WO2012131124A1

WO2012131124A1 - Method for the geometric design of spatial meshes using nurbs surfaces

Info

Publication number: WO2012131124A1
Application number: PCT/ES2012/000083
Authority: WO
Inventors: César Antonio OTERO GONZALES; Cristina Manchado Del Val; Rubén ARIAS FERNANDEZ
Original assignee: Universidad De Cantabria
Priority date: 2011-03-28
Filing date: 2012-03-28
Publication date: 2012-10-04
Also published as: ES2363546A1; ES2363546B2

Abstract

A method for obtaining a design with the geometry of a spatial mesh formed by a plurality L of linear entities 1', characterized in that it comprises the steps of: choosing three points (C₀₀ C_0n C_n0) in a plane that defines a triangle; choosing a point (O) outside said plane; choosing an order of complexity n, in which n is a natural number greater than or equal to 1; calculating a set of control points C_¡j in said triangle, which depend on n and on the three points chosen (C₀₀ C_0n C_n0); choosing weights λ_¡j,in which each one of these weights is connected with a control point C¡j; transforming the set of control points C_¡jinto a plurality of points P_¡jthat define a control polyhedron; obtaining a NURBS surface from said control polyhedron; within the plane defined by said points (C₀₀ C_0n C_n0), defining an interlattice with a plurality L of sides 1; transforming said plurality L of sides 1 into a plurality of linear entities 1' constituting said spatial mesh. Meshed structure. Computer program.

Description

MECHAS GEOMETRIC DESIGN METHOD

SPACES THROUGH NURBS SURFACES

FIELD OF THE INVENTION

The present invention belongs to the field of civil engineering and architecture, and more specifically to the design of space structure meshes.

BACKGROUND OF THE INVENTION

In the context of architecture and civil engineering, a spatial mesh is defined as a spatial structure composed of bars, which are responsible for stability and therefore bear the loads of the structure. Figure 1 shows an example of a spatial mesh, seen from various perspectives.

Likewise, in the context of graphic computing, a NURBS surface (in English, Non Uniform Rational B-splines) is defined as a type of surface defined mathematically by the division of two complete polynomials into two independent variables u and v. These polynomials can be written in the Berstein base, so that the components of a NURBS in that base are formed by their control points and their respective weights.

NURBS surfaces have a great application in computer, industrial, aeronautical and other design, due to the multitude of forms and complexity to which these surfaces can be adapted. These fields of knowledge need the most exactly possible materialization of the surface, unlike what happens in architectural structures, where a discrete approach is obtained, normally due to problems of scale and construction costs. Although not only the construction field needs discrete structures, based on continuous surfaces.

The geometric design of a spatial mesh tries to define the position and length of each and every one of the bars that make up a structure. These bars are idealized as straight segments so that, in later stages of a project, they can be structurally designed, manufactured and finally put into work. Therefore, geometric design is an essential step in obtaining a real structure.

There are several procedures to design different types of spatial meshes to different types of surfaces, based on geometric properties of the surfaces to which they adapt, and they are all procedures required by designers when technically specifying a project. In no case the mathematical surface is obtained, to which the mesh adapts, but a discrete approximation of it.

An example of this type of structures is defined in US patent US 2682235. In this document a spatial mesh is defined that adapts to a sphere, by means of a subdivision thereof according to maximum arcs.

Another unique work was that carried out by Joseph D. Clinton (NASA CONTRACTOR REPORT, CR-1734, Advanced Structural Geometry Studies, Part I - Polyhedral Subdivision Concepts for Structural Applications) that performs a different subdivision to define another spatial mesh on a sphere.

On the other hand, César Otero and others in CR-Tangent Meshes (Journal of the International Association for Shell and Spatial Structures: IASS, Vol. 41 (2000) n. 132) describe a method for geometrically designing a spatial structure from a plane. Specifically, the method allows the design for elliptical quadric surfaces.

The main problem derived from the attempt to build a spatial mesh is that it is necessary to define the basic elements of the bar type: position, angle, length, etc. This definition requires a complicated three-dimensional geometric study, to make meshes adapted to complex surfaces. On the other hand, the application of methods already known, which can offer a solution, limits the design of the meshes to certain types of surface studied.

Trying to select new, and more complex, surface shapes makes the geometric difficulty grow, so this can be a serious reason for the designer not to opt for the design he wants, but for the known solution.

On the other hand, the article Weighted radial displacement: A geometric look at Bézier conics and quadrics. Computer Aided Geometric Design Vol. 17, 2000, pp 267-289, relates the quadrics as NURBS surfaces of order 2 with a plane, by means of an artifice described by the authors, Javier Sánchez-Reyes and Marco Paluszny, within the field of geometric design computer assisted Its objective is to obtain a procedure to identify patches on quadrics.

