EP2029822B1 - Supporting structure for freeform surfaces in buildings - Google Patents

Abstract

The structure (8) has support elements (4) combined to form 4 to 6 sided polygons, which span envelope geometry of a building and enclose a planar surface element (5). The support elements form a common node region (3). The support elements comprise a longitudinal axis that extends in a straight line between another two node regions and runs parallel to an imaginary intersection line of surface element planes, where cross section of the support elements normal to the longitudinal axis has a relative twist angle of 0 degree along the entire longitudinal axis of the support element. An independent claim is also included for a method for providing a supporting structure for curved envelope geometries in buildings.

Description

The invention relates to a support structure for curved envelope geometries of structures consisting of support elements, each of which is composed of the envelope geometry spanning N-corners (N = 3, 4, ...), each defining a planar surface element, and the support elements of adjacent N Corner each forming a common node region in which collide the support members, and lie in at least a portion of the envelope geometry, the plane of a surface element and the respective planes of the surface elements of two adjacent in non-parallel spatial directions N-corners in different planes, according to the preamble of Claim 1.

The invention further relates to a method for establishing a support structure for curved envelope geometries in structures consisting of support elements, in which a predetermined, curved envelope geometry by a seamless network of N-corners (N = 3,4, ...), the each defining a planar surface element is approximated, wherein each adjacent N-corner have a common node, and in at least a portion of the envelope geometry, the plane of a surface element and the respective planes of the surface elements of two non-parallel spatial directions adjacent N-corners in different planes lie, according to the preamble of claim 6.

Curved envelope geometries of this type are used in the construction industry for the realization of free-form surfaces, in which the curvature is different in two different spatial directions, for example in dome structures, or more complex surface shapes. Free-form surfaces of this type are also referred to as non-developable surfaces, and in the course of architectural planning initially designed in the computer model as continuous surfaces. In the structural implementation, the continuous freeform surfaces are to be approximated by a plurality of individual surface elements, which are held in a supporting structure. So it is about possible, even complex freeform surfaces with eg multilayer planar glass elements to be realized, which are mounted above, between or below a support structure made of steel, for example. The support structure is formed from individual support elements, which are each to N-corners, so as triangles, squares, hexagons, etc., are composed. The N-corner span the support structure, with the support members colliding in node areas where they are secured together.

One way of approximating freeform surfaces by individual surface elements is to approximate the freeform surface by curved surface elements, which are, however, each held by planar support elements for cost reasons. An example of this is, for example, the bubble-shaped construction of the "Kunsthaus Graz", where plastically deformable material, namely "Plexiglas", was used, which was given the required curvature for the individual surface elements by thermal deformation. The support structure was realized in terms of construction technology by support members made of molded tubes, wherein the respectively required space curves of the support elements were generated via a rotation in the nodal region of adjoining molding tubes.

However, such a procedure has some disadvantages. For example, the choice of material for the surface elements is subject to restrictions due to the necessary deformability. Specifically, for example, in the case of the "Kunsthaus Graz" mentioned above, it became necessary to create an additional, fire-resistant layer due to the thermal deformation of the stretched plexiglass in order to neutralize the poor fire properties of the plexiglass. Furthermore, it has been shown after the realization that the thermal behavior due to the very high coefficient of thermal expansion (about 6 times that of steel) of the stretched Plexiglas under sunlight leads to "sagging" of the Plexiglas plates, and therefore own supports had to be provided to ensure the shape retention. Also about irregular reflections in the Plexiglas skin you can perceive permanent changes in shape in the form of bumps and dents.

Another, fundamental disadvantage of a realization of free-form surfaces by means of individual, curved surface elements also results in all cases where a multi-layered construction of the building envelope is required to accommodate about necessary building infrastructure such as pipelines and the like. The user perceives only the optically visible layers, ie the inner and outer building envelope, but for the functionality of the building, a plurality of structural physical layers between the optically visible layers while maintaining the shape of the outer shell of the building are provided. For this infrastructure, planar intermediate layers are necessary, which must be tediously inserted between the optically visible, curved free-form surfaces. This concept is also being pursued, for example, in the buildings of Frank O'Gehry, as a result of which his complex designs were first economically constructed.

Another fundamental disadvantage also arises in practice from the considerable amounts of data that have to be processed in the course of planning and, for example, exchanged between architect and specialist planners. To describe a free-form surface, the spatial position and the shape of support and surface elements must be reproduced in a spatial coordinate system, with hardly any possibilities of data reduction due to the sometimes different shape of each individual support and surface element. In practice, "point clouds", ie individual data points in large numbers, are processed, which causes problems in particular when using different CAD or FEM software packages, for example when transferring data between individual specialist planners, eg from architect to structural engineer. If a free-form surface is instead realized using planar surface elements, local coordinate systems can be defined for the individual surface elements, in which, for example, the boundary points of the surface element are already defined by two Coordinates, such as an x- and y-coordinate, are clearly definable, and can be determined by the surface normal to the surface element casual z-coordinate. These local coordinate systems are uniquely defined to the global coordinate system of the forest. This allows a simple data exchange, for example, by means of an output file that displays the position of the normals of the support structure in the node area and the associated local coordinates of the planar area elements.

Another advantage of using planar surface elements is the unrestricted choice of material for the surface and rod-shaped elements, since no special elastic properties or plastic deformability are required. In addition, the cutting of planar surface elements is easier to accomplish than in the case of curved surface elements. This significantly reduces the overall construction costs for designs with free-form surfaces.

The implementation of a freeform surface demanded by the architect into a structurally executable design with the aid of planar surface elements which approximate the freeform surface as well as possible is, however, associated with difficulties. The object is to find a technically and economically realistic distribution of support elements that reproduces the given free-form surface gapless and with the appearance of an aesthetically continuous course after the onset of flat surface elements.

According to the prior art, this is usually done by selecting a support structure in which the support elements are arranged in the form of triangles, since the modeling of free-form surfaces using a gap-free network of triangles, in each of which planar surface elements are held, is relatively easy to solve mathematically and structurally comparatively simple is to be realized.

