CN113591356B - Construction method of non-uniform irregular spline basis function retaining sharp features - Google Patents

Construction method of non-uniform irregular spline basis function retaining sharp features Download PDF

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CN113591356B
CN113591356B CN202110911630.4A CN202110911630A CN113591356B CN 113591356 B CN113591356 B CN 113591356B CN 202110911630 A CN202110911630 A CN 202110911630A CN 113591356 B CN113591356 B CN 113591356B
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curved surface
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CN113591356A (en
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冯毅飞
申立勇
李新
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University of Chinese Academy of Sciences
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/04—Constraint-based CAD

Abstract

The embodiments of the specification discloseA method of constructing a non-uniform irregular spline basis function that retains sharp features, comprising: step S1, according to the shape of a curved surface to be processed; s2, extracting and defining an initial C0 basis function by using a Bezier curved surface method based on the quadrilateral control grid; s3, calculating a tangential plane; s4, calculating a connection function; step S5, local step-by-step optimization; s51, optimizing vertex control points; s52, optimizing an edge control point; s53, optimizing a surface control point; solving a least square problem with linear equation constraint during step-by-step solution, namely:s.t.mp=b converts the least squares problem into a solution to a system of linear equations:the application provides a construction method of spline basis functions, which enables generated splines to have a curved surface for generating global G1 continuity on a control grid with arbitrary topology.

Description

Construction method of non-uniform irregular spline basis function retaining sharp features
Technical Field
The application relates to the technical field of numerical control machining, in particular to a construction method of a non-uniform irregular spline basis function retaining sharp features.
Background
Catmull-Clark surfaces are widely used in the animation field, while NURBS dominates over CAD industrial designs. Each curved surface format has its own unique advantages: the Catmull-Clark surface can generate a smooth surface on a control network of arbitrary topology, which is advantageous for animation design. And the NURBS curved surface can realize local modification, and is more suitable for high-precision industrial models. Many sets of surface representations have been developed that modify the shape of a surface by imparting pitch to the edges of the control mesh. If no singular points exist, the surfaces are consistent with NURBS surface expression; if all the node distances are 1, the curved surface is a Catmull-Clark curved surface.
Subdivision schemes are suitable for rendering applications, but are not suitable for CAD designs. Because CAD design analysis flows typically require models to be passed between numerous software packages, and most of these software are NURBS-based. However, subdivision methods are not backward compatible with NURBS, because at the singular point attachment, subdivision methods produce infinite series of hyperboloids, whereas NURBS can only introduce limited truncation. The hole filling-based approach avoids this compatibility problem by replacing infinite sequences around the singular point with a small number of patches. While the previous has been directed to a conventional Catmull-Clark surface, i.e., a node distance ratio of 1, it is straightforward to modify these methods to handle non-uniform node distances. However, the modified surface exhibits the same problem as the subdivision result in the vicinity of the singular point where the pitch ratio is greater than 3.
The isogeometric analysis IGA is an emerging surface analysis technique with the ability to analyze directly on spline models with a high degree of accuracy compared to finite element methods. However, since the general NURBS basis function is a rational polynomial, the problem of boundary intersection is difficult to solve while maintaining tensor properties.
There is a need for a spline generation method such that the generated spline has a curved surface that generates global G1 continuity on a control grid of arbitrary topology.
Disclosure of Invention
The embodiments of the present specification provide a method of constructing a non-uniform irregular spline basis function that preserves sharp features such that the generated spline has a surface that generates global G1 continuity on a control grid of arbitrary topology.
