Longitudinal Boundary Conditions Research Articles

Hybrid macroparticle algorithm for modeling space charge

A space charge algorithm is constructed that is a hybrid between envelope and multiparticle models. The transverse dynamics is simplified by tracking the transverse envelope of each macroparticle. The equations of motion are derived from the Hamiltonian-fluid formulation of the Vlasov Poisson system. A novel electrostatic self-field solver is derived that solves directly for the self-consistent on-axis potential and linear defocusing force, including longitudinal and beam pipe boundary conditions. The implementation of the algorithm, hyper3d, is presented. It is tested against the particle-in-cell code warp, using various configurations of a periodic focusing structure with rf cavities. The required number of macroparticles is reduced substantially compared to standard particle-tracking codes. This model is well adapted to cases where the transverse dynamics is linear and where the details of the longitudinal dynamics are the principal point of interest; an example is where a dc beam is converted to a bunched beam.

Two Cannulated Screws Provide Sufficient Biomechanical Strength for Prophylactic Fixation in Adult Patients With an Aggressive Benign Femoral Neck Lesion.

Background: Two cannulated screws were proposed for prophylactic fixation in adult patients with an aggressive benign femoral neck lesion in recent literature. However, the biomechanical properties of this intervention have not yet been investigated. Methods: After the evaluation of the heterogeneity of bone mineral density and geometry via quantitative computed tomography, 24 embalmed adult human cadaver femurs were randomized into the control, inferior half of the anterior cortical (25%) bone defect, entire anterior cortical (50%) bone defect, and the 50% bone defect and two cannulated screw group. Biomechanical analysis was conducted to compare the stiffness and failure load among the four groups when mimicking a one-legged stance. A CT-based finite element analysis (FEA) was performed to mimic the cortical and cancellous bone defect and the implantation of two cannulated screws of the four groups. Measurements of the maximal displacement and von Mises stress were conducted with the longitudinal load force and boundary conditions being established for a one-leg-standing status. Results: We noted a significant improvement in the failure load after the insertion of two 6.5 mm cannulated screws in femurs with 50% bone defect (+95%, p = 0.048), and no significant difference was found between the screw group and the intact femur. Similar trends were also found in the measurements of stiffness (+23%, p > 0.05) via biomechanical testing and the von Mises stresses (−71%, p = 0.043) by FEA when comparing the screw group and the 50% bone defect group. Conclusion: Our findings suggest that two cannulated screws provided sufficient biomechanical strength for prophylactic fixation in adult patients with an aggressive benign femoral neck lesion even when the entire anterior cortical bone is involved.

Topology in Shallow-Water Waves: A Spectral Flow Perspective

In the context of topological insulators, the shallow-water model was recently shown to exhibit an anomalous bulk-edge correspondence. For the model with a boundary, the parameter space involves both longitudinal momentum and boundary conditions, and exhibits a peculiar singularity. We resolve the anomaly in question by defining a new kind of edge index as the spectral flow around this singularity. Crucially, this edge index samples a whole family of boundary conditions, and we interpret it as a boundary-driven quantized pumping. Our edge index is stable due to the topological nature of spectral flow, and we prove its correspondence with the bulk Chern number index using scattering theory and a relative version of Levinson's theorem. The full spectral flow structure of the model is also investigated.

Nonlinear free and forced vibration of carbon nanotubes conveying magnetic nanoflow and subjected to a longitudinal magnetic field using stress-driven nonlocal integral model

In this paper primary and secondary resonance of carbon nanotube conveying magnetic nanofluid and subjected to a longitudinal magnetic field resting on viscoelastic foundation with different boundary conditions is investigated. To investigate the small scale effects, stress driven nonlocal integral model has been used and to show the more correctness of stress driven nonlocal integral model response, in studying the behavior of carbon nanotube with different boundary conditions, its results are compared with strain gradient model. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Kármán strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The nonlinear natural frequencies are obtained from the perturbation method and the divergence and flutter instability due to the increase in nanofluid velocity is investigated. Then the frequency response for primary, subharmonic and superharmonic resonance is obtained using the method of multiple scales.Finally, the effects of length small scale parameters, longitudinal magnetic field, magnetic nanofluid and boundary conditions on nonlinear free and forced vibration of carbon nanotube are investigated. As the most important results, as the intensity of the magnetic field increases, the critical flow velocity increases and divergence and flutter occur later. But the critical flow velocity decreases with increasing the intensity of the magnetic field for a carbon nanotube conveying magnetic nanofluid. In forced vibration, increasing the intensity of the magnetic field increases the amplitude of the response for all boundary conditions in primary and secondary resonance.

