PRESENTATION: Structural Forms in Architecture

By 2050, 70% of the world’s population will live in cities. We envision, design and construct structures that those city dwellers depend on daily. The construction industry is one of most resource‐intensive sectors, and yet our urban infrastructure continues to be built in the massive tradition in which strength is pursued through material mass. In contrast, I have focused my research on structural systems that derive their performance from their curved shape, dictated by the flow of forces. As a result, these structures can be extremely thin, cost‐effective, and have a smaller carbon footprint. My core research question is  ‘What is the relationship between form and efficiency in structures?’ In my lecture, I will focus on the design, optimization and realisation of structural forms for long-span shells, large-scale raised and submerged flexible net barriers, and adaptive building facades.  Some of these systems are inspired by systems that have evolved in biology, art or craft.


I am delighted to present our research on tension-only structures to architecture and engineering students in Prof. Ochsendorf’s Structures course at MIT. Our advances on the mechanics of suspended rope bridges, inflateable sea barriers and hanging networks will be discussed.

International Day of Women and Girls in Science

According to the UNESCO Institute for Statistics, women make up less than 30% of the world’s researchers. Today, Thursday 11th February 2021, is the International Day of Women and Girls in Science. A big shout out to all the women and girls changing the world through science!


It is with great joy that I can inform you that I have been named Fellow of the Structural Engineering Institute (SEI) of ASCE (American Society of Civil Engineering). The SEI Fellow grade distinguishes members of ASCE as leaders and mentors in the profession. I am being recognized for “her seminal contributions to the understanding of the mechanics of large-scale shells and membranes through the development of methods for form-finding, analysis and optimization, and for her leadership in bridging art and engineering to bring innovative solutions to the 21st century challenges of the built environment.” My thanks go out to all my students, collaborators, colleagues and all the other people (you know I mean you) that make my work possible every day!


How can art inspire engineering systems? We have been working with bobbin lace and textile artists to find out. Our work is exhibited at the 2021 Joint Mathematics Meetings. Inspired by a traditional bobbin lace pattern, ‘torchon ground’, elastic strips are interlaced to create a gradient of out-of-plane behavior. The key structural element is a bigon which consists of two strips deformed so as to intersect each other at a fixed angle at each end. Numerical modeling of each strip as a Kirchhoff rod shows that the 3D shape of a bigon depends on the width-to-thickness ratio of the component strips and the angle between strips. In this sculpture, the strip width and thickness are constant and the angle is varied along one axis. Intersecting twisted pairs in this torchon ground pattern form curved polygon cells with non-vanishing and tunable Gaussian curvature through the coupling of geometry and elasticity

PRESENTATION: DigitalFUTURES- Shell Structures

To start off the year in a good spirit, I cordially invite you to Virtual Panel and Lecture on Shell Structures on January 16th 2021 (9AM EST, 3PM CET, 10PM China) hosted by Philip Yuan on the DigitalFUTURES platform.  Besides myself, Philippe Block ( and Chris Williams ( will be offering thoughts and design approaches to shells.  This event explores a new way of connecting.

As the date comes closer, I will provide you with more details.

And now we welcome the new year, full of things that have never been.

And now we welcome the new year, full of things that have never been. – Rainer Maria Rilke (1875-1926)

Our best wishes to you from us at the Form Finding Lab.

JOURNAL PUBLICATION: Human–robot collaboration: a fabrication framework for the sequential design and construction of unplanned spatial structures

Robots in traditional fabrication applications act as passive participants in the process of creation—simply performing a set of predetermined actions to materialize a completed design. We propose a novel bottom-up design
framework in which robots are instead given the opportunity to participate centrally within a creative design process. This paper describes how two 6-axis industrial robotic arms were used to cooperatively aggregate a collection of solid spherical units. The branching spatial structure being constructed is unplanned at the outset of this process, and is instead designed in pseudo-real time during construction. This ‘design-as-you-build approach relies on robotic input, in the form of path-planning constraints in tandem with human evaluation and decision-making. The resulting structure emerges from a human–robot design collaboration operating within the specified physical domain.