However, the use of triangles as elementary basic structure of the support structure also has disadvantages. In particular, it is not possible, by means of triangles, to find a distribution of the carrier elements in which the carrier elements do not have to be subjected to torsion in the geometrical sense during the assembly between two nodal regions, ie a rotation of the longitudinal axis approximately in the nodal region. Only support elements with a circular cross-section can be strung together "torsion-free", in a geometric sense. When using non-circular cross-sections, a torsion (in the geometrical sense) has previously been created in the support structure in the account area. This leads to aesthetically and statically unsatisfactory nodal areas. Rather, this results in the problem that multi-layer structures are not or only with considerable additional effort to realize. It must therefore be provided for each layer a separate support system, which in turn increases the cost of materials and assembly costs many times over. The publication DE-A-2303060 discloses a support structure incorporating the features of the preamble of claim 1.

It is therefore the object of the invention to find a structural implementation of free-form surfaces, which reduces the technical and economic requirements and satisfies aesthetic demands. In particular, assembly costs and costs should be kept as low as possible. Another object of the invention is that the support structure for the approximation of free-form surfaces also offers the possibility of a problem-free multi-layer structure, ie the parallel offset assembly of several flat surface elements. These objects are achieved by the measures of claim 1.

Claim 1 initially relates to a support structure for curved envelope geometries of a structure consisting of support elements, each of which is composed of the envelope geometry spanning N-corners (N = 3,4, ...), each defining a planar surface element, and the support elements of contiguous N-corner each form a common node region in which the support elements collide, and in at least a portion of the envelope geometry, the plane of a surface element and the respective planes of the Surface elements of two adjacent in non-parallel spatial directions N-corners lie in different planes. The fact that the plane of a surface element and the respective planes of the surface elements of two N-corners adjoining in non-parallel spatial directions lie in different planes reflects the fact that the support structure is provided for the realization of free-form surfaces in which the curvature of the to be approximated freeform surface is different in two different spatial directions. According to the invention, it is provided that the support elements in this section in each case form a 4- or 6-corner, and the support elements each have a longitudinal axis which extends in a straight line between two node areas and extends parallel to the imaginary section line of the surface element planes assigned to it, the cross section the carrier elements each have the relative twist angle of 0 ° along its longitudinal axis along the entire longitudinal axis of the carrier element. The "associated surface element planes" of a carrier element are the planes of those surface elements which are carried by the respective carrier element.

The choice of four or six corners as an elemental basic shape of the support structure in contrast to the triangular network structure known in the prior art is of crucial importance, since the Applicants have recognized that a four- or six-cornered network structure for approximation of free-form surfaces has remarkable mathematical properties , which are for a structural implementation of freeform surfaces of great advantage. In particular, a form of geometric approximation of free-form surfaces can be found for four- or six-corner network structures, which ensures a parallel displacement of the surface elements, wherein the respective parallel displaced surface elements have boundary lines which are parallel to the corresponding original boundary lines, and in turn result in a complete total area , as will be shown below. The practical consequences of a parallel shift ("offset") and their meaning for the construction technology were not recognized in the prior art.

In addition, building costs can be reduced by using a four or hex network structure compared to triangular mesh structures, since the trimming of triangular area elements causes more expense than that for e.g. square surface elements. Furthermore, a supporting structure of quadrangular or hexagonal basic shapes requires less material, since Applicants have shown that, for example, with equivalent approximations of free-form surfaces using triangular and quadrilateral mesh structures, the realization using quadrilateral mesh structures requires a smaller number of support elements than those with triangular basic shapes ,

Due to the parallel displaceability made possible by the choice of four- or hexagons as the basic form of the support structure, it is possible to provide a further feature according to the invention with regard to the support elements, namely support elements which extend in a straight line between the nodal regions and without torsion (in the geometric sense). This is expressed by the feature that the support members each have a longitudinal axis extending straight between each two nodal regions and parallel to the imaginary cutting line of its associated surface planes, and the cross section of the support members normal to its longitudinal axis along the entire longitudinal axis of the Carrier element each having a relative drill angle of 0 °. In a carrier element, which is subjected to a torsion in the geometric sense, in contrast, the cross section along the longitudinal axis on a rotation, which thus has a twist angle not equal to 0 °. By means of this feature according to the invention, the carrier elements can be made higher transversely to the plane of the surface elements, so that a multi-layer structure on one and the same carrying structure is made possible. The support elements can be made sufficiently high to allow room for building infrastructure between the delimiting outer layers create. Due to the higher executable support elements and the use of four- or hexagonal surface elements also the installation of building services and a building physical multilayer structure is facilitated. Furthermore, the use of linear support elements without torsion (in the geometric sense) and bending facilitates assembly, which reduces assembly costs.

One possibility for ensuring the parallel displacement of planar surface elements in a supporting structure approximating a free-form surface is that the angle sum of respectively opposite angles at the point of intersection of the imaginary cut lines is equal to four surface plane adjoining surface areas. It should be noted that the result of a geometric approximation of a free-form surface specified by the architect is first a network of lines, the quadrilaterals being represented by a quadrangular polygon, hereinafter also referred to as "mesh", and two adjoining polygonal ones, respectively have common boundary line; and each four adjacent polygons have a common node in which intersect the respective common boundary lines, as will be described in more detail. In the structural implementation, of course, the support elements are to be set along the common boundary lines, so that the surface elements held in the support elements generally no longer physically collide to share a common boundary line, but are spaced apart from one another. Nevertheless, an imaginary cutting line can be formed, which is defined by the imaginary extension of the respective surface element planes. This imaginary line of intersection corresponds to the abovementioned common boundary line of the geometric approximations of the polygons. The point of intersection of the imaginary lines of intersection of four surface-level planes adjacent to each other in a nodal area corresponds to the above-mentioned node of four polygons. In the nodes, there are four angles between the four adjacent polygons. wherein according to claim 2, the support structure is to be designed so that the sum of two opposite angles is equal. This is a sufficient condition for the parallel displaceability of the surface elements held in the support structure, and the support elements can be arranged without torsion (in the geometric sense), as will be explained in more detail. In this case, the underlying basis of such a support structure polygon mesh is also referred to as a "conical mesh", as will also be explained in more detail.

