freeform architecture and discrete differential …€¢ develops discrete equivalents of notions...
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Freeform Architecture and Discrete Differential Geometry Helmut Pottmann, KAUST
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Freeform Architecture
Motivation:
• Large scale architectural projects, involving complex freeform geometry
• Realization challenging and costly; available digital design technology is not adapted to the demands in this area.
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Research Goals
• Make geometrically complex architectural structures affordable through novel computational tools by linking design, function and fabrication
• Provide new methodology for computational design, especially through links to discrete & computational differential geometry and optimization
• Develop new tools to explore the variety of feasible / optimized designs through links to the geometry of shape spaces
• Advance the theory (discrete differential geometry, shape spaces, …) through novel concepts motivated by applications
• Contribute to and learn from real-world projects (Evolute GmbH)
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Overview
• Planar quad meshes
• Planar quads and beam layouts in real projects
• Single curved panels
• Paneling
• Design of self-supporting surfaces
• Future research
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Planar Quad Meshes
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quad meshes in architecture
• work by Schlaich & Schober
hippo house, Berlin Zoo
quad meshes with planar faces (PQ meshes) only special shapes; what about freeform shapes?
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key insight on PQ meshes
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• PQ meshes reflect curvature behavior
• relation to discrete differential geometry: PQ meshes are discrete versions of conjugate curve networks
• any optimization has to be initialized respecting this fact
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Computing PQ meshes
• Computation of a PQ mesh is based on numerical optimization:
• Optimization criteria • planarity of faces
• aesthetics (fairness of mesh polygons)
• proximity to a given reference surface
• Requires initial mesh, found via a careful evaluation of the curvature behavior!
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subdivision & optimization
• refine a coarse PQ mesh by repeated application of subdivision and PQ optimization
• can be combined with surface fitting
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Opus (Zaha Hadid Architects)
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OPUS (Zaha Hadid Architects)
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Discrete Differential Geometry
• Develops discrete equivalents of notions and methods of classical differential geometry
• The latter appears as limit of the refinement of the discretization
• Basic structures of DDG related to the theory of integrable systems
• A. Bobenko, Y. Suris: Discrete Differential Geometry: Integrable Structure, AMS, 2008
• Discretize the theory, not the equations!
• Several discretizations; which one is the best?
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Conical meshes
• panels as rectangular as possible
• a discrete counterpart of network of principal curvature lines
• circular mesh
• for architecture, even better:
conical mesh
• PQ mesh is conical if all vertices of valence 4 are conical: incident oriented face planes are tangent to a right circular cone
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normals of a conical mesh
• neighboring cone axes (discrete normals) are coplanar
• conical mesh has precise offsets and a torsion-free support structure
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nodes in the support structure
• triangle mesh: generically nodes of valence 6; `torsion´: central planes of beams not co-axial
torsion-free node
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Conical mesh
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Discrete curvature theory
• The study of meshes with offsets led to a new curvature theory for discrete surfaces based on parallel meshes (Bobenko, P., Wallner, Math. Annalen, 2010)
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• meshes M, M* with planar faces are parallel if they are combinatorially equivalent and corresponding edges are parallel
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Discrete curvature theory
• Gaussian image mesh S of M is parallel to M and approximates the unit sphere
• Offset mesh at distance d: M+d S
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Discrete curvature theory
• Examples:
• conical mesh: faces of the Gaussian image are tangent to the unit sphere
offsets at constant face-face distance
• circular mesh: vertices of Gaussian image lie on unit sphere; corresponding vertices of base mesh and offset at constant distance
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Discrete curvature theory
• surface area of the offset of the mesh M relative to the Gauss image
• analogous to Steiner`s formula
• define curvatures in face F
mixed area
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Discrete curvature theory
• Discrete minimal surface:
• valid for polyhedral surfaces (different from triangle meshes)
• extends to relative differential geometry, where Euclidean sphere is replaced by another convex surface
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Planar quads and beam layouts in real projects
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Museum of Islamic Arts at Louvre Architects: Mario Bellini Architects, Rudy Ricciotti
Construction: Waagner Biro
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Museum of Islamic Arts
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triangle mesh
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planar quad mesh for Louvre
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another planar quad mesh
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Solution: hybrid mesh from planar quads and triangles
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Yas Island Marina Hotel Abu Dhabi Architect: Asymptote Architecture Steel/glass construction: Waagner Biro
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steel beam layout
• Faces non-planar: there is no elegant exact solution
• node axes should be nearly normal to surface
• node axes as solution of an optimization problem
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Single Curved Panels
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developable surfaces in architecture
• (nearly) developable surfaces
F. Gehry, Guggenheim Museum, Bilbao
F. Gehry, Walt Disney Concert Hall, Los Angeles
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developable surface strips
• Refinement of a PQ strip (iterate between subdivision and PQ optimization)
Limit: developable surface strip
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D-strip models
One-directional limit of a PQ mesh:
developable strip model (D-strip model)
semi-discrete surface representation initiated research on semi-discrete surface representations
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Design from single-curved panels based on subdivision modeling
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Multi-layer structure
D-strip model on top of a PQ mesh