Free downloads here for the first 50 downloads.

JOURNAL PUBLICATION: Robotic vault: a cooperative robotic assembly method for brick vault construction

Geometrically complex masonry structures built with traditional techniques typically require either temporary scaffolding or skilled masons. This paper presents a novel fabrication process for the assembly of full-scale masonry vaults without the use of falsework. The fabrication method is based on a cooperative assembly approach in which two robots alternate between placement and support to first build a stable central arch. Subsequently, the construction is continued individually by the robots – building out from the central arch based on an interlocking diagonal brick sequence. This proposed method is validated through its successful implementation in a full-scale vault structure consisting of 256 glass and concrete standardized bricks. The paper includes strategies for developing the design, sequencing, and robotic assembly methods used to build the vault.


The Form Finding Lab in the Department of Civil and Environmental Engineering (CEE) at Princeton University invites applications for post-doctoral and more senior research position to support research on the mechanics of curved crease origami. The research will be conducted under the direction of Professor Sigrid Adriaenssens. Responsibilities include conducting experimental and computational mechanics research as it relates to developing novel curved crease origami systems; overseeing day-to-day graduate and possibly undergraduate student research as it relates to project goals and meeting project milestones and go/no-go points; organizing and leading a curved crease origami workshop with our collaborating origami artist and mathematicians and designers; presenting at conferences and leading journal publication efforts; leading outreach effort of designing and constructing demonstrators; developing and writing research proposals. This is a 1-year position. Minimum qualifications: doctoral degree in mechanical or civil engineering, physics or applied mathematics field, appropriate experience in computational mechanics and mechanics of slender systems, prototyping and artistic flair. Strong project management skills and a superior publication record. Interested candidates should submit an application online at

Applications should include a CV, a brief statement of research experience and interests, and the contact information for three references. Please send inquiries to This position is subject to the University’s background check policy. Princeton University is an Equal Opportunity/Affirmative Action Employer and all qualified applicants will receive consideration for employment without regard to age, race, color, religion, sex, sexual orientation, gender identity or expression, national origin, disability status, protected veteran status, or any other characteristic protected by law.

GUEST SPEAKER: Rethinking Concrete, rethinking material conventions in the Anthropocene

Rethinking Concrete: Material Conventions in the Anthropocene, an interdisciplinary conference organized by Forrest Meggers and Lucia Allais, will discuss new approaches to the lifespan, material dynamics, cultural history, and design potential of reinforced concrete. Once conceived as a quintessentially modernist material, a “liquid stone” that announced the arrival of an eternal present, reinforced concrete is in fact a highly dynamic technological system, subject to inevitable failure through carbonation and other processes. Speakers will problematize the material conventions embedded in reinforced concrete, and expose its role as a complex agent of the anthropocene.

Conversations and presentations among speakers from architecture, engineering, material science, conservation, and design will include Daniel Abramson, Sigrid Adriaenssens, Ueli Angst, Dorit Aviv, Philippe Block, Brandon Clifford, Aude-Line Dulière, Branko Glisic, Tsz Yan Ng, John Ochsendorf, Evan Oskierko-Jeznacki, Antoine Picon, Sarah Nichols, Elisabeth Marie-Victoire, and Claire White, among others.

