Programme

Graphic statics aims to find funicular grids in equilibrium under different loads. Other objectives, such as panel planarity when designing gridshells, may also be sought. In this presentation, we propose to approach the problem from an engineer’s angle: the manufacturing and financial constraints on the project may be strong, the loads are varied, the site is given and imposes support conditions, the architect has intentions regarding the design of the grid or the shape… In this situation, obtaining a single solution is not necessarily the objective, but rather, it is to get a space of solutions in which we can easily navigate in search of a suitable design. We’ve constructed such a space: one in which all the grids have flat panels, and whose shape and forces are controlled by a limited number of parameters. We can then search for the most funicular grid in this space, with several possible metrics. The difference between the funicular load and the applied load is used as an example.

The theory of transformations of surfaces that preserve “good properties” in classical surface theory has been extensively studied through the methods of integrable systems and is also known as integrable geometry.

On the other hand, there are cases where transformations naturally induced from the equilibrium conditions of forces and moments under loads acting on a structure can be understood within the context of integrable geometry.

In this talk, we will introduce shape generation of membrane and truss structures based on integrable geometry and variational principle, considering both mechanical properties and constructability.

In this talk, we consider a class of plane curves called the log-aesthetic curves (LAC) and their generalizations which have been developed in industrial design as the curves obtained by extracting the common properties among thousands of curves that car designers regard as aesthetic. We consider these curves in the framework of similarity geometry (Klein geometry associated with $CO^+(2, \mathbb{R}) \simeq SO(2) \ltimes \mathbb{R}^+$) and characterize them as invariant curves under the integrable deformation of plane curves governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation [1, 2]. We then extend the LAC to space curves by considering the integrable deformation of space curves under similarity geometry. The deformation is governed by the coupled system of the modified KdV equation satisfied by the similarity torsion and a linear equation satisfied by the curvature radius. The curves also allow the deformation governed by the coupled system of the sine-Gordon equation and associated linear equation. The space curves corresponding to the travelling wave solutions of those equations would give a generalization of the LAC to space curves. We also consider the surface constructed by the family of curves obtained by the integrable deformation of such curves. A special class of surfaces corresponding to the constant similarity torsion yields quadratic surfaces (ellipsoid, one/two-sheeted hyperboloid and paraboloid) and their deformations, which may be regarded as a generalization of the LAC to surface. We discuss the construction of such curves and surfaces together with their mathematical properties, including integration scheme of the frame by symmetries, and present various examples of curves and surfaces.

Finally we discuss the self-affinity of plane curves that has been proposed in the area of industrial design as a characteristic property of the LAC. After some investigations and extending the definition[3], we propose a new class of “aesthetic curves” with self-affinity, which includes the logarithmic spiral (special case of the LAC) and quadratic curves (parabola, hyperbola and ellipse) under the framework of equiaffine geometry (Klein geometry associated with $\mathrm{SL}(2, \mathbb{R})$). It may be an interesting problem to investigate the similar class of curves in Möbius geometry.

References

1. Jun-ichi Inoguchi,Kenji Kajiwara, Kenjiro T.Miura, Masayuki Sato, Wolfgang K.Schief and Yasuhiro Shimizu, Log-aesthetic curves as similarity geometric analogue of Euler’s elasticae, Comp. Aided Geom. Design, 61 (2018) 1–5, https://doi.org/10.1016/j.cagd.2018.02.002.
2. Jun-ichi Inoguchi, Kenji Kajiwara, Kenjiro T. Miura, Yoshiki Jikumaru and Wolfgang K. Schief, Log-aesthetic curves: similarity geometry, integrable discretization and variational principles, Comput. Aided Geom. Design 105(2023) 102233, https://doi.org/10.1016/j.cagd.2023.10223.
3. Shun Kumagai and Kenji Kajiwara, Self-affinities of planar curves: towards unified description of aesthetic curves, arXiv:2407.17008v1, https://doi.org/10.48550/arXiv.2407.17008, to appear in Japan J. Indust. Appl. Math. (2025).

In this talk Marina will discuss conceptual structural design methods for lightweight architectural forms. Specifically, said frameworks hinge on: graphic statics based on JC Maxwell’s approach; reciprocal stress functions in three- and four-dimensions; the force density method; projective geometry duality principles and polarities. Several examples and typologies will be presented including tensile structures, strut-and-tie demonstrators, and gridshells. Lastly, current research and collaborations will be discussed focusing on developing an integrated design workflow for discrete tensile structures encompassing net topologies and their assessment in terms of load path and curvature.

Inflatable pads, such as those used as mattresses or protective equipment, are structures made of two planar membranes sealed according to periodic patterns, typically parallel lines or dots. In this work, we propose to treat these inflatables as metamaterials. By considering novel sealing patterns with 3-fold symmetry, we are able to generate a family of inflatable materials whose macroscale contraction is isotropic and can be modulated by controlling the parameters of the seals. We leverage this property of our inflatable materials family to propose a simple and effective algorithm based on conformal mapping that allows us to design the layout of inflatable structures that can be fabricated flat and whose inflated shapes approximate those of given target freeform surfaces.

Spectral geometry is a mathematical field that links geometrical properties to eigenvalues of differential operators on surfaces. Although it is a well-established tool in geometry processing and has been used in many contexts, the structural engineering and architectural geometry communities have not yet adopted this framework for shape modeling. In this talk, we will explore spectral methods for applications in architectural geometries. A novel methodology for generating anisotropic Laplacian operators based on regions of interest defined by the user is proposed. The potential of spectral methods in structural design is illustrated through design problems expressed on meshes and graphs.

