Workshop on Nonlinear Bending II - Programme

The goal of the first part of this talk is to propose, analyze, and demonstrate experimentally an entirely new optical effect in which the centroid of a coherent optical beam can be designed to propagate along a curved trajectory in free space. This is accomplished by tailoring the spatial distribution of linear polarization across the transverse beam profile. The beam is viewed as a transport problem. Under this model, the complex electric field amplitude is governed by the Transport of Intensity Equation (TIE) which indicates the conservation of optical intensity. On the other hand, the phase gradient is seen to obey a momentum balance type equation. In the conservative form, the resulting model resembles Euler type equations with a third order dispersion term. Remarkably, a similar model arises in a completely different physical framework of Quantum Hydrodynamics (QHD). We will discuss well-posedness of this problem. To design a numerical method, we also derive an energy for this problem and establish that the energy is conserved. Direct numerical simulations for this problem are quite expensive due to the third-order dispersion term and, in some cases, even impossible. Keeping this difficulty in mind, we consider a relaxation of the bending problem, where the third-order term is replaced by a first-order term along with an additional Poisson equation that defines a new phase-field parameter. We propose a finite volume–finite difference numerical scheme and will also examine its merits and demerits through numerical experiments. Our numerical results are validated by the real lab experiments.

The second part of the talk continues the discussion of 'bending' waves, however, the motivation here is to design clocking of devices. The problem is formulated as an optimal control problem constrained by full time dependent Maxwell’s equations. The control problem seeks to find a localized current density to negate the impact of a localized electromagnetic source. The time-discrete problem is analyzed, and the necessary and sufficient optimality conditions are rigorously derived both at the discrete and continuous levels. The efficacy of the proposed approach is illustrated via various numerical examples.

Motivated by physical experiments, we study a variational problem that models the transition between flat and wrinkled regions in a thin elastic sheet. We begin by describing the model and reviewing some related results. Our main focus is an asymptotic analysis, via $\Gamma$-convergence, in the limit as the sheet thickness tends to zero.

The limiting problem is scalar and convex, but subject to constraints and formulated in terms of measures. To establish the lower bound, we first pass to quadratic variables, which renders the constraint linear, and then apply Reshetnyak’s theorem. For the upper bound, we construct a recovery sequence by mollifying the quadratic variables and carefully selecting multiple construction parameters.

Finally, for the limiting problem, we prove the existence of a minimizer and establish energy equipartition across frequencies. This talk is based on joint work with Roberta Marziani (Siena).

Liquid crystal elastomers (LCEs) combine the large deformation response of a cross-linked polymer network with the nematic order of liquid crystals pendent to the network. The focus of this talk is the actuation of LCE sheets where the nematic order, modeled by a director field, is specified heterogeneously in the plane of the sheet. Heating such a sheet leads to a large spontaneous deformation, coupled to the director design through a well-known metric constraint. We derive a plate theory for the pure bending deformations of patterned LCE sheets in the limit that the sheet thickness tends to zero using the framework of $\Gamma$-convergence. After dividing the bulk energy by the cube of the thickness to set a bending scale, we show that all limiting midplane deformations with bounded energy at this scale satisfy the aforementioned metric constraint. We then identify the energy of our plate theory as an ansatz-free lower bound of the limit of the scaled bulk energy, and construct a recovery sequence that achieves this plate energy for all smooth enough midplane deformations. We then apply the plate theory to some known examples. This is joint work with David Padilla-Garza (Einstein Institute of Mathematics) and Paul Plucinsky (University of Southern California).

We study the length-preserving elastic flow in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolu- tion equation with nonlinear higher-order boundary conditions. We discuss how a suitable non-flatness assumption ensures global existence and subconvergence to critical points.

This is joint work with Manuel Schlierf.

Topology and analysis have always been connected. Especially, maintaining the topology of a manifold while deforming it proves to be a quite delicate task.

We have used fractional curvature energies in order to encode self-repulsiveness into a Riemannian metric on the manifold of embedded curves.

In my talk, I will introduce the tangent-point energy and another knot-energy. From there, I will demonstrate how we used the special features of the TP-Energy in order to design a self-repulsive Riemannian metric. The self-repulsiveness w.r.t. the footpoint induces a strong non-linearity in the operator governing the metric. I finish by illustrating some technical difficulties and how we were able to overcome them.

