JP7548604B2 - Method for connecting lines, method for displaying second-order real symmetric tensors, and program for displaying second-order real symmetric tensors - Google Patents

Description

The present invention relates to a method for connecting line segments, a method for displaying real symmetric second-order tensors, and a program for displaying real symmetric second-order tensors.

Generally, physical quantities that appear in physics problems are mathematically described using tensors. Among tensors, zero-order tensors are called scalars, and first-order tensors are called vectors. A description of a scalar at each point in a specific region as a function of the position of each point and time is called a scalar field, and a description of a vector at each point in the region as a function of the position of each point and time is called a vector field.

Visualization of scalar and vector fields has been practiced for a long time, whether it is a problem involving a region that has a two-dimensional spatial extent (two-dimensional region) or a problem involving a region that has a three-dimensional extent (three-dimensional region), and these visualization techniques are implemented in a variety of numerical analysis and visualization tools.

A "second-order tensor field in a two-dimensional domain" is a field that expresses physical quantities (e.g., strain, stress, etc. in a continuum) expressed using second-order tensors as a function of the position and time of each point in a domain that has a two-dimensional spatial extent. There are many visualization techniques for second-order tensor fields in two-dimensional domains. For example, Non-Patent Document 1 discloses a visualization technique using principal stress lines that connect the principal axis vectors of the stress field (a second-order symmetric tensor field in a two-dimensional domain) that occurs in a plate of a linear elastic body in a balanced state in an environment where a load acts statically.

Photoelasticity Journal Vol. 4, No. 1 & 2, (1982) pp. 20-27, "Computer methods for principal stress analysis and tracing of principal stress lines" (by Masataka Nishida, Makoto Kotani, Kiyoshi Nakajima, Setsuzou Honda, and Hirotake Shinke)

A visualization method has been proposed in which the directions of the principal axis vectors of the stress tensor at each point discretely placed within a two- or three-dimensional region are represented by line segments. Also proposed is a visualization method in which the lengths of the line segments in the directions of the principal axis vectors of the stress tensor at each point discretely placed within a two- or three-dimensional region correspond to the principal stress values in the directions of the respective principal axis vectors, and the positive and negative principal stresses are represented by the directions of the arrows attached to the line segments.

However, a "second-order tensor field in a three-dimensional domain" is a field that expresses physical quantities (e.g., strain, stress, etc. in a continuum) expressed using second-order tensors as a function of the position and time of each point in a domain that has a three-dimensional spatial extent. A second-order tensor field in a three-dimensional domain is expressed by six components for each point in the domain in the case of a second-order symmetric tensor, and by nine components in the case of a second-order asymmetric tensor. In such a second-order tensor field in a three-dimensional domain, there have been few proposals for a method of visualizing information such as the directions of the principal axes and the magnitudes of the corresponding principal values at each point in the domain, and there is a need for a display method that makes it easy to correctly and intuitively grasp this information.

The present invention has been made in consideration of the above circumstances, and has as its exemplary objective the provision of a method of connecting lines for displaying in a three-dimensional real second-order symmetric tensor field the directions of the principal axes and the magnitudes of the corresponding principal values at each point within the domain in a manner that allows them to be correctly and intuitively grasped, a method of displaying a real second-order symmetric tensor using the method of connecting lines, and a program for displaying a real second-order symmetric tensor.

In order to solve the above problems, a method for connecting lines as an exemplary aspect of the present invention is a method for connecting lines in a three-dimensional domain, comprising the following steps (1) to (7).

(1) Three first line segments OA, OB, and OC are defined that extend along the principal axis directions corresponding to three principal values of a real symmetric second-order tensor at an origin O selected as an arbitrary point within a three-dimensional region. Here, the three first line segments are all line segments of a predetermined infinitesimal length that extend from the origin O, and are line segments that constitute a selected coordinate system, which is either a right-handed coordinate system or a left-handed coordinate system, centered on the origin O, and are three-dimensionally orthogonal to each other.
(2) Three first line segments OA', OB', and OC' of predetermined small lengths are defined, extending from the starting point O. Here, the first line segment OA' is a line segment that extends in the opposite direction to the first line segment OA on the same line, the first line segment OB' is a line segment that extends in the opposite direction to the first line segment OB on the same line, and the first line segment OC' is a line segment that extends in the opposite direction to the first line segment OC on the same line.
(3) One line segment is selected from the six first line segments OA to OC and OA' to OC' as a first selected line segment, and the end point other than the starting point O is defined as an end point P.
(4) Three second line segments Pa, Pb, and Pc are defined, which extend along the principal axis directions corresponding to the three principal values of the second-order real symmetric tensor at the end point P. Here, the three second line segments are all line segments of a predetermined infinitesimal length extending from the end point P, which constitute the selected coordinate system centered on the end point P and are three-dimensionally orthogonal to each other.
(5) Three second line segments Pa', Pb', and Pc' of predetermined small lengths are defined, extending from the end point P. Here, the second line segment Pa' is a line segment that extends in the opposite direction to the second line segment Pa on the same line, the second line segment Pb' is a line segment that extends in the opposite direction to the second line segment Pb on the same line, and the second line segment Pc' is a line segment that extends in the opposite direction to the second line segment Pc on the same line.
(6) Based on the result of a calculation process using the six second line segments Pa-Pc, Pa'-Pc' and the first selected line segment, one second line segment that can be most smoothly connected to the first selected line segment is selected as a second selected line segment from among the six second line segments Pa-Pc, Pa'-Pc' as a second selected line segment, and the second selected line segment is connected to the first selected line segment, and the remaining second line segments are discarded.
(7) Of the two end points of the second selected line segment, the end point connected to the first selected line segment is read as a starting point O, the opposite end point is read as an end point P, and the second selected line segment is read as the first selected line segment, and then steps (4) to (6) are repeatedly executed until a specified termination condition is satisfied.

In addition, a method for displaying a second-order real symmetric tensor according to another exemplary aspect of the present invention includes connecting the first selected line segment of the predetermined infinitesimal length and the second selected line segment of the predetermined infinitesimal length using the above-mentioned line segment connecting method;
The connected selected line segments are displayed as lines or curves drawn within the three-dimensional region.

Furthermore, as yet another exemplary aspect of the present invention, a display program for second-order real symmetric tensors causes a computer to execute the above-mentioned display method for second-order real symmetric tensors.

Further objects and other features of the present invention will become apparent from the preferred embodiments described below with reference to the accompanying drawings.

According to the present invention, in a second-order real symmetric tensor field in a three-dimensional domain, the directions of the principal axes and the magnitudes of the corresponding principal values at each point in the domain can be displayed correctly and intuitively in a manner that is easy to understand.

