Dimitry Demin has studied Architecture at the Academy of Architecture and Arts in Rostov on Don, Russia. In Germany he studied at the Bauhaus Universität Weimar and at the ASl Universität Kassel. He recieved a Bachelor and Diploma Degree. In the Past he worked for SEVKAVNIPIAGROPROM, Rostov on Don in Russia and Bollinger-Grohmann, Frankfurt am Main, Germany. Currently he works for Schneider+Schumacher Architekten, Frankfurt am Main, Germany. post.ddemin@gmail.com Sébastien Turcaud studied in Strasbourg, France and at the Ecole Nationale de Pontes et Chaussées (ENPC), Paris, France. During his studies he worked at the RWTH Aachen. Since 2008 he works for Bollinger-Grohmann Ingenieure, Frankfurt am Main, Germany. sebastien.turcaud@gmail.com

DESIGN MODELLING SYMPOSIUM Universität der Künste Berlin 2009

Packing of soap bubbles - Geometrical solutions of soap bubble clusters Dimitry Demin, Sébastien Turcaud

From the intensive work of Frei Otto (Otto, 1987) to the more recent implementation in the design of the Aquatic Centre by PTW Architects and Arup (Weaire, 2008, p. 107-110), soap bubble clusters are a source of inspiration for complex geometrical solutions. This research was initiated by the future Saraceno presentation at the Biennale d’Arte di Venezia in 2009 „making worlds“, which is based on a physical model imitating a cluster of coalescing bubbles. The present work results from the confrontation between an artistic vision related to Tomas Saraceno’s work and a scientific engineering approach. These two paths complement one another into a new kind of emergent de- sign. The Equation of Laplace-Young enables to apprehend the principle governing bubbles and to generate interesting geometries in the software environment of Rhino. The parametric models were enabled through scripting tools and mathematical considerations. Efficient from-finding techniques can be based either on the analytical and analogous understanding of the behaviour of soap bubbles. Further, the described solutions can serve for inner-architectural design as clustered space, facade tessellations or efficient minimal structures. Introduction „Now let me pass a cruel, cruel space, Without one hope, without one faintest trace Of mitigation, or redeeming bubble Of colour’d phantasy; for I fear ’twould trouble Thy brain to loss of reason: and next tell How a restoring chance came down to quell One half of the witch in me.“ (John Keats Endymion III) A cluster of soap bubbles consist of a conglomeration of tiny coalescing polygonal cells of different sizes and shapes connected by thin soap films. This soap forth is controlled by its surface energy or surface tension of the water stabilized by the soap molecules. A close-up look reveals that the curvature of the interfaces between the bubbles is governed by the pressure difference between the cells. This subtle Equilibrium of unseen forces is controlled by the detailed geometry of the cluster. The idea is to apprehend the rules of the game as described in the Laplace-Young Equation and Plateau’s rules through an implementation in a three dimensional CAD software. In this paper we describe how to simulate clusters of bubbles in Rhino using Grasshopper plug-in and VB.NET programming. The ambition is to create a CASE STUDIES

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Figure 1: Representation of the principal curvature lines of a periodic structure.

DESIGN MODELLING SYMPOSIUM Universit채t der K체nste Berlin 2009

Demin, Turcaud: Packing of soap bubbles parametrical design environment based on the physics of bubbles with the possibility to change the number of bubbles, their radii and orientation in space through hidden matrix transformations. At the crossroad of science, architecture and programming knowledge, this interdisciplinary approach tackles a wide range of potential applications from minimal space partition problems (Weaire, 2008, p. 87-105) to topological simulation of polycristalline metal behaviour. The first part adumbrates a state of the art of the understanding of the hidden rules describing bubble clusters. We then depict the geometrical description of N-bubble clusters according to the available analytical and analogous knowledge. Eventually, the implementation into Rhino is described and the results are presented. Understanding Bubbles Analytical laws Laplace-Young Equation The Laplace-Young Equation (Eq. 1) relates the pressure difference through a surface at the interface between two static fluids to the surface tension and the local curvature.

