Presence Of Geometrical Constraints Research Articles

Crystallography of honeycomb formation under geometric frustration

As honeybees build their nests in preexisting tree cavities, they must deal with the presence of geometric constraints, resulting in nonregular hexagons and topological defects in the comb. In this work, we study how bees adapt to their environment in order to regulate the comb structure. Specifically, we identify the irregularities in honeycomb structure in the presence of various geometric frustrations. We 3D-print experimental frames with a variety of constraints imposed on the imprinted foundations. The combs constructed by the bees show clear evidence of recurring patterns in response to specific geometric frustrations on these starter frames. Furthermore, using an experimental-modeling framework, we demonstrate that these patterns can be successfully modeled and replicated through a simulated annealing process, in which the minimized potential is a variation of the Lennard-Jones potential that considers only first-neighbor interactions according to a Delaunay triangulation. Our simulation results not only confirm the connection between honeycomb structures and other crystal systems such as graphene, but also show that irregularities in the honeycomb structure can be explained as the result of analogous interactions between cells and their immediate surroundings, leading to emergent global order. Additionally, our computational model can be used as a first step to describe specific strategies that bees use to effectively solve geometric mismatches while minimizing cost of comb building.

Schrödinger Equation with Geometric Constraints and Position-Dependent Mass: Linked Fractional Calculus Models

We investigate the solutions of a two-dimensional Schrödinger equation in the presence of geometric constraints, represented by a backbone structure with branches, by taking a position-dependent effective mass for each direction into account. We use Green’s function approach to obtain the solutions, which are given in terms of stretched exponential functions. The results can be linked to the properties of the system and show anomalous spreading for the wave packet. We also analyze the interplay between the backbone structure with branches constraining the different directions and the effective mass. In particular, we show how a fractional Schrödinger equation emerges from this scenario.

Phase-field modeling of constrained interactive fungal networks

Fungi develop structures that interact with their surroundings and evolve adaptively in the presence of geometrical constraints, finding optimal solutions for complex combinatorial problems. The pathogenic fungus Ophiocordyceps constitutes a perfect model for the study of constrained interactive networks. Modeling these networks is challenging due to the highly coupled physics involved and their interaction with moving boundaries. In this work, we develop a computational phase-field model to elucidate the mechanics of the emerging properties observed in fungal networks. We use a variational approach to derive the equations governing the evolution in time of the mycelium biomass and the nutrients in the medium. We present an extensive testing of our model, reproduce growing and decaying phenomena, and capture spatial and temporal scales. We explore the variables interplay mechanism that leads to different colony morphologies, and explain abrupt changes of patterns observed in the laboratory. We apply our model to simulate analogous processes to the evolution of Ophiocordyceps as it grows through confined geometry and depletes available resources, demonstrating the suitability of the formulation to study this class of biological networks.

Diffusion in Porous Media: Phenomena and Mechanisms

Two distinct but interconnected approaches can be used to model diffusion in fluids; the first focuses on dynamics of an individual particle, while the second deals with collective (effective) motion of (infinitely many) particles. We review both modeling strategies, starting with Langevin’s approach to a mechanistic description of the Brownian motion in free fluid of a point-size inert particle and establishing its relation to Fick’s diffusion equation. Next, we discuss its generalizations which account for a finite number of finite-size particles, particle’s electric charge, and chemical interactions between diffusing particles. That is followed by introduction of models of molecular diffusion in the presence of geometric constraints (e.g., the Knudsen and Fick–Jacobs diffusion); when these constraints are imposed by the solid matrix of a porous medium, the resulting equations provide a pore-scale representation of diffusion. Next, we discuss phenomenological Darcy-scale descriptors of pore-scale diffusion and provide a few examples of other processes whose Darcy-scale models take the form of linear or nonlinear diffusion equations. Our review is concluded with a discussion of field-scale models of non-Fickian diffusion.

Коррекция полета тяжелой материальной точки в среде с сопротивлением при наличии геометрических ограничений на дополнительные управления

Коррекция полета тяжелой материальной точки в среде с сопротивлением при наличии геометрических ограничений на дополнительные управления

Task constrained motion planning for 7-degree of freedom manipulators with parameterized submanifolds

PurposeThis paper aims to use the redundancy of a 7-DOF (degree of freedom) serial manipulator to solve motion planning problems along a given 6D Cartesian tool path, in the presence of geometric constraints, namely, obstacles and joint limits.Design/methodology/approachThis paper describes an explicit expression of the task submanifolds for a 7-DOF redundant robot, and the submanifolds can be parameterized by two parameters with this explicit expression. Therefore, the global search method can find the feasible path on this parameterized graph.FindingsThe proposed planning algorithm is resolution complete and resolution optimal for 7-DOF manipulators, and the planned path can satisfy task constraint as well as avoiding singularity and collision. The experiments on Motoman SDA robot are reported to show the effectiveness.Research limitations/implicationsThis algorithm is still time-consuming, and it can be improved by applying parallel collision detection method or lazy collision detection, adopting new constraints and implementing more effective graph search algorithms.Originality/valueCompared with other task constrained planning methods, the proposed algorithm archives better performance. This method finds the explicit expression of the two-dimensional task sub-manifolds, so it’s resolution complete and resolution optimal.

