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Supporting Structure for Curved Envelope Geometries - EVOLUTE GMBH
Supporting Structure for Curved Envelope Geometries
United States Patent Application 20110179723
A support structure for curved envelope geometries, such as buildings and shipbuilding, that at least sectionally approximates a freeform surface, includes longitudinal connection elements and surface elements spanned by the connection elements. The surface elements are implemented as single-curved strip elements whose curvature runs in the longitudinal direction of the strip elements. Adjacent strip elements are connected to one another along their longitudinal edges via the longitudinal connection elements.
Jacques, Raynaud (Paris, FR)
12/975937
EVOLUTE GMBH (Perchtoldsdorf, AT)
RFR S.A.S. (Paris, FR)
E04B1/32; E04B1/34
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1. A support structure for curved envelope geometries, the curved envelope geometry at least sectionally approximating a freeform surface, comprising: longitudinal connection elements; and surface elements spanned by the connection elements, wherein the surface elements are implemented as single-curved strip elements whose curvature runs in the longitudinal direction of the strip elements, and wherein pairs of strip elements are connected to one another along their longitudinal edges via the longitudinal connection elements.
2. The support structure according to claim 1, wherein the strip elements are implemented such that a family of generatrixes exists in a transverse direction of the strip elements, the generatrixes each enclosing the same angle with the two longitudinal edges of the strip element.
3. The support structure according to claim 2, wherein at least two envelope geometries which are spaced apart from one another are provided, a strip element of a second envelope geometry being formed by parallel displacement of a strip element of a first envelope geometry.
4. The support structure according to claim 3, wherein the longitudinal connection elements are implemented as cuboid, and their transverse extension perpendicular to their longitudinal axis corresponds to a distance of two longitudinal edges lying one above another.
5. The support structure according to claim 1, wherein the strip elements are implemented such that a family of generatrixes exists in a transverse direction of the strip element, wherein one generatrix of two adjacent strip elements that actually intersect or intersect in their imaginary extension, encloses the same angle with tangents on an actual or imaginary shared longitudinal edge in their point of intersection.
6. The support structure according to claim 5, wherein at least two envelope geometries which are spaced apart from one another are provided, a strip element of a second envelope geometry being formed by parallel displacement of a strip element of a first envelope geometry.
7. The support structure according to claim 5, wherein the longitudinal connection elements are implemented as cuboid, and their transverse extension perpendicular to their longitudinal axis corresponds to a distance of two longitudinal edges lying one above another.
8. A method for establishing a support structure for curved envelope geometries comprising longitudinal connection elements and surface elements spanned by the connection elements, the curved envelope geometry at least sectionally approximating a freeform surface, the method comprising: implementing the surface elements as single-curved strip elements that adjoin one another along their respective longitudinal edges; and implementing the longitudinal connection elements running along shared longitudinal edges of two adjoining strip elements so that the longitudinal connection elements follow a course of the respective shared longitudinal edge.
9. The method according to claim 8, wherein the strip elements are implemented such that a family of generatrixes exists in the transverse direction of the strip element, the generatrixes enclosing the same angle with two longitudinal edges of the strip element.
10. The method according to claim 9, wherein at least two envelope geometries, which are spaced apart from one another, are established, a strip element of a second envelope geometry being formed by parallel displacement of a strip element of a first envelope geometry.
11. The method according to claim 9, wherein the longitudinal elements are implemented as cuboid, their height corresponding to a distance of two longitudinal edges lying one above another.
12. The method according to claim 8, wherein the strip elements are implemented such that a family of generatrixes exists in a transverse direction of the strip elements, wherein one generatrix of two adjacent strip elements that actually intersect or intersect in their imaginary extension, encloses the same angle with tangents on an actual or imaginary shared longitudinal edge in their point of intersection.
13. The method according to claim 12, wherein at least two envelope geometries, which are spaced apart from one another, are established, a strip element of a second envelope geometry being formed by parallel displacement of a strip element of a first envelope geometry.
14. The method according to claim 12, wherein the longitudinal elements are implemented as cuboid, their height corresponding to a distance of two longitudinal edges lying one above another.
This application is a continuation of International Application No. PCT/EP2009/057805, filed on Jun. 23, 2009, entitled “Support Structure for Curved Envelope Geometries,” which claims priority under 35 U.S.C. §119(a)-(d) to Application No. AT 1007/2008 filed on Jun. 24, 2008, entitled “Support Structure for Curved Envelope Geometries,” the entire contents of which are hereby incorporated by reference.