On the confinement of massless Dirac fermions in topological Möbius strips

In this work, we investigate some aspects of the confinement problem in Möbius strips in the context of the low-energy approximation of graphene, aiming to provide an analytical prescription for studying graphene-based nanoribbons in response to the peculiar background proper to such objects. Based on a topological treatment, we model the confinement of massless Dirac fermions in such nanostructures by proposing boundary conditions capable of incorporating their typical topological character, and examine how such configuration affects the physical properties of graphene. Energy eigenvalues are found to be sensitive to the transverse confinement and the positions along the transverse direction, exhibiting an unexpected spatial constraint. In addition, a dual electronic behavior associated with this aspect is also suggested under certain limits within our solution, which includes the presence of a length-dependent gap that responds to the longitudinal boundary conditions. At the same time, we look at the problem from a more mathematical perspective and sketch out some insights into the role played by the notion of self-adjointness in such boundary problem. A peculiar position constraint acts on our solution and manifests itself as partial restriction on the proposed set, pointing to the definition of incomplete domain self-adjoint extensions as well as to the nonexistence of conventional Möbius periodicities compatible with infinite mass confinement.

Efficient three-dimensional Poisson solvers in open rectangular conducting pipe

Three-dimensional (3D) Poisson solver plays an important role in the study of space-charge effects on charged particle beam dynamics in particle accelerators. In this paper, we propose three new 3D Poisson solvers for a charged particle beam in an open rectangular conducting pipe. These three solvers include a spectral integrated Green function (IGF) solver, a 3D spectral solver, and a 3D integrated Green function solver. These solvers effectively handle the longitudinal open boundary condition using a finite computational domain that contains the beam itself. This saves the computational cost of using an extra larger longitudinal domain in order to set up an appropriate finite boundary condition. Using an integrated Green function also avoids the need to resolve rapid variation of the Green function inside the beam. The numerical operational cost of the spectral IGF solver and the 3D IGF solver scales as O(Nlog(N)), where N is the number of grid points. The cost of the 3D spectral solver scales as O(NnN), where Nn is the maximum longitudinal mode number. We compare these three solvers using several numerical examples and discuss the advantageous regime of each solver in the physical application.

Computational simulation of carotid stenosis and flow dynamics based on patient ultrasound data – A new tool for risk assessment and surgical planning

PurposeThere is nowadays extensive experimental and computational investigation on the pathophysiology of atherosclerosis, searching correlations between its focal nature and local hemodynamic environment. The goal of this work is to present a methodology for patient-specific hemodynamics study of the carotid artery bifurcation based on the use of ultrasound (US) morphological and blood flow velocity patient data. Materials/methodsSubject-specific studies were performed for two patients, using a developed finite element code. Geometrical models were obtained from the acquisition of longitudinal and sequential cross-sectional ultrasound images and boundary conditions from Doppler velocity measurements at the common carotid artery. ResultsThere was a good agreement between ultrasound imaging data and computational simulated results. For a normal and a stenosed carotid bifurcation the velocity, wall shear stress (WSS) and WSS descriptors analysis illustrated the extremely complex hemodynamic behavior along the cardiac cycle. Different patterns were found, associated with morphology and hemodynamic patient-specific conditions. High values of time-averaged WSS (TAWSS) were found at stenosis site and for both patients TAWSS fields presented low values within areas of high oscillating shear index and relative residence time values, corresponding to recirculation zones. ConclusionSimulated hemodynamic parameters were able to capture the disturbed flow conditions in a normal and a stenosed carotid artery bifurcation, which play an important role in the development of local atherosclerotic plaques. Computational simulations based on clinic US might help improving diagnostic and treatment management of carotid atherosclerosis.