JOURNAL PUBLICATION: Machine learning generative models for automatic design of multi-material 3D printed composite solids

Mechanical metamaterials are artificial structures that exhibit unusual mechanical properties at the macroscopic level due to architected geometric design at the microscopic level. With rapid advancement of multi-material 3D printing techniques, it is possible to design mechanical metamaterials by varying spatial distributions of different base materials within a representative volume element (RVE), which is then periodically arranged into a lattice structure. The design problem is challenging however, considering the wide design space of potentially infinitely many configurations of multi-material RVEs. We propose an optimization framework that automates the design flow. We adopt variational autoencoder (VAE), a machine learning generative model to learn a latent, reduced representation of a given RVE configuration. The reduced design space allows to perform Bayesian optimization (BayesOpt), a sequential optimization strategy, for the multi-material design problems. In this work,we select two base materials with distinct elastic moduli and use the proposed optimization scheme to design a composite solid that achieves a prescribed set of macroscopic elastic moduli. We fabricated optimal samples with multi-material 3D printing and performed experimental validation, showing that the optimization framework is reliable.


I am so honored to be featured in the “Shakers and Movers” interview series. In the spirit of the key theme of ‘Inspiring the next generation,’ this series aims to reach out and encourage young people to enter the field of spatial structures, as well as to motivate everyone involved in this field. Read more in the link below

JOURNAL PUBLICATION: Adjoint optimization of pressurized membrane structures using automatic differentiation tools

This paper presents an adjoint-based method for solving optimization problems involving pressurized membrane structures
subject to external pressure loads. Shape optimization of pressurized membranes is complicated by the fact that, lacking bending stiffness, their three-dimensional shape must be sustained by the internal pressure of the inflation medium. The proposed method treats the membrane structure as an immersed manifold and employs a total Lagrangian kinematic description with an analytical pressure–volume relationship for the inflating medium. To demonstrate the proposed method, this paper considers hydrostatically loaded inflatable barriers and develops an application-specific shape parametrization based on the analytical inhomogeneous solution for the inflated shape of cylindrical membranes. Coupling this shape parametrization approach with the adjoint method for computing the gradients of functionals enables a computationally efficient optimization of pressurized membrane structures. Numerical examples include minimization and minimax problems with inequality and state constraints,which are solved considering both plane strain and general plane stress conditions. The numerical implementation leverages the
high-level mathematical syntax and automatic differentiation features of the finite-element library FEniCS and related librarydolfin-adjoint. The overall techniques generalize to a broad range of structural optimization problems involving pressurized membrane and thin shell structures.

JOURNAL PUBLICATION: Shape optimization of arches for seismic loading

In this paper a novel method for the shape optimization of tapered arches subjected to in-plane gravity (selfweight) and horizontal loading through compressive internal loading is presented. The arch is discretized into beam elements, and axial deformation is assumed to be small. The curved shape of the tapered arch is discretized into a centroidal B-spline curve with beam elements. Constraints are imposed for allowable axial force and bending moment in the arch so that only compressive stress exists in the section. The computational cost for optimization is reduced, and the convergence property is improved by considering the locations of the control points of B-spline curves as design variables. The height of section is also modeled using a B-spline function. A section update algorithm is introduced in the optimization procedure to account for the contact and separation phenomena and to further speed up the computation process. The objective function to minimize the total strain energy of the arch under self-weight and horizontal loading. Numerical examples are presented that demonstrate the effectiveness of the proposed method. To validate the findings, the properties of the obtained optimal shapes are compared to shapes obtained by a graphic statics approach.

JOURNAL PUBLICATION: A data-driven computational scheme for the nonlinear mechanical properties of cellular mechanical metamaterials under large deformation

Cellular mechanical metamaterials are a special class of materials, whose mechanical properties are primarily determined by their geometry. But capturing the nonlinear mechanical behavior of these materials, especially with complex geometries and under large deformation can be challenging due to the inherent computational complexity. In this work, we propose a data-driven multiscale computational scheme to as a possible route to resolve this challenge. We use a neural network to approximate the effective strain energy density as a function of cellular geometry and overall deformation. The network is constructed by “learning” from the data generated by finite element calculation of a set of representative volume elements at cellular scales. This effective strain energy density is then used to predict the mechanical responses of cellular materials atlarger scales. Compared with direct finite element simulation, the proposed scheme can reduce the computational time up to two orders of magnitude. Potentially, this scheme can facilitate new optimization algorithms for designing cellular materials of highly specific mechanical properties.