The invention relates to a support structure for curved envelope geometries, in particular in buildings and shipbuilding, the curved envelope geometry at least sectionally approximating a freeform surface, comprising connection elements and surface elements spanned by the connection elements and to methods for fixing such a support structure.
Curved envelope geometries of this type are used in construction or in shipbuilding to implement freeform surfaces, in which the curvature is different in two different spatial directions, for example, in buildings with domes, or also more complex surface shapes. Freeform surfaces of this type are also referred to as non-developable surfaces, or as double-curved or triple-curved surfaces, and are initially drafted as continuous surfaces in the computer model in the course of the architectonic drafting. In the construction implementation, the continuous freeform surfaces are to be approximated by a plurality of individual surface elements, which are held in a support structure. Thus, for example, it is also possible to implement complex freeform surfaces having multilayered level glass elements, for example, which are fastened above, between, or below a support structure made of steel, for example.
For the embodiment of the surface elements, there are two fundamental possibilities. On the one hand, the attempt can be made to implement the individual surface elements as planar, i.e., as level surface elements. In this case, the support structure is formed from individual connection elements, which are each assembled into polygons, for example, triangles, squares, hexagons, etc. The polygons span the support structure, the connection elements being implemented as girders which meet in node areas, where they are fastened to one another. A particularly advantageous embodiment of such a support structure is described in Austrian Patent Number 503,021, in which a torsion-free implementation of the individual girders is made possible in particular. This embodiment has the disadvantage, however, that sometimes very many connection elements are required, which increase the costs, on the one hand, and also have aesthetic disadvantages, in particular in buildings, on the other hand, since the visual impression of the “lightness” of the envelope geometry is lost due to the plurality of connection elements.
Another possibility for approximating freeform surfaces by individual surface elements further comprises approximating the freeform surface by curved surface elements. In this case, methods of approximating a freeform surface with the aid of double-curved surface elements are known, which have substantial disadvantages in practice, however, so that the material selection for the surface elements is subject to restrictions in this case because of the required deformability in double-curved surfaces, for example. Furthermore, it is typically not possible to find a distribution of the connection elements with the aid of double-curved surface elements in which the connection elements implemented as girders do not have to be subjected to torsion in the geometrical meaning in the course of the installation between two node areas, i.e., a twisting of the longitudinal axis in the node area, for example. Only connection elements having circular cross-section may be arrayed on one another “torsion-free,” in the geometrical meaning. If noncircular cross-sections are used, up to this point a torsion (in the geometrical meaning) has arisen in the node area up to this point. This results in aesthetically and statically unsatisfactory node areas. Rather, the problem also results therefrom that multilayer structures cannot be implemented or can only be implemented with substantial additional outlay. Therefore, a separate support system must be provided for each layer, which in turn multiplies the material costs and the installation effort.
It is therefore the object of the invention to find a structural implementation of freeform surfaces which reduces the technical and economic requirements and satisfies aesthetic demands. In particular, installation effort and costs are to be kept as low as possible. A further object of the invention is that the support structure for the approximation of freeform surfaces also offers the possibility of a problem-free multilayer structure, i.e., the installation of multiple surface elements offset in parallel. Furthermore, the number of connection elements is to be reduced in comparison to known support structures based on planar surface elements in triangular, square, or hexagonal form.
Described herein is a support structure for curved envelope geometries, in particular in buildings and shipbuilding, the curved envelope geometry at least sectionally approximating a freeform surface, comprising connection elements and surface elements spanned by the connection elements. According to the invention, surface elements are provided in this case which are implemented as single-curved strip elements, whose curvature runs in each case in the longitudinal direction of the strip elements, each two strip elements being connected to one another along their longitudinal edges via longitudinal connection elements. A buckle results in the area of the longitudinal connection elements between each two adjoining strip elements, i.e., in the mathematical meaning, a discrete transition which is kept as small as possible during the design of the strip elements, in order to have the best possible approximation of the envelope geometry of the freeform surface, and thus provide the impression of a continuous envelope geometry. An uninterrupted array of strip elements is thus implemented, which approximate the freeform surface in their entirety. The way in which suitable strip elements are ascertained will be described in greater detail hereafter. The curvature of the strip elements can be described by lines of curvature, which therefore run in the longitudinal direction of the strip elements. Because of the single-curved embodiment of the strip elements, generatrixes running transversal to the lines of curvature further exist, which are linear. Transverse connection elements may optionally be additionally provided transversely to the strip elements, for example, along generatrixes of the strip elements, the strip element also being able to be implemented as interrupted in the area of the transverse connection elements, so that a strip element is divided into individual panels.