FIG. 1 is a conceptual diagram of a computer S for realizing the operation of a tensor display program PG according to the present invention. FIG. 2 is a diagram showing an arbitrary point (origin O) in a three-dimensional region. 13 is a diagram illustrating a state in which an end point P of a first selected line segment is defined. FIG. 13 is a diagram showing how line segments are successively connected to a first selected line segment OA starting from a starting point O. FIG. 1 is a diagram showing a state in which straight lines or curves extend in six directions from a starting point O. 13 is a diagram showing a second selected group of line segments that form a right-handed coordinate system centered on point A. FIG. FIG. 2 is a diagram showing a three-dimensional domain V1 according to the first embodiment. This is an example of a display showing a real symmetric second-order tensor corresponding to the stress in a three-dimensional region V1 when a tensile force of 100 MPa is applied to the three-dimensional region V1 in the vertical direction. 13 is another example of a display showing a real symmetric second-order tensor corresponding to the stress in a three-dimensional region V1 when a tensile force of 100 MPa is applied to the three-dimensional region V1 in the vertical direction. This is yet another example of a display showing a real symmetric second-order tensor corresponding to the stress in a three-dimensional region V1 when a tensile force of 100 MPa is applied to the three-dimensional region V1 in the vertical direction. FIG. 11 is a diagram showing a three-dimensional domain V2 according to the second embodiment. This is an example of a display showing a real symmetric second-order tensor corresponding to the stress in a three-dimensional region V2 when a tensile force of 100 MPa is applied to the three-dimensional region V2 in the vertical direction. 13 is another example of a display showing a real symmetric second-order tensor corresponding to the stress in a three-dimensional domain V2 when a tensile force of 100 MPa is applied to the three-dimensional domain V2 in the vertical direction. This is yet another example of a display showing a real symmetric second-order tensor corresponding to the stress in a three-dimensional region V2 when a tensile force of 100 MPa is applied to the three-dimensional region V2 in the vertical direction. FIG. 11 is a diagram showing a three-dimensional domain V3 according to a third embodiment. This is an example of a display showing a real symmetric second-order tensor corresponding to the stress in a three-dimensional domain V3 when a tensile force of 100 MPa is applied to the three-dimensional domain V3 in the vertical direction.

[Embodiment 1]
Hereinafter, a display program for a real symmetric second-order tensor (hereinafter referred to as a tensor display program) PG according to a first embodiment of the present invention will be described with reference to the drawings. FIG. 1 is a conceptual diagram of a computer S for implementing the operation of the tensor display program PG. The tensor display program PG causes the computer S to execute a display method for a real symmetric second-order tensor (hereinafter referred to as a tensor display method) including steps 1 to 7 described below. The tensor display method is implemented by displaying, on a screen D of the computer S, line segments connected based on the line segment connection method described in steps 1 to 7.

The computer S has a calculation processing unit (CPU) 2 and a memory 4 as its main components. The memory 4 is a storage medium, and in addition to the tensor display program PG, various data and setting items are stored as data therein. The tensor display program PG may of course be stored in an external storage device (e.g., portable storage or a cloud server, etc.), or may be stored in the memory 4 in the computer S from the beginning.

The CPU 2 is a calculation device that constitutes the main part of the computer S. When the tensor display program PG is started, the CPU 2 executes the tensor display method based on its operating instructions.

The tensor representation method uses a line segment connection method that successively connects line segments of a predetermined infinitesimal length extending from each point in a three-dimensional region. Each point in the three-dimensional region corresponds to each infinitesimal region when the three-dimensional region is divided into predetermined infinitesimal regions.
Therefore, a line segment of infinitesimal length is a line segment that connects two points in two adjacent infinitesimal regions.

Here, a three-dimensional region means a solid object that physically exists in three-dimensional space, regardless of its shape. A specified infinitesimal region is conceptually a three-dimensional shape divided into extremely small parts, and a specified infinitesimal length is conceptually an extremely short length. A specified infinitesimal length is a finite length, and this length is set according to the convenience of the calculation process. In general, a shorter length improves the calculation accuracy but increases the amount of calculation processing and the calculation processing load, and a longer length reduces the calculation accuracy but reduces the amount of calculation processing and the calculation processing load.

Figure 2 is a diagram showing an arbitrary point O within a three-dimensional region. The shape of the three-dimensional region is not particularly limited, and may be, for example, a cube, a cylinder, a triangular pyramid, a sphere, or a three-dimensional shape having other shapes. Point O is a point arbitrarily selected to start the calculation process. Therefore, the first point selected will be specifically referred to as the starting point O. Starting point O may be located anywhere within the three-dimensional region, and is selected based on the judgment of the person executing the calculation process (hereinafter referred to as the user).

(Step 1) Define three first line segments OA, OB, and OC (hereinafter referred to as first line segments OA-OC) that extend along the principal axis vector directions (principal axis directions) corresponding to the three principal stress values (principal values) of the second-order real symmetric tensor at the starting point O (see Figure 2). Here, the three first line segments are all line segments of a predetermined infinitesimal length that extend from the starting point O, and are line segments that are three-dimensionally mutually orthogonal, constituting a selected coordinate system that is either a right-handed coordinate system or a left-handed coordinate system centered on the starting point O.

In FIG. 2, the first line segments OA-OC are line segments that extend in mutually perpendicular directions in three-dimensional space, and their direction is the principal axis vector direction at starting point O. It does not matter which of the first line segments OA-OC is the first principal axis vector direction, the second principal axis vector direction, or the third principal axis vector direction. The first line segments OA-OC are all line segments of finite length, and their lengths are predetermined infinitesimal lengths. Note that in this embodiment 1, a case will be described in which the selected coordinate system is a right-handed coordinate system, and the first line segments OA-OC constitute the right-handed coordinate system in this order. The selected coordinate system is not limited to a right-handed coordinate system, and may be a left-handed coordinate system.

(Step 2) Next, three first line segments OA', OB', OC' (hereinafter referred to as first line segments OA'-OC') of a predetermined minute length extending from starting point O are defined. Here, first line segment OA' is a line segment that extends in the opposite direction on the same line as first line segment OA, first line segment OB' is a line segment that extends in the opposite direction on the same line as first line segment OB, and first line segment OC' is a line segment that extends in the opposite direction on the same line as first line segment OC. As a result, as shown in Figure 2, six first line segments OA-OC, OA'-OC' are defined that extend in six axial directions with starting point O as the center. Points A-C and A'-C' are all end points of the respective first line segments (points that are not starting point O).

(Step 3) One line segment is selected from the six first line segments OA-OC, OA'-OC' as the first selected line segment, and the end point that is not the starting point O is defined as the end point P. In this embodiment 1, a case will be described in which the first line segment OA is selected as the first selected line segment. Therefore, point A is defined as the end point P.