In order to solve the above technical problems, the embodiments of the present specification are implemented as follows: the embodiment of the specification provides a construction method of a non-uniform irregular spline basis function retaining sharp features, which comprises the following steps:
s1, inputting quadrilateral control grids with arbitrary topology according to the shape of a curved surface to be processed;
s2, extracting and defining an initial C0 basis function by using a Bezier curved surface method based on the quadrilateral control grid, wherein each surface of the quadrilateral control grid is represented by using a bicubic Bezier curved surface; for a face point F on each face of the quadrilateral control grid i With four control points P on each face i Is represented by a linear combination of (a); edge point E of the quadrilateral control grid i And the vertex V is the face point F i Is a linear combination of (a);
s3, calculating a tangential plane;
for a singular point with the degree of n, the control point adjacent to the singular point is E i ,F i Node distance length d i ,a i ；
Calculating new control pointsLet p= [ V, E 0 ,…,E n-1 ,F 0 ,…,F n-1 ] T ,/> Write subdivision rule +.>
Defining a NURSS format subdivision matrix as M and an Eigen-polyhedron-based subdivision matrix as N;
defining a circle of Bezier control points around the singular point asIs point V, { E i },{F i Linear combination of }:
s31, calculating limit points C based on Eigen-polyhedron subdivision;
by L 0 The unitization of the eigenvector corresponding to the eigenvalue of matrix N of 1 is represented, and the limit point is defined as c=l 0 M T P；
S32, calculating a tangent plane based on Eigen-polyhedron subdivision;
defining two matrices of size 2nx2n Let-> Lambda is the matrix->Main feature value of (2), then->Write-> Λ is a diagonal matrix of singular values; let i be 1 ,i 2 Is such that Λ (i 1 ,i 1 )＝Λ(i 2 ,i 2 ) Index =λ, let ∈ ->In addition to->Other diagonal matrix with zero position, thus obtaining +.>Further define vector set +.>
Step S33, defining Bezier control points around the singular points;
bezier control points around singular points
S4, calculating a connection function;
s41, defining a weight k for each angle for singular points with degree n i ：
S42, defining angles according to the weights;
for theta i Summing, if the summation result is not equal to 2 pi, normalizing the angle to 2 pi;
definition (i, j) =sin (θ) i )sin(θ i+2 )…sin(theta j )(i<j)；
If n=2k+1 is an odd number:
if n=2k is even:
let S be n numbersIs set of->Is the value of the most frequently occurring set, if each number appears only once, let +.>
Step S5, local step-by-step optimization;
dividing constraint conditions on each side into two parts, wherein the first two equations become vertex constraints, and the rest are called side constraints; solving a new basis function according to the following sequence;
s51, optimizing vertex control points;
each singular point vertex constraint with the degree of n comprises 3n+1 control points and 2n linear constraint equations, and the 3n+1 control points meeting the vertex constraint are obtained by solving;
s52, optimizing an edge control point;
after the vertex control points are determined, the control points of the rest edge constraint designs are controlled on each edge to become local problems, and the control points are used for solving the problem that each edge meets the edge constraint;
s53, optimizing a surface control point;
solving a least square problem with linear equation constraint during step-by-step solution, namely:
s.t.MP＝b
converting the least squares problem into a solution problem of a system of linear equations:
s54, designing the shape of the curved surface to be processed according to the result obtained by solving in the step S53.
One embodiment of the present specification achieves the following advantageous effects:
the application provides a construction method of spline basis functions, which enables generated splines to have a continuous curved surface of global G1 generated on a control grid with any topology, and further can be directly used for CAD design and CAE analysis, thereby accelerating the integrated flow of processing design analysis.
Drawings
In order to more clearly illustrate the embodiments of the present description or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described below, it being obvious that the drawings in the following description are only some embodiments described in the present application, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a schematic diagram illustrating a control grid in an embodiment of the present disclosure;
FIG. 2 is a schematic diagram illustrating points and vertices in an embodiment of the present disclosure;
FIG. 3 is a schematic diagram illustrating a face point in an embodiment of the present description;
FIG. 4 is a schematic diagram illustrating NURSS subdivision formats in embodiments of the present specification;
FIG. 5 is a schematic diagram illustrating a bisection plane in an embodiment of the present description;
FIG. 6 is a schematic view for explaining a definition angle around a singular point in the embodiment of the present specification;
FIG. 7 is a schematic diagram illustrating the step up of a singular surface to a bi-quintic Bezier surface to satisfy the G1 continuity condition in an embodiment of the present disclosure;
FIG. 8 is a schematic diagram for explaining the singular point isolation in the technical scheme of the embodiment of the present specification;
FIG. 9 is a first graph of test results of the examples of the present specification;
FIG. 10 is a second graph of test results of the examples of the present specification;
FIG. 11 is a third graph of test results of the examples of the present specification;
FIG. 12 is a fourth graph of test results of the examples of the present specification.