Such an approximation of freeform surfaces with the aid of single-curved surface elements which are uninterruptedly arrayed on one another is also referred to hereafter as the “strip model.” In the context of the mathematical modeling of such strip models, the curves along which the strip elements are arrayed on one another are also referred to as “edge curves.” These edge curves correspond to the longitudinal edges of the strip elements in their structural implementation, along which the longitudinal connection elements are situated according to the invention.
In the scope of the invention, it has surprisingly been established that such strip models may also be found for complicated freeform surfaces, and allow an implementation of a support structure according to the invention. In the scope of the present invention, the term “freeform surface” is therefore to be understood in particular as a surface which meets the following conditions. It is
a double-curved surface, and
it has no kinematic generation by movement (possibly including scaling) of a curve in such a way that discrete layers of the moved curve may be used as edge curves of a strip model.
As an example of surfaces which are therefore not freeform surfaces according to the invention, surfaces of revolution may be mentioned. A further example are the sliding surfaces, generated by displacement of a curve along another curve.
According to an advantageous embodiment of the invention, it can further be provided that the strip elements are implemented so that a family of generatrixes exist in the transverse direction of the strip element, the generatrixes enclosing the same angle in each case with the two longitudinal edges of the strip element. The design of the strip elements is performed via mathematical optimization methods, which are also referred to hereafter as the “circular model,” because a circle, which touches the two longitudinal edges in the endpoints of the generatrix, results upon fulfillment of this condition in the tangential plane of the generatrix. This will be described in greater detail hereafter.
Alternatively or additionally, it can also be provided that the strip elements are implemented so that a family of generatrixes exist in the transverse direction of the strip element, in each case a generatrix of two adjacent strip elements, which actually intersect or intersect in their imaginary extension, enclosing the same angle with the tangent at the actual or imaginary shared longitudinal edge in their point of intersection. The design of the strip elements is performed via mathematical optimization methods, which are also referred to hereafter as the “conical model,” because a right cone results upon fulfillment of this condition in the tangential plane of the generatrix, whose tip lies in the point of intersection of the two generatrixes, and the two adjacent strip elements touch along the intersecting generatrixes. This will also be discussed in greater detail hereafter. The simultaneous fulfillment of the two mentioned conditions is also conceivable, in that they may be required as “soft” secondary conditions, mathematical optimization methods of this type also being referred to hereafter as “approximative curved strip models.”
According to an advantageous refinement of the invention, at least two envelope geometries which are spaced apart from one another may be provided, a strip element of a second envelope geometry being formed by parallel displacement of a strip element of a first envelope geometry. Specifically, if one of the two or also both of the above-mentioned conditions are fulfilled, an uninterrupted network of strip elements according to the invention can be displaced in parallel to form a further uninterrupted network of strip elements. This parallel displaceability is also referred to hereafter as “offset.” Corresponding generatrixes of two parallel displaced strip elements, and corresponding longitudinal edge tangents, are parallel.
With respect to the embodiment of the longitudinal connection elements, it can be provided that the longitudinal connection elements are implemented as cuboid, their height corresponding to the distance of two longitudinal edges lying one above another. Specifically, corresponding longitudinal edges of two strip elements according to the invention which are displaced in parallel may be connected by single-curved surfaces, on the basis of which corresponding longitudinal connection elements may be easily manufactured. Transverse connection elements may also be implemented as cuboid between generatrixes, which are assigned to one another, of two strip elements displaced in parallel, the height of the connection elements corresponding to the distance of the respective generatrixes. These transverse connection elements are not curved, and the occurring nodes between longitudinal girder elements and transverse connection elements are torsion-free. This allows an embodiment of the support structure according to the invention which is simpler overall.
Furthermore, the invention relates to a method for establishing a support structure for curved envelope geometries, in particular in buildings and shipbuilding, comprising connection elements and surface elements spanned by the connection elements, the curved envelope geometry being at least sectionally approximated by a freeform surface. In this case, it is provided according to the invention that the surface elements are implemented as single-curved strip elements adjoining one another along their respective longitudinal edges, and the longitudinal connection elements running along the shared longitudinal edges of two strip elements adjoining one another are implemented so that they follow the course of the respective shared longitudinal edge. The establishment of the strip elements can concretely be performed, for example, in that sequences of lines of curvature of the freeform surface are selected, which may each be connected by single-curved strip elements, whose curvature runs in the longitudinal direction of the strip elements in each case, and, for the best possible approximation to the freeform surface, the angles between the generatrixes, which intersect one another in their shared longitudinal edge, of a family of generatrixes in the transverse direction of two strip elements adjoining one another are minimized. This will be described in greater detail hereafter.
