MODELLING OF VIRTUAL COMPRESSED STRUCTURES THROUGH PHYSICAL SIMULATION P.Brivio1, G.Femia1, M.Macchi1, M.Lo Prete2, M.Tarini1;3 1Dipartimento di Informatica e Comunicazione, Universita` degli Studi dell’Insubria, Varese, Italy - [email protected] 2DiAP - Dipartimento di Architettura e Pianificazione, Politecnico Di Milano, Milano, Italy 3Visual Computing Group, Istituto Scienza e Tecnologie dell’Informazione, C.N.R., Pisa, Italy KEY WORDS: (according to ACM CCS): I.3.5 [Computer Graphics]: Physically based modeling I.3.8 [Computer Graphics]: Applications ABSTRACT This paper presents a simple specific software tool to aid architectural heuristic design of domes, coverings and other types of complex structures. The tool aims to support the architect during the initial phases of the project, when the structure form has yet to be defined, introducing a structural element very early into the morpho-genesis of the building shape (in contrast to traditional design practices, where the structural properties are taken into full consideration only much later in the design process). Specifically, the tool takes a 3D surface as input, representing a first approximation of the intended shape of a dome or a similar architectural structure, and starts by re-tessellating it to meet user’s need, according to a recipe selected in a small number of possibilities, reflecting different common architectural gridshell styles (e.g. with different orientations, con- nectivity values, with or without diagonal elements, etc). Alternatively, the application can import the gridshell structure verbatim, directly as defined by the connectivity of an input 3D mesh. In any case, at this point the 3D model represents the structure with a set of beams connecting junctions. User defined custom constraints (reflecting physical ones) can be imposed over this resulting structure in an quick and intuitive way. Points on the surface can be constrained to never leave a specific position, line or plane. A physical simulation is then run, with the objective of obtaining a surface that, while preserving to a certain degree the initial shape, and fulfilling the constraints, represents a valid compressed structure, i.e. one composed by sub-elements which are only subject to compression, without residual bending forces. It is a well established fact, in architecture, that this property allows for elegant structures which can be realized with lighter and cheaper materials (as testified for example by the pioneering work of Spanish Architect Antoni Gaud`ı, or, more recently, of the Deutsch architect Frei Otto). During or after the simulation, constraints can be made weaker or lifted, and a few simulation parameters can be reedited, in an interactive way: resulting changes in the final 3D shape of the structure are made visible on the screen. The user can thus easily explore through a set of viable solutions all sharing with the aforesaid structural property. The final result is exported in standard 3D digital formats for further processing. In summary the proposed standalone application represents a quick and focussed tool to address a specific task. 1 INTRODUCTION man architect and structural engineer Frei Otto, when he devised the procedure he used to build the Multihalle at Curved and self-portant shapes have an important role in Mannheim. Firstly, he designed a scaled catenary model, architectural design since the studies on catenary systems which was then reproduced in place as a static skeleton of the Spanish architect Antoni Gaud`ı. In order to design of beams and junctions. Once completed, every beam has buildings such as the chapel of the Colonia Guell and the been let move freely around its junctions, so that the whole arches of the Casa Mila, he resorted to catenaries in a pre- structure deformed under its own weight. During this phase, liminary study: the shape of the structures was determined any residual internal force has been compensated and the by building a physical model in scale, composed by a net final shape was then in equilibrium. Finally, the beam po- of strings and additional weights, hanging it with hooks to sitions have been frozen and the whole structure has been a ceiling, and letting it deform. When inverted (flipped covered, again ensuring compressibility of the result. on the vertical direction), the resulting structure acts in pure compression, ideally requiring no additional support Similar results can be clearly obtained through computer to preserve its shape. In this paper, we refer to this char- physical simulations. acteristic as compressibility, and we call such a structure a compressible structure1. Earlier in the design process of a new structure, e.g. when the concepts are being considered, architects tend to focus More recently, this approach was extended by the Ger- more on factors (aesthetic, function, etc) other than struc- 1Compressible structures are also commonly referred to as “funicular” tural