However, currently the designer of a space mesh does it manually. There is no known tool that allows custom design of space meshes.

SUMMARY OF THE INVENTION

The present invention tries to solve the aforementioned drawbacks related to the design of structures, by means of a method that simplifies the three-dimensional problem of a spatial mesh, to a framework defined in a plane. The method offers a solution for NURBS surfaces of order n, that is, of any order. In comparison with NURBS surfaces of order 2, those of order n allow greater freedom of design.

Specifically, in a first aspect of the present invention, a method is provided to obtain a design of the geometry of a spatial mesh formed by a plurality of linear entities. The method comprises the steps of: -Choose three points on a plane that defines a triangle;

-Choose a point outside that plane;

-Choose an order of complexity n, where n is a natural number greater than or equal to 1; - calculate a set of control points in said triangle that depend on n and the three points chosen;

-Choose some weights, each of which is related to a control point;

-transform the set of control points into a plurality of points that define a control polyhedron;

- obtain a NURBS surface from that control polyhedron;

- within the plane defined by the three points chosen in the first stage, define a fabric as a plurality of sides;

-transform that plurality of sides into a plurality of linear entities constituting the spatial mesh object of the design.

Preferably, the step of calculating a set of control points in the triangle is given by the expression:

In a possible embodiment, the point outside the plane can be finite, that is, defined by its coordinates. Alternatively, that point can be at infinity, that is, defined by one direction.

If that point is finite, the weights are freely chosen, while if the point is infinite, the weights are worth all 1.

Preferably, the step of transforming the set of control points C into a plurality of points Ρ _υ · that define a control polyhedron is given by the expression:

L

if point O is finite

Preferably, the step of obtaining a NURBS surface from said control polyhedron is given by the expression:

where (u, v) are the baricecentric coordinates of the triangle defined by the points of the initial plane, C ₀₀ C _0n C _n o.

The step of transforming the plurality of sides into a plurality of linear entities constituting the spatial mesh object of the geometric design is carried out as follows: for each side:

- get the coordinates (u v) of its first end;

-from these coordinates (u v), calculate the point Ny (u, v);

- get the coordinates (u v) of its second end;

-from said coordinates (uv), calculate the point Nf _j (u, v);

-being the linear entity formed by the points obtained, the transformation of the side.

In a particular embodiment, n> 2 is chosen. And in an even more particular embodiment, n> 3 is chosen. Also, the invention provides a meshed structure obtained from the above method.

Finally, the invention provides a computer program comprising computer program code means adapted to perform the steps of the method described above, when the program is executed on a computer, a digital signal processor, a programmable field gate arrangement, a specific application integrated circuit, a microprocessor, a microcontroller, and any other form of programmable hardware.

The advantages of the invention will become apparent in the following description.

BRIEF DESCRIPTION OF THE FIGURES

In order to help a better understanding of the characteristics of the invention, in accordance with a preferred example of practical realization thereof, and to complement this description, a set of drawings is attached as an integral part thereof, whose character is Illustrative and not limiting. In these drawings:

Figure 1 shows an example of a spatial mesh, seen from various perspectives.

Figure 2 shows a reference plane defined by three points C ₀₀ C ₀₂ C ₂₀ and how that plane can be transformed into another one defined by three other points P ₀₀ P ₀₂ P ₂₀ from a point O outside the original plane.

Figure 3 shows an example of control points obtained, in the case of n =, given a reference plane defined by three points A B C.

Figure 4 shows a plane containing a structure composed of a set of vertices connected by sides. Figure 5 shows two different spatial meshes, obtained with the method of the present invention.

Figure 6 is an example of a space mesh adapted to a NURBS of order n = 5.

Figure 7 shows the generality of the method (order of complexity n). Specifically, the use of the method for surfaces of order 2, 3, and 4 is represented.

DETAILED DESCRIPTION OF THE INVENTION

In this text, the term "comprises" and its variants should not be understood in an exclusive sense, that is, these terms are not intended to exclude other technical characteristics, additives, components or steps.

In addition, the terms "approximately", "substantially", "around", "ones", etc. they should be understood as indicating values close to which these terms accompany, since due to calculation or measurement errors, it is impossible to achieve those values with total accuracy.

The following preferred embodiments are provided by way of illustration, and are not intended to be limiting of the present invention. In addition, the present invention covers all possible combinations of particular and preferred embodiments indicated herein. For those skilled in the art, other objects, advantages and features of the invention will be derived partly from the description and partly from the practice of the invention.