A further possibility for ensuring the parallel displaceability of planar surface elements in a support structure approximating a free-form surface consists in that the angle sum of respectively opposite angles between the surface normals of two adjoining surface element planes is equal to four surface element planes adjoining one another in a node region. In this case, the underlying basis of such a support structure polygon mesh is also referred to as a "dual-isothermal network", as will also be explained in more detail.

According to claim 4 it is provided that the carrier elements have a rectangular cross-sectional shape, or in a rectangular cross-sectional shape are inscribed. The longitudinal axis of these support elements extends between each two node areas, and the transverse axis is along the entire longitudinal extension of the support member both normal to the longitudinal axis, as well as normal to the imaginary line of intersection of its associated surface element planes. The longitudinal axis does not necessarily have to be an axis of symmetry of the carrier element, but instead it is decisive that there is a straight line along the carrier element between two node regions and parallel to the imaginary cutting line of the surface element planes assigned to it. Rectangular cross-sections also have an aesthetic advantage, as they appear slimmer than approximately circular cross-sections, and can also be made slimmer, since only the beam height is required for the bending stress due to the load acting through the surface elements along the transverse axis normal to the imaginary cutting line of its associated surface element levels is crucial. Furthermore, however, depending on the technical requirements, other forms of the carrier elements may also be advantageous, for example carrier elements with an I-shaped cross-sectional shape.

According to claim 5 it is provided that at least two surface elements are held on the support elements. Here, the advantage of the features according to the invention is utilized that a multi-layer structure by parallel offset surface elements on one and the same support structure is easily possible. Between the at least two surface element planes provided according to claim 5, space is created for facilities of additional building infrastructure, such as pipelines or building physical layers. Furthermore, the interlayer region can fulfill tasks for the air conditioning properties of the design, such as a circulation of air masses for ventilation and thermal insulation.

Claim 6 refers to a method for establishing a support structure for curved envelope geometries in structures consisting of support elements, in which a predetermined, curved envelope geometry through a gap-free network of N-corners (N = 3,4, ...), each one define planar mesh, is approximated, wherein each adjacent N-corner have a common node, and lie in at least a portion of the envelope geometry, the mesh plane of an N-corner and the respective mesh planes of two adjacent in non-parallel spatial directions N-corner in different planes. According to the invention, it is provided that the approximation of the predetermined, curved design takes place with the aid of a first, continuous network of 4 or 6 corners, which are parallel to one another in a direction normal to the mesh plane of the respective 4- or 6-corner into a further, gap-free network can be converted by 4- or 6-corners, wherein in each case two adjacent N-corner have a common boundary line, which determines the course of the longitudinal axis of these N-corners associated support member, and the dimension of a Carrier element is defined perpendicular to this boundary line by the distance of the corresponding boundary line of the first network to that of the other, parallel shifted network. A parallel displacement of a network of, for example, four corners means that each mesh of a 4-corner of the first mesh is displaced in a direction normal to the mesh plane of the respective 4-edge in parallel. According to the invention, the network of 4 corners, which has been moved in parallel in this way, must again produce a gap-free network of 4 corners, each with planar mesh planes. This has the advantage that the distance of a boundary line of two adjoining 4-corner of the first network can be used to determine the dimension of a support element perpendicular to this boundary line to the resulting by parallel displacement from her boundary line of further, parallel shifted network. Since the course of the longitudinal axis of this support member according to the invention is also defined by this boundary line, ie parallel to it, thus resulting by the inventive determination of the support frame straight support elements without torsion (in the geometric sense).

Claim 7 again relates to the definition of a support structure by means of a conical net on which it is based, in that the angle sum of respectively opposing angles between the boundary lines of four contiguous 4 corners in their common node is the same.

Claim 8 refers to the further possibility of defining a supporting frame by means of a dual-isothermal network on which it is based, in that the angle sum of respectively opposite angles between the surface normals of two adjacent mesh planes is equal to four plane planes adjacent to one another in a node.

Claim 9 in turn aims at a multi-layer structure of the curved design by at least a second, continuous network of 4 corners, each defining a planar mesh plane, is defined in a portion of the support structure, wherein the 4-corner of the second network are formed by parallel displacement of the 4-corner of the first network in a direction normal to the mesh plane of the respective 4-corner.

• Fig. 1 eine Darstellung zweier paralleler Vierecksnetze mit jeweils planaren Maschenebenen,
• Fig. 2 eine Darstellung eines Abschnittes einer erfindungsgemäßen Tragstruktur in der architektonischen Anwendung,
• Fig. 3 die Konstruktion eines parallel verschobenen Netzes N₀ aus einem Basisnetz N und dem Parallelnetz p(N) in zweidimensionaler Ansicht,
• Fig. 4 die Konstruktion eines parallel verschobenen Netzes N₀ aus einem Basisnetz N und dem Parallelnetz p(N) in dreidimensionaler Ansicht,
• Fig. 5a eine Darstellung eines konischen Knotens,
• Fig. 5b eine Darstellung von zwei benachbarten konischen Knoten,
• Fig. 6a eine Darstellung einer Schramm'schen Kreispackung auf der Kugel und das isotherme Netz p(N),
• Fig. 6b eine Darstellung eines dual-isothermen Knotens,
• Fig. 6c eine Darstellung zur geforderten Winkelbeziehung in dual-isothermen Netzen,
• Fig. 7 ein Vierecksnetz (im Hintergrund) und ein verfeinertes Vierecksnetz mit jeweils planaren Maschenebenen (im Vordergrund),
• Fig. 8 ein Beispiel einer architektonischen Anwendung des Vierecksnetzes gemäß Fig. 7,
• Fig. 9-12 ein Ausführungsbeispiel zur Festlegung von Trägerelementen anhand eines gemäß dem erfindungsgemäßen Verfahren generierten Vierecksnetzes,
• Fig. 13 eine schematische, zweidimensionale Darstellung der rechteckigen Querschnittsform des Bauraums im Knotenbereich,
• Fig. 14 eine schematische, dreidimensionale Darstellung der rechteckigen Querschnittsform des Bauraums im Knotenbereich,
• Fig. 15 a-d mögliche architektonische Anwendungen einer Tragstruktur,
• Fig. 16 eine schematische Darstellung zur Erläuterung der Konstruktion der Trägerelemente,
• Fig. 17 Trägerelemente, die in einem Knotenbereich eines dual-isothermen Netzes zusammenlaufen,
• Fig. 18 eine Darstellung eines Mehrschichtaufbaus am Beispiel eines dual-isothermen Netzes,
• Fig. 19a-b eine Darstellung der geometrischen Stützstruktur eines hexagonalen Netzes mit Stütztrapezen konstanter Höhe, und die
• Fig. 20a-c in der Fig. 20b ein Offset-Paar N, N_o mit konstantem Eckenabstand und konstantem Flächenabstand, das aus der Stützstruktur von Fig. 20a gewonnen wurde, und aus dem das dazugehörige, ebene Flächentragsystem in Diagrammform in der Fig. 20c gezeigt ist.