A Method for Computing the Analytical Solution of the Steady-State Heat Equation in Multilayered Media

The computation of the analytical solution of the steady temperature distribution in multilayered media can become numerically unstable if there are different longitudinal (i.e., the directions parallel to the layers) boundary conditions for each layer. In this study, we develop a method to resolve these computational difficulties by approximating the temperatures at the junctions step-by-step and solving for the thermal field separately in only the single layers. First, we solve a two-layer medium problem and then show that multilayered media can be represented as a hierarchy of two-layered media; thus, the developed method is generalized to an arbitrary number of layers. To improve the computational efficiency and speed, we use varying weighting coefficients during the iterations, and we present a method to decompose the multilayered media into two-layered media. The developed method involves the steady-state solution of the diffusion equation, which is illustrated for 2D slabs using separation of variables (SOV). A numerical example of four layers is also included, and the results are compared to a numerical solution.

On the longitudinal boundary condition of the Sverdrup stream function: A new derivation

The vanishing of the Sverdrup transport at the eastern coastline of any oceanic basin is a classic result of physical oceanography. It has been obtained in several ways: here it is rederived in the framework of the homogeneous model, whatever wind forcing, dissipation operator and additional boundary conditions may be taken into account. The considered method applies, in particular, to a class of model solutions singled out recently.

Packing and melting of mesoscopically confined two-dimensional Coulomb crystals in straight narrow channels

The microstructure and melting dynamics of the two-dimensional mesoscopic Coulomb crystal with 1/r-type mutual interaction force and parabolic transverse confining potential, under different degrees of incommensurability, are investigated through molecular-dynamics simulation. To tune the degree of incommensurability, N(a) extra particles are added into the commensurate uniform triangular lattice which has seven-layer structure and 40 particles in each layer with the periodic longitudinal boundary condition, until the system reaches another commensurate packing with eight-layer structure at N(a)=40. It is found that the increasing incommensurability with the increasing N(a) or 40-N(a) gradually deteriorates the structural order with the presence of intrinsic defects and the anisotropic bond-length distribution, except for the defect-free configurations at a few magic N(a)'s. The system prefers the seven- and the eight-layer single structures through the entire crystal for the low- and the high-N(a) regimes, respectively, and the configurations with seven- and eight-layer domain mixtures for 18≤N(a)≤24. The increasing strain or the worse local particle interlocking around the intrinsic defects with the increasing incommensurability also causes the easier structural rearrangement associated with the easier particle hopping and the earlier onset of melting transition. The transverse confinement suppresses the transverse motion, induces nonuniform melting, and sustains the layered structure after melting.

The influence of boundary conditions and axial deformability on buckling behavior of two-layer composite columns with interlayer slip

This paper presents a detailed analysis of the influence of boundary conditions and axial deformation on the critical buckling loads of the geometrically perfect elastic two-layer composite columns with interlayer slip between the layers. An investigation is based on the extension of our preliminary analytical study of slip-buckling behavior of two-layer composite columns. It is proved that the boundary conditions of composite columns with interlayer slip are interrelated in longitudinal and transverse directions. The parametric analysis reveals that the influence of different longitudinal boundary conditions on critical buckling load is significant and can be up to 20%, while, on the other hand, the influence of axial deformation is negligible.

Weighted Graph Colorings

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.

Casimir Force of Piston Systems with Arbitrary Cross Sections under Different Boundary Conditions

We study the Casimir force between two pistons under different boundary conditions inside an infinite cylinder with arbitrary cross section. It is found that the attractive or repulsive character of the Casimir force for a scalar field is determined only by the boundary condition along the longitudinal direction and is independent of the cross section, transverse boundary conditions and the mass of the field. Under symmetric Dirichlet–Dirichlet, Neumann–Neumann and periodic longitudinal boundary conditions the Casimir force is always attractive, but is repulsive under non-symmetric Dirichlet–Neumann and anti-periodic longitudinal boundary conditions. The Casimir force of the electromagnetic field in an ideal conductive piston is also investigated. This force is always attractive regardless of the shape of the cross section and the transverse boundary conditions.

Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice

We present exact calculations of the Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on $n$-vertex strip graphs $G$ of the honeycomb lattice for a variety of transverse widths equal to $L_y$ vertices and for arbitrarily great length, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^m$, where $m$ denotes the number of repeated subgraphs in the longitudinal direction. We give general formulas for $N_{Z,G,j}$ for arbitrary $L_y$. We also present plots of zeros of the partition function in the $q$ plane for various values of $v$ and in the $v$ plane for various values of $q$. Explicit results for partition functions are given in the text for $L_y=2,3$ (free) and $L_y=4$ (cylindrical), and plots of partition function zeros are given for $L_y$ up to 5 (free) and $L_y=6$ (cylindrical). Plots of the internal energy and specific heat per site for infinite-length strips are also presented.

3-D Full-Vectorial Analysis of Strong Optical Waveguide Discontinuities Using Pade Approximants

A full-vectorial 3-D numerical method applicable to high-index contrast optical waveguide discontinuities is presented. Rigorous treatment of the longitudinal boundary condition is incorporated in the formulation. The square root of the characteristic matrix is approximated using Padeacute approximants which results in an efficient implementation. The biconjugate gradient stabilized method is utilized to iteratively calculate the reflected and transmitted fields. A preconditioner is proposed which results in reduced number of iterations. The proposed method is applied to various optical waveguide facets exhibiting strong transverse and longitudinal refractive index discontinuities. In all cases, the modal reflectivities of the fundamental TE-Like and TM-Like modes are calculated for both the full-vectorial and the semi-vectorial formulations. Significant difference in the calculated modal reflectivity is seen between the full and semi-vectorial models. The error in the power balance remains low in the full-vectorial case irrespective of the waveguide dimensions. However, in the semi-vectorial case, the error in the power balance is found to increase when the waveguide width is reduced

Dynamic analysis of 3-D beam elements including warping and shear deformation effects

In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced transverse, longitudinal or torsional vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.

Dynamic Analysis Of Beams IncludingWarping And Shear Deformation Effects:Application In Bridge Deck Analysis

In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected to arbitrarily distributed dynamic twisting, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14x14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. In order to account for shear deformations, the concept of shear deformation coefficients is used, where these factors are defined by a strain energy approach. Eight boundary value problems are formulated and solved employing a pure BEM approach, which uses only boundary discretization. Free and forced, transverse, longitudinal or torsional vibrations are considered, taking into account both rotatory inertia and damping resistance. The influence of the warping effect especially in members of open form cross section is analyzed demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy arising from ignoring the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.

Exact Potts Model Partition Functions for Strips of the Triangular Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^{m-1}. We give general formulas for N_{Z,G,j} and its specialization to v=-1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus {\cal B}, arising as the accumulation set of partition function zeros as m\to\infty, in the q plane for fixed v and in the v plane for fixed q.

Flow polynomials and their asymptotic limits for lattice strip graphs

We present exact calculations of flow polynomials F(G,q) for lattice strips of various fixed widths Ly≤4 and arbitrarily great lengths Lx, with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face fl in the infinite-length limit. We study the zeros of F(G,q) in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros ℬ in the infinite-length limit for the various families of strips. The function fl is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case Ly=1, the maximal point, qcf, where ℬ crosses the real axis, is universal on cyclic and Mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value qcf=3 for the infinite square lattice.

Potts model partition functions for self-dual families of strip graphs

We consider the q-state Potts model on families of self-dual strip graphs G D of the square lattice of width L y and arbitrarily great length L x , with periodic longitudinal boundary conditions. The general partition function Z and the T=0 antiferromagnetic special case P (chromatic polynomial) have the respective forms ∑ j=1 N F, L y , λ c F, L y , j ( λ F, L y , j ) L x , with F= Z, P. For arbitrary L y , we determine (i) the general coefficient c F, L y , j in terms of Chebyshev polynomials, (ii) the number n F ( L y , d) of terms with each type of coefficient, and (iii) the total number of terms N F, L y , λ . We point out interesting connections between the n Z ( L y , d) and Temperley–Lieb algebras, and between the N F, L y , λ and enumerations of directed lattice animals. Exact calculations of P are presented for 2⩽ L y ⩽4. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), W( q). Generalizing q from Z + to C , we determine the continuous locus B in the complex q plane where W( q) is singular. We find the interesting result that for all L y values considered, the maximal point at which B crosses the real q-axis, denoted q c , is the same, and is equal to the value for the infinite square lattice, q c =3. This is the first family of strip graphs of which we are aware that exhibits this type of universality of q c .