Positive Gaussian Curvature Research Articles

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

Inverse design of bistable composite laminates with switching tunneling method for global optimization

Bistability enables adaptive designs with tunable deflections for applications including morphing wings, robotic grippers, and consumer products. Composite laminates may be designed to exhibit bistability due to pre-strains that develop during the processing of the polymer matrix, enabling fast reconfiguration between two stable shapes. Unfortunately, designing bistable laminates is challenging because of their highly nonlinear behavior. Here, we propose the Switching Tunneling Method to address this challenge by alternating between gradient-based local minimization and tunneling search phases, with the enhancement of objective expression switching to improve numerical conditioning. Results demonstrate high effectiveness compared to existing optimizers; the Switching Tunneling Method achieves a 99% success rate in finding all energy minima across general composite layups. Additionally, our method facilitates the inverse design of variable pre-strain fields, enabling bioinspired, positive Gaussian curvatures, which are not possible with conventional pre-strain laminates. Validations through both finite element analysis and 3D printed samples confirm the optimal designs.

Shape prediction in continuous roll forming of sheet metal based on rigid rolls

The continuous roll forming based on rigid rolls (CRFRR) is an innovative process for rapid fabrication of doubly curved surfaces of sheet metal. The curved parts with different shapes and sizes can be obtained by adjusting the generatrix radius of the two rigid rolls or controlling the rolling reduction. In this paper, the principle of CRFRR is introduced and the process that a 3D surface of sheet metal with positive or negative Gaussian curvature is formed using two rigid rolls is analyzed based on the distribution of exit velocity. The exit velocity of material after the sheet metal passes through the roll gap is decomposed into two portions: a linearly distributed velocity and a very small additional velocity. On the basis of exit velocity decomposition and shallow shell theory, an analytical method for predicting the shape of formed surface in CRFRR process is proposed for the first time. The primary part of curvature of the formed 3D surface in longitudinal direction is determined by the linear portion of exit velocity and that in transverse direction is determined by the generatrix radius of the two rigid rolls. The curvature change in the final shape of 3D surface is induced by the additional portion of exit velocity and it is solved from equilibrium equation of shallow shell. The final curvature of the formed surface is acquired through superimposing the primary curvature and the curvature change. Numerical simulations and forming experiments have been carried out on convex and saddle-shaped parts, and the results confirm the validity and accuracy of presented method.

Fermion-antifermion pairs in a magnetized space-time with non-zero cosmological constant

We analyze a fermion-antifermion pair within the (2+1)-dimensional Bonnor-Melvin magnetic (BMM) spacetime featuring a nonzero cosmological constant (Λ>0). To do this, we rigorously investigate analytical solutions to the corresponding form of a fully-covariant two-body Dirac equation. We derive a non-perturbative wave equation and obtain an exact closed-form analytical solution for the system under consideration. In this magnetic background spacetime, described by a surface with positive Gaussian curvature, we noticed that fermion-antifermion pairs display a collective behavior resembling that of a single quantum oscillator, carrying a total rest mass of the particles and oscillating with a frequency (ωo∝Λ), even in the absence of mutual interaction between the particles. This indicates a compelling correlation between the cosmological constant and the energy spectrum of fermion-antifermion systems, implying connections between quantum realms and cosmology.

A physics-based tessellation algorithm for particle assemblies on arbitrary surfaces

Interacting particle assemblies embedded on a surface are often used to model biophysical systems and to study new colloidal materials. The configurations resulting from the particles interacting with each other and with their substrate affect the system's physical properties, which depend on local symmetries and defects. It is therefore important to identify the nature and location of defects in the particle assembly. This task is often achieved using either the Voronoi tessellation algorithm or order parameters. Although very useful, they present limitations, especially when the particle assemblies are embedded on irregular or highly deformed 3D surfaces. In this work, we present a novel algorithm to generate the tessellation of particle assemblies on 3D surfaces of arbitrary geometry. The algorithm is based on the particles' physical interactions and does not require a priori information on the surface geometry. The resulting cells in the tessellation represent each particle's interactions with its first ring of neighboring particles. The algorithm is tested using 2D and 3D surfaces, with or without periodic boundary conditions, with holes or fully covered by particles, of regular geometry or highly deformed shape, and in the presence of positive, zero, and negative Gaussian curvature. In all cases, the presented algorithm is capable of generating a tessellation representing the particle interactions and highlighting the location and nature of the assembly defects. Finally, the proposed algorithm is compared with both Voronoi and hexatic order parameters in several representative cases to highlight similarities with existing methods as well as the advantages of the newly proposed algorithm.

Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.

A multistable composite hinge structure

A composite tape-spring (CTS) structure is a thin-walled open slit tube with fibres oriented at ±45°, which is stable in both extended and coiled configurations. The governing factors of its bistability include composite constitutive behaviour, initial geometrical proportions, and geometrically non-linear structural behaviour. Its bistable principle can be employed to produce a flexible multistable hinge structure with tailorable stability. This is achieved by introducing variable stiffness design within a cylindrical shell structure, where folding stability is dependent on central functional patch region, and then connected to linking ploy regions. Thus, a novel multistable composite hinge structure can be designed with positive Gaussian curvature deformation, and its multistability is highly tailorable: a lengthy one-dimensional mechanical arm can be designed to coil and fold multiple times to enable large folding ratio. An analytical model was established based on the strain energy principle, in order to determine effects from functional tape length; the typical structural stability and stable configurations were then predicted with respect to regional length of the functional layer. It is found that the stability of a multistable composite hinge structure is dependent on geometry and combination of both the functional patch region, and connecting ploy region; the stable criteria are then proposed and show good agreement with experimental observations and FE analysis. These enrich the diversities of functional deployable structures to benefit novel requirements for various deployable mechanisms, and enable customised design, as well as smart driving for flexible and multifunctional mechanical composite hinge applications.

Negative Gaussian curvature regulated pattern evolution on curved bilayer system

Diverse wrinkling patterns may occur on curved bilayer system under equi-biaxial compression. Here we study the wrinkling pattern evolution on curved bilayer system with negative Gaussian curvature via theoretical analysis and numerical simulations. Different with curved system with zero Gaussian curvature or positive Gaussian curvature, critical buckling analysis demonstrates that negative Gaussian curvature doesn't delay the occurrence of the surface wrinkling. Fourier spectral method, finite element simulations and post buckling analysis based on an upper-bound method have revealed three different evolutionary paths of the wrinkling patterns with the increase of the excess stress. Selection of the evolutionary path is determined by the dimensionless curvature and curvature anisotropy. The results in this work help understand the profound effect of curvature on surface wrinkling of bilayer systems, and provide a new means to regulate the wrinkling pattern and its evolutionary path on soft materials.

Chirality effects in molecular chainmail.

Motivated by the observation of positive Gaussian curvature in kinetoplast DNA networks, we consider the effect of linking chirality in square lattice molecular chainmail networks using Langevin dynamics simulations and constrained gradient optimization. Linking chirality here refers to ordering of over-under versus under-over linkages between a loop and its neighbors. We consider fully alternating linking, maximally non-alternating, and partially non-alternating linking chiralities. We find that in simulations of polymer chainmail networks, the linking chirality dictates the sign of the Gaussian curvature of the final state of the chainmail membranes. Alternating networks have positive Gaussian curvature, similar to what is observed in kinetoplast DNA networks. Maximally non-alternating networks form isotropic membranes with negative Gaussian curvature. Partially non-alternating networks form flat diamond-shaped sheets which undergo a thermal folding transition when sufficiently large, similar to the crumpling transition in tethered membranes. We further investigate this topology-curvature relationship on geometric grounds by considering the tightest possible configurations and the constraints that must be satisfied to achieve them.

Electro-Mechanical Buckling of Shallow Segments of Functionally Graded Porous Toroidal Shell Covered by Piezoelectric Actuators

The current work deals with the mechanical buckling behavior of saturated porous toroidal shell segments sandwiched by piezoelectric actuator layers. Material properties of the porous shells with nonuniform distributed porosities are assumed to vary as a specific function of the thickness and porosity parameters. The nonlinear equations of the sandwich toroidal shell having either positive or negative Gaussian curvature are obtained on the basis of Biot’s poroelasticity theory. The energy method as the minimum potential energy principle is applied to derive the governing equations of the sandwich shells. The buckling equations of the piezo-porous sandwich shells under various types of mechanical loading are established by employing the adjacent equilibrium criterion. The governing equations as a set of the coupled partial differential equations are solved using an analytical method including the Airy stress function. Closed-form solutions are derived for shallow segments of the sandwich toroidal shell with positive/negative Gaussian curvature under lateral pressure, axial compression, and hydrostatic pressure loadings. Novel numerical results are presented in this work to show the effects of important factors such as the piezoelectric layer thickness, applied voltage, shell geometrical ratio, and variation of the porosity coefficient on the buckling behavior of these structures.