ones. It is usually only later in the process that struc- and “compression-only” (or “tension-only” when referring to the model tural properties like compressibility are fully taken into before vertical flipping). account. Any virtual model must be carefully evaluated and validated, before realizing any real-world structure, us- • the parameters of the simulation, ing specialized tools like those based on FEM (Ekkehard, 2004). • the set of constraints imposed by the user. We propose a way to embed structural consideration in the architectural design process as early as the design of the Importantly, any surface produced in output by the sys- very concept of the structure of interest. To achieve this, tem is guaranteed to be compressible (in the sense defined we resort to a quick and direct tool capable of visualiz- in Sec. 1 above). Manipulating any of the factors in the ing a “compressible version” of a preliminary 3D struc- above list through a simple GUI (Sec. 5.2), the user navi- ture taken as input (and to export it for later uses). Basic gates among a set of possible compressible structures. The editing capabilities allow the user to navigate between al- factors can be changed even before the simulation is fully ternative plans, all equally compressible, thus navigating converged, allowing for a more interactive search among through a space of compressible structures. This way, later possible solutions. stages (when structural characteristic must be enforced) Note that, depending on several choices, it is not always the will hopefully impose smaller changes in the final output. case that the resulting compressible structure is close to the During this preliminary process, accuracy is traded for speed source one. For example, the Hausdorff distance between and simplicity: at this stage the model is still conceptual, the two cannot be easily bounded a priori. Howevereven consisting of approximate shapes defined by few elements. surfaces which are far away to the source one can be useful Also, some approximation and heuristic are adopted in or- as viable design alternatives to be considered. der to speed up the process. 1.2 Structure of this paper To compute catenary-based surfaces, we adopt a particle- spring model taking inspiration from the computer graph- First, in the next section, we show the particle-and-spring ics physical simulation techniques. Such systems efficiently explicit solver which we will be using. In Sec. 3.3 we show simulate physical behavior with a sufficient level of accu- how this system can be used to find the shape of a proper racy, as testified by the numerous scopes where it has been catenary curve embedded in a 2D plane. Then (Sec. 4) successfully adopted: hair, human muscle, aquatic plants, we extend the same concept to surfaces embedded in 3D and many other simulation (Nealen et al., 2006, Gibson space. Sec. 5 describes the interactive software tool that is and Mirtich, 1997). We show that these systems can be used be make this approach adoptable by the final user in adapted to simulate closely the behavior of catenary curves the early stage of dome-like structure design. In Sec. 6 we and surfaces. show a small gallery of results which were obtained with our tool and end with a brief discussion. This way, we take advantage of a mature technology (de- veloped for computer-games). Off-the-shelf physical sim- 2 PARTICLE AND SPRINGS SYSTEMS ulator specialized hardware, like AGEIA PhysX accelera- tor, could be used for the task. With respect to the typical computer-game scenario, our task is simplified, because Particle-and-spring systems are an intuitive way for repre- we are only interested in the final state of the simulation senting a deformable model physics (Nealen et al., 2006). rather than in its step-by-step evolution (this allows for fur- For more than two decades they have been widely exploited ther optimization, see section 2). in Computer Graphics. Due to their simplicity and adap- tivity to many different contests, they have been especially We implemented a tool (as a stand-alone software) based exploited for achieving realistic animations of human bod- on this approach to aid the early stages if designing archi- ies, human skin, soft organic tissues, clothes and garments, tectural domes and covering. etc. 1.1 Solution approach A particle-and-spring system is composed by a net of par- ticles connected by springs that evolve in time. Springs are Technically, our goal is to produce a compressible 3D struc- linear weightless elements, elastic along their axial direc- ture which is similar to a given arbitrary one. The idea is to tion, which connect particles. Each spring is characterized deform the source structure into a new one which resem- at least by a rest length and a stiffness coefficient, which bles it but acts in pure compression. determine an axial reaction force whenever the spring is stressed or stretched, as stated by Hooke’s law.