Rod Model Research Articles

The shape of the mitral annulus: A hypothesis of mechanical morphogenesis

This paper investigates the role of mechanics in the morphogenesis of the annulus of the mitral valve. We represent the annulus in its embryonic stage as an elastic ring and we perform a mechanical simulation of the development process applying a distributed torque on the rod: because of the mechanical action of the other growing cardiac chambers on the atrio-ventricular region, it departs from a planar circular shape. The numerical integration of the mathematical rod model subject to a bending load yields a shape very near to the one reported in the medical literature as anatomical reference for healthy patients. To make the comparison quantitative, we illustrate a numerical approach to match two curves in 3D defining their distance in a proper mathematical way. Such a methodology is first applied to compare the annular shape resulting from the mechanical model with an anatomical reference “master” shape and it is then applied to set to clinical data extracted from MRI for a cohort of healthy patients. The good agreement among anatomical master description, numerical mechanical model, and clinical data supports our speculation about a possible role of mechanics in determining the shape of the mitral valve.

Spatially-dependent model for rods and cones in the retina

We develop a mathematical model for photoreceptors in the retina. We focus on rod and cone outer segment dynamics and interactions with a nutrient source associated with the retinal pigment epithelium cells. Rod and cone densities (number per unit area of retinal surface) are known to have significant spatial dependence in the retina with cones located primarily near the fovea and the rods located primarily away from the fovea. Our model accounts for this spatial dependence of the rod and cone photoreceptor density as well as for the possibility of nutrient diffusion. We present equilibrium and dynamic solutions, discuss their relation to existing models, and estimate model parameters through comparisons with available experimental measurements of both spatial and temporal photoreceptor characteristics. Our model compares well with existing data on spatially-dependent regrowth of photoreceptor outer segments in the macular region of Rhesus Monkeys. Our predictions are also consistent with existing data on the spatial dependence of photoreceptor outer segment length near the fovea in healthy human subjects. We focus primarily on the healthy eye but our model could be the basis for future efforts designed to explore various retinal pathologies, eye-related injuries, and treatments of these conditions.

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation (slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.

Inverse design of polymorphic Dirac-like cone dispersion relationship in photonic crystals

Dirac-like cone linear dispersion relations in photonic crystals (PhCs) often endow them with unique properties, yet searching for such relations can be challenging. We introduce a generalized inverse design system that, given the dielectric constants and lattice of two-dimensional PhCs, can efficiently determine its structural parameters to obtain its Dirac-like cone dispersion. Employing this inverse design strategy, we design three types of Dirac cone PhCs, including triple degenerate, quadruple degenerate, and triple degenerate under dual polarization with the same frequency. Further investigations reveal a systematic relationship between the radius of the dielectric rods in these PhCs and their corresponding Dirac frequencies across varying dielectric constants. The zero refractive index characteristic is validated in two of the three PhCs studied, as confirmed through numerical simulations. Additionally, by leveraging our proposed inverse design method, we introduce an innovative shell dielectric rod model, which encapsulates a dielectric material, achieving a quadruple degenerate dispersion structure with dual Dirac cones. This research provides a potent tool for the inverse design of PhCs and expands its application in Dirac cone dispersion design.

Stiffness modelling and analysis of soft fluidic-driven robots using Lie theory

Soft robots have been investigated for various applications due to their inherently superior deformability and flexibility compared to rigid-link robots. However, these robots struggle to perform tasks that require on-demand stiffness, that is, exerting sufficient forces within allowable deflection. In addition, the soft and compliant materials also introduce large deformation and non-negligible nonlinearity, which makes the stiffness analysis and modelling of soft robots fundamentally challenging. This paper proposes a modelling framework to investigate the underlying stiffness and the equivalent compliance properties of soft robots under different configurations. Firstly, a modelling and analysis methodology is described based on Lie theory. Here, we derive two sets (the piecewise constant curvature and Cosserat rod model) of compliance models. Furthermore, the methodology can accommodate the nonlinear responses (e.g., bending angles) resulting from elongation of robots. Using this proposed methodology, the general Cartesian stiffness or compliance matrix can be derived and used for configuration-dependent stiffness analysis. The proposed framework is then instantiated and implemented on fluidic-driven soft continuum robots. The efficacy and modelling accuracy of the proposed methodology are validated using both simulations and experiments.