According to an advantageous refinement of the method according to the invention, it can be provided that the strip elements are implemented so that a family of generatrixes exists in the transverse direction of the strip element, the generatrixes each enclosing the same angle with the two longitudinal edges of the strip element.
Alternatively or additionally thereto, however, it can also be provided that the strip elements are implemented so that a family of generatrixes exists in the transverse direction of the strip element, one generatrix of two adjacent strip elements, which actually intersect or intersect in their imaginary extension, enclosing the same angle in each case with the tangent on the actual or imaginary shared longitudinal edge in their point of intersection.
If one of these two, or also both conditions are fulfilled, the invention according to the method can also be refined so that at least two envelope geometries which are spaced apart from one another are established, a strip element of a second envelope geometry being formed by parallel displacement of a strip element of a first envelope geometry.
Finally, it can also be provided in the scope of the method according to the invention that the longitudinal connection elements are implemented as cuboid, their height corresponding to the distance of two longitudinal edges lying one above another.
The invention is explained in greater detail hereafter on the basis of embodiments with the aid of the appended drawings. In the figures:
FIG. 1 shows a detail of a strip model to explain fundamental concepts,
FIG. 2a shows the angle conditions and normals along the edge curves in a circular model,
FIG. 2b shows the angle conditions and normals along the edge curves in a conical model,
FIG. 3 shows a detail of a strip model having offset,
FIGS. 4a-4c show figures to explain a geodetic model,
FIGS. 5a-5c show figures to explain a cylindrical model for various generatrix directions, the generatrixes only being visualized on every second strip for better visibility,
FIG. 6 shows a layer structure made of partially curved, cuboid elements,
FIG. 7 shows a multilayer structure, which implements the main support structure via a square network having level planar elements, and places strip models thereon, optionally on both sides,
FIG. 8 shows a connection of two strip models, the discrete direction and the continuous direction being selected differently in the two layers,
FIG. 9 shows a perspective view of a detail of a support structure having connection elements, which are implemented as I-beams,
FIG. 10 shows a detail view of the implementation of I-beams in the scope of a support structure according to the invention,
FIG. 11 shows a view of a girder element along an edge curve of a geodetic model,
FIG. 12 shows a girder element having a core part made of multiple rails,
FIG. 13 shows a view of a strip model for the use of level panels in weakly curved areas,
FIG. 14 shows a view of a strip model for the use of double-curved panels in strongly curved areas, and
FIG. 15 shows an exemplary view of a freeform surface according to the invention.
The implementation of a freeform surface, as is shown in FIG. 15, for example, in a support structure according to the invention of a building, for example, requires two fundamental steps. Firstly, the double-curved freeform surface specified by the planner is to be converted according to the invention into single-curved strip elements S, which each adjoin one another continuously along a shared edge curve K. These single-curved strip elements S are also referred to as “developable,” because they may be imaged without distortion on a plane. In a second step, the result of this conversion, which is also referred to hereafter as the “strip model,” or also “curved strip model” of the freeform surface, is finally to be converted into a support structure which can be structurally implemented, the calculated strip elements S occurring as surface elements of the support structure, and the edge curves K as longitudinal edges L of these strip elements S. Finally, the connection elements 1, 2 are to be added, longitudinal connection elements 1 running along the longitudinal edges L of the strip elements S, and optionally transverse connection elements 2 running transversely to the longitudinal connection elements 1. The two steps will be explained in greater detail hereafter, the developable strip model established according to the invention first being discussed.
Developable strip models may be understood as semi-discrete surface representations in the following meaning: a family of parameter lines, the edge curves K, occurs discretely, i.e., in a finite number; these curves are smooth. The polygons formed by the generatrixes E of the strip elements S may be understood as the second family of parameter curves, as is obvious on the basis of FIG. 1. This second family is dense, but the curves are “discrete,” i.e., polygons.
From a theoretical viewpoint, these semi-discrete representations lie precisely between the discrete representations and the smooth surfaces. Concretely, a square network having level meshes, as is described in Austrian Patent Number 503,021, is a discrete version of a strip model. On the continuous side, a so-called conjugated curve network results on a smooth surface. This state of affairs can be used for the actual calculation in that, depending on the stated object, the optimization can be initialized via the discrete version (square network having level meshes) or the smoothed version (conjugated network). It is also still described hereafter that a square network having level meshes and a strip model may be embedded in a single architectonic structure.