1

1

p =σ( + ) R1 R 2

(1)

Local Curvature: Any two-dimensional surface is characterized by the functions followed by its two principal radii of curvature defined at every point of the surface. The principal radii of curvature are the minimum and maximum radii of curvature of the surface and indicate the principal directions which are orthogonal to each other and to the normal of the surface (Fig. 1). A useful measure of the curvature of a surface is given by the mean curvature H (Eq. 2).

H=

1 1 1 ( + )

2 R1 R 2

(2)

Surface Tension: The surface tension is defined as the force per unit length which acts between two parts of a surface separated by an imaginary line. A surface will be in a state of uniform tension if the surface tension stays perpendicular to any line drawn on the surface and remains constant in magnitude at every location on the surface. In the case of soap films it is convenient to introduce the film tension, which is the force per unit length of film and Equals twice the surface tension (Eq. 3). σf = 2 *σ

(3)

This is mathematically described by a diagonal constant surficial stress tensor (Eq. 4).

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σ 0 σ = (4) 0 σ The origin of this tension derives from Van der Waals interactions between the molecules. In the case of soap solutions the molecular structure is composed of water and amphipathic soap molecules. Pressure difference: The pressure is commonly defined as the surficial normal force to a given surface. The Laplace-Young Equation (Eq. 1) expresses the Equilibrium between the pressure forces acting perpendicularly to the surface and the tension forces within the surface through the curvature. Energetic considerations and minimal surface The thermodynamic Equilibrium under constant temperature of a closed system is characterized by the minimization of its free energy, which is defined as the potential of pressure work. If the film tension is assumed constant, the free energy is directly proportional to the surface area (Eq. 5). F = σf * A

(5)

Thus, the Equilibrium of the system is Equivalent to the minimization of its surface area. Analogous laws Finding the form of a surface with a constant film tension and no pres- sure difference is analytically Equivalent of finding the minimal surface area enclosed in an adapted boundary. Analogue observations can therefore pro- vide a guide for analytical solution research. Joseph Plateau made three observations, which have been derived from mathematical analysis of minimal surfaces subsequently (Isenberg, 1992, p. 83-84). The first two apply for any soap froth, whereas the third is only for soap films contained in a framework. • Three smooth surfaces of a soap film intersect along a line. • The angle between any two tangent planes to the intersecting surfaces, at any point along the line of intersection of three surfaces, is 120°. • Only four edges can join at a vertex, each formed by the intersection of three surfaces, forming together the tetrahedral angle 109°2811611 between any pair of adjacent lines. The constant value of 120° for any plane intersection is directly linked to the hypothesis of uniform tension.

DESIGN MODELLING SYMPOSIUM Universität der Künste Berlin 2009

Demin, Turcaud: Packing of soap bubbles Playing with Bubbles Interactions Single Bubble One single bubble tends to be spherical as it is the minimum possible area for the volume of gas it contains. The soap molecules actually decrease the surface tension of the water, but stabilize the bubble via an action known as the Marangoni effect: stretched regions of the sphere present a lower concentration of soap molecules leading to a gradient in the surface tension, which in turn causes the surface tension to increase, preventing the stretching of the sphere. Coalescing Bubbles The force acting between two bubbles on a water surface resembles the interatomic forces in a closed-packed crystal. The bubbles are subject to a weak attraction when their centres are separated by a distance large com- pared to their diameter. This force increases and reaches its maxima as the centres are one diameter apart. As the distance is decreased further, the attraction rapidly changes into repulsion due to the coalescing of the bubbles. This is used to model the microstructural behaviour of a crystal lattice and is known as the Bragg’s raft. An Equivalent Lennard-Jones potential can be assumed to reproduce both long-ranged attraction and short-range repulsion between the bubbles. Cluster of Bubbles A cluster of bubbles consist in a conglomeration of spherical films and caps. The precise geometry of a cluster corresponds to its minimal free energy, which depends on the total surface area and the respective pressures inside the bubbles. The excess pressure in a single spherical bubble is given by the following Equation (Eq. 6).

2σf pf = r

(6)

The geometry of groups of bubbles can be deduced from the analytical laws to a certain extent. In the case of two coalescing bubbles, three excess pressure arise from the three different pressure regions, namely inside each bubble and the exterior pressure. The relation between these excess pressures and the corresponding radii of curvature is expressed in the following relation (Eq. 7).