It should be noted that which first line segment is selected as the first selected line segment is based on the user's judgment. If first line segment OA is selected as the first selected line segment corresponding to starting point O, point A is the end point P, but of course if another first line segment (OB-OC, OA'-OC') is selected as the first selected line segment, the other point (B-C, A'-C') becomes the end point P. Also, as will be described later, each time line segments are connected, the end point of the first selected line segment selected each time is defined as the end point P. Figure 3 is a diagram explaining the state in which point A is defined as the end point P.

(Step 4) Define three second line segments Pa, Pb, and Pc (hereinafter referred to as Pa to Pc) that extend along the principal axis vector directions (principal axis directions) corresponding to the three principal stress values (principal values) of the second-order real symmetric tensor at the end point P (point A) (see Figure 3). Here, the three second line segments are all line segments of a predetermined infinitesimal length that extend from the end point P, and are line segments that are three-dimensionally mutually orthogonal, constituting a selected coordinate system centered on the end point P.

Note that here, since the end point P = point A, the second line segments Pa to Pc can be expressed as the second line segments Aa to Ac. In FIG. 3, the second line segments Aa to Ac are line segments that extend in mutually perpendicular directions in a three-dimensional space, and their direction is the principal axis vector direction at point A. It does not matter which of the second line segments Aa to Ac is the first principal axis vector direction, the second principal axis vector direction, or the third principal axis vector direction. The second line segments Aa to Ac are all line segments of finite length, and their lengths are a predetermined infinitesimal length. Note that in this embodiment 1, since the selected coordinate system is a right-handed coordinate system, the second line segments Aa to Ac are in this order in the right-handed coordinate system.

(Step 5) Define three second line segments Pa', Pb', and Pc' (hereinafter referred to as Pa'-Pc') of a predetermined minute length extending from the end point P (point A). Here, the second line segment Pa' is a line segment that extends in the opposite direction on the same line as the second line segment Pa, the second line segment Pb' is a line segment that extends in the opposite direction on the same line as the second line segment Pb, and the second line segment Pc' is a line segment that extends in the opposite direction on the same line as the second line segment Pc. The second line segments Pa'-Pc' can also be expressed as the second line segments Aa'-Ac'. As a result, as shown in FIG. 3, six second line segments Aa-Ac, Aa'-Ac' are defined that extend in six axial directions with point A as the center. Points a-c and a'-c' are all end points (points other than point A) of the respective second line segments.

(Step 6) Based on the result of the calculation process using the six second line segments Aa-Ac, Aa'-Ac' and the first selected line segment OA, one second line segment that can be most smoothly connected to the first selected line segment OA is selected from the six second line segments Aa-Ac, Aa'-Ac' as the second selected line segment, and the second selected line segment is connected to the first selected line segment and the remaining second line segments are discarded. In this embodiment 1, the case where the second selected line segment is Aa will be described. That is, the second selected line segment Aa is connected to the first selected line segment OA, and the remaining second line segments Aa, Ac, Aa'-Ac' are discarded. Here, discarding the remaining second line segments means that the second line segments other than the second selected line segment are not connected to the first selected line segment.

The specific process of (Step 6), i.e., a specific example of the calculation process used to select one second line segment that can be most smoothly connected to the first selected line segment as the second selected line segment, will be described later.

(Step 7) Of the two end points of the second selected line segment Aa, the end point (point A) connected to the first selected line segment is read as the new starting point O, the opposite end point (point b) is read as the new end point P, and the second selected line segment Aa is read as the new first selected line segment OA. Then, steps (step 4) to (step 6) are repeatedly executed until a predetermined termination condition is satisfied. The second selected line segment is connected to the first selected line segment, and then the second selected line segment is read as the new first selected line segment. A new second selected line segment to be connected to the new first selected line segment is selected based on the calculation processing of step 6, and the selected new second selected line segment is connected to the new first selected line segment. Each time a new second selected line segment is connected, the new second selected line segment is read as a new first selected line segment from a new starting point O to a new end point P, and a new second selected line segment is selected.

In this way, by repeating (step 4) to (step 6), the infinitesimal line segments starting from the starting point O are connected one after another to form a straight line or a curve. FIG. 4 is a diagram showing how the selected line segments Aa, ax1, x1x2, x2x3, etc. are connected one after another to the first selected line segment OA starting from the starting point O by repeating (step 4) to (step 6). By repeating (step 4) to (step 6), a straight line or a curve smoothly connected to the first selected line segment OA can be formed. As shown in FIG. 4, the tensor display method is realized by displaying the straight line or the curve formed by connecting the infinitesimal line segments on the screen D of the computer S. The predetermined termination condition will be described later.

(Specific example of step 6)
A specific example of the calculation process for selecting, as the second selected line segment, one second line segment that can be most smoothly connected to the first selected line segment OA from the six second line segments Aa to Ac and Aa' to Ac', will be described.
An arbitrary second line segment (here, the second line segment Aa) is selected from the six second line segments Aa to Ac and Aa' to Ac'. A direction cosine value CSa between the first selected line segment OA and the second line segment Aa is calculated. Next, another second line segment Ab is selected, and a direction cosine value CSb between the first selected line segment OA and the second line segment Ab is calculated. In this manner, the direction cosine values CSa to CSc and CSa' to CSc' between the first selected line segment OA are calculated for all six second line segments Aa to Ac and Aa' to Ac'. The second line segment corresponding to the maximum of these six direction cosine values CSa to CSc and CSa' to CSc' is selected as the second selected line segment. For example, when the direction cosine value CSa is the maximum of the six direction cosine values, the second line segment Aa is selected as the second selected line segment.

It is possible to select the second line segment that forms the most obtuse angle with the first selected line segment (i.e., that connects most smoothly with the first selected line segment) as the second selected line segment.

(Predetermined termination conditions)
Next, a description will be given of a predetermined termination condition for stopping the repetition of steps 4 to 6 and terminating the line segment connection in step 7. For example, if the number of line segment connections, i.e., the number of repetitions of the arithmetic processing from step 4 to step 6, is set in advance by the user to a finite number such as 1000 or 10000, the line segment connection is terminated when the set number is reached. The line segment connection is also terminated when the user presses a stop switch in the middle of the arithmetic processing and inputs a command to interrupt the arithmetic processing. Terminating the line segment connection means stopping the execution of the process.