PRESENTATION: Amortized Finite Element Analysis for Fast PDE-Constrained Optimization

Optimizing the parameters of partial differential equations (PDEs), i.e., PDE-constrained optimization (PDE-CO), allows us to model natural system from observations or perform rational design of structures with complicated mechanical, thermal, or electromagnetic properties. However, PDE-CO is often computationally prohibitive due to the need to solve the PDE—typically via finite element analysis (FEA)—at each step of the optimization procedure. In this paper we propose a mortized finite element analysis (AmorFEA), in a neural network learns to produce accurate PDE solutions, while preserving many of the advantages of traditional finite element methods. This network is trained to directly minimize the potential energy from which the PDE and finite element method are derived, avoiding the need to generate costly supervised training data by solving PDES with traditional FEA. As FEA is a variational procedure, AmorFEA is a direct analogue popular amortized inference approaches in latent variable models, with the finite element basis acting as the variational family. AmorFEA can perform PDE-CO without the need to repeatedly solve the associated PDE, accelerating optimization when compared to a traditional workflow using FEA and the adjoint method.

KEYNOTE: A different perspective on shells

I am very grateful that the organizers of the 1st Italian Workshop on Shell and Spatial Structures went on-line and gives us the opportunity to connect and discuss our research in a keynote.

FUNDING: Fabrication-Informed Design: Building Efficient Structures with Cooperative Robotic Fabrication Methods

The aim of the collaborative research is to develop design and construction techniques to build geometrically complex, but materially efficient, structural forms. Designing structures with cooperative robotic approaches in mind has the potential to reduce the amount of waste (e.g. scaffolding and off-cuts) that is generated during the construction process of these efficient but complex form-found structures.

MEDIA: The Times “Hi-tech sleuths reveal dome truths”

In one of today’s leading articles of the Times, our study on the role of the herringbone brick spirals in the stability of the Italian Renaissance Domes, like the Duomo in Florence, is discussed.

Generations of pilgrims, tourists and architecture students have marvelled at the splendour of Italy’s Renaissance domed churches and cathedrals. But as the visitors wondered, the secret of how these majestic buildings came to be has remained elusive.

Despite their fabulous wealth, popes and merchant princes did not care for the expense of traditional building methods, which would have used elaborate wooden frames to support brickwork spans — and in any case, timber was in short supply. So how were they built? Modern technology has finally provided the answer: it is all down to geometry.

Engineers have used computer analysis to show how the architect Antonio da Sangallo perfected a complex double-helix design to build domes without the usual temporary timber centering. more here

JOURNAL PUBLICATION: Statics of self-balancing masonry domes constructed with a cross herringbone spiraling pattern

The Brunelleschi herringbone pattern was certainly known by the Sangallo in the 16th century, who developed their own self-balanced construction technology for masonry domes based on the cross-herringbone spiraling pattern. Such technology was used for over one century in Italy to build masonry domes without shoring and formwork. However today it is not well known how this cross-herringbone spiraling pattern enables equilibrium states of self-balancing masonry domes. Therefore, in this study we demonstrate how this pattern permits equilibrium states of an octagonal masonry dome using two analysis approaches (i.e. Discrete Element Model and Limit State Analysis). The Discrete Element Model analysis has been performed to show the existence of the plate-bande resistance within the pattern. Even in the construction stages, these plate-bande resistance systems are capable of preventing sliding and overturning of the masonry dome. With the global self-balanced static equilibrium state proven, a Limit State Analysis is then adopted to estimate a possible thrust configuration needed to achieve equilibrium of the plate-bandes and the whole dome at each construction stage. It is shown that the value of the mortar friction has little influence on the static behavior of the dome. The study of the cross herringbone spiraling pattern does not merely serve historical or conservation purposes. It has practical application for the development of dry self-balanced robotic masonry construction technologies, particularly suited for unmanned aerial vehicles.