Конечно-элементная модель перегрузочного крана КПП 16/32т для перевалки грузов в речных и морских портах

Transshipment cranes of the checkpoint of the Lenpodyemtransmash plant are widely used in river and sea ports, at industrial enterprises for grappling transshipment of bulk cargo and for working with piece loads or containers, as well as for transshipment of scrap metal and rolled metal using lifting electromagnets. It is proposed to eliminate the uncertainties caused by the incompatibility of one-dimensional (rod) and two-dimensional (plate) finite elements having a different number of degrees of freedom, 12 and 20, when developing digital twins of portal cranes of the Lenpodyemtransmash plant on the example of the CHECKPOINT 16/32t crane by approximating axisymmetric shells of the portal head with rod finite elements. The method is based on the equality of the potential shear energy of the finite element model of the plate and the potential energy of the approximating rod model, the elements of which work on tension/compression, which made it possible to move from the finite element model of the GEARBOX 16/32t reloading crane, built on the basic finite elements of Kirchhoff plates, to the finite element model of the crane based on the core base finite elements. A mathematical model of a matrix equation of static equilibrium with many degrees of freedom is proposed, the numerical solution of which by the Gauss method allows us to proceed to six-component internal forces in each finite element due to the corresponding vector of external loads reduced to the degrees of freedom of the calculated finite element model. The possibility of inter-pretation of bearing elements of spatial metal structures of portal cranes by both one-dimensional rod and two-dimensional plate finite elements is investigated, a re-evaluation of traditional views on the computational analysis of the stress-strain state of metal structures of portal cranes in the field of reliable determination of their bearing capacity during operation in ports, especially in the field of assessing post-repair risk analysis and the resource of reloading cranes is proposed, those who have completed the standard service life in the port and have undergone capital repairs. A calculated rod finite element idealized model of a PPC 16/32t gantry crane for transshipment of cargo in river and sea ports is constructed, demonstrating the transition from a real design to a calculation model based on geometric principles of discretization of both rod and continuum elements by the basic rod end elements of the crane, as a result of which the digital calculation model should be detailed and complex, like a system with many degrees of freedom. The algorithm of application of the digital calculation model of the crane is given on the example of static finite element calculation analysis of the KPP 16/32t crane.

Smoothing Splines for Discontinuous Signals

Smoothing splines are twice differentiable by construction, so they cannot capture potential discontinuities in the underlying signal. In this work, we consider a special case of the weak rod model of Blake and Zisserman that allows for discontinuities penalizing their number by a linear term. The corresponding estimates are cubic smoothing splines with discontinuities (CSSD) which serve as representations of piecewise smooth signals and facilitate exploratory data analysis. However, computing the estimates requires solving a non-convex optimization problem. So far, efficient and exact solvers exist only for a discrete approximation based on equidistantly sampled data. In this work, we propose an efficient solver for the continuous minimization problem with non-equidistantly sampled data. Its worst case complexity is quadratic in the number of data points, and if the number of detected discontinuities scales linearly with the signal length, we observe linear growth in runtime. This efficient algorithm allows to use cross-validation for automatic selection of the hyperparameters within a reasonable time frame on standard hardware. We provide a reference implementation and supplementary material. We demonstrate the applicability of the approach for the aforementioned tasks using both simulated and real data. Supplementary materials for this article are available online.

Analysis of flatness and critical crown of hot-rolled strip based on thermal–mechanical coupled residual stress analytical model

Residual stress is introduced during hot-rolled strip production due to various factors, such as temperature distribution, crown change, and metal transverse flow. Excessive residual stress can cause flatness defects in the strip, while insufficient residual stress may result in deformation during subsequent cutting. To explore the mechanism of residual stress formation in hot-rolled strip and how to control it, an analytical model for the coupling of temperature, crown and residual stress is established. Firstly, based on the symplectic space Hamiltonian system, an analytical solution for the strip cross section temperature field in each inter-stand zone is obtained. Then the residual stress analytical model is derived based on the fiber rod model combined with the law of volume invariance, lateral metal flow function and high temperature stress relaxation phenomenon. Meanwhile, to verify the accuracy of the analytical model, a multi field coupled finite element model based on the measured data validation is established. The calculation results of the analytical model are in good agreement with the finite element model. Furthermore, the buckling model of strip in symplectic space Hamiltonian system is derived, and the critical buckling stress of different strip flatness defects is directly calculated. Finally, the residual stress analytical model is combined with critical buckling stress to calculate the critical exit crown ratio to keep the strip flat.