(Display of principal stress value (principal value))
In addition, each infinitesimal line segment extending in the direction of the principal axis vector at each point in the three-dimensional region has a corresponding principal stress value (principal value). By displaying the infinitesimal line segments in different colors according to the principal stress values (principal values), it is possible to display not only the direction of the principal axis vector but also its magnitude on the screen D in a manner that makes it easy to intuitively grasp it. If the principal stress values (principal values) are classified and color-coded according to the magnitude and positive/negative of the principal stress values (principal values) for a predetermined numerical range, the user can easily grasp the principal stress value (principal value) at each point.

In this embodiment 1, the first principal axis vectors, the second principal axis vectors, and the third principal axis vectors at each point in the three-dimensional region are not necessarily connected to each other. According to this embodiment 1, line segments in the principal axis vector direction that the user feels is smooth and that is easy to intuitively grasp are connected to each other. In addition, the straight line or curve resulting from the connection of multiple line segments will change color midway along the line.

[Embodiment 2]
In the above embodiment 1, the case where the first line segment OA is selected as the first selected line segment has been described. In embodiment 2, all of the six first line segments OA-OC, OA'-OC' are selected as the first selected line segments. As a result, as shown in FIG. 5, the second selected line segments are sequentially connected to all of the first line segments OA-OC, OA'-OC', and straight lines or curved lines extend in six directions from the starting point O. It is possible to display a second-order real symmetric tensor that extends in six directions from the starting point O and is easy to understand intuitively. Note that the procedures other than those described above are the same as those in embodiment 1, so that the description will be omitted.

[Embodiment 3]
In the above embodiment 1, a case where the first line segment OA is selected as the first selected line segment based on the user's judgment is described. In embodiment 3, a case where only two of the three principal stress values (principal values) of the second-order real symmetric tensor at the starting point O match and the remaining principal stress values (principal values) do not match is described.

For example, when the principal stress value (principal value) sA and the principal stress value (principal value) sB are the same value among the three principal stress values (principal values) sA, sB, and sC at the starting point O (principal stress values (principal values) sA, sB, and sC correspond to the first line segments OA, OB, and OC, respectively), and only the principal stress value (principal value) sC is different, the first selected line segment is not selected based on the user's judgment. The first line segment OA corresponding to the principal stress value (principal value) sA or the first line segment OB corresponding to the principal stress value (principal value) sB is not selected as the first selected line segment, and the first line segment OC corresponding to the principal stress value (principal value) sC or the first line segment OC' on the same straight line as the first selected line segment is selected as the first selected line segment. Note that the procedures other than those described above are the same as those in embodiment 1, so explanations will be omitted.

[Modification 1]
(Another specific example 1 of step 6)
We will now explain another specific example (other specific example 1) of the calculation process for selecting one second line segment from the six second line segments Aa to Ac, Aa' to Ac' that can be most smoothly connected to the first selected line segment OA as the second selected line segment.
First, difference values da to dc between the principal stress value (principal value) (here, principal stress value (principal value) sA) corresponding to the first selected line segment (here, first line segment OA) and the principal stress values (principal values) sa to sc (at end point P) corresponding to the second line segments Aa to Ac, respectively, are calculated. From the difference values da to dc, the one with the smallest absolute value (e.g., da) is selected, and the second line segment corresponding to the difference value with the smallest absolute value (difference value da) is defined as a candidate second line segment (e.g., second line segment Aa).

Calculate the direction cosine value (direction cosine value CSa) between the candidate second line segment (second line segment Aa) and the first selected line segment (first line segment OA). Calculate the direction cosine value (direction cosine value CSa') between the second line segment (second line segment Aa') that is collinear with the candidate second line segment and extends in the opposite direction, and the first selected line segment (first line segment OA). Select the second line segment (second line segment Aa) that corresponds to the larger direction cosine value (for example, direction cosine value CSa) of the two direction cosine values (direction cosine values CSa, CSa') as the second selected line segment.

It is possible to select, as the second selected line segment, the second line segment that forms the most obtuse angle with the first selected line segment (i.e., that is most smoothly connected to the first selected line segment) from among the second line segments that have the smallest difference with the principal stress value (principal value) corresponding to the first selected line segment. Note that the steps other than those described above are the same as those in embodiment 1, and therefore will not be described here.

[Modification 2]
(Another specific example 2 of step 6)
We will now explain yet another specific example (another specific example 2) of the calculation process for selecting, as the second selected line segment, one second line segment that can be most smoothly connected to the first selected line segment OA from among the six second line segments Aa to Ac, Aa' to Ac'.
First, any three second line segments are selected from the six second line segments Aa to Ac and Aa' to Ac'. These three second line segments are the three that make up a selected coordinate system (here, a right-handed coordinate system) centered on the end point P (point A), and this set of three second line segments is defined as a set of second selected line segments.

Figure 6 is a diagram showing the second selected group of line segments that form a right-handed coordinate system centered on point A. For example, when the end points of the second line segments are endpoints a, b, and c, the second selected group of line segments is a set made up of second line segments Aa, Ab, and Ac, in that order (A-abc in Figure 6), a set made up of second line segments Ab, Ac, and Aa, in that order (A-bca in Figure 6), and a set made up of second line segments Ac, Aa, and Ab, in that order (A-cab in Figure 6), for a total of three sets.

Similarly, there are three pairs of end points for the second line segment when it is end points b, a', c (A-ba'c, A-a'bc, A-cba' in Fig. 6), three pairs of end points a', b', c (A-a'b'c, A-b'ca', A-ca'b' in Fig. 6), three pairs of end points a, c, b' (A-acb', A-cb'a, A-b'ac in Fig. 6), three pairs of end points b, a, c' (A-bac', A-ac'b, A-c'ba in Fig. 6), three pairs of end points a', b, c' (A-a'bc', A-ba'c', A-c' in Fig. 6), a'b.), three sets when the end points are a', c', b' (A-a'c'b', A-c'b'a', A-b'a'c' in Figure 6), and three sets when the end points are a, b', c' (A-ab'c', A-b'c'a, A-c'ab' in Figure 6). A total of 24 sets of second selected line segment groups are defined.

The three first line segments that include the first selected line segment (here, the first line segment OA) and that form a selected coordinate system (here, a right-handed coordinate system) centered on the starting point O are selected as a first selected line segment group (here, a group consisting of the first line segments OA, OB, and OC). Note that if this is the first time that the first selected line segment group is selected (i.e., the first time that step 6 is executed), the user may arbitrarily select the first selected line segment group that includes the first selected line segment OA and forms the right-handed coordinate system (for example, it may be a group consisting of the first line segments OA, OC, and OB'). The case in which the first selected line segment group is selected for the second or subsequent time (i.e., the second or subsequent time that step 6 is executed) will be described later.