2σf pi , f = rα

(7)

where i = 1, 2, 3 the respective regions and α = A, B, C the corresponding films. As the sum of all the excess pressures compensate is Equal to zero the curvature of the cap is related to the curvature of the two-coalescing bubbles (Eq. 8).

1 rC

1 rA

1 rB

= − CASE STUDIES

(8)

137

Figure 2: Grasshopper tree organization for the parametric generation of the geometry

Figure 3: Three dimensional Rhino model of a two-bubble cluster

DESIGN MODELLING SYMPOSIUM Universit채t der K체nste Berlin 2009

Demin, Turcaud: Packing of soap bubbles This standard reciprocal relation can also be deduced from Plateau’s rules independently. Only the case with two coalescing bubbles has been proved mathematically while cases with a higher number of coalescing bubbles rely on conjectures assuming the Plateau’s analogous laws and generalize the Equation obtained (Eq. 8). For N-clusters with N > 2, the difficulty is to predict the respective loci of the bubbles as the commonly assumed hypothesis of symmetries has not yet been proved to minimize the total surface area (Durikovic, 2005, part 2.3). Rhino modelling The implementation of the consequences of Laplace- Young Equation and Plateau’s rules into a set of algorithmic formulas, including interactive parametrical interaction, was enabled by the new intelligent Grasshopper plug-in for Rhinoceros 4 and the VB.NET scripting component. The Rhino standard NURBS sphere was used to simulate the bubbles separately with the possibility to change their number trough a slider inter- face and to fix the boundaries of their randomly determined radii within a typical Grasshopper tree organization (Fig. 2). The calculation of the interfacing planes was made using the VB.NET component standard matrix class functions (OnMatrix) in a Cartesian coordinate system. The planes were calculated according to the origin plane defined by O= (0, 0, 0) and a normal vector z in correspondence to mentioned mathematical Equations of the new plane latter allocated to the corresponding array of the bubble-spheres. According to the radii and the position of each bubble, the curved polyhedral surface between the coalescing bubbles was calculated. Double Bubble The radius of the spherical cap (Fig. 3) at the coalescing interface is linked by Eq. 8 to the numerical values of the randomly chosen radii of the two initial spherical bubbles (Eq. 9).

(9)

Triple Bubble As for the double bubble case the reciprocal Eq. 8 links the radii of curvature of the spherical caps (Fig. 4) to the initial randomly chosen radii of the spherical bubbles (Eq. 10).

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Figure 4: Three dimensional Rhino model of a three-bubble cluster

Figure 5: Three dimensional Rhino model of a four-bubble cluster DESIGN MODELLING SYMPOSIUM Universit채t der K체nste Berlin 2009

Demin, Turcaud: Packing of soap bubbles

(10)

N-Bubble The previous principle is generalizable to an unlimited number of bubbles. This way, bubble froth in accord with Plateau’s rules and the Laplace-Young Equation can be generated in the Rhino environment as illustrated with a supplementary bubble in Fig. 5. Outlook Since an efficient method to generate well-formed parametrical three dimensional bubble-clusters in Rhino has led to the aimed result, further goals can be depicted: • Verify the minimal area property of the generated clusters with the open-source software Surface Evolver (Brakke). • Implement dynamic phenomena in order to model the evolution of the clusters (Marangoni effect, Lennard-Jones potential). • Make a structural analysis of a given cluster in appropriated software (Ansys or Dlubal). Conclusion The built parametrical models suits for a wide range of applications. It can be used to design building facades, load-bearing structures and architectural units with a constant appeal for further development and implication of any programmable constraints. Bubbles clusters also are of commercial interest due to the inherent seek of minimal surfaces. They belong to field of emergent structures and can be used at every scale of buildings. Further computing work is required to simulate complex multi-bubble cluster at a detail scale. This three dimensional simulation also gives the opportunity to apply an integrative design method through software merging. The design of an interface between Rhino and Ken Brakke’s Surface Evolver (Brakke) could enhance the interdisciplinary of this design process. References [1] Aste, T.; Weaire, D., 2008: The Pursuit of Perfect Packing. New York [2] Brakke, K.: The Surface Evolver. http://www.susqu.edu/brakke/ [3] Durikovic, R., 2005: Computer Animation: Animation of Soap Bubble Dynamics, Cluster Formation and Collision. Jamsi [4] Isenberg, C., 1992: The Science of Soap Films and Soap Bubbles. New York [5] Otto, F., 1987: IL18. Seifenblasen. Stuttgart Cyril CASE STUDIES