Concentric Tube Robot Redundancy Resolution via Velocity/Compliance Manipulability Optimization.

Concentric Tube Robots (CTR) have the potential to enable effective minimally invasive surgeries. While extensive modeling and control work have been proposed in the past decade, limited efforts have been made to improve the path tracking performance from the perspective of manipulability, which can be critical to generate safe motion and feasible actuator commands. In this paper, we propose a gradient-based redundancy resolution framework that optimizes velocity/compliance manipulability-based performance indices during path tracking for a kinematically redundant CTR. We efficiently calculate the gradients of manipulabilities by propagating the first- and second-order derivatives of state variables of the Cosserat rod model along the CTR arc length, reducing the gradient computation time by 68% compared to the finite difference method. Task-specific performance indices are optimized by projecting the gradient into the null-space of path tracking. Simulation results show that the proposed method is able to accomplish the required tasks while commonly used redundancy resolution approaches underperform or even fail.

Regularity of a Parabolic Differential Equation on Graphs

In this paper, motivated by a biodegradable stent problem, we consider a parabolic differential equation on graphs. This kind of equation describes the stent material degradation in time. Since stents are mesh-like structures with thin struts, they can be modelled using a simple structure called the one-dimensional curved rod model. In this way, we obtain a graph-like domain for our parabolic equation. Here, we prove the regularity estimate for the unique solution of the equation, together with corresponding estimates.

The Effect of the Traction Rod on the Vertical Vibration Behavior of the Railway Vehicle Carbody

Although research has shown that through the additional rigidity introduced in the secondary suspension, traction rods can affect the vertical dynamic performance of railway vehicles, this topic has been less studied by researchers in the field. In this paper, the effect of a traction rod on the vertical vibration behavior of a carbody of a railway vehicle is analyzed, using the results obtained through numerical simulations. Numerical simulation applications are developed based on a vehicle model, where the vehicle carbody is modeled using a free-free equivalent beam Euler–Bernoulli, and the bogie chassis and wheelsets are represented by rigid bodies linked together by Kelvin-Voigt systems that model the secondary suspension and the primary suspension. The novelty element of this paper is found in the model of the traction rod. This includes traction rod damping, which has been neglected in previous research. The stiffness and damping of the traction rod are represented by a longitudinal Kelvin–Voigt system integrated into a secondary suspension model. The effect of the traction rod on the vertical vibration behavior of the vehicle carbody is analyzed based on the power spectral density of the acceleration, the root mean square of acceleration, and the ride comfort index, for three cases for analysis: a ‘without traction rod’ case, a ‘with traction rod—with damping’ case, and a ‘with traction rod—without damping’ case. The conclusions of the paper highlight the influence of the stiffness and damping of the traction rod on the vibration level of the carbody, especially in its middle. Depending on the stiffness of the traction rod, significant increases in the ride comfort index are obtained, which at high velocities can exceed 300%. Damping of the traction rod reduces the ride comfort index by up to 10%.

Analytical Computation of the Contact Force Jacobian for MRI-Actuated Robotic Catheter.

Contact force Jacobian relates the changes in the contact force to the changes in the actuation of a robotic catheter in contact with a surface. In this paper, we present an analytical method for calculating the contact force Jacobian for the Cosserat rod model of an MRI-actuated robotic catheter. First, the Cosserat rod model of the MRI-actuated robotic catheter under tip contact position constraint is introduced. For the analytical derivation of contact force Jacobian, the initial value problem parameter derivatives are defined and calculated analytically. Finally, simulation results show that the presented analytical method calculates the contact force Jacobian in significantly shorter computation time with comparable accuracy, compared to direct numerical computation.