This presentation reviews recent advances in the form-finding of tension-compression mixed shells. Although purely compressive stress states are traditionally considered ideal for shell structures, I propose that allowing a mix of tension and compression can expand the range of feasible shell geometries. The key challenge lies in the fact that the equilibrium problem becomes a hyperbolic boundary value problem, which is notoriously difficult to solve. I point out that the introduction of Airy’s stress function reveals that the equilibrium equation is a bilinear partial differential equation (PDE). I then indicate that this PDE can be solved using the Variable Projection (VarPro) method—developed specifically for bilinear problems. I also demonstrate that the alignment of stress and curvature directions is governed by a bilinear PDE, which can be solved concurrently with the equilibrium equation using the VarPro method.

We will discuss mesh-types such as sphere meshes and s-nets which appear in applications and fit naturally to Möbius geometry. Sphere meshes are interesting from a simplified manufacturing viewpoint when compared to other double curved face shapes. We investigate the generation of sphere meshes which allow for a geometric support structure and characterize all such meshes with triangular combinatorics in terms of non-Euclidean geometries. The existence of a torsion free support structure that is formed by the planes of circular edges implies that the mesh simultaneously defines a second mesh whose faces lie on the same spheres as the faces of the first mesh. The two surfaces are considered as the two envelopes of a discrete sphere congruence, i.e., a two-parameter family of spheres. We relate such sphere congruences to torsal parameterizations of associated line congruences. By broadening polyhedral meshes to sphere meshes we exploit the additional degrees of freedom to minimize intersection angles of neighboring spheres enabling the use of spherical panels that provide a softer perception of the overall surface. S-nets are surface parametrizations where the isolines run symmetrically to the principal directions. Our discretization of a principal symmetric mesh comes naturally with a family of spheres, the so-called Meusnier spheres. We obtain Weingarten surfaces including constant mean curvature surfaces and minimal surfaces. We describe two discrete versions of these special nets which are dual to each other and show their usefulness for various applications in the context of fabrication and architectural design. We illustrate the potential of Weingarten surfaces for paneling doubly curved freeform facades by significantly reducing the number of necessary molds. Moreover, we have direct access to curvature adaptive tool paths for cylindrical CNC milling with circular edges as well as flank milling with rotational cones.

Compliant shell mechanisms are creased and corrugated thin-walled structures that can drastically change shape to move, deploy, or adapt to a changing environment. They have found use cases in the context of recent space programs and in other domains ranging from biomedical technology to architecture. Not unlike slender beams, thin shells prefer bending over stretching. Ideally, thin shells deform isometrically should isometric deformations exist. The problem of finding, or disproving the existence of, isometric deformations for various surfaces preoccupied many mathematicians and mechanicians. The most noteworthy results undoubtedly pertain to three broad categories of surfaces: developable surfaces, convex surfaces, and axisymmetric surfaces. In the modern context of computer graphics, discrete differential geometry and “Origami science,” more focus has been directed towards tri- and quad-based polyhedral surfaces. In this lecture, we report on recent results that characterize the isometric deformations of periodic surfaces, be them smooth or piecewise smooth with straight or curved creases.

In this talk I will discuss computational design strategies for different classes of deployable gridshells. I will show how geometric analysis, physics-based simulation, and numerical optimization can be combined to create effective design tools for gridshells with straight or curved flexible beams, as well as for segmented gridshells that are assembled using weaving techniques. In several design studies, large-scale demonstrators, and commissioned installations I will highlight the unique benefits of deployable gridshells, but also indicate challenges that motivate future research.

Circular meshes are discrete quadrilateral surfaces composed of faces whose vertices lie on circles. Interest in these meshes for architectural gridshell design arises from the existence of torsion-free nodes and vertex offsets. In this talk, we shall explore the solution space of circular meshes foliated by a family of planar or spherical parameter lines. The proposed method, called ’lifted-folding’, creates these meshes using 2D data and a folding function.

If the 2D data is provided by specific discrete holomorphic maps built from elastic curves, the generated meshes are isothermic. This class then includes discrete minimal and CMC nets with a family of spherical parameter lines.

Finally, we will discuss how these discrete differential geometric ideas may lead to results in related fields as smooth differential geometry and kinematics: namely, to a novel construction method for smooth isothermic surfaces with spherical curvature lines and to snapping structures.

will discuss the pipeline of remeshing existing freeform shapes into meshes that adhere to constraints that are prevalent and desired in architectural geometry and fabrication: developability, valid printing or deposition, flexibility, and coplanarity. The mutual ingredient is reducing the constraints into a functional design of directional fields on triangle meshes, that upon integration obey the constraints infinitesimally, allowing for a simple and robust discrete optimization.

Developable surfaces as a rule are easier to make physically than double-curved, general surfaces. However their mathematical understanding and interactive geometric modeling poses many difficulties. The sheer number of different computational approaches to the problem testifies to this. In this presentation I am going to review some of the many properties of developables and talk about their discretization. This subject has origins in numerical analysis (R. Sauer, 1930’s) and also in convex geometry (A.D. Alexandov 1942). I will focus on a developability criterion proposed by V. Ceballos Inza et 2023 which applies to general quad meshes and has properties making it easy to incorporate in CAD software.

The Codazzi equations for the shape operator or normal curvature tensor and the tangential components of the membrane equilibrium equations have exactly the same form. It is well known that the Codazzi equations lead to a unique spacing of the principal curvature lines on a Weingarten surface, and this therefore also applies to the principal membrane stress trajectories if we have a predefined relationship between the principal membrane stresses. This produces interesting results for the formfinding of shells when combined with the large-strain analysis of isotropic hyperelastic membranes.