A variant of Caffarelli’s contraction theorem for probability distributions of negative powers

Let μ be a probability measure on Rd with barycenter at the origin, which is supported on an open, bounded and convex set K having smooth boundary and positive Gauss curvature. Suppose that dμx=ϱx−αdx, with α>d+1 and ϱ:K→0,+∞ is C∞-smooth, satisfies ɛ0-convex condition for some ɛ0>0 such that ϱ and its first derivatives are bounded in K. In this paper, we prove that for any convex solution ψ:Rd→0,+∞ to the following Monge–Ampère equation (1)[ϱ∇ψx]−αdet(∇2ψx)=(ψx)−α,x∈Rdthe function x↦logψx has second derivatives to be bounded by a constant Cɛ0 depending on ɛ0, α, ψ(x0) and values of first, second and fourth derivatives of ψ at a fixed point x0∈Rd. This estimate is as a variant case of the Caffarelli contraction theorem by replacing target densities of e−W by densities of ϱ−α that Cauchy distributions (i.e., ϱx−α=1+|x|2−α) are typical examples.

Numerical and experimental investigation of a variable-curvature shell class I flextensional transducer.

Class I flextensional transducers cannot form broadband characteristics because of the deep valley between their first- and second-order response curves. This valley arises from the sound pressure cancellation produced by the vibration in different areas of the shell. This paper examines the cause of the deep valley by analyzing the three-dimensional equivalent radiated sound field of the transducer. By solving the three-dimensional vibration equation of the shell of revolution and reducing its dimensionality, it can be concluded that its vibration state is related to the positive and negative curvatures in both main directions. Therefore, a variable-curvature shell class I flextensional transducer is proposed, and the second-order vibration of the transducer is changed by the variable-curvature structure composed of positive and negative Gaussian curvature surfaces. The proposed transducer avoids sound pressure cancellation and achieves broadband characteristics. After optimization, a transducer prototype is fabricated, and a pool test is conducted. The test results show that the transducer couples the first three vibration modes, attaining broadband characteristics with an in-band fluctuation of less than 8 dB and a bandwidth of more than 5.4 kHz in the axial and circumferential directions.

Level set function based on conformal geometry theory: An efficient approach for optimization of shell structure with arbitrary Gaussian curvature

Topology optimization based on the level set (LS) theory is of great significance to shell structure design. However, the Gaussian curvature of the shell structure can turn complex in many cases, and the establishing process of the LS function will become extremely cumbersome, which consumes overmuch time to solve the Hamilton-Jacobi(H-J) equation. In order to improve the efficiency of optimization, an LS function based on the conformal geometry theory (LSF-CGT) is proposed in this paper. Firstly, the shell structure with arbitrary Gaussian curvature is conformally mapped into 2D Euclidean space to generate the orthogonal mesh of the preset initial LS function. Then the barycentric interpolation method is used to reveal the transformation relationship between the orthogonal and shell structure mesh. Finally, the LS function of the orthogonal mesh is assigned to the mapped shell structure, and the LS function of the 3D shell structure is subsequently obtained. Three typical shell structures with positive, zero, and negative Gaussian curvature are used to verify the effectiveness of LSF-CGT in topology optimization. Results show that LSF-CGT can effectively reduce the strain energy of shell structures with arbitrary Gaussian curvatures by more than 30%. As a method for optimizing shell structures with arbitrary Gaussian curvatures, LSF-CGT can not only stably and quickly converge to the optimal structure in the 2D domain but also effectively maintain the initial geometric characteristics of the shell structure.