Calculate the direction cosine values between one set of the first selected line segment group and each of the 24 sets of the second selected line segment group. The direction cosine values are calculated for corresponding lines between the first selected line segment group and the second selected line segment group. For example, if the first selected line segment group is composed of OA, OB, and OC, and the second selected line segment group is composed of Aa, Ab, and Ac, calculate the direction cosine value between the first line segment OA and the second line segment Aa, the direction cosine value between the first line segment OB and the second line segment Ab, and the direction cosine value between the first line segment OC and the second line segment Ac.

The sum of the three direction cosine values calculated for one set of second selected line segments in this way is defined as the total index value. One total index value is calculated by calculation processing between one set of first selected line segments and one set of second selected line segments. 24 total index values are calculated by performing calculation processing on 24 sets of second selected line segments. The second selected line segment group corresponding to the largest of the 24 total index values is defined as the connected line segment group.

Of the three second line segments constituting the connecting line group, the second line segment (here, Aa) that corresponds to the first selected line segment (here, first line segment OA) is selected as the second selected line segment. Note that if the first selected line group is selected for the second time or later (i.e., (step 6) is executed for the second time or later), the user does not arbitrarily select the first selected line group, but instead replaces the connecting line group with the first selected line group and executes (step 6). Note that the steps other than those described above are the same as those in embodiment 1, and therefore will not be described here.

[Modification 3]
(Other Predetermined End Condition 1)
In (Step 7), the repetition of (Step 4) to (Step 6) is stopped, and another predetermined end condition (another predetermined end condition 1) for ending the connection of the line segments is described. As a result of repeating (Step 4) to (Step 6), when the defined end point P reaches the surface of the three-dimensional area, that is, not inside the solid, but a part of any face constituting the surface area of the solid, or when it reaches outside the three-dimensional area (outside the solid), the connection of the line segments is ended. Note that the procedures other than those described above are the same as those in the first embodiment, and therefore description thereof will be omitted.

[Modification 4]
(Predetermined End Condition 2)
In (Step 7), another predetermined end condition (another predetermined end condition 2) for stopping the repetition of (Step 4) to (Step 6) and terminating the connection of the line segments will be described. In (Step 4), if only two of the three principal stress values (principal values) sa to sc at the end point P (here, principal stress values (principal values) sa and sb) match and the remaining principal stress values (principal values) (here, principal stress values (principal values) sc) do not match, all of the six second line segments Pa to Pc, Pa' to Pc' are not defined. Instead, a second line segment α (here, second line segment Ac) corresponding to the remaining principal stress value (principal value) sc, and a second line segment β (here, second line segment Ac') and a second line segment γ extending in the opposite direction on the same straight line as the second line segment α are defined. Here, the second line segment γ is a line segment of a predetermined infinitesimal length defined on a plane PL that passes through the end point P (point A) and is perpendicular to the second line segment α, and is the line segment that has the largest direction cosine value with the first selected line segment among the multiple line segments extending radially from the end point P (point A).

(Step 6) is performed using three second line segments, namely, the second line segment α corresponding to the remaining principal stress value (principal value) sc, the second line segment β on the same straight line as the second line segment α in the opposite direction, and the second line segment γ, and the first selected line segment (here, OA). If, as a result of performing (Step 6), the line segment selected as the second selected line segment is the second line segment γ, the connection of the lines is terminated. Note that the steps other than those described above are the same as those in embodiment 1, and therefore will not be described here.

[Modification 5]
(Predetermined End Condition 3)
In (step 7), the repetition of (step 4) to (step 6) is stopped, and further other predetermined termination conditions (other predetermined termination conditions 3) for terminating the connection of the line segments are described. In (step 7), when all three principal stress values (principal values) of the second-order real symmetric tensor at the point newly designated as the starting point O are identical, the connection of the line segments is terminated. Also, when all three principal stress values (principal values) of the second-order real symmetric tensor at the starting point O initially selected by the user before the start of (step 1) are identical, the connection of the line segments is terminated (in this case, the method of connecting the line segments is not started). Note that the procedures other than those described above are the same as those in embodiment 1, and therefore will not be described again.

[Modification 6]
(Predetermined End Condition 4)
In (Step 7), the repetition of (Step 4) to (Step 6) is stopped, and further other predetermined end conditions (other predetermined end conditions 4) for terminating the connection of the line segments are described. When all three principal stress values (principal values) of the second-order real symmetric tensor at the end point P defined in (Step 3) are the same, the connection of the line segments is terminated. Note that the procedures other than those described above are the same as those in the first embodiment, and therefore description thereof will be omitted.

[Modification 7]
(Predetermined End Condition 5)
In (Step 7), a further predetermined end condition (another predetermined end condition 4) for stopping the repetition of (Step 4) to (Step 6) and terminating the connection of the line segments will be described. The connection of the line segments is terminated when the distance between any two of the multiple starting points O generated by the reinterpretation in (Step 7) becomes equal to or less than the predetermined infinitesimal length. In other words, the connection of the line segments is terminated when the point newly designated as the starting point O in (Step 7) is at a distance equal to or less than the length of the line segment (predetermined infinitesimal length) from any of the multiple points that were previously the starting points O (if the point exists within a sphere with an infinitesimal radius centered on any of the multiple points that were previously the starting points O).

The multiple points that were previously the starting point O are points included in any of the multiple line segments that are connected to the line segment that includes the point that was newly set as the starting point O in (step 7). Note that the steps other than those described above are the same as those in embodiment 1, so the explanation will be omitted.

[Example 1]
7 is a diagram showing a three-dimensional region V1 according to Example 1. In this Example 1, the three-dimensional region V1 has a cubic shape with each side measuring 10 cm, and has a hollow N with a diameter of 3 cm at its center.

The three-dimensional region V1 has a Young's modulus of 200 GPa, a Poisson's ratio of 0.25, and the number of divisions when the three-dimensional region V1 is divided into predetermined microregions (tetrahedral regions) is 94,947,160. The length of one side of the predetermined microregion is a maximum of 0.92 mm, a minimum of 0.075 mm, and an average of 0.5 mm. The maximum value of the length of one side of each microregion is the maximum value of the length of the line segment in each microregion (predetermined microlength).

As shown in FIG. 7, the state of the second-order real symmetric tensor when a tensile force of 100 MPa is applied to the three-dimensional region V1 in the vertical direction is shown in FIG. 8. FIG. 8 shows a display example in which the position of the initial starting point O at the start of the calculation process is arranged at a lattice-like position that divides one side of the three-dimensional region V1 into eight equal parts (excluding the surface of the three-dimensional region V1). The method described in the first embodiment is used for (Step 6). In addition, as shown in FIG. 8, the principal stress value (principal value) at each point is displayed in multiple gradations from black to white, ranging from a range of less than -2.0 x 10 ⁶ Pa (negative values indicate compression) to a range of 2.0 x 10 ⁶ Pa or more (positive values indicate tension).