In the calculation of an approximation of a given freeform surface by a strip model, it is to be ensured in particular that the edge curves K avoid the asymptotic directions of the freeform surface, because these directions are self-conjugated. Transversal generatrix polygons are thus obtained, and therefore practically usable strip models.
Methods and algorithms for the calculation of strip models are described in greater detail hereafter. Firstly, several practically important classes of strip models will be discussed.
Curved strip models are the semi-discrete counterpart to the network made of lines of curvature k on a smooth surface, or also to the known discrete versions of these networks, for example, the circular or conical square networks, as are described in Austrian Patent Number 503,021. In a curved strip model, the generatrix polygons are approximately perpendicular to the edge curves K. If one has a smooth underlying freeform surface, it can be constructed in that a sequence of lines of curvature k on the freeform surface are selected, and these are then connected by developable strip elements S. One can also start from a discrete variant and make this into a curved strip model by refining and optimization. The following discrete identifications of curved strip models are suitable for their calculation, reference also being made to FIGS. 2a and 2b for this purpose.
Circular model. Each generatrix E of a strip element S meets the two edge curves K at an equal angle here. Therefore, a circle lying in the tangential plane of the generatrix E results, which touches the two edge curves K in the endpoints of the generatrix E.
Conical model. One has the same angle in every point p of an edge curve K between the tangents here on the edge curve K and the two generatrixes E of strips ending there. Therefore, a cone having tip p results, which touches the adjoining strip elements S along generatrixes E.
Approximative curved strip model. For many practical applications, it is sufficient to incorporate the conditions (a) and (b), or a combination thereof, as “soft” secondary conditions using penalty terms in an optimization algorithm. The obtained strip models have similar properties as the models in (a) or (b).
One advantage of curved strip models is the existence of offsets. To generate an offset of a model M, a parallel spherical model MS can first be calculated. This is a strip model which approximates a sphere S (of radius 1) and has a unique correspondence to M, so that corresponding generatrixes E and edge curve tangents of M and MS are parallel. In a circular model M, the edge curves K of MS are on the sphere S. For a conical M, the strips of the parallel model MS of the sphere S are redrawn touching. In the case of an approximative curved strip model, MS approximates the sphere S well.
The offsets may now be easily described analytically. In this case, S is centered in the origin of the employed coordinate system. If p refers to the coordinate vector of a point of an edge curve K of M and pS to the associated point of MS, then pd=p+d pS is the corresponding point of the edge curve K of the offset model Md of M at the distance d. The vector pS is used as the normal vector of M in p. In a conical model, this vector is in the axis of the above-described touching right cone.
The constant offset distance d is measured as follows: in a circular model, it occurs between corresponding points p and pd on edge curves K of the starting model and the offset. In a conical model M, corresponding generatrixes E and tangential planes of the strip elements S of M and Md are at the constant distance d (see also FIG. 3).
The ruled surface strips formed by the connection sections of corresponding points p and pd between corresponding edge curves K (“normal surfaces”) are developable. The construction advantages resulting therefrom are described in greater detail hereafter. This is also true for approximative curved strip models.
Furthermore, it is to be noted that offsets and developable “normal surfaces” (connection surfaces of edge curves K on base and offset) can also be calculated for developable arbitrary strip models by optimization. However, stronger deviations will possibly result in the distances occurring between base and offset, or the normal surfaces will no longer be approximately perpendicular to base and offset.
The geodetic model, as is shown in FIG. 4a, represents a special case of a strip model. A geodetic model is a strip model whose developable strips follow the geodetic lines of an underlying smooth surface. As a consequence, the development of the strip elements S is nearly linear. Such models may also be directly identified as follows: in every point p of an edge curve K, the osculating plane of the edge curve K forms equal angles with the tangential planes of the adjoining strip elements S (see also FIG. 4b in this regard). Therefore, the edge curve K has the same absolute value of the geodetic curvature for the two strip elements S and the edge curve K is imaged on congruent curves in opposite directions in the development of the two strip elements S (see FIG. 4c). Geodetic models are well suitable for the coverage of curved freeform surfaces having long panels, for example, made of wood, which have an almost linear development. It can be necessary to cover a freeform surface using multiple geodetic models of varying direction.
Cylindrical models have strip elements S made of general cylindrical surfaces (see also FIG. 5a). These models can be constructed by merging touching redrawn cylinders of a given freeform surface, as shown in FIGS. 5a-5c. The optimization algorithms described hereafter are also usable for this purpose.