Preferred embodiments of the invention are explained in more detail below with reference to the accompanying drawings. It show here the

• Fig. 1 a representation of two parallel quadrilateral meshes each with planar mesh planes,
• Fig. 2 a representation of a portion of a support structure according to the invention in the architectural application,
• Fig. 3 the construction of a parallel shifted network N ₀ from a base network N and the parallel network p (N) in two-dimensional view,
• Fig. 4 the construction of a parallel shifted network N ₀ from a base network N and the parallel network p (N) in three-dimensional view,
• Fig. 5a a representation of a conical node,
• Fig. 5b a representation of two adjacent conical nodes,
• Fig. 6a a representation of a Schramm circle packing on the sphere and the isothermal network p (N),
• Fig. 6b a representation of a dual-isothermal node,
• Fig. 6c a representation of the required angular relationship in dual-isothermal networks,
• Fig. 7 a quadrangle net (in the background) and a refined quadrilateral mesh, each with planar mesh planes (in the foreground),
• Fig. 8 an example of an architectural application of the quadrangle network according to Fig. 7 .
• Fig. 9-12 an exemplary embodiment for fixing carrier elements on the basis of a quadrilateral mesh generated according to the method according to the invention,
• Fig. 13 a schematic, two-dimensional representation of the rectangular cross-sectional shape of the space in the node area,
• Fig. 14 a schematic, three-dimensional representation of the rectangular cross-sectional shape of the installation space in the node area,
• Fig. 15 ad possible architectural applications of a supporting structure,
• Fig. 16 a schematic representation for explaining the construction of the support elements,
• Fig. 17 Carrier elements which converge in a node region of a dual-isothermal network,
• Fig. 18 a representation of a multi-layer structure using the example of a dual-isothermal network,
• Fig. 19a-b a representation of the geometric support structure of a hexagonal network with constant support heights of constant height, and the
• Fig. 20a-c in the Fig. 20b an offset pair N , N _o with constant corner distance and constant area distance, which consists of the support structure of Fig. 20a and from which the associated, flat surface carrying system in diagram form in the Fig. 20c is shown.

In the following, it will be shown how, with the aid of the method according to the invention, a support structure according to the invention 8 can be implemented from the beginning planning of a free-form surface to the realized design. The starting point is a computer-generated model of a free-form surface, whereby the architect has to pay attention to aesthetic and well-proportioned shapes. The architectural design of the freeform surface will also include their structure of individual surface elements 5 and the design of the support structure 8. For the architect, the question of the design of the individual surface elements 5 in type, size and shape will primarily arise in the context of the visual impression of the overall structure, although of course the technical and economic feasibility must be taken into account. The freeform surface is to be built up of individual surface elements 5, which compose the freeform surface gapless.

In the computer model, therefore, the task is first of all to solve a given, continuous Freiformfläche subdivide so in N-corner, that the freeform surface is reproduced without gaps. The boundary lines 2 of the N-corner are polygons that limit a surface area, which is referred to as mesh 1. The mesh plane should be planar, and represents the plane of the future surface element 5, such as a glass plate. In the following, therefore, the term "mesh" 1 is used in connection with the geometric approximation of a freeform surface through a network of N-corners, and the Term "surface element" 5 in the context of the physical cover element in the structural implementation, which is inserted into the support elements 4 and extends in the corresponding mesh plane of the geometric model.

As is well known, only the plane or a single curved surface can be divided into a plurality of equal triangles, quadrilaterals and polygons. As soon as the area for the formation of space has a second curvature, ie becomes a free-form surface, a mesh of, for example, the same triangles is only possible in a few special cases. The manner of subdivision of a sphere, that is, a geometrically comparatively easy to describe shape, is one of the oldest engineering tasks. One possible solution for the sphere is, for example, Buckminster Fuller's geodesic domes, which are an example of an area distribution of a sphere with identical hexagons. It is much more difficult to find suitable solutions for the approximation of more complex free-form surfaces, which are required in view of the arbitrarily shaped, multiply curved surfaces of contemporary architecture.

The invention relates to approximations of free-form surfaces by four- or six-cornered nets with flat meshes 1. In the following, quadrilateral mesh structures will first be discussed. Such a quadrilateral mesh is made up of flat quadrilaterals so that along each inner edge of the mesh exactly two quadrilaterals collide. In an inner corner of the net, in general, exactly four squares collide; such a corner is referred to below as node X and is called regular, otherwise the node X is called singular. By edges or boundary lines 2 of edge polygons is only a square mesh 1. At regular corners of Randpolygonen encounter only one or two stitches 1 together. In the following, when we speak of a "net" without any other specification, we always mean a quadrilateral net with flat stitches 1.

In the following, it will now be explained how within the scope of the invention a quadrilateral mesh N can be found as an approximation of a surface F , usually a free-form surface, which has the advantages according to the invention in the structural realization.

