure of the conventional advancing front method.
In the flowchart of
FIG. 9
, first the surface of an analytic domain is partitioned into a triangular mesh at step S
1
. Next, at step S
2
, the mesh surfaces of the triangular mesh created at step S
1
are registered as unprocessed triangular surfaces. Step S
3
is iterated as long as an unprocessed surface exists. Step S
3
includes selecting one unprocessed triangular surface and extracting a tetrahedral element having this triangular surface as one side face thereof. (This step corresponds to means
7
in
FIG. 8
for extracting a tetrahedral element having a triangular surface as one side face.)
In order to construct the tetrahedral element, a grid point must be given in addition to the triangular surface. Grid points given inside the analytic domain also are utilized in addition to the grid points of the triangular mesh on the domain surface. The grid points inside the analytic domain can all be given in advance or the grid points can be generated at appropriate positions when the tetrahedral element is extracted.
As the extraction of the tetragonal elements proceeds, the domain that is to be meshed becomes smaller in size. The list of unprocessed triangular surfaces is updated at step S
3
of
FIG. 9
in such a manner that unprocessed triangular surfaces become a set of triangular surfaces on the periphery of the domain that is to be meshed. In other words, among side faces of a tetrahedral element that has been extracted, surfaces that have been registered in the list of unprocessed surfaces are deleted from the list and triangular surfaces that have been created anew are registered in the list.
FIG. 8
shows the relationship between means provided in a data processing unit
1
and data preserved in a memory device
2
in order to execute the processing described above. This advancing front scheme comprises the processing unit
1
, the storage (memory) device
2
and a storage medium
3
. The data processing unit
1
includes means
4
for generating a triangular mesh on a surface, initial setting means
5
for setting the surface of a domain on which a mesh is to be generated, and means
7
for extracting a tetrahedron having a triangular surface as one side face thereof. The storage device
2
includes a storage unit
9
for storing a three-dimensional analytic domain, a storage unit
10
for storing a triangular surface mesh, a storage unit
11
for storing an unprocessed triangular surface, and a storage unit
13
for storing a tetrahedral mesh.
In semiconductor simulation, an analytic domain almost always comprises a plurality of areas of different material qualities. This can be dealt with by generating a triangular mesh on the boundary surface between different materials and then applying the processing from step S
2
(
FIG. 9
) onward to each domain of each material.
An overview of an approach for applying the advancing front method to the generation of a Delaunay mesh will be described with reference to FIGS.
10
(A)-
10
(C). For the sake of simplicity, we will discuss the case of a two-dimensional example in which a grid point is not placed inside a domain for which a mesh is to be generated. A property of a Delaunay mesh is that grid points are not enclosed by the circumcircle of a mesh element. The advancing front method is applied as set forth below utilizing this property.
In FIG.
10
(A) it is assumed that a grid point exists only a vertex of a polygon of a domain
14
a
for which a mesh is to be generated. One side
15
a
on the perimeter of the domain
14
is selected and a grid point (the

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