Since cylinders, in particular right cylinders, are producible more easily in certain materials (such as glass) than general developable panels, one will sometimes attempt to employ panels which are as cylindrical as possible. It is not necessary for this purpose to use a cylindrical model and to provide a single cylindrical surface per strip element S. Rather, a general developable strip element S is decomposed into panels and each panel is approximated by a cylinder (in particular a right cylinder). This is well possible in particular if the recession points of the strip element S are far away from the strip element S. The approximation using right cylinders is a standard task of geometrical data processing and can be performed using known algorithms for surface approximation and registration. The latter method is to be used if only a finite fixedly predefined number of possible cylinder radii is to be maintained. This method can be performed perfectly as long as the approximation lies within the manufacturing tolerances (or the tolerances for cold bending of glass).
Conical models have strip elements S made of general conical surfaces. Similar conditions apply here as in the cylindrical models. Conical models can be constructed by merging touching redrawn cones of a given freeform surface.
If one wishes to use right cones as panels for manufacturing reasons, general developable strip elements S can in turn be decomposed into panels and an approximating right cone can be calculated per panel using known methods. It is known from the theory of the developable surfaces (for example, because of the existence of the so-called curved cone), that a segmentation into right cones is well possible.
Finally, models having level edge curves K will be discussed briefly. Upon the implementation of the edge curves K of a strip model in connection elements 1, 2, specific advantages result during the manufacturing of the connection elements 1, 2 if they lie in planes. The planarity of edge curves K can be incorporated into the below-mentioned optimization. However, the possibility also results of connecting a sequence of level sections of a given freeform surface by developable strip elements S. It is to be noted that a strip model represents a semi-discrete version of a conjugated curve network. Therefore, level sections are to be avoided, which touch osculating directions of the freeform surface (these generate turning points in the level sections).
Partially level edge curves K are still simpler to achieve. A very simple and practical solution is to approximate the edge curves K of an arbitrary strip model by so-called arc splines and then optionally incorporate new developable strip elements S which are adapted to the modified edge curves.
Possible optimization algorithms for calculating strip models are described hereafter. The calculation of a strip model is performed with the aid of a numeric optimization algorithm. The ith edge curve K is estimated as a third-degree B spline curve, for example, and having uniform nodes, as
pi(u)=ΣjB3(u−j)bi,j
Sequential edge curves K are connected by linear interpolation of the two curve representations to form a strip element S:
xi(u,v)=(1−v)pi(u)+vpi+1(u)| xi(u,v)=(1−v)pi(u)+vpi+1(u)| xi(u,v)=(1−v)pi(u)+vpi+1(u)| xi(u,v)=(1−v)pi(u)+vpi+1(u)|
The system thus described of strip elements S made of ruled surfaces is subjected to an optimization, which describes the developability, the proximity to a reference surface, and the smoothness of the resulting model. A target function of the following form thus results for the optimization
F=w1fAbw+w2fFlaeche+w3fRand+w4fglatt|
The individual terms are defined as follows:
The developability of the ith strip is achieved by minimization of
fAbw=Σi∫δpi,pi+1(u)2du|
The integrand identifies the squared distance of the diagonals in the polygon
pi,pi+λi{dot over (p)}i,pi+1+μi{dot over (p)}i+1,pi+1|
Dots indicate derivatives according to the curve parameter u therein. These polygons must result as level in the case of a developable surface. In order that the deviation receives a practical significance for the design, the points on the tangents are selected equal to the length of the generatrix distance, i.e., the following condition is set:
λi=∥pi+1−pi∥/∥{dot over (p)}i∥, μi=∥pi+1−pi∥/∥{dot over (p)}i+1∥
The proximity to a reference surface R is controlled by
fFlaeche=Σkdist(xk,Tk)2
The points xk are selected sufficiently densely on the strip model therein and Tk refers to the tangential plane of the reference surface R in the point of the reference surface R closest to xk. Squared distances to tangential planes of the reference surface R are thus minimized. This does not have to be performed precisely in this manner, but converges more rapidly than squared distances to closest points of the reference surface R. To maintain boundaries, tangents tk are used on the given boundaries of the reference surface R instead of tangential planes,
fRand=Σkdist(xk,tk)2
Finally, it is important above all in an application as in architecture, where aesthetics play a large role, to also achieve the best possible smoothness (aesthetics) of the strip model. The term responsible for this in the optimization is a combination of a term which is responsible for the smoothness of the edge curves K,
fglatt/1=Σi∫∥{umlaut over (p)}i(u)∥2du
and a term which positively influences the aesthetics of the generatrix polygons transversal thereto,
fglatt/2=∫(Σi∥pi+1−2pi+pi−1∥2)du
Linearized bending energies, i.e., integrals over the squares of the second derivatives or their numeric approximations, were used here. However, third derivatives may also be used, or, at best, combinations of second and third derivatives.