We propose a variational model for two-phase surfactant films separating aqueous and oily fluids. Considering two species of surfactant molecules we describe a phase seperation within the film. The analysis builds on a model for single species lipid biomembranes proposed by Peletier and Röger [ARMA 2009]. We prove a Gamma-convergence result in the limit of vanishing surfactant length and show that the limit inherits a phase separation and a bending energy.

The talk will be about regularity and structure results for $p$-elasticae in $\mathbb{R}^n$, with arbitrary $p\in (1,\infty)$ and $n\geq2$.  In particular, we show that every non-planar $p$-elastica is analytic and three-dimensional, with the only exception of flat-core solutions of arbitrary dimensions. Subsequently, we classify pinned $p$-elasticae in $\mathbb{R}^n$ and, as an application, establish a Li--Yau type inequality for the $p$-bending energy of closed curves in $\mathbb{R}^n$.  This extends previous works for $p=2$ and $n\geq2$ as well as for $p\in (1,\infty)$ and $n=2$.

In order to capture non-classical size effects in bending problems, a frame indifferent Cosserat shell model with curvature has been derived in several works by Neff et al. The model proposes an energy minimization problem for a deformation and an independent field of microrotations. Thus, the solutions take their values in a nonlinear space, namely $\mathbb{R}^3\times SO(3)$.

As the nonlinearity of the target poses challenges for standard finite elements, geometric finite elements have been introduced by Sander et al. for Riemannian manifolds in general, and deployed for the discretization of the Cosserat shell model. A generalization of the standard Galerkin-based numerical analysis for second-order elliptic problems has been provided for the general methods, but the Cosserat energy does not fulfill the strong assumptions needed for the theory to be applied directly.

In this talk I will give an overview of geometric finite element theory and show optimal a priori error estimates for the Cosserat shell model.

An elastic surface resists not only changes in curvature but also tangential stretches and shears. In classical plate and shell theories, e.g., due to von Karman, the latter two strain measures are approximated infinitesimally. Such models are inadequate for describing the behavior of highly deformable elastic surfaces. In particular, we motivate our approach via the phenomenon of wrinkling in stretched elastomer sheets. We postulate a novel, physically reasonable class of stored-energy densities, and we prove various existence theorems based on the direct method of the calculus of variations.

In this talk, we are interested in the numerical approximation of a both non-local and non-smooth convex minimization problem that allows to determine the optimal distribution, given by $h\colon \partial\Omega\to [0,+\infty)$, of a given amount $m\in \mathbb{N}$ of insulating material attached to the boundary $ \partial\Omega$ of a thermally conducting body $\Omega\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$. The non-local and non-smooth convex minimization problem is obtained as a model reduction (in the sense of $\Gamma$-convergence) of an extenstion of the physical setting in the work on the 'thin insulation' case by Buttazzo (1988) to polyhedral domains. This is joint work with Harbir Antil and Keegan L. A. Kirk (George Mason University)

In order to tackle the non-local and non-smooth character of the problem, we resort to a (Fenchel) duality~framework:

(a) At the continuous level, using (Fenchel) duality relations, we derive an a posteriori error identity that can handle arbitrary admissible approximations of the primal and dual formulations of the non-local and non-smooth convex~minimization~problem;

(b) At the discrete level, using discrete (Fenchel) duality relations, we derive an a priori error identity that applies to a Crouzeix-Raviart discretization of the primal formulation and a Raviart-Thomas discretization of the dual formulation. The proposed framework leads to error decay rates that are optimal with respect to the specific regularity of a minimizer. Since the discrete dual formulation can be written as a quadratic program, it is solved using a primal-dual active set strategy interpreted as semi-smooth Newton scheme. Then, a solution of the discrete primal formulation is reconstructed from the solution of the discrete dual formulation by means of an inverse generalized Marini formula.

When the simple support boundary value problem on a domain with curved boundary parts is approximated by a sequence of problems on polygonal domains, convergence to the wrong limit may occur. A possible remedy for this so-called Babuška's paradox is to relax the boundary condition to only be satisfied in corner points. For this strategy, we derive error estimates in the energy norm, which are in this generality applicable to conforming and nonconforming elements. We further discuss potential extensions to the nonlinear folding model.

The bending energy of a surface is the square of the $L^2$ norm of its second fundamental form. It arises naturally in plate and shell theories. Notably, a “very thin” non-Euclidean shell will assume a configuration that minimizes its bending energy among all its isometric immersions in $R^3$. A natural question arises: given the /intrinsic/ geometry of a surface, bound from below its /extrinsic/ bending energy. In this lecture I will report a new lower bound in the form of a negative Sobolev norm of the (intrinsic) Gaussian curvature. Applications to cones and E-cones will be demonstrated, as well as analysis of surfaces endowed with a geometry describing an edge-dislocation (corresponding to a curvature dipole).