A minimization principle for deformation-diffusion processes in polymeric hydrogels: Constitutive modeling and FE implementation

This paper presents a recently developed, innovative minimization principle for coupled deformation-diffusion processes applied to hydrogels and compares this new structure with the classical saddle point formulation in both variational foundation and finite element implementation. First, balance equations and boundary conditions associated with dissipative fluid transport in solids undergoing large deformation are shown to be rooted in a canonical minimization formulation. This two-field principle determines the deformation rate and the fluid flux, constitutively governed by the scalar free energy and the dissipation potential functions. It can be used to derive the well-known saddle point formulation by a Legendre transformation of the dissipation potential. Next, the variational potential is transformed to its incremental counterpart by means of a discretization in time, which offers an intuitive and unconstrained discretization within the finite element method. To this end, vectorial Raviart-Thomas shape functions are chosen for flux degrees of freedom in order to fulfill the required H(Div,B) conformity. The need for this ansatz space can be interpreted as a counterpart to the LBB condition that arises within the saddle point principle and is usually addressed by mixed element formulations. However, we are able to demonstrate equivalent or superior performance of the minimization principle in several representative boundary value problems. Phenomena specific to hydrogels like diffusion-induced large volume change with instability patterns in the presence of geometrical constraints are successfully modeled. The proposed variational framework can thus be validated and its significance complementary to classical approaches is underlined, with the inherent symmetry of the coupled problem as a key feature and consequence of the minimization formulation.

Working under confinement

We analyze the performance of a Brownian ratchet in the presence of geometrical constraints. A two-state model that describes the kinetics of molecular motors is used to characterize the energetic cost when the motor proceeds under confinement, in the presence of an external force. We show that the presence of geometrical constraints has a strong effect on the performance of the motor. In particular, we show that it is possible to enhance the ratchet performance by a proper tuning of the parameters characterizing the environment. These results open the possibility of engineering entropically-optimized transport devices.

Strain hardening characteristics of deep drawing quality steels with geometric constraints

Strain hardening characteristics of two rolled deep drawing quality (DDQ) steels, namely, interstitial free (IF) and aluminium killed (AK), have been studied with and without the presence of geometric constraints at room temperature. Strain hardening exponent and rate are investigated for plain, centre holed and double notched sheet specimens for both longitudinal and transversal orientations. Four empirical relations based on total and incremental stress and strain components are considered to get a deeper insight into the deformation and workhardening behaviour of annealed sheets. Strain hardening exponents and yield strengths for steel with ultralow carbon contents (IF) are found in the range of 0·22–0·30 and 145–170 MPa respectively. The steel grade (AK) with higher carbon content (0·028%) and yield strength (174–197 MPa) exhibits strain hardening exponent around 0·18–0·27. Single and two-stage workhardening is observed for the IF and AK grades. Workhardening as a function of plastic strain follows the power law relation of the form workhardening rate (WHR) = α(ϵT)β . A transition from high to low hardening rate tends to occur for strains ranging from 0·05 to 0·10. An analytical model is formulated for theoretical estimation of strain hardening exponent from true strain data. The experimentally determined and the theoretically computed strain hardening exponent data match fairly well.On a étudié les caractéristiques d’écrouissage de deux aciers laminés de qualité emboutissage (DDQ), nommément sans interstitiel (IF) et calmé à l’aluminium (AK), avec et sans la présence de contraintes géométriques, à la température de la pièce. On examine l’exposant et la vitesse d’écrouissage pour des échantillons de tôle lisse, à trou central et avec entaille double, tant dans l’orientation longitudinale que transverse. On considère quatre relations empiriques basées sur les composantes de contrainte et de déformation, totales et incrémentales, afin d’obtenir une compréhension plus profonde du comportement de déformation et de durcissement des tôles recuites. La valeur des exposants d’écrouissage et les limites d’élasticité pour l’acier à teneur extrêmement basse en carbone (IF) se trouvent dans la gamme de 0·22–0·30 et 145–170 MPa, respectivement. La nuance d’acier (AK) avec une teneur plus élevée en carbone (0·028%) et une limite d’élasticité plus élevée (174–197 MPa) exhibe un exposant d’écrouissage autour de 0·18–0·27. On observe le durcissement simple ou en deux étapes pour les nuances sans interstitiel ou calmé à l’aluminium. Le durcissement en fonction de la déformation plastique suit une relation de loi de puissance de la forme WHR = α(ϵT)β. Une transition d’une vitesse élevée à basse du durcissement tend à se produire pour les déformations allant de 0·05–0·10. On formule un modèle analytique pour l’estimation théorique de l’exposant d’écrouissage à partir de véritables données de déformation. Les données d’exposant d’écrouissage déterminées expérimentalement et calculées théoriquement se rapprochent relativement bien.