For special models, such as circular or conical models, a term must also be added, of course, which contains the characterizing property of the respective model. This is noted as an example for the circular model,
fzirk=∑i∫〈pi+1-pi,p.ip.i+p.i+1p.i+1〉2u
A very similar procedure is used in the other cases.
Various known methods of nonlinear optimization may be used for the actual numeric optimization. This is a function of the desired precision at which the individual requirements are to be achieved. In many cases, it is sufficient to follow the penalty approach described here. A suitable control of the weights wi in the course of the iterative method is to be ensured. However, methods of restricted optimization may also be used.
A suitable initialization of the optimization is very important. This can be performed from the discrete side or from the continuous side. Depending on the model type, in the first-mentioned case, the starting version of the strip model is established using a discrete version thereof (square network having level meshes, conical network, etc., as described in Austrian Patent Number 503,021). In the second-mentioned case, a continuous version is used (conjugated network, network of the lines of curvature k, etc.).
With respect to the implementation of a strip model as an architectonic support structure, for example, or in boat building or shipbuilding, there are various possibilities for using strip models. These are a function of the employed materials, costs, existing construction technologies, and also aesthetic and structural considerations. Some possibilities are listed hereafter, mixed forms are conceivable and are not explicitly described.
To implement a first variant of a multilayered structure, for example, a strip model M (at best a curved strip model) is used as a starting point, as well as an offset Md of this model. Corresponding edge curves K of base model and offset may be connected by developable surfaces. A sequence of generatrix polygons may also be selected on the strip model M, and connected to the corresponding generatrix polygons on Md by plane parts. A layered structure made of partially curved “cuboid” elements thus results, only planes and developable surfaces occurring as the lateral surfaces thereof. The occurring nodes are torsion-free. Such a multilayered structure is shown in FIG. 6. Although FIG. 6 only shows two layers, more layers may also be implemented in this way, of course.
In modern constructions, however, various functions (aesthetics, structure, water tightness, insulation, ventilation, etc.) are often separated in various technical elements. It is not necessary to curve all of these elements. Therefore, a second variant of a multilayered structure is proposed, in which the main support structure is implemented via a square network having level meshes, and strip models are placed thereon, optionally on both sides. The natural close relationship between strip models and square networks having level meshes, which is founded in the geometrical state of affairs, makes it obvious to connect these two geometries in a single structure, as shown in FIG. 7. The calculation can be performed starting from a square network P having level meshes, as was described, for example, in Austrian Patent Number 503,021, and an offset Pd of P. A strip model M is constructed from Pd by the above-described optimization approach, the nodes of Pd being interpolated (optionally within a given tolerance). The optimization can also be used so that constant distances result between node points of P and the corresponding contact points of the strip model M. The main support structure can now be implemented by the torsion-free design of linear connection elements 1, 2, which are connected to P.
In a third variant of a multilayered structure, in the above multilayered structure, the network P can also be left out in the final embodiment, the strip models being designed in various directions on both sides of P (see FIG. 8). It is of interest in this case that the linear elements are situated externally and internally in various directions, and a structurally noteworthy, completely novel construction thus results.
The construction of connection elements 1, 2 and panel fixations, which consider the edge curves K and developable surfaces connected thereto, will be discussed in greater detail hereafter. The edge curves K of the strip models occur in this case as longitudinal edges L of the support structure. For the construction of the support structure for a freeform surface based on a strip model, for example, in an architectonic application, the geometric properties of the strip model can be used for the design and manufacturing of the connection elements 1, 2, for example, as girders. This is essentially based on developable surfaces, which are linked to the edge curves of a strip model.
One possibility for implementing the connection elements 1, 2 is in the form of I-beams, for example. An I-beam comprises three parts, namely an upper flange 3 and a lower flange 4, which absorb axial pressure above all, and a core 5, which bears shear load above all (see FIGS. 9 and 10). It is possible to manufacture a curved girder separately from these three parts. It is advantageous if all three parts are developable. First the development of the connection element 1, 2 is cut out of the desired material (e.g., steel), and it is then bent into the final shape. The moment of the three connected parts is higher than that of each individual part, and the separate manufacturing is therefore advantageous.