Detailed Description
For the purposes of making the objects, technical solutions and advantages of one or more embodiments of the present specification more clear, the technical solutions of one or more embodiments of the present specification will be clearly and completely described below in connection with specific embodiments of the present specification and corresponding drawings. It will be apparent that the described embodiments are only some, but not all, of the embodiments of the present specification. All other embodiments, which can be made by one of ordinary skill in the art based on the embodiments herein without undue burden, are intended to be within the scope of one or more embodiments herein.
Catmull-Clark surfaces are widely used in the animation field, while NURBS dominates over CAD industrial designs. Each curved surface format has its own unique advantages: the Catmull-Clark surface can generate a smooth surface on a control network of arbitrary topology, which is advantageous for animation design. And the NURBS curved surface can realize local modification, and is more suitable for high-precision industrial models. Many sets of surface representations have been developed that modify the shape of a surface by imparting pitch to the edges of the control mesh. If no singular points exist, the surfaces are consistent with NURBS surface expression; if all the node distances are 1, the curved surface is a Catmull-Clark curved surface.
The node distance ratio is defined as the maximum node distance to the minimum node distance of the connected edges of each singular point. Solves the problem that all published Catmull-Clark curved surfaces which support any node distance appear. This problem arises first in non-uniform Catmull-Clark surfaces and NURCCs, where the quality of the surface near the singular point decreases as the pitch ratio of the singular point increases. Cashman et al thin in one direction until the pitch ratio is less than 2, and then thin in both directions to obtain a curved surface. The expansion subdivision surface forces the node distance ratio of all singular points to be 1, so that good effects are achieved under simple conditions, but better results cannot be obtained under other conditions.
Subdivision schemes are suitable for rendering applications, but are not suitable for CAD designs. Because CAD design analysis flows typically require models to be passed between numerous software packages, and most of these software are NURBS-based. However, subdivision methods are not backward compatible with NURBS, because near singular points subdivision methods produce infinite sequences of hyperboloids, whereas NURBS can only introduce limited truncations. The hole filling-based approach avoids this compatibility problem by replacing infinite sequences around the singular point with a small number of patches. While the previous methods were all directed to a conventional Catmull-Clark surface, i.e., a node distance ratio of 1, it is straightforward to modify these methods to handle non-uniform node distances. However, the modified surface exhibits the same problem as the subdivision result in the vicinity of the singular point where the pitch ratio is greater than 3.
Iso-geometric analysis IGA is an emerging surface analysis technique that has the ability to analyze directly on spline models with a high degree of accuracy compared to finite element methods. However, since the general NURBS basis function is a rational polynomial, the problem of boundary intersection is difficult to solve while maintaining tensor properties. The spline provided has the advantages that a global G1 continuous curved surface is generated on a control grid with any topology, and the spline can be directly used for CAD design and CAE analysis, so that the integrated flow of processing design analysis is accelerated.
The technical proposal of the embodiment of the application provides a frame for generating acceptable surface near the singular point with large pitch ratio. The framework is based on the hole filling method, so it is compatible with NURBS back and forth. The method can be directly applied to the current CAD software, can be transmitted between various CAD analysis design software packages without loss, can directly support an isogeometric analysis method, and connects the CAD design with CAE analysis.
The technical idea of the technical scheme of the embodiment of the application is explained first.
The input of the technical scheme of the embodiment of the application is a quadrilateral control grid with arbitrary topology, wherein the singular points can be at any position except the boundary, the singular points support any degree, and the singular points can be directly connected. The method of (2) allows for a non-uniform control network, the node lengths may not be equal while supporting designation of arbitrary edges, including continuous or discontinuous folds across the edges of the singular point.
Output of
The algorithm output is a polynomial parametric surface, which is C2 continuous in the normal region and G1 continuous on the singular edge. The curved surface maintains good contouring in non-uniform conditions and can produce folds along a designated sharp edge.
The technical scheme of the embodiment of the application is described in detail below.