Extrinsic curvature, geometric optics, and lamellar order on curved substrates

When thermal energies are weak, two-dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We present broad design principles to engineer the geometry of the underlying substrate so that a desired lamellar pattern can be obtained by self-assembly. Two distinct physical effects are identified as key factors that contribute to the interaction between the shape of the underlying surface and the resulting lamellar morphology. The first is a local ordering field for the direction of each individual layer, which tends to minimize its curvature with respect to the three-dimensional embedding. The second is a nonlocal effect controlled by the intrinsic geometry of the surface that forces the normals to the (nearly incompressible) layers to lie on geodesics, leading to caustic formation as in optics. As a result, different surface morphologies with predominantly positive or negative Gaussian curvature can act as converging or diverging lenses, respectively. By combining these ingredients, as one would with different optical elements, complex lamellar morphologies can be obtained. This smectic optometry enables the manipulation of lamellar configurations for the design of materials.

Electrochemical Waves on Patterned Surfaces: Propagation through Narrow Gaps and Channels

The dissolution and corrosion of iron and certain types of steel can involve the propagation of electrochemical waves. Low-carbon steel plates covered by nitric acid solutions self-organize pseudo-two-dimensional wave patterns which share many features with reaction−diffusion waves in excitable systems. Here, we investigate the dynamics of these electrodissolution waves in the presence of geometrical constraints. A simple procedure for the preparation of insulating, millimeter-scale obstacles is presented, which allows the controlled, local protection of the metal surface. This method is used in the measurement of the material consumption rate per wave [(850 ± 50) nm/wave or 6−7 g/(m2 wave)] as well as for the study of front propagation through narrow gaps and long channels. In channels, the wave velocity decreases with decreasing width from 10 to 4 mm/s. Propagation failures occur within the channel if its width is smaller than 0.2−0.4 mm. These results indicate that the obstacles have sink-like boundaries. For the estimation of the critical nucleation size in this system, we prepared short and narrow gaps that induce wave blockage for openings smaller than 0.65 ± 0.1 mm.

Dynamics of surface directed mesophase formation in block copolymer melts

The dynamic mean-field density functional method is adapted to describe phase separation in the presence of geometrical constraints. We observe that inclusion of small filler particles (such as rods) already has a dramatic effect on the morphology of polymer melts. The effect is comparable to the effect of applied simple steady shear. Mesostructures in the presence of large filler particles such as plates are totally governed by the geometry of the particle. Effects of polymer–surface interactions on morphology formation are investigated in detail.

A Separating Channel Decomposition Algorithm for Nonconvex Polygons With Application in Interference Detection

An algorithm for efficient decomposition of interface channels between nonconvex polygons in a Computer-Aided Design (CAD) environment is presented. This algorithm forms the computational basis for the solution of several design automation problems. In this paper, the channel decomposition algorithm is presented and applied to the problem of interference detection between nonconvex polygons. The resulting interference detection algorithm does not require preprocessing of the data and uses a simple data structure. In a companion paper (Ku and Ravani, 1989), the rigid channel decomposition algorithm is applied to the problem of model-based rigid-body guidance in presence of geometric constraints.

Molecular dynamics study of a lipid bilayer and a polymer liquid

We present molecular dynamics simulation of a lipid bilayer of 2 × 10 freely jointed chain molecules of 16 beads each. All beads interact with each other (via a Lennard-Jones potential) and with an aqueous medium. Terminal beads are bound near the bilayer surfaces by a harmonic potential. An algorithm for integrating the equations of motion of a molecular system in the presence of geometrical constraints, such as fixed bond lengths, is described. All results are compared with those obtained for an isotropic polymer liquid with the same Lennard-Jones interaction and density as for the bilayer. Equilibrium properties of the chains such as end-to-end distance, radius of gyration, molecular order parameters and pair correlation functions are computed. Dynamic properties considered include the time autocorrelation functions of various variables, the translational motion of chain beads and the space-time van Hove correlation functions. It is found that compared with the polymer liquid, the molecular motions in ...