JOURNAL PUBLICATION: Occupant-centered optimization framework to evaluate and design new dynamic shading typologies

Dynamic solar shading has the potential to dramatically reduce the energy consumption in buildings while at the same time improving the thermal and visual comfort of its occupants. Many new typologies of shading systems that have appeared recently, but it is difficult to compare those new systems to existing typologies due to control algorithm being rule-based as opposed to performance driven. Since solar shading is a design problem, there is no single right answer. What is the metric to determine if a system has reached its optimal kinematic design? Shading solutions should come from a thorough iterative and comparative process. This paper provides an original and flexible framework for the design and performance optimization of dynamic shading systems based on interpolation of simulations and global minimization. The methodology departs from existing rule-based strategies and applies to existing and to complex shading systems with multiple degree-of-freedom mobility. The strategy for control is centered on meeting comfort targets for work plane illuminance while minimizing the energy needed to operate space. The energy demand for thermal comfort and work plane daylight quantity (illuminance) are evaluated with Radiance and EnergyPlus based on local weather data. Applied to a case study of three typologies of dynamic shading, the results of the methodology inform the usefulness and quality of each degree-of-freedom of the kinematic systems. The case study exemplifies the iterative benefits of the methodology by providing detailed analytics on the behavior of the shades. Designers of shading systems can use this framework to evaluate their design and compare them to existing shading systems. This allows creativity to be guided so that eventually building occupants benefit from the innovation in the field.


The research goal of the network is to develop an overarching methodology of systems, methods and processes based on the integration of mechanics, robotics, physics, material and computer science principles. Collectively, the network’s methodological insights and research findings are expected to uncover comprehensive approaches and contribute to a robust sustainable built environment.  The key components of our network are: (i) exchange opportunities for graduate students and faculty, (ii) a seminar series and workshop, (iii) co- teaching of a new undergraduate course “Origami and morphing structures”, (iv) joint journal and conference publications and (v) an annual industry advisory board meeting.

JOURNAL PUBLICATION: How and Why Laurent Ney Finds Steel Structural Forms

The talent, knowledge and approaches of the structural designer Laurent Ney (1964-present) are increasingly recognized by engineering, architecture and construction awards. Most of the writing on his work has focused on his design philosophy or on individual projects. The aim of this paper is threefold: 1) to provide a social, historic and geological context for his work, 2) to showcase how he masters digital and numerical shape finding and optimization approaches to inform his design and construction decisions and 3) to illustrate how his works revive underutilized public spaces and augment people’s happiness and well-being. The three chosen case studies are all large-span steel structures: one beam bridge (Centner) and two shell structures (steel/glass gridshell over the courtyard of the Dutch Maritime Museum and the hanging steel shell of the Knokke Lichtenlijn footbridge). The scholarship presented in this paper forms the basis for one of the contemporary lectures of CEE262 “Structures and the Urban Environment,” a course first taught by Prof. Billington in 1974 at Princeton University.


Connecting the technical and conceptual, the work of Anne Tyng stands out within and beyond the field of architecture. Through independent projects, in addition to her work with architects Louis Kahn and Pier Luigi Nervi, Tyng explored geometry as it relates to natural form and construction. She approached design as a process and profession through teaching and writing, addressing the social, psychological, and experiential dynamics of creativity and collaboration; her work has influenced other practitioners as well as models of practice. At the center of this conference is the question, “How do we position the legacy of an architect whose interests and methods remain relevant in contemporary discourse?” Anne Tyng: Ordered Randomness reconsiders established histories by tracing Tyng’s design approach through built and unbuilt works, and further explores continuing resonances of her work as found in current architectural and engineering practices.

Organized by Women in Design and Architecture (WDA), a graduate student group formed in 2014 at Princeton University School of Architecture, this annual conference celebrates the work and legacy of a pivotal female architect or designer with contributions from international historians and scholars, in addition to artists, curators, and practitioners.