General procedure
And using and popularizing a Bezier extraction algorithm to obtain a curved surface consistent with the definition of the B spline in a rule area. A common tangent plane is defined near the singular point. Then defining continuous connection functions of each singular edge G1 according to node distances around the singular points. And finally, solving the optimization problem with constraint by local distribution, and calculating to obtain a basis function meeting G1 constraint to obtain a final curved surface.
(1) Bezier extraction
As shown in FIG. 1, bezier extraction is used to define the original C0 basis functions, and each face of the control grid is represented using bicubic Bezier surfaces.
As shown in fig. 2, for each of the above-surface points F i And (3) calculating:
F 2i+j ＝(1-α i )[1-γ j )P 0,0 +γ j P 0,1 ]+α i [(1-γ j )P 1,0 +γ j P 1,1 ]
wherein the method comprises the steps of
As shown in fig. 3, for each edge point E i And calculating with the vertex V:
for singular point of degree n, face point F i And the edge point E i The calculation is the same as that of the regular area,for vertex V:
(2) Tangential plane definition
When the node distance ratio around the singular point is very large, the base function obtained by the generalized Bezier extraction can have multiple peaks. The surrounding surface points and edge points of the singular point are defined here in such a way that they lie in the same tangential plane, using a subdivision format.
NURSS subdivision format:
as shown in FIG. 4, for a degree n of singular point V, the adjacent control point is E i ,F i Node distance length d i ,a i . To calculate new control pointsLet-> The subdivision rule may be written as +.>
And (5) pastry:
edge points:
M i ＝ω i E i +(1-ω i )V
vertex:
f i ＝d i-1 d i+2 ,m i ＝f i +f i-1
eigen-polyhedron subdivision format
Definition of the definition At R 2 Define Point set +.>
Eigen-polyhedron based subdivision can be written in a formatVertex point
The calculation of vertex coordinates is the same as NURSS subdivision rule, and letIs to make V, E i ,F i Use->Instead of the calculated points.
Pastry
Wherein alpha is i,1 ,α i,2 Is the only solution to the following equation:
edge point
Wherein the method comprises the steps of
P i,1 ＝(1-α i-1,1 )V+α i-1,1 E i-1 ,P i,2 ＝(1-α i,2 )V+α i,2 E i+1
P i,3 ＝(1-α i-1,1 )E i +α i-1,1 F i-1 ,P i,4 ＝(1-α i,2 )E i +α i,2 E i
β i,1 ,β i,2 Is the only solution to the following equation:
here, theIs to make V, E i ,F i Use->And (5) replacing and calculating.
Tangent plane calculation
As shown in figure 5 of the drawings,is defined as point V, { E i },{F i Linear combinations of eigen-polyhedron subdivision formats are used herein as guidance.
Calculating limit point C based on eigen-polyhedron subdivision
By L 0 The unitization of the eigenvector corresponding to the eigenvalue of matrix N of 1 is represented, and the limit point is defined as c=l 0 M T P；
Computing a tangent plane based on eigen-polyhedron subdivision
Defining two matrices of size 2nx2n Let-> Lambda is the matrix->Main feature value of (2), then->Can be written as +.> Λ is a diagonal matrix of singular values. Let i be 1 ,i 2 Is such that Λ (i 1 ,i 1 )＝Λ(i 2 ,i 2 ) Index =λ, let ∈ ->In addition to->Other diagonal matrices with zero positions can be obtainedFurther define vector set +.>
Bezier control points around defining singular points
Bezier control points around singular points
Scale optimization
The points of the face, edges, surrounding the singular point need to be on a given tangential plane, and therefore the distance between each point and the singular point needs to be determined. The final control point is defined as follows:
the unfolding can be obtained:
it is desirable that all basis functions are non-negative, so s i Defined as follows:
angle calculation
As shown in fig. 6, in order to obtain a good quality curved surface in the non-uniform case, an angle is now defined near the singular point. Included angle of one circle on each singular pointAnd should be equal to 2 pi, and the larger the edges should correspond to the closer the angle
Defining a weight k i ：
Defining an angle according to the weight:
in general θ i The summation is not equal to 2pi, at which time the angle is normalized to 2pi.
Connection function definition
As shown in FIG. 7, the singular surface is stepped up to a bi-five Bezier surface to satisfy the G1 continuity condition, and a parameter b is defined for each edge i . Let the degree of singular point be n.