This is a joint work with Cy Maor and David Padilla-Garza.

Soft machines take advantage of their compliance to transmit motion and perform mechanical work through a monolithic design, without the need for assembling multiple rigid parts. However, when it comes to motion control, this compliance presents challenges, as control algorithms become increasingly complex in nonlinear systems. Nevertheless, embracing the nonlinear compliance of soft machines enables the replacement of software-based control with physical control, where not only the machine but also the controller is composed of soft mechanisms. In particular, nonlinear bending phenomena that lead to reversible elastic instabilities (e.g., buckling, snapping, kinking, ballooning, etc.) serve as excellent building blocks for designing simple physical controllers capable of generating low-level, self-regulating behaviors. These physical controllers can produce oscillatory patterns to enable soft robots to walk, encode actuation sequences for playing a piano, or close reactive sensorimotor loops that trigger state changes.

We consider a thin elastic sheet with a finite number of disclinations in a variational framework in the Föppl-von Kármán approximation. Under the non-physical assumption that the out-of-plane displacement is a convex function, we prove that minimizers display ridges between the disclinations. We prove the associated energy scaling law with upper and lower bounds that match up to logarithmic factors in the thickness of the sheet. One of the key estimates in the proof that we consider of independent interest is a generalization of the monotonicity property of the Monge-Ampère measure.

Joint work with Peter Gladbach

Tape springs, which are slender metallic strips featuring a pre-curved cross-section, serve as an appealing structural option and hinge mechanism for deployable structures due to their lightweight design, affordability, and overall simplicity. For these reasons, the study of these devices has been particularly prolific in recent years.

In this talk, we model tape spring devices using shell theory and analyze their asymptotic behavior as the width of the cross-section approaches zero.

The talk is based on ongoing work with Marco Picchi-Scardaoni.

We present a surface model for the evolution of biological thin structures, that couples energy dissipation via viscous stress and a bending energy, under the constraints of local inextensibility and conservation of enclosed volume. Additionally, self-intersection of the surface is penalized with a non-local tangent-point energy. We discuss the numerical implementation for higher order surface finite elements and their benefits for the treatment of the non-local terms. We present simulations in the low volume regime, specifically inversion of a sphere induced by localized area-growth, which relates to the biological process of gastrulation.

The elastic flow for curves is one of the most important examples of fourth order geometric flows and it has been extensively studied in the past years. From a numerical point of view it is well known that detrimental grid deformations might occur as the curve undergoes strong deformations. In this talk I will revisit the definition of elastic flow and propose an alternative formulation whose FEM discretization provides good grid properties while being amenable for error analysis. After  addressing  important analytical aspects of the  flow, I will present  the numerical scheme and provide ideas for the error analysis.

Deforming a given object into another equivalent configuration while maintaining its topological structure throughout this process is a central task that arises for instance in engineering and computer science. In particular, one seeks to avoid self-intersections, pull-tights and pinch-offs.

As a model case for this situation we study the manifold of embedded curves. We propose a Riemannian metric which not only ensures that curves maintain the topology on paths of finite length but also characterizes optimal deformations as pathlength minimizers. This metric is inspired by the so-called tangent-point energy which grants impermeability, e.g., in physical simulations with self-contact.

In the case of infinite dimensional manifolds, metric completeness does not necessarily imply the existence of pathlength minimizers. Consequently, we will give a separate argument to show that there are minimizing geodesics between every pair of curves in the same path component. (joint work with Elias Döhrer and Henrik Schumacher)

In this talk, an overview is given over two works regarding the dimension reduction of models involving nonlinear elasticity to obtain an effective bending model for rods. We discuss two problems that involve a time-dependence: elastoplasiticty and viscoelasticity. These two problems are representative for the study of rate-independent and rate-dependent gradient systems in this context. The main achievement is the construction of a recovery sequence. In essence, for the gradient systems a recovery sequence is needed that interpolates between two limiting states which is not trivial as the state space in the bending regime is not a vector space. To regularize the finite energy, we introduce suitable strain gradient terms, which vanish in the limit.

