Rod Formulation Research Articles

An objective FE-formulation for Cosserat rods based on the spherical Bézier interpolation

A generalization of the spherical linear interpolation (or slerp) for the finite rotations to the case of more than two control variables on SO(3) is introduced to design an objective FE-formulation for the non-linear space Cosserat rod model. The interpolation uses the De Casteljau’s algorithm.In this way, the same interpolation degree can be used for the placement of the centroid curve and for the finite rotation of the cross-section. A recursive formula is obtained for the interpolation of the rotations. Similar recursive formulas are derived for the spin and the curvature vector, leading to a generalization of the Bézier basis functions on the manifold SO(3).The rod formulation so obtained is invariant under a rigid rotation (objective), in the sense that the patch-test with respect to the rigid body motion is satisfied. Furthermore, an optimal path-independence is achieved as verified by several numerical investigations.

An efficient isogeometric/finite-difference immersed boundary method for the fluid–structure interactions of slender flexible structures

In this contribution, we present a robust and efficient computational framework capable of accurately capturing the dynamic motion and large deformation/deflection responses of highly-flexible rods interacting with an incompressible viscous flow. Within the partitioned approach, we adopt separate field solvers to compute the dynamics of the immersed structures and the evolution of the flow field over time, considering finite Reynolds numbers. We employ a geometrically exact, nonlinear Cosserat rod formulation in the context of the isogeometric analysis (IGA) technique to model the elastic responses of each rod in three dimensions (3D). The Navier–Stokes equations are resolved using a pressure projection method on a standard staggered Cartesian grid. The direct-forcing immersed boundary method is utilized for coupling the IGA-based structural solver with the finite-difference fluid solver. In order to fully exploit the accuracy of the IGA technique for FSI simulations, the proposed framework introduces a new procedure that decouples the resolution of the structural domain from the fluid grid. Uniformly distributed Lagrangian markers with density relative to the Eulerian grid are generated to communicate between Lagrangian and Eulerian grids consistently with IGA. We successfully validate the proposed computational framework against two- and three-dimensional FSI benchmarks involving flexible filaments undergoing large deflections/motions in an incompressible flow. We show that six times coarser structural mesh than the flow Eulerian grid delivers accurate results for classic benchmarks, leading to a major gain in computational efficiency. The simultaneous spatial and temporal convergence studies demonstrate the consistent performance of the proposed framework, showing that it conserves the order of the convergence, which is the same as that of the fluid solver.

Preliminary Results and a Treatment Algorithm for Osteomyelitis of the Distal Tibia and Talus

Category: Hindfoot; Ankle Introduction/Purpose: Management of hindfoot osteomyelitis involves eradication of infection, deformity correction, and bony stabilization in the setting of bone loss and poor bone quality. This case series of 20 patients presents a two-stage technique: debridement, deformity correction, joint surface preparation, intramedullary reaming with antibiotic-impregnated polymethyl methacrylate rod and ring external fixation, followed by a second stage of external fixator removal and exchange of antibiotic rod for definitive hindfoot fusion rod. The purpose of the study is to highlight techniques and preliminary results that can improve the success of obtaining stable fixation, remission of infection and ultimately fusion. Techniques highlighted are ring fixator placement, antibiotic rod formulation, intramedullary canal preparation, rod exchange and use of blocking screws to enhance compression and axial loading of the fusion. Methods: Twenty patients were treated with this two-stage protocol. The most common indication was post-traumatic osteomyelitis followed by neuropathic (Charcot) joint. All were treated with an initial debridement of nonviable tissue including osteomyelitic bone. The tibiotalar and talocalcaneal joint surfaces were prepared for fusion at the initial debridement. Provisional stabilization and compression were obtained using a ring external fixator for patients without rotational stability. The polymethyl methacrylate-antibiotic impregnated rod (PMMA with gentamycin and vancomycin) was introduced through the calcaneus, talus, and the intramedullary space of the tibia. The patients were kept in the fixator until the skin was healed and all incisions were dry. At this point, the fixator was removed, and the antibiotic rod was exchanged for the definitive hindfoot fusion rod. Results: Cultures were positive for Staphylococcal or Streptococcal species in 60% of cases, Fingeoldia magna 10%, Cutibacterium Acnes 2%, and Gram-negative organisms in 30%. On average patients spent 80 days in the fixator while wounds were healing, and were receiving systemic antibiotics. One patient died while in the frame. There have been no amputations in the series. Of the patients that have completed the protocol, all have gone on to either bony fusion or stable fibrous union. All are able to walk unaided and 25% used an AFO to control and protect the foot and equalize leg length. Secondary procedures are described. Conclusion: This technique of staged fusion for hindfoot osteomyelitis offers the following advantages. The first stage involves thorough debridement and deformity correction. Oftentimes, there is extensive scarring, and the soft tissue envelope is not conducive to internal fixation. This technique allows the soft tissues to heal before placing the definitive internal fixation. It also allows for graduated compression at the fusion site with the ring fixator. The antibiotic rod not only delivers high concentration of local antibiotics, but it prevents deformity recurrence during the compression phase. The definitive rod provides support of pathologic bone and maintains deformity correction.