Definition (i, j) =sin (θ) i )sin(θ i+2 )…sin(theta j )(i<j)
If n=2k+1 is an odd number:
if n=2k is even:
let S be n numbersIs set of->Is the value of the most frequently occurring set, let +.>
The G1 continuity constraint is not the same in the case of isolated singular points and connected singular points
As shown in fig. 8, in the singular point isolation case:
definition α(s) =a i *a i-1 ,β(s)＝a i *ai +1 ,γ(s)＝b i (1-s) 2 The G1 constraint is obtained as follows:
in the case of singular point connection:
definition of the definitionThe G1 constraint is obtained as follows:
local step-wise optimization
In order to process the condition that a plurality of singular points exist on one surface and avoid the constraint condition of global optimization solution, a local step-by-step optimization algorithm is adopted.
The constraint on each edge is divided into two parts, the first two equations become vertex constraints, and the rest are called edge constraints. The new basis functions are solved in the following order:
optimizing vertex control points
Each degree n singular point vertex constraint comprises 3n+1 control points and 2n linear constraint equations, and the 3n+1 control points meeting the vertex constraint are obtained by solving.
Optimizing edge control points
Once the vertex control points are determined, the control points of the remaining edge constraint designs are controlled on each edge, which becomes a local problem, which is the control point that solves for each edge to satisfy the edge constraint.
Optimized surface control point
The surface control points do not affect the continuity between the singular edges, so that in order to obtain a smoother basis function, the surface control points on the singular surfaces can minimize the integral of the second derivative and make the curved surface smoother.
When solving the distribution, a least square problem with linear equation constraint needs to be solved, namely:
s.t.MP＝b
this optimization problem can be converted into a solution to a system of linear equations:
crease mark
The spline definition for the rule area is consistent with that of the B-spline, where a method of node insertion may be used to insert heavy nodes along a given edge, so that a crease can be constructed along that edge.
For folds passing through singular points, the G1 constraint equation corresponding to the singular edge is removed, so that the continuity of the basis function on the edge can be reduced by one step. (2) Meanwhile, in order to control sharp features, edge control points corresponding to appointed singular edges are independent, namely the control points are linear combinations of control points in the grid originally, and each edge control point is added into the grid control points and can be directly controlled, so that the shape of folds is controlled. (3) The basis function of the singular point is modified to a linear combination of the surrounding circle of newly added control points.
If a plurality of folds pass through the same singular point, when two folds are designated to be continuous, the direct angle theta of the two sides is modified i …θ j So thatKeep->If these folds are not continuous, the angle remains unchanged.
Test results of the technical scheme of the embodiment of the application.
The constructed basis function has good modeling capability under the non-uniform condition, and the curved surface near the singular point keeps unimodal smooth.
For singular points with degree of 5, the distance between partial edge nodes is set to be 6, and the rest is set to be 1, so that the algorithm can obtain continuous basis functions of a single peak G1 as shown in the figure.
In the practical model, the node distance of part of the edge in the ring model is 2, the rest is 1, and the edge of part of the node is set as continuous crease, so that the result as shown in the figure can be obtained.
The application provides a construction method of spline basis functions, which enables generated splines to have a continuous curved surface of global G1 generated on a control grid with any topology, and further can be directly used for CAD design and CAE analysis, thereby accelerating the integrated flow of processing design analysis.
The above embodiments are merely illustrative of the principles of the present application and its effectiveness, and are not intended to limit the application. Modifications and variations may be made to the above-described embodiments by those skilled in the art without departing from the spirit and scope of the application. Accordingly, it is intended that all equivalent modifications and variations of the application be covered by the claims, which are within the ordinary skill of the art, be within the spirit and scope of the present disclosure.