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DESIGN MODELLING SYMPOSIUM Universit채t der K체nste Berlin 2009

APPENDIX

355

PARTICIPANTS

Ackermann, Frank Dr. Aish, Robert Dr. Ammon, Sabine Angulo Garcia, Daniel Auweck, Roman Azevedo, Gonçalo Back, Thomas Graf v. Ballestrem, Matthias Barcucci, Sabina Bauer, Hans-Georg Bentscheff, Ilija Bildsøe, Thomas Prof. Dr. Bobenko, A. Brakelmann, Stephan Brehm, Verena Brockmann, Bruce Budig, Michael Burkhardt, Uwe Cannaerts, Corneel Chen, Shouheng Christensen, Jesper Thøger David, Michel Dechert, Felix Dehio, Romuald Demin, Dimitry Dempsey, Alan Deuss, Mario Dimcic, Milos Dimitric, Aleksandar Dines Petersen, Mads Doherty, Ben Dölling, Max Dondit, Steffi Dony, Hans-Jörg Drew, Williamson Drobnik, Michael Droste, Stephan Eckardt, Matthias Eickhoff, Jan Eigensatz, Michael Facklam, Ferdinand

Georg Ackermann GmbH Autodesk, Inc. TU Berlin TU Berlin agp UdK Berlin TU Berlin

UdK Berlin Aalborg University TU Berlin Dassault Systèmes Deutschland GmbH Leibniz Universität Hannover Transcat PLM GmbH Institut für experimentelle Architektur sbp GmbH Sint-Lucas Architectur Shouheng Design and Technology Inc. Aalborg University Helixator UdK Berlin UdK Berlin ETH Zurich NEX / Architectural Association ETH Zürich Stuttgart University RTKL-UK Ltd Aalborg University Oxford Brookes TU Berlin Dassault Systèmes Deutschland GmbH Dassault Systèmes Deutschland GmbH McBride Charles Ryan RPM Architekten Universität Stuttgart UdK Berlin Mehnert Corporate Design GmbH ETH Zürich

DESIGN MODELLING SYMPOSIUM Universität der Künste Berlin 2009

Geschäftsführer

Künstlerischer Mitarbeiter Architect Student Wissenschaftlicher Mitarbeiter Architect Architect Student Student

Student Wissenschaftlicher Mitarbeiter Phd Candidate Principal Student CTO Student Student Student Director Student Student Architectural Designer Student Student Student

Student Architect Akademischer Mitarbeiter Student Geschäftsführer Student Architekt und Gestalter

Fleischmann, Moritz Foerster, Florian Fornes, Marc Forstlechner, Franz Xaver Frankenberg, Judith Füssler, Urs Prof. Dr.-Ing. Gengnagel, C. Gießen, Sibylle Gmelin, Sebastian Greiner, Switbert Prof. Dr.-Ing. Grohmann, M. Güttgemanns, Jörg Hasselberg, Paal Eide Hegedues, Krisztian Heil, Daniel Heinzelmann, Florian Hempel, Jan Hey, Christoph Prof. Dr. Höhl, Wolfgang Hong, June Hoppermann, Marc Hudert, Markus Jacobi, Markus Jäger, Ariane Janitz, Steffen Just, Linda Prof. Kahane, Josiah Kaldenhoff, Max Dr. Kilian, Axel Kleffing, Uta Kleine, Holger Kluibenschedl, Andreas Kondziela, Andrea König, Abraham Koren, Benjamin Prof. Kufus, Axel Kuhnen, Johannes Kuroda, Ken Latein, Tine Lautenbach, Ulf Lee, Royce