Nonlocal finite element analysis for axial vibration of embedded love–bishop nanorods

In this paper, nonlocal free vibration of axial rods embedded in elastic medium is examined by utilizing Love–Bishop rod theory with finite element method for the first time in the literature. Firstly, kinematic relations and dynamic equilibrium of this rod formulation are constituted. Equation of motion and boundary conditions are attained with the help of first variation of total potential of nano rod and solved using separation of variables. The frequency equations of four different nanorod types are obtained. On the other hand, a size–dependent finite element formulation is presented based on Weighted Residual Method. The nondimensional frequency parameters of nanorods are calculated by using analytical and finite element methods for different parameters such as mode number, nonlocal parameter, rod length, length–to–diameter ratio, finite element number and a detailed discussion is performed in the numerical results. In order to understand the characteristics of this rod theory, several comparative results that include nonlocal simple rod formulation are given as well.

A three-dimensional model of flagellar swimming in a Brinkman fluid

We investigate three-dimensional flagellar swimming in a fluid with a sparse network of stationary obstacles or fibres. The Brinkman equation is used to model the average fluid flow where a flow-dependent term, including a resistance parameter that is inversely proportional to the permeability, captures the effects of the fibres on the fluid. To solve for the local linear and angular velocities that are coupled to the flagellar motion, we extend the method of regularized Brinkmanlets to incorporate a Kirchhoff rod, discretized as point forces and torques along a centreline. Representing a flagellum as a Kirchhoff rod, we investigate emergent waveforms for different preferred strain and twist functions. Since the Kirchhoff rod formulation allows for out-of-plane motion, in addition to studying a preferred planar sine wave configuration, we also study a preferred helical configuration. Our numerical method is validated by comparing results to asymptotic swimming speeds derived for an infinite-length cylinder propagating planar or helical waves. Similar to the asymptotic analysis for both planar and helical bending, we observe that with small amplitude bending, swimming speed is always enhanced relative to the case with no fibres in the fluid (Stokes) as the resistance parameter is increased. For regimes not accounted for with asymptotic analysis, i.e. large amplitude planar and helical bending, our model results show a non-monotonic change in swimming speed with respect to the resistance parameter; a maximum swimming speed is observed when the resistance parameter is near one. The non-monotonic behaviour is due to the emergent waveforms; as the resistance parameter increases, the swimmer becomes incapable of achieving the amplitude of its preferred configuration. We also show how simulation results of slower swimming speeds for larger resistance parameters are actually consistent with the asymptotic swimming speeds if work in the system is fixed.

Meshfree analysis of structures modeled as extensible slender rods

The analysis of members that can be modeled as extensible elastic slender rods is investigated. A meshfree formulation using a Local Radial Point Interpolation Method (LRPIM) is developed that utilizes radial basis functions in curvilinear coordinates. This approach bypasses the need to utilize more conventional element meshes and significantly reduces the number of equations needed for the numerical solution. The slender rod formulation presented allows for tension variation, axial stretch, incremental loading and distributed load variation along the rod. It is well suited for nonlinear problems that involve large deflections and rigid rotations. The position and tangent vectors are expressed using Hermite-type approximations, and radial basis functions, while the interpolation of tension variation and distributed loads are described using polynomials. The solution procedure of weighted residuals Galerkin weak formulation combined with an incremental iterative numerical scheme is introduced to address the incremental loading and large deflection issues for static and quasi-static problems. The implementation of the analytical formulation and the numerical procedure are illustrated using three nonlinear problems. The first two examples provide insight into the validity, accuracy, and efficiency of the methodology. The third example presents the case of a moving boundary condition problem which models a cable entangled by fishing boat-trawling equipment.

Granular materials interacting with thin flexible rods

In this work, we develop a computational model for the simulation of problems wherein granular materials interact with thin flexible rods. We treat granular materials as a collection of spherical particles following a discrete element method (DEM) approach, while flexible rods are described by a large deformation finite element (FEM) rod formulation. Grain-to-grain, grain-to-rod, and rod-to-rod contacts are fully permitted and resolved. A simple and efficient strategy is proposed for coupling the motion of the two types (discrete and continuum) of materials within an iterative time-stepping solution scheme. Implementation details are shown and discussed. Validity and applicability of the model are assessed by means of a few numerical examples. We believe that robust, efficiently coupled DEM–FEM schemes can be a useful tool to the simulation of problems wherein granular materials interact with thin flexible rods, such as (but not limited to) bombardment of grains on beam structures, flow of granular materials over surfaces covered by threads of hair in many biological processes, flow of grains through filters and strainers in various industrial segregation processes, and many others.

Effect of higher order constitutive terms on the elastic buckling of thin-walled rods

In this paper we revisit an elastic constitutive equation proposed in two previous works and extend it in order to include all higher-order terms on the deformations. Our purpose is to assess the influence of these terms on the elastic buckling of thin-walled rods. The resulting material model was incorporated into a geometrically exact rod formulation and implemented into a nonlinear finite element code. By means of simple numerical examples we show that the higher order terms may play a significant role on the values of the buckling loads and on the post-buckling behavior of thin-walled beams and columns.

An atomic interaction‐based rod formulation for modelling Gecko adhesion

AbstractWe present a nonlinear rod model which describes the adhesion mechanism used by the Gecko to cling to inclined and overhanging surfaces. The rod formulation is used to model the fine, hair–like setae, which form the microstructure of the Gecko toes. A simple geometric representation, based on hyperbolas, is used to capture the branching of the seta. The adhesion between seta and surface is modelled by an atomic interaction potential. The seta model is cast into a nonlinear finite element framework, which is used to simulate and study the pull–off behavior of a single seta. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A geometrically exact beam element based on the absolute nodal coordinate formulation

In this study, Reissner’s classical nonlinear rod formulation, as implemented by Simo and Vu-Quoc by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.