In the following, a quadrilateral mesh N is always intended as an approximation of a surface F , usually a free-form surface. One can refine a network so that the side surfaces are becoming smaller and move closer and closer to F. As a limit, this results in a curve network K on F. If, in the refinement, the evenness of the meshes 1 is obtained, a so-called conjugate curve network K results on F in the boundary.

Important methods for generating square meshes 1, and in particular those having properties which are advantageous for the architecture, are based on the concept of parallel (parallel) networks. Parallel meshes M , N are those in which the meshes 1 of the one mesh M can be mapped onto the meshes 1 of the other mesh N while preserving all neighborhood relationships so that the planes of corresponding meshes 1 are parallel. Since meshes 1 with a common edge 2 are again imaged onto meshes 1 with a common edge 2 during this transformation or parallel displacement, corresponding edges 2 in meshes M and N are parallel because of the parallelism of corresponding mesh planes (see FIG Fig. 1 ). It can happen that all meshes 1 of the one mesh, say M , are convex, but there are N meshes 1 with self-intersections in the parallel mesh.

One can start from a net M and construct new nets N by parallel displacement of the side surfaces in compliance with the corner conditions. The resulting degrees of freedom can be used for the design. The Fig. 1 illustrates a construction method which first establishes the parallel relation of two polygons (bold); the rest of N (right figure in the Fig. 1 ) necessarily results from parallel pulling to corresponding edges of M (left illustration in FIG Fig. 1 ).

For the construction of special nets, those nets N are important, to which there is a parallel network p ( N ), which approximates a convex surface S (eg a sphere). This regularizes the mesh N in the sense that too small corner angles in the stitches 1 and thus narrow squares are avoided. The deeper reason for this is that, with refinement in the sense given above and keeping the flat meshes 1 of N and p ( N ), the following happens: N has a curve network K on a surface F , p ( N ) has a boundary position a curve network p ( K ) on the surface S. The meshes K and p ( K ) are related to each other in parallel, and therefore K is the mesh of the relative curvature lines of F with respect to the "relative sphere" S.

If S is a sphere, K is the network of ordinary curves and therefore rectangular. The network of curvature lines describes the directions of strongest and weakest normal curvature of a surface. It is appropriate to give the viewer a good idea of the form, which enhances the importance of the architecture. In the following, a network N with a parallel network p ( N ), which approximates a convex surface S , is also called a general curvature network .

According to the invention, the concept of parallel displacement of a net N, M is used to create a so-called "offset" for defining a supporting structure and for realizing multilayer structures of shells in architecture. An offset N _o to a quadrilateral network N is intended to be a parallel quadrilateral network, whereby depending on the application, certain requirements are placed on the spacings of corresponding meshes 1, edges 2 or nodes X.

The Fig. 2 shows a section of a support structure according to the invention 8. There are two offsets, which on a net mesh 1, in the Fig. 2 As surface element 5 can be seen, are explained. On the one hand, there is a quadrangle of the second layer (eg glass construction) running in a parallel plane to the flat quadrangle of the base mesh 1, in which also the tension 7 runs. By means of the spacer elements 6, a third plane is defined at a slightly greater distance, which is indicated by the connecting lines between the end points of the spacer elements 6. The FIG. 2 also shows a typical construction of the support elements 4, in which the glass sheets are inserted. Geometrically, the following situation occurs in the case of these carrier elements 4, wherein the carrier elements 4 in the geometric model first appear as two-dimensional support quadrilaterals, which are also referred to below as support trapeids: Corresponding (parallel) edges of base network N and offset network N _o are represented by flat quadrilaterals (FIG. Supporting quadrilaterals); the link connecting corresponding nodes X and X _o of base and offset is a common edge n ( X ) of the four in these nodes converging support quadrilaterals. The common edge n ( X ) can be thought of as a counterpart to the surface normal of a smooth surface. The set of all support quadrilots is referred to below as a support structure . In the construction of offsets and support structures, it is convenient to use oriented meshes that distinguish between two sides (a skin and an inner skin). Thus, the normals n ( X ) are uniformly oriented (outwards or inwards). The node areas 3 of the structural implementation are located at the positions of the nodes X of the geometric model.

Now consider a quadrilateral mesh N with plane meshes 1 parallel to a similar mesh p ( N ), where p ( N ) approximates a convex surface S. Within S , a point Z can be chosen, which plays the role of a center and regulates the distribution of the distances of the offsets. The convexity of S is not absolutely necessary, but it does exist that there exists a point Z from which all surfaces of p ( N ) are visible. Now, offsets N _o of N with respect to p ( N ) are constructed as follows: Each mesh plane Q of the net N is shifted parallel to the new orientation Q _o in the sense of the given orientation. The distance from Q to Q _o must be equal to the distance from Z to p ( Q ). Here, p ( Q ) denotes the plane of that mesh 1 of the parallel network p ( N ) belonging to Q (see Fig. 3 ). Based on the Minkowski sum of convex polyhedra, one can look at the resulting offset network N _o as the sum of N and p ( N ); we briefly write N _o = N ⊕ p ( N ). In summation, all distances can be multiplied by a uniform factor λ (or preceded by a scaling of p (N)). Writing the scaling of p ( N ) with factor λ as λ · p ( N ), the offsets constructable with Z and p ( N ) are the nets N _o = N ⊕ λ · p ( N ).

It should be noted that the distances of corresponding nodes X and X _o from base and offset are given by the distances from Z to the nodes of the parallel network p ( N ), and the same applies to corresponding edges.

Obviously, in this offset construction, some freedom in the distribution of distances are still possible. These are the choice of the center Z. For example, if all the mesh planes of p ( N ) are the same distance from Z , we obtain offsets at a constant area spacing. However, the output network N must not be arbitrary, but must already be parallel-related to a suitable network p ( N ).

If corresponding quadrilateral faces of base and offset are to have a constant spacing, the parallel mesh p ( N ) must have quadrilateral faces at a constant distance from Z. Therefore, p ( N ) of a sphere S must be circumscribed with center Z , ie all mesh planes of p ( N ) must touch the sphere S. For example, the value of 1 can be assumed for the radius of the sphere S.