Universität Stuttgart Buro Happold Berlin theverymany New York TU Graz TU Berlin

Research Associate, Phd Candidate

Student Student Architect

UdK Berlin Dassault Systèmes Deutschland GmbH Architecture School of Aarhus PhD Student ArtEngineering GmbH Geschäftsführer Universität Kassel Yacht Projects & Design Dipl.-Ing. Architekt Snøhetta Architect Autodesk GmbH Customer Success Engineer Weicken Architekten Shau Partner Universität Kassel Student UdK Berlin Student Hochschule für Medien u. Kommunikation UdK Berlin Student UNStudio Architect EPFL studioB Principle UdK Berlin Arup GmbH Ingenieur Kirkegaard Associates Holon Institute of Technology Vice President UdK Berlin Student TU Delft Westsächsische Hochschule Zwickau Lehrbeauftragte CAD HOLGER KLEINE ARCHITEKTEN mbH Bernard Ingenieure ZT GmbH Geschäftsbereichsleiter MA Architect Autodesk GmbH Industry Product Manager One UdK Berlin Universität Kassel Student agp Tine Latein UdK Berlin Student BVN Architecture Practice Director Modus

357

Leifels, Marina Lind, Adam Dr. Lordick, Daniel Mainz, Till Mangelsdorf, Wolf Marine, Bagneris Marsh, Andrew Mehrtens, Peter Prof. Menges, Achim Müller, Dirk Nakajima, Atsuo Prof. Neumann, Holger Niedzwiecki, Mieszko B. Nolte, Astrid Nolting, Nils Prof. Dr. Nytsch-Geusen, C. Ott, Inka Ouyang, Liwen Prof. Oxman, Rivka Palz, Norbert Paulisch, Stefan Prof. Pauly, Mark Pérez, Carlos Perktold, Martin Piacentino, Giulio Pochat, Jérôme Prof. Pongratz, Christian R. Prof. Dr. Pottmann, Helmut Prokop, Ines Proll, Mischa Quinn, Greg Rädler, Jörg Reinhardt, Martin Reisigl, Mathias Reitz, Judith Riegas, Steffen Riether, Gernot Sadowski, Michal Salzmann, Peter Schamun, Kathrin Scheible, Florian

Bollinger + Grohmann Ingenieure Nyréns Arkitektkontor TU Dresden Kunsthochschule Berlin Buro Happold London SLA AEC-Simulation Autodesk, Inc. Planungsbüro Universität Stuttgart - ICD gmp architekten AppliCraft UdK Berlin ATP parametric design GmbH Universität Kassel

Dozent

Dr. Eng M.Sc. Architektur Professor Architect

Managing Director Student

UdK Berlin UdK Berlin Technion, Haifa Royal Danish Academy of Fine Arts Westsächsische Hochschule Zwickau ETH Zürich McNeel Europe agp McNeel Europe Guscetti & Tournier SA Pongratz Perbellini Architects TU Wien UdK Berlin Universität Kassel UdK Berlin UdK Berlin Universität Kassel agp BFR LAB/RWTH Aachen UNStudio Georgia Institute of Technology UdK Berlin Virtual Shape Research UdK Berlin Knippers Helbig Advanced Engineering

DESIGN MODELLING SYMPOSIUM Universität der Künste Berlin 2009

Innenarchitektin Student

Dozent CAD

Product Manager Architecture Principal Dozentin Student WM WM Systemadministrator Partner Architect Assistant Professor Student CTO Associate