ІНФІНІТЕЗИМАЛЬНІ ДЕФОРМАЦІЇ ПОВЕРХОНЬ ІЗ ЗАДАНОЮ ЗМІНОЮ ТЕНЗОРА РІЧЧІ

In three-dimensional Euclidean space, we study the problem of the existence of an infinitesimal first-order deformation of single-connected regular surfaces with a predetermined change in the Ricci tensor. It is shown that for surfaces of nonzero Gaussian curvature, this problem is reduced to the study and solution of a system of seven equations (including differential equations) with respect to seven unknown functions, each solution of which determines a vector field that is a univariate function (with an accuracy of a constant vector) and can be interpreted as a moment-free stress state of equilibrium of a loaded shell. For regular surfaces of non-zero Gaussian and mean curvatures, the problem is reduced to finding solutions to one second-order partial differential equation with respect to two unknown functions. Given one of these functions, the resulting equation will in general be a nonhomogeneous second-order partial differential equation (nonhomogeneous Weingarten differential equation). It is proved that any regular surface of positive Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor in a sufficiently small region. In this case, the tensor fields will be represented by an arbitrary and predefined regular function. By considering the Neumann problem, it is shown that a single-connected regular surface of elliptic type of positive Gaussian and negative mean curvature with a regular boundary under a certain boundary condition admits, in general, an infinitesimal first-order deformation with a predetermined change in the Ricci tensor. In this case, the tensor fields will be determined uniquely. For surfaces of negative Gaussian and non-zero mean curvature, the resulting inhomogeneous partial differential equation with second-order partial differentials will be of hyperbolic type with known coefficients and right-hand side. The Darboux problem is considered for this equation. It is proved that any regular surface of negative Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor. Tensor fields are expressed through a given function of two variables and through two arbitrary regular functions of one variable. Keywords: infinitesimal deformation, Ricci tensor, tensor fields, Gaussian curvature, mean curvature.

Vibration of Conjugated Shell Systems Under Combined Static Loads

We propose a mathematical model of vibration of elastic systems formed by conjugated shells of revolution with different geometry in the field of combined static axisymmetric loads. The model is based on the concepts of geometrically nonlinear theory of mean bending and realized within the framework of the classical Kirchhoff–Love theory with the use of contemporary methods of applied mathematics and numerical analysis. The spectral picture of a shell structure with elements of positive, zero, and negative Gaussian curvature is constructed. This picture enables us to detect resonance situations under specific dynamic actions and determine dangerous combinations of static loads in the analysis of stability of the equilibrium states of the structure.

Programmable electric-field-induced bending shapes of dielectric liquid crystal elastomer sheets

The nematic director of dielectric liquid crystal elastomers (DLCEs) can be rotated by electric field, which makes the DLCE macroscopically deformed. In this paper, the electric-field-induced bending of DLCE sheets with director orientation gradient in thickness are investigated theoretically. The results reveal that the flat DLCE sheets in response to electric fields can deform to surfaces with zero, positive, or negative Gaussian curvature. Bending shapes of DLCE sheets can be encoded by patterning the nematic director field. Specifically, bending shapes with zero and positive Gaussian curvatures caused by unidirectional and multidirectional bending, respectively, are encoded by the splay–bend and splay–twist–bend director alignments, while the bending shape with negative Gaussian curvature is encoded by the twist or splay–twist–bend​ director alignments. Moreover, for each of these bending shapes, the Gaussian curvature and principal curvatures can also be maximized and optimized by adjusting the initial director alignments. The results imply that it is possible to program more complex bending shapes such as human face, in surface topography engineering, and this work provides a theoretical basis for promoting the application of LCEs as the electrically-triggered actuators and sensors.

Why we should study protein binding to bicontinuous inverted cubic phases, to better understand membrane trafficking.