Each parallel to such a network p (N) network N is hereinafter referred to as conical network. This designation derives from the following geometric identification: The four lattice planes colliding in a regular inner node X of N touch a sphere S _x of radius 1, which is obtained from the inscribed sphere S of p ( N ) by suitable parallel displacement. Therefore, these four planes through X also touch that cone with tip X , which is described touching the sphere S _x (see Fig. 5a and 5b ). In the following, we also speak of a conical node X in this case. Contains the axis of rotation n (X) of the cone, the node X X _o corresponding node of the offsets _o N = N ⊕ d · p (N) at a distance d.

Conical nets thus have the practical property of having offsets at constant area spacing. Since p ( N ) approximates a sphere S , conical nets are also considered as approximations of curvature nets. In addition to the construction of p ( N ), the following optimization method is particularly suitable for the calculation.

• Input ist ein Vierecksnetz N, dessen Maschen nicht zu stark von ebenen Maschen abweichen. Solche Netze können automatisch oder interaktiv aus konjugierten Kurvennetzen der unterliegenden Fläche F generiert werden.
• Output ist ein Vierecksnetz mit ebenen Maschen, welches möglichst nahe an F liegt.

For the realization of the invention an algorithm is proposed, which accomplishes the following:

• Input is a quadrilateral mesh N whose mesh does not deviate too much from flat mesh. Such networks can be generated automatically or interactively from conjugate curve networks of the underlying area F.
• Output is a quadrilateral mesh with flat mesh that is as close to F as possible.

The algorithm works with a numerical optimization, which runs iteratively. It is done by gradually shifting node X while preserving connectivity. At the beginning of the iteration, the following objective function f = f _planar + w ₁ f _smooth + w ₂ f _{close is} minimized by a penalty method. The meaning of these functions is the following: Since a quadrangle is plane and convex if and only if the sum of the inner angles α ₁ + α ₂ + α ₃ + α ₄ = 2π, then f is _planar as the sum of terms (α ₁ + α ₂ + α ₃ + α ₄ -2π) ² , the sum extending over all quadrangles of the network. The function f _smooth should accept small values for smooth and aesthetic networks, whereby in principle any known smoothing function can be used. The function f _near holds the network during the optimization close to the given reference surface F or the input network N. The simplest way is to use the sum of the distances of corresponding nodes X of the current and the improved network. However, the well-known tangential distance method proves to be better. The weights w ₁ and w ₂ must be reduced in the course of the optimization in order to increase the influence of the planarity term f _planar . In order to obtain a numerically high flatness of the meshes 1, following the penalty procedure, a Lagrange-Newton iteration is used which minimizes the target function w ₁ f _smooth + w ₂ f _close to a set of constraints. The secondary conditions are the equations α ₁ + α ₂ + α ₃ + α ₄ = 2π for the individual square meshes 1.

The algorithm can only be used meaningfully if the input network reflects the geometry of a conjugate curve network, ie in particular has been obtained from such a network.

To make it easier to have an input that meets these requirements, one can combine the proposed optimization method with a subdivision algorithm that works on quadrilateral networks. Provide a coarse and simple quadrilateral mesh with (almost) even meshes 1 (see example in the background of Fig. 7 ), and then switches between subdivision and optimization. The net in the foreground of Fig. 7 was generated using such a procedure, which also serves as the basis of a specific design of a public transport stop according to Fig. 8 is.

Furthermore, the following labeling of a conical node X is to be built into the optimization: The four corner angles ω ₁ , ω ₂ , ω ₃ , ω ₄ occurring around a conical node X fulfill the equation $${\mathrm{\omega }}_{}$$$${\mathrm{}}_{\mathrm{1}}\mathrm{+}{\mathrm{<figure-callout\; id="3"\; label="\omega "\; filenames="imgf0002.png,imgf0008.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{3}}\mathrm{=}{\mathrm{<figure-callout\; id="2"\; label="\omega "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{2}}\mathrm{+}{\mathrm{<figure-callout\; id="4"\; label="\omega "\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_{\mathrm{4}}\mathrm{,}$$

In the penalty method, a term (ω ₁ + ω ₃ -ω ₂ -ω ₄ ) ^{2 is} added to the function f _planar per node X. In the optimization with constraints, the constraint (K) is added per node X. The net in the foreground of Fig. 7 was calculated by alternating between Catmull-Clark subdivision and optimization with the addition of condition (K) per X-node.

If one wishes to use support elements 4 with constant profiles for a support frame 8 and to do this with optimized node X (4 planes through an axis), then offsets at a constant edge distance (height of the support elements 4) are necessary. This is a stronger constraint than the requirement for constant area spacing or constant corner spacing, and therefore one can no longer approximate arbitrary shapes with such meshes. Nevertheless, there are a variety of such networks, which are obtained in the following way: Because of the constant spacing of the edges 2 of the basic network N and the offset network N _o , there must be a parallel network p ( N ) whose edges 2 are always the same distance from Z. So all the edges of p ( N ) touch a sphere S with center Z. Each square mesh of p ( N ) lies in one Level E , whose circle intersects with S all sides of the stitch 1. So every stitch 1 of p ( N ) has an inscribed circle lying on the sphere S. Adjacent meshes 1 create touching incursions. Overall, therefore, the amount of incircle forms a circle pack on the sphere (see Fig. 6a ). It is a so-called Schramm circle packing, since four common touch tangents go through a common point (node X of p ( N )). The network p ( N ) is known as an isothermal network.

Nets N parallel-related to such a quadrilateral network p ( N ) are called dual-isothermal nets because they represent the Laguerre-geometric (and thus in a sense dual) counterparts to the isothermal nets. However, it also follows that with such nets only those surfaces can be approximated in which the Gaussian image of the curvature lines is an isothermal network. This means a certain limitation in the design. The class of surfaces includes in particular the minimal surfaces. Constant edge meshes approximating minimal surfaces are known in the art, but without the present offsets property and its importance to the architecture being recognized. Important for the practical implementation is the well-known construction of the nets p (N), which are also called Koebe polyhedra.

