Abstract. Several theoretical and practical geometry applications are based on polygon meshes with planar faces. The planar panelization of freeform surfaces is a prominent example from the field of architectural geometry. One approach to obtain a certain kind of such meshes is by intersection of suitably distributed tangent planes. Unfortunately, this simple tangent plane intersection (TPI) idea has a number of limitations. It is restricted to the generation of hexagon-dominant meshes: as vertices are in general defined by three intersecting planes, the resulting meshes are basically duals of triangle meshes. Furthermore, the explicit computation of intersection points requires dedicated handling of special cases and degenerate constellations to achieve robustness on freeform surfaces. Another limitation is the small number of degrees of freedom for incorporating design parameters.Using a variational re-formulation, we equip the concept of TPI with additional degrees of freedom and present a robust, unified approach for creating polygonal structures with planar faces that is readily able to integrate various objectives and constraints needed in different applications scenarios. We exemplarily demonstrate the abilities of our approach on three common problems in geometry processing.

Blechumformung ist traditionell ein hochaufwendiger Prozess, der teurer Werkzeuge und Formen wie Gesenke oder Patrizen und zugehöriger Stempel bedarf, die erst bei hohen produzierten Stückzahlen rentabel werden. Aus diesem Grund beschränkt sich im Bauwesen der Einsatz umgeformter Stahlprodukte bisher auf die Verwendung von Halbzeugen wie z. B. Walzprofilen. Die inkrementelle Blechumformung (IBU), ein neuartiges Umformverfahren, arbeitet auf ganz andere Weise. Statt eines einstufigen Pressvorgangs, bei dem das Blech in Form gebracht wird, fährt ein Umformwerkzeug die Bauteilkontur in einem sequenziell ablaufenden Prozess mit lokaler Verformung ab, der sich mit einfachen Mitteln an die individuell herzustellenden Blechformen anpassen lässt. Die IBU stellt insofern einen hochflexiblen Prozess dar, der es möglich macht, Serien von geometrisch unterschiedlichen Bauteilen bei hoher Effizienz zu fertigen. Sie verspricht ein ideales produktionstechnisches Mittel für die Umsetzung des Leichtbauprinzips von Raumfaltwerken und mehrlagigen Faltungen zu sein, auf dessen Basis selbsttragende Hüll-und Fassadenkonstruktionen aus Feinblech für die Architektur und den Ingenieurbau gebaut werden können. Die tatsächlichen Eigenschaften und Bedingungen der konstruktiven und praktischen Umsetzung einer mit Hilfe der IBU hergestellten, frei geformten zweilagigen Hüllfläche wurde im Rahmen eines Forschungsprojekts zwischen dem Lehrstuhl für Tragkonstruktionen (TRAKO) und dem Institut für Bildsame Formgebung (IBF) an der RWTH Aachen untersucht.Free-formed, self-supporting folding structure made of metal sheets using the incremental sheet forming (ISF). Sheet forming is traditionally a costly process that requires expensive tools as well as forms such as winds or dies with customized punches, which are only profitable when producing high quantities. However, the incremental sheet forming (ISF), a novel forming technique, works highly differential. In contrast to the common single-stage jacking process, the incremental metal forming is a sequential process in which a forming instrument circumscribes the part contour and warps the sheet repeatedly. This process can easily be adjusted to a variety of individual thin sheet forms. In view of this fact, the ISF is a highly flexible process that allows to craft series of geometrically different components efficiently. ISF appears to be an ideal production-related instrument to realize the principle of lightweight constructions in space folding structures and multilayer folding structures that form the basis of self-supporting building envelopes or facades made of thin sheet. The actual characteristics and conditions of the constructive realization of a free-shaped, two-layered envelope formed with the incremental metal forming procedure has been analyzed during a research project

Unter dem Begriff “Starre Faltungen” sind neben Faltwerken und ähnlichen Konstruktionen des Bauwesens vor allem Anwendungen des Strukturformprinzips “Faltung” zu verstehen, wie sie z. B. Geodätische Kuppeln darstellen. Ihre facettenartigen dreieckigen oder rautenförmigen Blechelemente bilden einerseits die geometrischen Grundelemente eines regelmäßig geformten Körpers, nämlich der Kugel, andererseits die Bausteine eines Leichtbausystems, mit dem selbsttragende dreidimensional gekrümmte Raumflächen erzeugt werden. Auf Basis moderner Computermethoden lassen sich auch unregelmäßige, frei geformte Flächen‐ und Hüllformen unabhängig von mathematisch‐funktionalen Beschreibungen generieren, denen verschiedenartige Faltmuster — abhängig von der Gestaltung, der Fassadentechnik und dem statischem Verhalten — eingeprägt werden können. Diese Faltungen lassen sich geometrisch bis hin zum einzelnen Bauteil beschreiben, ausdetaillieren und in unterschiedlichen Materialien durchkonstruieren. Starre Faltungen repräsentieren so ein Leichtbauprinzip, das auf freie Formen angewendet, neue, noch unerschlossene bautechnische Dimensionen eröffnet. Rigid Folded Plate Construction as a Principle for Lightweight Buildings. The term “Rigid Folded Plate Construction” refers to conventional folded plate structures made of concrete as well as to applications of the morphological principle of folding as they are represented by thin‐walled‐structures like Geodesic Domes. Their facetted triangular elements on one hand form the geometrical basic elements of a regularly, mathematically defined body and on the other hand the structural components of a lightweight construction system. Based on modern computer methods irregular free‐form‐surfaces can also be generated independently from mathematical‐functional descriptions. These surfaces may be provided with different folding patterns, depending on the design or the requirements due to the façade engineering or static behavior. Those folded structures can be geometrically described down to the individual element and detailed out in different materials. This way rigid folded plate construction represents a principle of lightweight construction for free form facades and structures opening new and still unexplored structural dimensions.

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In the architecture and construction sector the trend for individualization is often expressed in complex-shaped freeform buildings. Due to missing universal and mature construction methods for freeform buildings, they are usually realized with customized solutions that often include massive, material-consuming substructures, while the visible skin has neither structural nor functional properties. In this context a new concept for self-supporting lightweight structures for the realization of free-form surfaces and the production of the corresponding components has recently been proposed. Taking into account the large part dimensions and the varying part geometries in this application a flexible production chain based on incremental sheet forming has been developed and optimized. It has been validated by producing six-sided large-scale pyramids in 140 similar variants which were assembled to a self-supporting free-form demonstrator. Two-point incremental sheet forming (TPIF) was used with a universal partial supporting tool with the goal to produce all variants without dedicated tooling. Although the majority of pyramids was produced successfully with the applied TPIF strategy, there was a small number of parts with a very asymmetric shape that showed severe buckling in the side walls. For a detailed analysis of this observation the asymmetry was quantified using a wall angle ratio. Subsequently, a single-point incremental sheet forming (SPIF) strategy was successfully applied as an approach to avoid buckling. The results confirm the assumption that the circumferential expansion in SPIF suppresses buckling due to tensile stresses in the side walls, whereas the circumferential compression in TPIF triggers buckling due to the compressive stresses in the side walls.