As is obvious from FIG. 10, the upper flange 3 and the lower flange 4 lie approximately tangentially on offsets of the fundamental strip model, and the core 5 lies normal thereto. The strip elements S have a coating 7, which is to be adapted to the required structural demands. It is possible that in the structural implementation of a support structure according to the invention, the strip elements S do not entirely abut one another, but rather two adjoining strip elements S are held in a receptacle 6 at a small distance from one another. The receptacle 6 also ensures the required water tightness, for example. In FIG. 10, the imaginary extensions S′ of two adjoining strip elements S are also shown by dashed lines, which intersect in the longitudinal edge L. This longitudinal edge L corresponds to the edge curve K of the fundamental strip model.
Curved strip models have an array of advantages for this purpose. Offsets exist, and a developable surface is obtained by connecting corresponding edge curves K of base and offset. This can be used as the core 5 of the connection elements 1, 2. The two flange parts 3, 4 are implemented approximately tangentially on suitable offsets of the strip model. It is advantageous for the bending of the flange parts 3, 4 that the generatrixes E of the fundamental strip model are approximately perpendicular to the edge.
If a connection element 1, 2 follows an edge curve K of a geodetic strip model, the two flange parts 3, 4 have a linear development. However, the core 5 can then only be implemented as developable if it does not run perpendicular to the flanges 3, 4. A connection of the two flange parts 3, 4 by a system of linear connection bars in a crossed configuration can then replace the core 5, as shown in FIG. 11. For a model having level edge curves K, the core 5 can also be selected in this plane in any case, when the angle of intersection between the fundamental freeform surface and the plane is not too flat at any point. This angle also occurs as the angle between the flange parts 3, 4 and the core 5.
Finally, the core 5 of a connection element 1, 2 can be manufactured from multiple rail parts 5a, which slide on one another, and which are first fixed in the final position (FIG. 12). Depending on the strain of the envelope geometry, a H-beam can also be used better than an I-beam. Completely similar considerations apply here. It is advantageous if the three parts of a H-beam do not have to be bent strongly, i.e., result as almost level. In particular in models having level edge curves K, this variant can be of interest.
A further support structure which can be linked well with strip models is given by tubular girders (having circular cross-section). The centerline of the tube is laid out at a constant distance to the model (for example, edge curve K on an offset). It is advantageous here if the centerline is formed by a smoothly arrayed sequence of circular segments, because these tubes are easily available standard parts.
Finally, the possibility is also to be discussed of using level panels 8 in weakly curved areas (see FIG. 13). Questions of cost play an important role in the selection of panel types for a given freeform geometry. It is therefore advisable to provide level panels 8 in those areas in which the developable panel deviates only very slightly from a plane. The close theoretical and algorithmic relationship between models made of developable strips and level square networks makes it very simple to install level panels 8 in the weakly curved areas. A criterion for the selection of a level surface element instead of a very weakly single-curved panel 10 is the buckle angle occurring along the longitudinal edges L of adjoining panels. If this is less than a barrier as a function of the aesthetic demands and the surface behavior of the material (reflection properties), level panels 8 will be provided. All above-mentioned multilayered structures and girder designs may also be accordingly applied in this hybrid case (level and single-curved panels).
In strongly curved areas, in contrast, double-curved panels 9 may also be used (see FIG. 14). The support structure according to the invention does not preclude double-curved panels 9 also being incorporated sectionally, if the curvature is so strong in both main directions in individual areas that the buckle angles occurring along the edge curves K of a strip model become too large. The geometry can fundamentally be taken directly from the underlying freeform geometry. If the manufacturing of the panels prefers specific types of double-curved panels 9, instead of the given freeform geometry, a double-curved panel 9 can also be used, which is of this easily producible type, and approximates the given geometry in the scope of the desired tolerances. Overall, all above-mentioned multilayered structures and girder designs are also to be accordingly applied in the hybrid case (level panels 8, single-curved panels 10, and double-curved panels 9).
It is therefore obvious how a support structure according to the invention can be applied in manifold ways for the structural implementation of freeform surfaces. The implementation of freeform surfaces according to the invention reduces the technical and economic requirements, and nonetheless satisfies aesthetic demands. In particular, installation effort and costs may be kept as small as possible. Furthermore, the possibility also exists of a problem-free multilayered structure, i.e., a parallel offset installation of multiple level surface elements. Furthermore, the number of connection elements 1, 2 is reduced in comparison to known support structures based on planar surface elements in triangular, square, or hexagonal form.
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