Prof. Dr.-Ing. Scheurer, F. designtoproduction Zürich Schieber, Roman Knippers Helbig Advanced Engineering Prof. Schmitz, Burkhardt UdK Berlin Schork, Tim RMIT University Schroth, Martin Akademie der Bildenden Künste Stuttgart Prof. Dr. Schwandt, Hartmut TU Berlin Schwitalla, Max Henn Studio B Sechelmann, Stefan TU Berlin Segas, Yosi TU Berlin Seifert, André hed-architekten Seurinck, Patrick Saint Lucas Ghent Shelden, Dennis R. Gehry Technologies Prof. Dr. Sommer, Inge UdK Berlin Sostmann, Katina UdK Berlin Spenling, Frank Archimedes Solutions GmbH Stadel, Ingo Virtual Shape Research Stauch, Susanne goldprodukt Stehling, Hanno designtoproduction Sternitzke, André UdK Berlin Strathaus, Thorsten Bollinger + Grohmann Ingenieure Strehlke, Kai Herzog & de Meuron, Basel Tamke, Martin CITA - Royal Academy of Fine Arts Dr.-Ing. Tessmann, Oliver Bollinger + Grohmann Ingenieure Thoelen, Jan UdK Berlin Troche, Christian BRT Turcaud, Sébastien Ecole des Ponts Paritech van der Heijden, Ramon UNStudio Prof. Wallisser, Tobias ABK Stuttgart Walmsley, Lowell Robert McNeel & Associates Dr. Wehdorn-Roithmayr, R. Wehdorn Architekten ZT Weid, Manfred Georg Ackermann GmbH Weiss, Christian Autodesk GmbH Wiertelarz, Kathrin Universität Kassel/TU Insbruck Woda, Angela RMIT University Wöllm, Dominik Dassault Systèmes Deutschland GmbH Wunderlich, Christian UdK Berlin Yun, Wai Wing TU Delft Zapel, Eric UdK Berlin Zinssmeister, Holger HCU Hamburg

Parametric Design Student Künstlerischer Mitarbeiter

Student BIM-Manager Student

Künstlerische Mitarbeiterin SW Entwickler Designer WM

Professor Student Student Architect

Architect Produktionsleiter EMEA Campaign Man. AEC WM Student Student Student Student Student

359

CONFERENCE VENUE Ground floor University of the Arts Berlin

Check-In

Entrance University of the Arts Hardenbergstrassse 33

DESIGN MODELLING SYMPOSIUM Universit채t der K체nste Berlin 2009

1st floor University of the Arts the Berlin

Key Lectures R 158

LOCATION

361

1st floor University of the Arts Berlin - detail front building

Entrance University of the Arts Hardenbergstrassse 33

R 101

Alte Bibliothek

DESIGN MODELLING SYMPOSIUM Universit채t der K체nste Berlin 2009

3rd floor University of the Arts Berlin - detail front building

WC

Workshop IV

R 314 R 326

Case Studies

R 312/ R 313

Exhibition

R 310

MC I

CafĂŠ R 333

Workshop III

R 309

MC II

R 306

Workshop I

R 305

Conference Office

Lift

WC

LOCATION

363

IMPRINT Proceedings of the Design Modelling Symposium Berlin 2009 Herausgeber: Christoph Gengnagel Redaktionelle Bearbeitung: Christoph Gengnagel Unter Mitarbeit von Jan Thoelen und Christian Wunderlich Gestaltung: Oliver Schleith, Jan Thoelen und Christian Wunderlich © Universität der Künste Berlin 2009 Satz: Universität der Künste Berlin Druck/Bindebearbeitung: Copy Print, Berlin ISBN: 978-3-89462-177-3 Bibliographische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliographie; detaillierte bibliographische Daten sind im Internet über http://dnb.d-nb.de abrufbar. Produktion Lehrstuhl für Konstruktives Entwerfen und Tragwerkslehre Studiengang Architektur Universität der Künste Berlin Hardenbergstr.33 D 10623 Berlin The papers are published in the form submitted by the authors, after revision by the Scientific Committee and re-layout by the Editors. The Editors cannot accept any responsibility for the content of the papers. Copyright Chair for Structural Design and Technology, Faculty of Architecture, Media and Design, University of the Arts Berlin, Germany All rights reserved. These proceedings may not be reproduced, in whole, or in part, in any form without permission.

DESIGN MODELLING SYMPOSIUM Universität der Künste Berlin 2009

Proceedings of the Design Modelling Symposium Berlin 2009 University of the Arts Berlin

1

Packing of soap bubbles

Published on Sep 7, 2012

From the intensive work of Frei Otto (Otto, 1987) to the more recent implementation in the design of the Aquatic Centre by PTW Architects an...

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