Many proteins bind to membranes in a curvature-sensitive manner. This includes some adaptors/effectors in the vesicle budding and fusion stages of membrane trafficking. Recently, others showed that some proteins bind preferentially to membranes with Gaussian curvature: they bind more strongly to spherical vesicles (with positive Gaussian curvature) than to tubules with the same mean curvature but zero local Gaussian curvature. Fusion and fission pores (which are intermediates in vesicle fusion & budding) have special curvature properties. Here I show that the surfaces of such pores have curvature that is qualitatively and quantitatively different than in the membrane systems conventionally used to assess curvature-sensitive protein binding (membrane vesicles and tubules). Hence, existing methods may miss proteins that have evolved to bind to pore-relevant curvatures. However, the curvature properties of bicontinuous inverted cubic (QII) phases are quite similar to those of fusion/fission pores. Specifically, both QII phases and fusion/fission pores have negative Gaussian curvature, which is true of neither vesicles nor tubules; have more negative mean curvature than many tubules; and gradients in mean curvature that are either absent or smaller in vesicles and tubules. QII phases can be equilibrated with peptide through the aqueous phase: it has been shown that proteins with molecular weights of several tens of kDa can enter the water channel networks and penetrate into aggregates of QII phase. Thus, we might find new adaptors/effectors of trafficking by studying peptide or protein binding to preparations of QII phase. Some features of QII phases make such experiments awkward. Only certain ranges of lipid compositions form QII phases, and, in phospholipids, temperature-cycling techniques or particular electrolyte compositions may be needed. Appropriate protocols will be proposed.

Wrinkling and buckling instabilities of contractile suspended actomyosin sheets.

In many biological processes tissues and cells present active shape deformations resulting in wrinkling and buckling through contraction that result from internal forces generated by motor proteins. In many cases, the contracting material is also coupled to an elastic substrate, presenting a major difficulty for studying contraction induced folding. Here, we present the study of contraction and buckling of active, initially homogeneous, thin elastic actomyosin gels isolated from bounding surfaces. We find that the gels exhibit dimensionally dependent contraction that starts at the system boundaries, proceeds into the bulk, and eventually leads to spontaneous buckling of the sheet at the periphery resulting in different three-dimensional shapes at steady state. While thin sheets end up in wrinkled structures of mean negative Gaussian curvature whose configurations are of a single wavy mode that refines with decreasing thickness, thick sheets end up in domes with mean positive Gaussian curvature. Note that regardless of their distinct contraction dynamics and final shape, both thin and thick sheets show similar volume reduction, inferring on similar material properties of the contracted gel. Our system provides a well-controlled platform for studying contraction-induced folding of actively self-organizing gel sheets, in the absence of confinement.

Disrupting bacterial cytoskeletal protein interaction partners modulates geometric localization and alters cell shape.

The shape a bacterial cell assumes is an essential component in its ability to interact with its physical environment. For many bacterial species, shape is genetically encoded in the action of a small number of cell shape determining proteins. These proteins guide the building of a rigid peptidoglycan cell wall of a given geometry. We use a computational imaging framework to reconstruct the 3D shape of individual cells from fluorescence microscopy images and determine where a secondary fluorescent probe is enriched. In the straight-rod bacterium Escherichia coli, the bacterial actin MreB is enriched in areas with low Gaussian curvature, avoiding the poles. Disrupting MreB's interactions with the transmembrane protein RodZ alters MreB's geometric localization and cells lose their characteristic rod-like morphology. In the helical-rod bacterium Helicobacter pylori, the bactofilin CcmA is preferentially enriched at regions of small positive Gaussian curvature along the outer or major helical axis. Disrupting CcmA interaction partners results in less CcmA on the membrane, altered geometric localization of CcmA, and loss of the characteristic helical morphology of cells. These results are consistent with a model of cell shape determination that includes proper geometric localization of bacterial cytoskeletal proteins.

Curvature-Regulated Multiphase Patterns in Tori.

Biological functions in living systems are closely related to their geometries and morphologies. Toroidal structures, which widely exist in nature, present interesting features containing positive, zero, and negative Gaussian curvatures within one system. Such varying curvatures would significantly affect the growing or dehydrating morphogenesis, as observed in various intricate patterns in abundant biological structures. To understand the underlying morphoelastic mechanism and to determine the crucial factors that govern the patterning in toroidal structures, we develop a core-shell model and derive a scaling law to characterize growth- or dehydration-induced instability patterns. We find that the eventual patterns are mainly determined by two dimensionless parameters that are composed of stiffness and curvature of the system. Moreover, we construct a phase diagram showing the multiphase wrinkling pattern selection in various toroidal structures in terms of these two parameters, which is confirmed by our experimental observations. Physical insights into the multiphase transitions and existence of bistable modes are further provided by identifying hysteresis loops and the Maxwell equal-energy conditions. The universal law for morphology selection on core shell structures with varying curvatures can fundamentally explain and precisely predict wrinkling patterns of diverse toroidal structures, which may also provide a platform to design morphology-related functional surfaces.