The same calculation results are shown in Figures 9 and 10, but with the arrows viewed in different directions. Figures 8 to 10 display the same calculation results, but the arrows are viewed in different directions, as shown by the reference XYZ coordinates in the upper right corner of the figures. Also, Figures 8 to 10 and Figures 12 to 14 and 16, which will be described later, are displayed in grayscale, meaning they are shown in gradations from black to white, but it is of course possible to display these figures in color. In that case, it would be possible to display the principal stress values (principal values) at each point in multiple colors, for example from blue to red, making the displayed figures even easier to understand intuitively.

[Example 2]
11 is a diagram showing a three-dimensional region V2 according to Example 2. In this Example 2, the three-dimensional region V2 has a cubic shape with each side measuring 10 cm, and has a sector-shaped crack Q1 with a radius of 2 cm and centered at the vertex Q on its bottom surface.

The three-dimensional region V2 has a Young's modulus of 200 GPa, a Poisson's ratio of 0.25, and the number of divisions when the three-dimensional region V2 is divided into predetermined microregions (tetrahedral regions) is 62,818,055. The length of one side of the predetermined microregion is a maximum of 1.05 mm, a minimum of 0.19 mm, and an average of 0.57 mm. The maximum value of the length of one side of each predetermined microregion is the maximum value of the length of the line segment in each microregion (predetermined microlength).

As shown in FIG. 11, the state of the second-order real symmetric tensor when a tensile force of 100 MPa is applied to the three-dimensional region V2 in the vertical direction is shown in FIG. 12. FIG. 12 shows a display example when the position of the initial starting point O at the start of the calculation process is arranged at a lattice-like position that divides one side of the three-dimensional region V2 into eight equal parts (excluding the surface of the three-dimensional region V2). The method described in the first embodiment is used for (Step 6). As shown in FIG. 12, the principal stress value (principal value) at each point is displayed in multiple gradations from black to white, ranging from a range value of less than 0 Pa (negative values indicate compression) to a range value of 1.5×10 ⁸ Pa or more (positive values indicate tension).

13 and 14 show examples of displaying the same calculation results, but with the line segments whose principal axis vector directions are in the horizontal plane (XY plane) hidden. Figures 13 and 14 show the same calculation results, but the arrow directions are different, as shown by the XYZ coordinates in the upper right corner of each figure. In Figure 14, the principal stress values (principal values) at each point are displayed in multiple gradations from black to white, ranging from values less than -1.0 x ¹⁰⁸ Pa (negative values indicate compression) to values equal to or greater than 1.0 x ¹⁰⁸ Pa (positive values indicate tension).

[Example 3]
15 is a diagram showing a three-dimensional region V3 according to Example 3. In this Example 3, the three-dimensional region V3 has a cubic shape with each side being 10 cm long, and has a band-shaped crack Q2 with a width of 2 cm from the vertex Q on its bottom surface, parallel to the side including the vertex Q and the vertex R.

The three-dimensional region V3 has a Young's modulus of 200 GPa, a Poisson's ratio of 0.25, and the number of divisions when the three-dimensional region V3 is divided into predetermined microregions (tetrahedral regions) is 62,818,055. The length of one side of the predetermined microregion is a maximum of 1.05 mm, a minimum of 0.19 mm, and an average of 0.57 mm. The maximum value of the length of one side of each predetermined microregion is the maximum value of the length of the line segment in each microregion (predetermined microlength).

As shown in FIG. 15, the state of the second-order real symmetric tensor when a tensile force of 100 MPa is applied to the three-dimensional region V3 in the vertical direction is shown in FIG. 16. FIG. 16 shows a display example when the position of the initial starting point O at the start of the calculation process is arranged at a lattice-like position that divides one side of the three-dimensional region V3 into eight equal parts (excluding the surface of the three-dimensional region V3). The method described in the first embodiment is used for (Step 6). As shown in FIG. 16, the principal stress value (principal value) at each point is displayed in multiple gradations from black to white, ranging from a range of less than 0 Pa (negative values indicate compression) to a range of 1.5×10 ⁸ Pa or more (positive values indicate tension).

Although the preferred embodiments of the present invention have been described above, the present invention is not limited to these, and various modifications and variations are possible within the scope of the gist of the present invention. For example, the present invention includes the following aims:

[Objective 1]
A method for connecting line segments in a three-dimensional area, comprising the following steps (1) to (7).
(1) Three first line segments OA, OB, and OC are defined that extend along the principal axis directions corresponding to three principal values of a real symmetric second-order tensor at an origin O selected as an arbitrary point within a three-dimensional region. Here, the three first line segments are all line segments of a predetermined infinitesimal length that extend from the origin O, and are line segments that constitute a selected coordinate system, which is either a right-handed coordinate system or a left-handed coordinate system, centered on the origin O, and are three-dimensionally orthogonal to each other.
(2) Three first line segments OA', OB', and OC' of predetermined small lengths are defined, extending from the starting point O. Here, the first line segment OA' is a line segment that extends in the opposite direction to the first line segment OA on the same line, the first line segment OB' is a line segment that extends in the opposite direction to the first line segment OB on the same line, and the first line segment OC' is a line segment that extends in the opposite direction to the first line segment OC on the same line.
(3) One line segment is selected from the six first line segments OA to OC and OA' to OC' as a first selected line segment, and the end point other than the starting point O is defined as an end point P.
(4) Three second line segments Pa, Pb, and Pc are defined, which extend along the principal axis directions corresponding to the three principal values of the second-order real symmetric tensor at the end point P. Here, the three second line segments are all line segments of a predetermined infinitesimal length extending from the end point P, which constitute the selected coordinate system centered on the end point P and are three-dimensionally orthogonal to each other.
(5) Three second line segments Pa', Pb', and Pc' of predetermined small lengths are defined, extending from the end point P. Here, the second line segment Pa' is a line segment that extends in the opposite direction to the second line segment Pa on the same line, the second line segment Pb' is a line segment that extends in the opposite direction to the second line segment Pb on the same line, and the second line segment Pc' is a line segment that extends in the opposite direction to the second line segment Pc on the same line.
(6) Based on the result of a calculation process using the six second line segments Pa-Pc, Pa'-Pc' and the first selected line segment, one second line segment that can be most smoothly connected to the first selected line segment is selected as a second selected line segment from among the six second line segments Pa-Pc, Pa'-Pc' as a second selected line segment, and the second selected line segment is connected to the first selected line segment, and the remaining second line segments are discarded.
(7) Of the two end points of the second selected line segment, the end point connected to the first selected line segment is read as a starting point O, the opposite end point is read as an end point P, and the second selected line segment is read as the first selected line segment, and then steps (4) to (6) are repeatedly executed until a specified termination condition is satisfied.