The following is detailed: first, how to define a NURBS of any order n, where n is any natural number; and secondly, how to obtain the design of the geometry of a spatial mesh that fits it. The spatial mesh is formed by a set of linear entities Γ. In a particular embodiment, the linear entities can be bars. Alternatively, the linear entities can be: columns, beams, cables or other structural element defined by their length, or any other entity that can form a spatial mesh. To do this, the following steps are followed:

First, three points C ₀₀ C _0n C _n0 are chosen in a plane that defines a triangle. Figure 2 shows a reference plane defined by three points C ₀₀ C ₀₂ C ₂₀ . Next, a point outside that plane is chosen. Figure 2 shows a point O outside the plane defined by the triangle. The three coplanar points and the non-plane point are also illustrated in Figure 4. Since the method allows to define a NURBS surface of any order, it is necessary to choose the order of complexity n, that is, the order of the required NURBS. This order n can be any natural number including n = l, although the method has its greatest advantages in designs with a high degree of complexity, at least n> l. In the example in figure 2, n = 2 has been chosen.

The point (O) outside the plane can be both finite, that is, defined by its coordinates, and be at infinity, that is, defined by an address. In the example in Figure 2, point (O) is finite.

The following shows how that plane defined by the three Coo C _0n C _{n0 points} can be transformed into another plane defined by another three points, from the point O outside the original plane.

For this, a set of control points C _and in the triangle defined by the three points C ₀₀ C _0n C _n0 is calculated. The control points depend on n and on the three points chosen Co ₀ C _0n C _n0 . Control points are chosen according to the following expression:

where n is the order of complexity previously chosen and the indices ij are positive integers:

Figure 2 shows the Coo Coi C ₀₂ C _í or CnC _{20 points} obtained for n = 2. Figure 3 shows another example of control points obtained, in the case of n = 4, given a reference plane defined by three points AB C.

Next, weights λ ,, (real values) are chosen for each control point C¡ _j . In the event that the point outside the plane (O) is finite, the weights λ¾ are freely chosen, that is, the designer is free to assign the value he wants. On the contrary, in case the point (O) is infinite, weights worth every one.

The next step is to transform the set of control points C¡ _j into a set of points Py that define a control polyhedron. This transformation is given by the expression:

Figure 2 illustrates the transformation of the control points C ₀₀ Coi C ₀₂ C _í or C1 1C20 into a set of points P ₀₀ P ₀ and Po ₂ P _i or Pn P20 for the example mentioned above.

Once these control points and these weights have been defined, the NURBS equation of order n can be obtained from the control polyhedron (surface to which the spatial mesh to be constructed is to be adapted), which is given by the expression where

and where (u, v) are the baricecentric coordinates of the triangle defined by the points

C ₀₀ C _0n CnO.

The next step to achieve the spatial mesh is, within the plane defined by the points C ₀₀ C _0n C „o, define a fabric as a set L of sides 1. This fabric represents the sides that, once spatially projected, will represent the linear entities that form the spatial mesh. This fabric can be as exhaustive as desired, that is, you can define as many sides 1 as the designer wants. It is also possible that some areas of the plane defined by points C ₀₀ With C _n o, are very densely populated from this network of sides, while other areas of that plane are empty or less densely populated from sides.

Figure 4 shows the framework (set L) of sides and one of those sides is highlighted 1. The vertices of side 1 have been designated A B.

This framework in the plane defined by the points C ₀ or C _0n C _n0 can also be described as a generic triangulation on said plane, which may affect the entire plane or only a region thereof. That is, from any point of said plane, a plurality of triangles is constructed. It can also contain "islands", areas without triangles. That is, it is not necessary that the triangulation on the plane be thorough and complete. The points that determine the framework or triangularization are chosen by the designer as desired (randomly, according to a certain criteria, etc.), depending on the mesh you want to obtain. Thus, The plane contains a structure composed of a set of vertices connected by sides. The set of sides we will call L and an element of it, any side, we will call 1. Each side 1 is formed by two points A and B which are its extreme points (vertices).

Finally, the set L of sides 1 is transformed into a plurality of linear entities Γ constituting the spatial mesh. This is also illustrated in Figure 4, which highlights the transformation of side 1 into the linear entity Γ, whose vertices have been called A 'B'. This transformation is performed as follows:

For each side 1 of the set L of sides that form the framework in the Coo Co _n plane

C _n0 :

- obtain the coordinates (u v) of its first end (A);

-from these coordinates (uv) of the first end, calculate the point Nj _j (u, v) called A ', which is the transformed point of vertex A on the NURBS surface previously obtained;

- obtain the coordinates (u v) of its second end (B);

-from these coordinates (uv) of the second end, calculate the point Ni _j (u, v) called B ', which is the transformed point of vertex B on the NURBS surface previously obtained;

The linear entity Γ formed by points A 'B' is the transform of side 1.