Identification of dual-isothermal networks can also be done by angle conditions. In each corner of the network, the outgoing edges 2 are on a turntable. The angle between the surfaces occurring along an edge 2 (angle between the normals measured in the interval (0, π)) is referred to below as the surface angle along this edge 2, where φ ₁ , φ ₂ , φ ₃ , φ _{4 are} the longitudinal ones If the edges 2 are surface angles occurring around a node X, then (see also Fig. 6b ) $${\mathrm{<figure-callout\; id="1"\; label="\phi "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{\mathrm{1}}\mathrm{+}{\mathrm{<figure-callout\; id="3"\; label="\phi "\; filenames="imgf0002.png,imgf0008.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{\mathrm{3}}\mathrm{=}{\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{\mathrm{2}}\mathrm{+}{\mathrm{<figure-callout\; id="4"\; label="\phi "\; filenames="imgf0008.png,imgf0009.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_{\mathrm{4}}\mathrm{,}$$

In every node of a dual-isothermal network, this condition must be met. But there are other angle conditions that are associated with the edges 2 of the network.

Let A and B be the endpoints of an edge 2, and if the angles occurring there to the adjacent edges 2 opening into A and B on both sides of this edge 2 are denoted α ₁ , α _r , β ₁ , β _r (see Fig. 6c ). Then the relationship applies between these angles $$\mathrm{<figure-callout\; id="1"\; label="tan\; \alpha "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">tan\; </figure-callout>}\left({\mathrm{\alpha }}_{}\right)\left({\mathrm{}}_{\mathrm{1}}\mathrm{/}\mathrm{2}\right)\mathrm{\cdot }\tan \left({\mathrm{\beta }}_{\mathrm{r}}\mathrm{/}\mathrm{2}\right)\mathrm{=}\tan \left({\mathrm{\alpha }}_{\mathrm{r}}\mathrm{/}\mathrm{2}\right)\mathrm{\cdot }\mathrm{<figure-callout\; id="1"\; label="tan\; \beta "\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">tan\; </figure-callout>}\left({\mathrm{\beta }}_{}\right)\left({\mathrm{}}_{\mathrm{1}}\mathrm{/}\mathrm{2}\right)\mathrm{,}$$

The validity of (D1) in each node X and the validity of (D2) in each edge 2 of a square-meshed quad mesh 1 are necessary and sufficient for the existence of a dual-isothermal network.

Fig. 14 shows a node region 3 of a dual-isothermal network, formed by carrier elements 4 with a rectangular cross-section. The longitudinal axes of the carrier elements 4 form the same angle with the node axis A (axis of the above-mentioned turntable). In the case of consistently elliptical (locally convex) node X in the geometric model (all edges 2 on the same half-cone) can then apply the glass directly on the edges of the support elements 4, since these edges themselves define planar square mesh 1.

If one wants to achieve a fixed distance between corresponding nodes X and X _o of base and offset, the nodes of the parallel network p ( N ) must lie on a sphere S. Since each square mesh 1 of p ( N ) lies in a plane E , the corners of the mesh 1 lie on the intersection circle of E and S. Each square mesh 1 of p ( N ) is thus a circular quadrangle, ie it has a perimeter. By shifting the sides of a circular quadrangle in parallel, the sides of a circular quadrangle are created again. Since the sides of each mesh 1 of N are parallel to the corresponding sides of the corresponding mesh 1 of p ( N ), each square in N must also be a quadrilateral, ie, have a perimeter. Therefore, N is also called a circular network . Since p ( N ) approximates a sphere, the circular nets are also considered as approximations of curvature-line nets. Your offsets N _o = N ⊕ d · p ( N ) at distance d have nodes X which are at a distance d from the corresponding nodes of N.

For calculating circular networks, the optimization method is as described above. One has only to replace the planarity condition α ₁ + α ₂ + α ₃ + α ₄ = 2π per mesh 1 by the two conditions α ₁ + α ₃ = π and α ₂ + α ₄ = π, which characterize a convex circular quadrilateral.

There are also nets whose offsets have both constant area spacing and constant corner spacing. They are parallel to nets p ( N ) whose surfaces touch a ball S ₁ with center Z , and whose corners are on a concentric ball S ₂ . This is the case if and only if the square meshes have 1 of p ( N ) orbits of fixed radius. One constructs such a circular pattern from a spherical rhombus net. In the Fig. 20b is such an offset pair N , N _{o shown} with constant corner distance and constant area spacing, which consists of the support structure of Fig. 20a was won. The Fig. 20c shows the associated flat surface carrying system in diagram form.

The statements about parallel-related meshes, their offsets and support structures apply to any meshes with flat meshes 1, even if the meshes are N- corners. However, they do become trivial for triangular meshes: two triangles with parallel sides are similar, and therefore parallel mesh triangular meshes are always similar. By contrast, the invention also the case of the hexagonal networks (hexagonal networks) of interest. As with a regular paving with hexagons (honeycomb), in a regular corner of such a mesh, three meshes 1 (and three edges 2) collide ( Fig. 19a ). Because only convex hexagons are generally of interest for practical application, only surfaces with positive Gaussian curvature can be approximated by hexagonal nets with flat meshes 1.

Of particular interest again are those networks where the offsets have special properties. A node X is always conical (because three planes always touch a turning cone) and therefore you always have offsets in constant area spacing. If one wants to achieve constant edge distance, then there must be a parallel-related hexagonal network p ( N ) whose edges 2 touch a sphere. Such a network can be constructed using known algorithms for Koebe polyhedra. The associated support structure of N again has the property that the trapezoids that occur have a constant height. In a supporting structure 8 according to the invention, therefore, carrier elements 4 with a constant transverse profile can be used. The longitudinal axes of the carrier elements 4 open at equal angles in the node axes A and the surface elements 5 can be mounted directly on the inner edges of the support structure 8.

The result is in each case a quadrilateral mesh with flat meshes 1, which approximates a free-form surface F. In the Fig. 9 Such a network is shown schematically, which should be about a conical network. In the following Fig. 10-12 will now be shown by way of example, how the course of the support elements 4 can be determined by such a network of a geometric model.