[Objective 2]
The selection of the first selected line segments in the step (3) may be performed for all of the six first line segments OA to OC and OA' to OC'.

[Objective 3]
In step (1), when only two of the three principal values of the second-order real symmetric tensor at the starting point O are the same and the remaining principal values are not the same, instead of defining the six first line segments OA to OC, OA' to OC', in step (3), either the first line segment of the specified infinitesimal length corresponding to the remaining principal values or the first line segment of the specified infinitesimal length collinear with and in the opposite direction to the first line segment may be set as the first selected line segment.

[Objective 4]
The step (6) may include the following step (61).
(61) The direction cosine values between the second line segment and the first selected line segment are calculated for each of the six second line segments Pa to Pc, Pa' to Pc', and the second line segment corresponding to the maximum of the six calculated direction cosine values is selected as the second selected line segment.

[Objective 5]
The step (6) may include the following steps (62) and (63).
(62) A difference value between a principal value corresponding to the first selected line segment at the starting point O and a principal value corresponding to the second line segment at the end point P is calculated for each of the three second line segments Pa to Pc, and the second line segment corresponding to the one having the smallest absolute value among the three calculated difference values is defined as a candidate second line segment.
(63) A second line segment corresponding to the larger of the direction cosine values between the candidate second line segment and the first selected line segment, or the direction cosine values between a second line segment extending in the opposite direction on the same straight line as the candidate second line segment and the first selected line segment, is selected as the second selected line segment.

[Objective 6]
The step (6) may include the following steps (64) to (69).
(64) Among the six second line segments Pa-Pc, Pa'-Pc', any three second line segments that constitute the selected coordinate system centered on the end point P are selected as a set of second selected line segment groups, thereby defining 24 selectable sets of second selected line segment groups.
(65) Three first line segments that include the first selected line segment and that form the selected coordinate system centered on the starting point O are selected as a set of first selected line segments. Here, when step (65) is executed for the first time, the set of first selected line segments that includes the first selected line segment is arbitrarily selected.
(66) A direction cosine value between one of the 24 sets of the second selected line segments and the one set of the first selected line segments is calculated, where the direction cosine values are calculated for corresponding line segments between the one set of the second selected line segments and the one set of the first selected line segments, and the sum of the calculated three direction cosine values is defined as a total index value.
(67) The step (66) is executed for all 24 pairs of second selected line segment groups, and the second selected line segment group for which the total index value is maximum is defined as a connected line segment group.
(68) Among the three second line segments included in the group of connecting line segments, the one corresponding to the first selected line segment is selected as a second selected line segment.
(69) The group of connecting lines is read as the first group of selected lines.

[Objective 7]
In the step (7), the predetermined termination condition may include the following condition 1:
(Condition 1) The starting point O reaches the surface of the three-dimensional region or outside the region.

[Objective 8]
In the step (7), the predetermined termination condition may include the following condition 2:
(Condition 2) In the step (4), when only two of the three principal values of the second-order real symmetric tensor at the end point P are the same and the remaining principal values are not the same, instead of defining the six second line segments Pa to Pc, Pa' to Pc', define the following three second line segments α, β, and γ:
When step (6) is executed using the three second line segments α, β, and γ and the first selected line segment instead of the six second line segments Pa to Pc and Pa' to Pc', the second selected line segment becomes the second line segment γ.
Second line segment α: A second line segment of the specified infinitesimal length corresponding to the principal value of the remainder. Second line segment β: A second line segment of the specified infinitesimal length extending in the opposite direction on the same straight line as the second line segment α. Second line segment γ: A line segment of the specified infinitesimal length defined on a plane that passes through the end point P and is perpendicular to the second line segment α, and which is the line segment among multiple line segments extending radially from the end point P that has the largest direction cosine value with the first selected line segment.

[Objective 9]
In the step (7), the predetermined termination condition may include the following condition 3:
(Condition 3) All three principal values of the second-order real symmetric tensor at the origin O updated in step (7) above are consistent.

[Objective 10]
In the step (7), the predetermined termination condition may include the following condition 4:
(Condition 4) The three principal values of the second-order real symmetric tensor at the end point P are all the same.

[Objective 11]
In the step (7), the predetermined termination condition may include the following condition 5:
(Condition 5) The distance between any two of the multiple starting points O generated by the reinterpretation in the step (7) is equal to or less than the predetermined infinitesimal length.

[Objective 12]
Using the above connection method, the first selected line segment of the predetermined small length is connected to the second selected line segment of the predetermined small length;
A method for displaying a real symmetric second-order tensor, the method comprising: displaying the connected selected line segments as lines or curves drawn within the three-dimensional region.

[Objective 13]
The method for displaying a real symmetric second-order tensor may include displaying the first selected line segment and the second selected line segment in different colors according to their corresponding principal values.

[Objective 14]
On the computer,
A program for displaying second-order real symmetric tensors that executes the above-mentioned method for displaying second-order real symmetric tensors.

Aa to Ac, Aa' to Ac': second line segment CSa to CSc, CSa' to CSc': direction cosine values da to dc: difference value D: screen N: cavity O: starting point OA to OC, OA' to OC': first line segment P: end point PG: tensor display program (program)
PL: Plane Q, R: Vertex Q1, Q2: Crack sA-sC, sa-sc: Principal stress value (principal value)
S: Computer V1, V2, V3: Three-dimensional area α, β, γ: Second line segment 2: CPU
4. Memory

Claims (14)