In other words, at this stage a relationship is established between the triangularization of the plane indicated above and a spatial mesh that adapts to a NURBS surface of order n.

This process is iterative, that is, it is repeated for each side 1 of the previously defined framework, and ends when it travels through all the required sides. So many sides 1 we have defined, so many bars Γ are obtained in the mesh structure. As indicated above, the method allows to define a NURBS surface of any order, including n = 1, although the method has its greatest advantages in designs with a high degree of complexity, at least n> 1. For this reason, the method has its greatest advantages for and more particularly, for n> 3.

Finally, the invention allows to design any meshed structure obtained from the method described above.

Below are several examples that illustrate the final result (meshed structure formed by linear entities).

In figure 5 two different spatial meshes are represented. In the windows "f 'and" g "in figure 5 the two meshes are seen in perspective. Both are NURBS of order 3 (n = 3). In window" a "two possible triangularizations of the plane are represented, which establishes the method described above, the one on the left corresponds to the window mesh "f '. The triangularization on the right corresponds to the window mesh "g". Window "b" warns us of this correspondence. In the "c" window we can see both meshes in a plan view -vertical- and appreciate their difference. However, in the "d" window, which shows a side view, we can see that both meshes also have similarities. The "e" window shows a different side view. It shows that the meshes are actually adapting to the same reference NURBS surface, which defines the method of the invention. Its coincidence is appreciated.

For this case the following data were used:

Coordinates C ₀₀ C ₀₃ C ₃₀ O

Pesos

Figure 6 is an example of a space mesh adapted to a NURBS of order 5 (n = 5): Window "a" shows its plan view. Windows "b" and "c" are side views, and window "d" is a general view.

For this case the following data were used:

Coordinates Coo C ₀₅ C ₅ or O

Pesos

As can be seen, one of the great advantages of the method of the invention is that the mathematical form of the structure is controlled.

Figure 7 shows the generality (n) of the method. This figure represents the use of the method, for surfaces of order 2, 3, and 4. Obviously, this is generalizable to anyone.

Claims

1. A method for obtaining a design of the geometry of a spatial mesh formed by a plurality L of linear entities Γ, characterized in that it comprises the steps of:

-Choose three points (Co ₀ C _0n C _n0 ) in a plane that defines a triangle;

-Choose a point (O) outside that plane;

-Choose an order of complexity n, where n is a natural number greater than or equal to 1; - calculate a set of control points Q _j in said triangle that depend on n and on the three points chosen (C ₀ or C _0n C _n0 )

-Choose λ¾ weights, where each of these weights is related to a control point C _and -;

-transforming the set of control points Q _j into a plurality of points Py defining a control polyhedron;

- obtaining a NURBS surface from said control polyhedron;

- within the plane defined by said points (Coo C _0n C _n0 ), define a fabric as a plurality L of sides 1;

- transforming said plurality L of sides 1 into a plurality of linear entities Γ constituting said spatial mesh.

2. The method according to claim 1, wherein said step of calculating a set of control points C _j in said triangle is given by the expression:

3. The method according to any of claims 1 or 2, wherein said point (O) outside the plane can be finite, that is, defined by its coordinates, or it can be at infinity, that is, defined by an address.

4. The method according to claim 3, wherein if said point (O) is finite, the weights λ

they are freely chosen, while if said point (O) is infinite, the weights are worth

all 1.

5. The method according to claim 4, wherein the step of transforming the set of points of Cy control a plurality of points Pj _j defining a polyhedron control is given by the expression:

6. The method according to claim 5, wherein said step of obtaining a NURBS surface from said control polyhedron is given by the expression:

where (u, v) are the baricecentric coordinates of the triangle defined by the points

7. The method according to claim 6, wherein said step of transforming said plurality L of sides 1 into a plurality of linear entities constituting said spatial mesh, is performed as follows:

-for each side 1:

- obtain the coordinates (uv) of its first end (A); -from these coordinates (uv), calculate the point Ny (u, v) called A ';

- obtain the coordinates (u v) of its second end (B);

-from these coordinates (u v), calculate the point Ny (u, v) called B ';

-being the linear entity Γ formed by points A 'B', the transformation of side 1;

8. The method according to any of the preceding claims, wherein n> 2.

9. Mesh structure obtained from the method according to any of the preceding claims.

10. A computer program comprising computer program code means adapted to perform the steps of the method according to any one of claims 1 to 8, when said program is run on a computer, a digital signal processor, an arrangement of Programmable field doors, a specific application integrated circuit, a microprocessor, a microcontroller, and any other form of programmable hardware.