3D Printed Voronoi Structures Inspired by Paracentrotus lividus Shells

The present paper investigates the mechanical behavior of a biomimetic Voronoi structure, inspired by the microstructure of the shell of the sea urchin Paracentrotus lividus, with its characteristic topological attributes constituting the technical evaluation stage of a novel biomimetic design strategy. A parametric design algorithm was used as a basis to generate design permutations with gradually increasing rod thickness, node count, and model smoothness, geometric parameters that define a Voronoi structure and increase its relative density as they are enhanced. Physical PLA specimens were manufactured with a fused filament fabrication (FFF) printer and subjected to quasi-static loading. Finite element analysis (FEA) was conducted in order to verify the experimental results. A minor discrepancy between the relative density of the designed and printed models was calculated. The tests revealed that the compressive behavior of the structure consists of an elastic region followed by a smooth plateau region and, finally, by the densification zone. The yield strength, compressive modulus, and plateau stress of the structure are improved as the specific geometric parameters are enhanced. The same trend is observed in the energy absorption capabilities of the structure while a reverse one characterizes the densification strain of the specimens. A second-degree polynomial relation is also identified between the modulus, plateau stress, and energy capacity when plotted against the relative density of the specimens. Distinct Voronoi morphologies can be acquired with similar mechanical characteristics, depending on the design requirements and application. Potential applications include lightweight structural materials and protective gear and accessories.

The Design Space of Kirchhoff Rods

The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length. We also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.

Optimization as generator of structural instability: Koiter’s worst imperfection

This paper is focused on the interaction of local modes in pin-jointed structures, with the aim to present an algorithm for the appraisal of the lowest critical load characterizing the structure under the effect of small imperfections. This interaction phenomenon is framed within Koiter’s theory of elastic instability as a quartic system. First of all, the paper discusses the theoretical results of Koiter’s theorem about the determination of the worst imperfection with respect to the critical load. In doing this, the characterization of the fold lines in the modal space is recalled and used for the purposes of the reformulation of the theorem. Then, a FEM code aimed at the determination of the most dangerous shape of the imperfection, and at performing the related sensitivity analysis, is presented. The basis of the code is a nonlinear model of a 3D rod equipped with a planar flexural deformation, within the context of Eulerian instability. Some numerical results having a practical interest are discussed.

MAGIC-GPS global variance reduction method for large-scale shielding calculation

We propose a Method of Automatic Generation Importance by Calculation with Growth Population Scheme (MAGIC-GPS) for efficient and accurate shielding calculation. The new method adopts a Growth Population Scheme (GPS) in the reciprocal iteration process of flux and importance for the Monte Carlo fixed-source calculation. Mathematical derivation is performed to prove that the GPS scheme is conducive to a rapid flux distribution convergence and can improve the Figure of Merit (FOM) of the Monte Carlo deep-penetration simulation. The effectivity of the MAGIC-GPS method is tested by a single concrete rod model, the H.B. Robinson-2 reactor benchmark (HBR2) and the China Fusion Engineering Test Reactor (CFETR). The results show that the MAGIC-GPS method can accelerate the Monte Carlo deep-penetration simulation, and the faster the population of particles increases, the better the acceleration. The MAGIC-GPS method is helpful for large-scale shielding calculation.

A comprehensive Cosserat rod drill-string model for arbitrary well geometry that includes the dynamics of the cutting and lateral contact