A method for connecting line segments in a three-dimensional area, comprising the following steps (1) to (7) executed by a computer .
(1) Three first line segments OA, OB, and OC are defined that extend along the principal axis directions corresponding to three principal values of a real symmetric second-order tensor at an origin O selected as an arbitrary point within a three-dimensional region. Here, the three first line segments are all line segments of a predetermined infinitesimal length that extend from the origin O, and are line segments that constitute a selected coordinate system, which is either a right-handed coordinate system or a left-handed coordinate system, centered on the origin O, and are three-dimensionally orthogonal to each other.
(2) Three first line segments OA', OB', and OC' of predetermined small lengths are defined, extending from the starting point O. Here, the first line segment OA' is a line segment that extends in the opposite direction to the first line segment OA on the same line, the first line segment OB' is a line segment that extends in the opposite direction to the first line segment OB on the same line, and the first line segment OC' is a line segment that extends in the opposite direction to the first line segment OC on the same line.
(3) One line segment is selected from the six first line segments OA to OC and OA' to OC' as a first selected line segment, and the end point other than the starting point O is defined as an end point P.
(4) Three second line segments Pa, Pb, and Pc are defined, which extend along the principal axis directions corresponding to the three principal values of the second-order real symmetric tensor at the end point P. Here, the three second line segments are all line segments of a predetermined infinitesimal length extending from the end point P, which constitute the selected coordinate system centered on the end point P and are three-dimensionally orthogonal to each other.
(5) Three second line segments Pa', Pb', and Pc' of predetermined small lengths are defined, extending from the end point P. Here, the second line segment Pa' is a line segment that extends in the opposite direction to the second line segment Pa on the same line, the second line segment Pb' is a line segment that extends in the opposite direction to the second line segment Pb on the same line, and the second line segment Pc' is a line segment that extends in the opposite direction to the second line segment Pc on the same line.
(6) Based on the result of a calculation process using the six second line segments Pa-Pc, Pa'-Pc' and the first selected line segment, one second line segment that can be most smoothly connected to the first selected line segment is selected as a second selected line segment from among the six second line segments Pa-Pc, Pa'-Pc' as a second selected line segment, and the second selected line segment is connected to the first selected line segment, and the remaining second line segments are discarded.
(7) Of the two end points of the second selected line segment, the end point connected to the first selected line segment is read as a starting point O, the opposite end point is read as an end point P, and the second selected line segment is read as the first selected line segment, and then steps (4) to (6) are repeatedly executed until a specified termination condition is satisfied. The method for connecting lines in a three-dimensional region according to claim 1, wherein the selection of the first selected line segments in step (3) is performed for all six of the first line segments OA-OC and OA'-OC'. The method for connecting lines according to claim 1, wherein in step (1), when only two of the three principal values of the second-order real symmetric tensor at the origin O match and the remaining principal values do not match, instead of defining the six first line segments OA-OC, OA'-OC', in step (3), the first selected line segment is either the first line segment of the predetermined infinitesimal length corresponding to the remaining principal values or the first line segment of the predetermined infinitesimal length that is collinear with the first line segment and in the opposite direction. 2. The method for connecting line segments in a three-dimensional area according to claim 1, wherein the step (6) includes the following step (61).
(61) The direction cosine values between the second line segment and the first selected line segment are calculated for each of the six second line segments Pa to Pc, Pa' to Pc', and the second line segment corresponding to the maximum of the six calculated direction cosine values is selected as the second selected line segment. 2. The method for connecting line segments in a three-dimensional area according to claim 1, wherein the step (6) includes the following steps (62) and (63).
(62) A difference value between a principal value corresponding to the first selected line segment at the starting point O and a principal value corresponding to the second line segment at the end point P is calculated for each of the three second line segments Pa to Pc, and the second line segment corresponding to the one having the smallest absolute value among the three calculated difference values is defined as a candidate second line segment.
(63) A second line segment corresponding to the larger of the direction cosine values between the candidate second line segment and the first selected line segment, or the direction cosine values between a second line segment extending in the opposite direction on the same straight line as the candidate second line segment and the first selected line segment, is selected as the second selected line segment. 2. The method for connecting line segments in a three-dimensional area according to claim 1, wherein the step (6) includes the following steps (64) to (69).
(64) Among the six second line segments Pa-Pc, Pa'-Pc', any three second line segments that constitute the selected coordinate system centered on the end point P are selected as a set of second selected line segment groups, thereby defining 24 selectable sets of second selected line segment groups.
(65) Three first line segments that include the first selected line segment and that form the selected coordinate system centered on the starting point O are selected as a set of first selected line segments. Here, when step (65) is executed for the first time, the set of first selected line segments that includes the first selected line segment is arbitrarily selected.
(66) A direction cosine value between one of the 24 sets of the second selected line segments and the one set of the first selected line segments is calculated, where the direction cosine values are calculated for corresponding line segments between the one set of the second selected line segments and the one set of the first selected line segments, and the sum of the calculated three direction cosine values is defined as a total index value.
(67) The step (66) is executed for all 24 pairs of second selected line segment groups, and the second selected line segment group for which the total index value is maximum is defined as a connected line segment group.
(68) Among the three second line segments included in the group of connecting line segments, the one corresponding to the first selected line segment is selected as a second selected line segment.
(69) The group of connecting lines is read as the first group of selected lines. 2. The line segment connecting method according to claim 1, wherein in the step (7), the predetermined termination condition includes the following condition 1:
(Condition 1) The starting point O reaches the surface of the three-dimensional region or outside the region. 2. The method for connecting line segments according to claim 1, wherein in the step (7), the predetermined termination condition includes the following condition 2:
(Condition 2) In the step (4), when only two of the three principal values of the second-order real symmetric tensor at the end point P are the same and the remaining principal values are not the same, instead of defining the six second line segments Pa to Pc, Pa' to Pc', define the following three second line segments α, β, and γ:
When step (6) is executed using the three second line segments α, β, and γ and the first selected line segment instead of the six second line segments Pa to Pc and Pa' to Pc', the second selected line segment becomes the second line segment γ.
Second line segment α: A second line segment of the specified infinitesimal length corresponding to the principal value of the remainder. Second line segment β: A second line segment of the specified infinitesimal length extending in the opposite direction on the same straight line as the second line segment α. Second line segment γ: A line segment of the specified infinitesimal length defined on a plane that passes through the end point P and is perpendicular to the second line segment α, and which is the line segment among multiple line segments extending radially from the end point P that has the largest direction cosine value with the first selected line segment. 2. The line segment connecting method according to claim 1, wherein in the step (7), the predetermined termination condition includes the following condition 3:
(Condition 3) All three principal values of the second-order real symmetric tensor at the origin O updated in step (7) above are consistent. 2. The line segment connecting method according to claim 1, wherein in the step (7), the predetermined termination condition includes the following condition 4:
(Condition 4) The three principal values of the second-order real symmetric tensor at the end point P are all the same. 2. The method for connecting line segments according to claim 1, wherein in the step (7), the predetermined termination condition includes the following condition 5.
(Condition 5) The distance between any two of the multiple starting points O generated by the reinterpretation in the step (7) is equal to or less than the predetermined infinitesimal length. The method for connecting lines according to any one of claims 1 to 11 is used to connect the first selected line segment of the predetermined small length and the second selected line segment of the predetermined small length;
The method comprises displaying the connected selected line segments as lines or curves drawn within the three-dimensional region by a computer . 13. The method for displaying a real symmetric second-order tensor according to claim 12, wherein the first selected line segment and the second selected line segment are displayed by a computer in a color-coded manner according to the principal values corresponding to the first selected line segment and the second selected line segment. On the computer,
A display program for a second-order real symmetric tensor, which causes execution of the display method for a second-order real symmetric tensor according to claim 12 or 13.