Claims (1)

1. A construction method of a non-uniform irregular spline basis function retaining sharp features is characterized in that the construction method is applied to the technical field of numerical control machining; the construction method comprises a frame, wherein the frame is used for generating an acceptable surface near a singular point with a large pitch proportion by adopting a hole filling method; the framework is compatible with NURBS back and forth, is applied to current CAD software, and can be transmitted between various CAD analysis design software packages without damage; supporting an isogeometric analysis method, connecting CAD design and CAE analysis to accelerate the integrated process of processing design analysis; the construction method comprises the following steps:
s1, inputting quadrilateral control grids with arbitrary topology according to the shape of a curved surface to be processed;
step S2, obtaining a curved surface consistent with spline definition by using a Bezier extraction algorithm based on the quadrilateral control grid, wherein the method specifically comprises the following steps of: extracting an initially defined C0 basis function by using a Bezier curved surface method based on the quadrilateral control grid, wherein each surface of the quadrilateral control grid is represented by using a bicubic Bezier curved surface; for a face point F on each face of the quadrilateral control grid i With four control points P on each face i Is represented by a linear combination of (a); edge point E of the quadrilateral control grid i And the vertex V is the face point F i Is a linear combination of (a);
step S3, calculating a tangential plane, which specifically comprises the following steps: defining the peripheral surface points and the edge points of the singular points by using the NURSS subdivision format and the Eigen-polyhedron subdivision format so that the points and the edge points are positioned on the same tangential plane;
for a singular point with the degree of n, the control point adjacent to the singular point is E i ,F i The node distance is d i ,a i ；
Calculating new control pointsLet p= [ V, E 0 ,…,E n-1 ,F 0 ,…,F n-1 ] T ,/> Write subdivision rule +.>
Defining a NURSS format subdivision matrix as M and an Eigen-polyhedron-based subdivision matrix as N;
defining a circle of Bezier control points around the singular point asIs the point V, E i ,F i Is a linear combination of (a); the Eigen-polyhedron subdivision format was used as a guide:
s31, calculating limit points C based on Eigen-polyhedron subdivision;
by L 0 The unitization of the eigenvector corresponding to the eigenvalue of matrix N of 1 is represented, and the limit point is defined as c=l 0 M T P；
S32, calculating a tangent plane based on Eigen-polyhedron subdivision;
defining two matrices of size 2nx2n Let-> Lambda is the matrix->Main feature value of (2), then->Write-> Λ is a diagonal matrix of singular values; i.e 1 ,i 2 Is such that Λ (i 1 ,i 1 )＝Λ(i 2 ,i 2 ) Index of =λ, letIn addition to->Other diagonal matrix with zero position, get +.> Further define vector set +.>
Step S33, defining Bezier control points around the singular points;
bezier control points around singular points
Step S4, calculating a connection function, wherein the connection function specifically comprises the following steps: the singular surface is stepped up to a bi-quintic Bezier curved surface to meet the G1 continuity condition;
s41, setting each angle for singular point with degree nSense weight k i ：
S42, defining angles according to the weights;
for theta i Summing, if the summation result is not equal to 2 pi, normalizing the angle to 2 pi;
definition (i, j) =sin (θ) i )sin(θ i+2 )…sin(θ j )(i<j)；
If n=2k+1 is an odd number:
if n=2k is even:
the set S is n numbersIs set of->Is the value of the most frequently occurring set, if each number appears only once, let +.>
Step S5, local step-by-step optimization;
dividing constraint conditions on each side into two parts, wherein the first two equations become vertex constraints, and the rest are called side constraints; solving a new basis function according to the following sequence; the new basis function comprises a hole filling mode compatible with NURBS front and back, and CAD design and CAE analysis are connected based on a NURBS transfer model;
s51, optimizing vertex control points;
each singular point vertex constraint with the degree of n comprises 3n+1 control points and 2n linear constraint equations, and the 3n+1 control points meeting the vertex constraint are obtained by solving;
s52, optimizing an edge control point;
after the vertex control points are determined, the control points of the rest edge constraint designs are controlled on each edge, and then the control points meeting the edge constraint on each edge are solved;
s53, optimizing a surface control point;
solving a least square problem with linear equation constraint during step-by-step solution, namely:
s.t.MP＝b；
converting the least squares problem into a solution problem of a system of linear equations:
s54, designing the shape of the curved surface to be processed according to the result obtained by solving in the step S53, and generating the spline, wherein the spline is provided with a curved surface which is used for generating global G1 continuity on a control grid with any topology and is used for CAD design and CAE analysis.
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