The starting point for the construction of the carrier elements 4 is a quadrilateral mesh N with flat meshes 1 (FIG. Fig. 9 ). In each node X of the network, there is a common line (node axis A) on which the corresponding nodes X of the offset networks lie ( Fig. 10 ). For a conical net N , the node axis A is the axis of the turntable which is contacted by the adjacent mesh planes. In a dual-isothermal network N , the node axis A is the axis of the turntable that contains the adjacent edges.

The quadrilateral network N and an associated offset network N _o determine a geometric support structure which is built up by flat quadrangles ( Fig. 11 ). Each of these squares is a trapezoid bounded by an edge XY of N , the corresponding parallel edge X _o Y _o of N _o and the links XX _o and YY _o corresponding node X of N and N _o . For the construction of the support structure, the support elements 4 are to be arranged along the geometric support structure. In order to create a construction space in which the support elements 4 are located, one can simply extrude the support trapezes by a desired width normal to their planes towards both sides, resulting in cuboid installation spaces for the support elements 4 with an oblique waste in the node region 3 ( Fig. 12 ). Depending on the static design of the support system can pass through a carrier layer, the second carrier layer is pushed against the first carrier layer. (see also Fig. 14 ).

The Fig. 13 shows a schematic, two-dimensional representation of the rectangular cross-sectional shape of the installation space. The Fig. 14 shows a corresponding three-dimensional representation of the installation space in the region of the node region 3, and the Fig. 15a-d possible architectural applications of a supporting structure 8 in steel, wood and concrete (scale-free). The space must not have a rectangular cross-sectional shape, but must not exceed the maximum possible space.

When creating the maximum dimensions for the cross sections of the support elements 4, the extreme positions of the angles between the surfaces of the base network and the planes of the support structure are to be considered (see Fig. 16 ).

In a dual-isothermal network, the heights of the support trapeze are constant and thus the distances of the line bearings to the support element 4. Another advantage of dual-isothermal networks is the following: One can the panels of the cover, so the surface elements 5, or other layers also Mount on the inner sides of the support elements 4, since these also form flat meshes 1 (see also Fig. 18 ). However, this only applies to convex nodes X. Fig. 17 shows a convex node region 3 of a dual-isothermal network, formed by carrier elements 4 with a rectangular cross-section. The longitudinal axes L of the support elements 4 form the same angle with the node axis A (axis of the above-mentioned turntable) (see also FIG Fig. 14 ).

Using the invention, it is thus possible to find a structural implementation of free-form surfaces using a support structure 8, which reduces the technical and economic requirements. In particular, assembly costs and costs can be kept as low as possible. Furthermore, it is possible for the support structure 8 to approximate free-form surfaces to also offer the possibility of a multilayer structure, that is to say the parallel offset mounting of a plurality of surface elements 5 on the carrier elements 4 of a single support structure 8.

Claims (9)

1. A supporting structure (8) for curved envelope geometries in buildings, consisting of support elements (4) which are each combined to form N-gons (N=3,4, ...) which span the envelope geometry and which each enclose a planar surface element (5), and the support elements (4) of adjacent N-gons each form a common node region (3) in which the support elements (4) abut, and the plane of a surface element (5) and the respective planes of the surface elements (5) of two N-gons which adjoin in non-parallel spatial direction lie in different planes in at least one section of the envelope geometry, characterized in that the support elements (4) form in this section 4-gons or 6-gons each, and the support elements (4) each have a longitudinal axis (L) which extends in a straight line between two node regions (3) each and runs parallel to the imaginary line of intersection of the surface element planes associated with the same, with the cross section of the support elements (4) normal to their longitudinal axis in each case having a relative twist angle of 0° along the entire longitudinal axis (L) of the support element (4).
2. A supporting structure according to claim 1, characterized in that the angular sum of respectively opposing angles is equal in the point of intersection of the imaginary lines of intersection of four surface element planes adjoining in a node region (3).
3. A supporting structure according to claim 1, characterized in that the angular sum of respectively opposite angles between the surface normals of two adjoining surface element planes of four surface element planes adjoining in a node region (3) is equal.
4. A supporting structure according to one of the claims 1 to 3, characterized in that the support elements (4) have a rectangular cross-sectional shape or can be inscribed into a rectangular cross-sectional shape.
5. A supporting structure according to one of the claims 1 to 4, characterized in that at least two surface elements (5) are held on the support elements (4).
6. A method for determining a supporting structure (8) for curved envelope geometries in buildings, consisting of support elements (4) in which a predetermined curved envelope geometry is approximated by a continuous network of N-gons (N=3, 4,...) which each define a planar mesh (1), with respectively adjoining N-gons having a common node (X), and the mesh plane of an N-gon and the respective mesh planes of two N-gons adjoining in non-parallel spatial directions lie in different planes in at least one section of the structural shape, characterized in that the approximation of the predetermined curved envelope geometry occurs with the help of a first continuous network of 4-gons or 6-gons which can be transferred to a further continuous network of 4-gons or 6-gons by parallel displacement in a direction normal to the mesh plane of the respective 4-gon or 6-gon, with two respective adjoining N-gons having a common boundary line (2) which determines the progression of the longitudinal axis of one support element (4) associated with said N-gons, and the dimensions of a support element (4) perpendicular to said boundary line (2) being determined by the distance (d) of the respective boundary line (2) of the first network to that of the further, parallel displaced network.
7. A method for determining a supporting structure (8) according to claim 6, characterized in that the angular sum of respectively opposite angles between the boundary lines (2) of four adjoining 4-gons is equal in their common node (X).
8. A method for determining a supporting structure (8) according to claim 6, characterized in that the angular sum of respectively opposite angles between the surface normals of two adjoining mesh planes of four surface element planes adjoining in a node (X) is equal.
9. A method for determining a supporting structure (8) according to one of the claims 6 to 8, characterized in that in a section of the supporting structure (8) at least one second continuous network of 4-gons or 6-gons is determined which each define a planar mesh plane, with the 4-gons or 6-gons of the second network being formed by parallel displacement of the 4-gons or 6-gons of the first network in a direction normal to the mesh plane of the respective 4-gon or 6-gon.

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