FUNDING: NODES – Net Topology and Dance Exploration Systems

When dancers dynamically interacted with manmade nets in our choreographic piece “In*Tension” (Seattle, June 2019), the nets exhibited counterintuitive stiffness properties.  They stiffened under increased dancer impact loading and this phenomenon substantially differed for the orthogonal and bias net topologies. Historically, structural engineering has eschewed the design of structural nets, implicating their unsatisfactory stiffness as the cause of disastrous resonance and fatigue failures in, for example, cablenet building facades and impact net barriers.  We propose to harness the stiffening effect found in net topologies when they are driven into large displacements through dance to generate novel 3D resilient flexible systems nets with adaptive stiffness properties. The core concept is that, under external loading, the flexible net undergoes large displacements and stiffens, partially as a function the elastic stiffness of the net’s individual strands, but more as a function of its mesh shape, size, orientation, and arrangement.  Current structural net design occurs according to prescriptive guidelines that heavily restrict large displacements, while choreographic design focuses on expanding possibilities and here would maximize the dynamic interactions between the dancer and the net. The originality of the proposed research lies in harnessing the stiffening effect found in net topologies that are displaced unusually and extremely by dynamic dancer loading as a strategy to create resilient structural systems with force-dependent stiffness properties.

Our goals are to create choreographic works that generate a new understanding of how  different net topologies rigidify when loaded and soften when unloaded.

JOURNAL PUBLICATION: A 3-dimensional elastic beam model for form- finding of bending-active gridshells

In this paper, we present a 3-dimensional elastic beam model for the form-finding and analysis of elastic gridshells subjected to bending deformation at the self-equilibrium state. Although the axial, bending, and torsional strains of the beam elements are small, the curved beams connected by hinge joints are subjected to large-deformation. The directions and rotation angles of the unit normal vectors at the nodes of the curved surfaces in addition to the translational displacements are chosen as variables. Based on the 3-dimensional elastic beam model, deformation of an element is derived from only the local geometrical relations between the orientations of elements and the unit normal vectors at nodes without resorting to a large rotation formulation in the 3-dimensional space. Deformation of a gridshell with hinge joints is also modeled using the unit normal vectors of the surface. An energy-based formulation is used for deriving the residual forces at the nodes, and the proposed model is implemented within dynamic relaxation method for form-finding and analysis of gridshells. The accuracy of the proposed method using dynamic relaxation method is confirmed in comparison to the results by finite element analysis. The results are also compared with those by optimization approach for minimizing the total potential energy derived using the proposed formulation.

PRESENTATION: Lightweight Structures for a resilient urban environment


PUBLICATION: Effect of Gravity on the Scale of Compliant Shells

Abstract: Thin shells are found across scales ranging from biological blood cells to engineered large-span roof structures. The engineering design of thin shells used as mechanisms has occasionally been inspired by biomimetic concept generators. The research goal of this paper is to establish the physical limits of scalability of shells. Sixty-four instances of shells across length scales have been organized into five categories: engineering stiff and compliant, plant compliant, avian egg stiff, and micro-scale compliant shells. Based on their thickness and characteristic dimensions, the mechanical behavior of these 64 shells can be characterized as 3D solids, thick or thin shells, or membranes. Two non-dimensional indicators, the Föppl–von Kármán number and a novel indicator, namely the gravity impact number, are adopted to establish the scalability limits of these five categories. The results show that these shells exhibit similar mechanical behavior across scales. As a result, micro-scale shell geometries found in biology, can be upscaled to engineered shell geometries. However, as the characteristic shell dimension increases, gravity (and its associated loading) becomes a hindrance to the adoption of thin shells as compliant mechanisms at the larger scales-the physical limit of compliance in the scaling of thin shells is found to be around 0.1 m.

Find our full paper here