In this paper, a comprehensive continuous Cosserat rod model is developed to study the dynamics of a drill-string. The model presents the following features: (1) it can simulate the 3-D dynamics of the system, including lateral, axial and torsional motion; (2) it includes damping effects by adopting a Kelvin–Voigt material; (3) it uses a velocity-independent bit-rock interaction formulation in which the forces and torques at the bit are obtained by considering the dynamics of the cutting blades and the evolution of the soil profile, which is simulated with the help of an advection equation; (4) lateral contact is also included. In this paper, two application cases are analysed. The first one deals with a vertical borehole. It includes four simulations that are used to further investigate the hypothesis that low-dimensional lumped formulations are not able to capture the dynamics of a drill-string and, while doing so, evaluate the effect of the damping in the system’s response. The predictions are compared with those given by two other models: the first one considers a 2-degrees-of-freedom (DOF) formulation, with 1-axial and 1-torsional DOF; the second one is a semi-discrete approach, with 1 DOF for the axial dynamics and a continuous formulation for the torsional one. Divergences between the predictions are found, associated with the presence of higher frequencies in the signals obtained with the Cosserat model, which reassures the hypothesis that the new continuous approach can capture aspects of the dynamics that cannot be modelled with low-dimensional representations. The efficiency of the drilling is also assessed for the simulations obtained with the new model. The highest performance was observed in those simulations where only axial oscillations occurred, while a performance drop was seen when those vibrations were accompanied by torsional stick–slip, suggesting that an optimum damping value associated with the best drilling performance could be found, were damping controllable. Finally, the second application case is used to illustrate the full potential of the new model to tackle the 3-D lateral, axial, and torsional dynamics in a curved borehole.

SoRoSim: A MATLAB Toolbox for Hybrid Rigid–Soft Robots Based on the Geometric Variable-Strain Approach

Soft robotics has been a trending topic within the robotics community for almost two decades. However, available tools for the modeling and analysis of soft robots are still limited. This paper introduces a user-friendly MATLAB toolbox, Soft Robot Simulator (SoRoSim), that integrates the Geometric Variable Strain (GVS) model of Cosserat rods to facilitate the static and dynamic analysis of soft, rigid, or hybrid robotic systems. We present a brief overview of the design and structure of the toolbox and validate it by comparing its results with those published in the literature. To highlight the toolbox's potential to efficiently model, simulate, optimize, and control various robotic systems, we demonstrate four sample applications. The demonstrated applications explore different actuator and external loading conditions of single-, branched-, open-, and closed-chain robotic systems. We think that the soft-robotics research community will significantly benefit from the SoRoSim toolbox for a wide variety of applications.

Towards Realtime

This paper introduces a hair simulator optimized for real-time applications, including console and cloud gaming, avatar live-streaming, and metaverse environments. We view the collisions between strands as a mechanism to preserve the overall volume of the hair and adopt explicit Material Point Method (MPM) to resolve the strand-strand collision. For simulating single-strand behavior, a semi-implicit Discrete Elastic Rods (DER) model is used. We build upon a highly efficient GPU MPM framework recently presented by Fei et al. [2021b] and propose several schemes to largely improve the performance of building and solving the semi-implicit DER systems on GPU. We demonstrate the efficiency of our pipeline by a few practical scenes that achieve up to 260 frames-per-second (FPS) with more than two thousand simulated strands on Nvidia GeForce RTX 3080.

Multiple equilibrium states of a curved-sided hexagram: Part II—Transitions between states

Curved-sided hexagrams with multiple equilibrium states have great potential in engineering applications such as foldable architectures, deployable aerospace structures, and shape-morphing soft robots. In Part I, the classical stability criterion based on energy variation was used to study the elastic stability of the curved-sided hexagram and identify the natural curvature range for the stability of each state with circular and rectangular rod cross-sections. Here, we combine a multi-segment Kirchhoff rod model, finite element simulations, and experiments to investigate the transitions between four basic equilibrium states of the curved-sided hexagram. The four equilibrium states, namely the star hexagram, the daisy hexagram, the 3-loop line, and the 3-loop "8", carry uniform bending moments in their initial states, and the magnitudes of these moments depend on the natural curvatures and their initial curvatures. Transitions between these equilibrium states are triggered by applying bending loads at their corners or edges. It is found that transitions between the stable equilibrium states of the curved-sided hexagram are influenced by both the natural curvature and the loading position. Within a specific natural curvature range, the star hexagram, the daisy hexagram, and the 3-loop "8" can transform among one another by bending at different positions. Based on these findings, we identify the natural curvature range and loading conditions to achieve transition among these three equilibrium states plus a folded 3-loop line state for one specific ring having a rectangular cross-section with a height to thickness ratio of 4. The results obtained in this part also validate the elastic stability range of the four equilibrium states of the curved-sided hexagram in Part I. We envision that the present work could provide a new perspective for the design of multi-functional deployable and foldable structures.