Energy methods are a vital tool in deriving estimates in PDEs, especially in higher-order problems where maximum principle techniques are not available. However, if the canonical energy associated with a gradient flow is always infinite, the formal "energy identity" provides no additional information. We discuss this phenomenon in the context of the elastic flow of curves—a gradient flow involving the Euler–Bernoulli bending energy. Our key idea is to reinterpret the length term as a measure of tangent alignment, leading to a new, geometrically meaningful energy that can remain finite even for non-compact curves. For initial data with finite modified energy, we establish global existence and sharp convergence results. We also discuss applications to the curve shortening flow.

This is joint work with T. Miura (Kyoto).

For spherical topology and for tori of revolution, the asymptotic behavior of the Willmore flow is well-understood below a critical initial-energy-threshold. In this talk, we focus on surfaces with boundary. First, we present results for the Willmore flow with Dirichlet boundary data if the initial surface is axially symmetric and topologically a cylinder. Here the critical initial-energy-threshold depends on the boundary data. Then we exemplarily study a free boundary problem to discuss asymptotic stability for Willmore flows with higher-order non-linear boundary data.

An important aspect of the numerical approximation of the elastic flow of inextensible curves is the discretization of the inextensibility constraint. A standard approach is to enforce this constraint only at the nodes of the interval decomposition which leads to suboptimal convergence rates for the semi-discrete elastic flow. In this talk we explain why this happens and provide an alternative discretization of the constraint that yields quasi-optimal convergence rates. We discuss the construction of test functions for these problems, derive an error estimate for the improved constraint and verify it experimentally using numerical simulations.

Mechanical metamaterials are artificial materials that exhibit unusual properties due to their internal architecture. Their overall unorthodox behavior is primarily determined by the geometry and properties of the elementary building blocks -- so-called unit cells. In lattice-based metamaterials, these unit cells form a periodic lattice, and traditionally, the lattice nodes are connected by straight beams. However, with recent advances in additive manufacturing, there is no longer a compelling reason to restrict the beams to straight shapes. As we demonstrate, replacing straight elements with curved ones opens up additional possibilities for programming the response of metamaterials in both static and dynamic regimes. Using AI-driven approaches and generative models, we showcase a wide variety of designs that leverage curved-beam geometries.

We present a hybrid method combining a minimizing movement scheme with neural operators for the simulation of phase field-based Willmore flow. The minimizing movement com-ponent is based on a standard optimization problem on a regular grid whereas the functional to be minimized involves a neural approximation of mean curvature flow proposed by Bretin et al. Numerical experiments confirm stability for large time step sizes, consistency and significantly reduced computational cost compared to a traditional finite element method. Moreover, applications demonstrate its effectiveness in surface fairing and reconstructing of damaged shapes. Thus, the approach offers a robust and efficient tool for geometry processing.

Natural material systems developed over 3.8 billion years of evolution within various living organisms. These systems are divers, complex and excellent adapted to their environment and serve as concept generators for bioinspired materials systems. Transferring the functions of animated nature into technical applications, enables the creation of novel functions immanent to materials system such as embodied intelligence, and embodied energy. A class of bioinspired systems draw their inspiration from various plant movement strategies as role models utilizing e.g. principles of carnivorous snap-trap plants for hinge-less movements in technical applications and principles of pine cone scales for environmental responsive autonomous shading systems. Within our project, novel materials systems are being developed that show dynamic, life-like and non-equilibrium motion features. The combination of bioinspired design principles and advanced fabrication techniques opens up new avenues for the development of autonomous systems that can operate in complex and dynamic environments, with potential applications in areas such as search and rescue, environmental monitoring and exploration. Overall, these bioinspired and biomimetic soft autonomous systems utilize natural motion principles for efficient and autonomous motion in technical system. These systems and the conceptual approach represent a way to learn from nature and go beyond towards the improvement of human made technical systems.

This talk presents some methods for computing 4th order elliptic PDEs and surfaces, as well as for computing curvature.

We give a quick review of the surface version of the Hellan-Herrmann-Johnson (HHJ) method for solving the surface Kirchhoff plate problem with numerical examples.  We also describe a post-processing technique based on the surface HHJ method that can be used to compute the surface Hessian of a Lagrange finite element function on discrete meshes.

Moreover, the scheme can be used to compute convergent approximations of the full shape operator from discrete meshes, even for piecewise linear surface triangulations.  Several numerical examples are given on non-trivial surfaces to illustrate the method.  We then apply this method in an optimization setting for smoothing surfaces and computing minimal bending energy shapes.  Several numerical examples show the efficacy of the method.