Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Coupling of energy from quantum emitters to the plasmonic mode of V groove waveguides: A numerical study

This work is a theoretical investigation into the coupling of a single excited quantum emitter to the plasmon mode of a V groove waveguide. The V groove waveguide consists of a triangular channel milled in gold and the emitter is modeled as a dipole emitter, and could represent a quantum dot, nitrogen vacancy in diamond, or similar. In this work the dependence of coupling efficiency of emitter to plasmon mode is determined for various geometrical parameters of the emitter-waveguide system. Using the finite element method, the effect on coupling efficiency of the emitter position and orientation, groove angle, groove depth, and tip radius, is studied in detail. We demonstrate that all parameters, with the exception of groove depth, have a significant impact on the attainable coupling efficiency. Understanding the effect of various geometrical parameters on the coupling between emitters and the plasmonic mode of the waveguide is essential for the design and optimization of quantum dot–V groove devices.

Unsteady natural convection within a differentially heated enclosure of sinusoidal corrugated side walls

Numerically investigation of natural convection within a differentially heated modified square enclosure with sinusoidally corrugated side walls has been performed for different values of Rayleigh number. The fluid inside the enclosure considered is air and is quiescent, initially. The top and bottom surfaces are flat and considered as adiabatic. Results reveal three main stages: an initial stage, a transitory or oscillatory stage and a steady stage for the development of natural convection flow inside the corrugated cavity. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. Investigation has been performed for the Rayleigh number, Ra ranging from 105 to 108 with variation of corrugation amplitude and frequency. Constant physical properties for the fluid medium have been assumed. Results have been presented in terms of the isotherms, streamlines, temperature plots, average Nusselt numbers, traveling waves and thermal boundary layer thickness plots, temperature and velocity profiles. The effects of sudden differential heating and its consequent transient behavior on fluid flow and heat transfer characteristics have been observed for the range of governing parameters. The present results show that the transient phenomena are greatly influenced by the variation of the Rayleigh Number with corrugation amplitude and frequency.

Towards determining soft tissue properties for modelling spine surgery : current progress and challenges

Current complication rates for adolescent scoliosis surgery necessitate the development of better surgical planning tools to improve outcomes. Here we present our approach to developing finite element models of the thoracolumbar spine for deformity surgery simulation, with patient-specific model anatomy based on low-dose pre-operative computed tomography scans. In a first step towards defining patient-specific tissue properties, an initial 'benchmark' set of properties were used to simulate a clinically performed pre-operative spinal flexibility assessment, the fulcrum bending radiograph. Clinical data for ten patients were compared with the simulated results for this assessment and in cases where these data differed by more than 10%, soft tissue properties for the costo-vertebral joint (CVJt) were altered to achieve better agreement. Results from these analyses showed that changing the CVJt stiffness resulted in acceptable agreement between clinical and simulated flexibility in two of the six cases. In light of these results and those of our previous studies in this area, it is suggested that spinal flexibility in the fulcrum bending test is not governed by any single soft tissue structure acting in isolation. More detailed biomechanical characterisation of the fulcrum bending test is required to provide better data for determination of patient-specific soft tissue properties.

Analysis of Rail Ends under Wheel Contact Loading

The effect of the discontinuity of the rail ends and the presence of lower modulus insulation material at the gap to the variations of stresses in the insulated rail joint (IRJ) is presented. A three-dimensional wheel – rail contact model in the finite element framework is used for the analysis. It is shown that the maximum stress occurs in the subsurface of the railhead when the wheel contact occurs far away from the rail end and migrates to the railhead surface as the wheel approaches the rail end; under this condition, the interface between the rail ends and the insulation material has suffered significantly increased levels of stress concentration. The ratio of the elastic modulus of the railhead and insulation material is found to alter the levels of stress concentration. Numerical result indicates that a higher elastic modulus insulating material can reduce the stress concentration in the railhead but will generate higher stresses in the insulation material, leading to earlier failure of the insulation material

Impact and energy absorption of road safety barriers by coupled SPH/FEM

Road safety barriers are used to minimise the severity of road accidents and protect lives and property. There are several types of barrier in use today. This paper reports the initial phase of research carried out to study the impact response of portable water-filled barrier (PWFB) which has the potential to absorb impact energy and hence provide crash mitigation under low to moderate speeds. Current research on the impact and energy absorption capacity of water-filled road safety barriers is limited due to the complexity of fluid-structure interaction under dynamic impact. In this paper, a novel fluid-structure interaction method is developed based on the combination of Smooth Particle Hydrodynamics (SPH) and Finite Element Method (FEM). The sloshing phenomenon of water inside a PWFB is investigated to explore the energy absorption capacity of water under dynamic impact. It was found that water plays an important role in energy absorption. The coupling analysis developed in this paper will provide a platform to further the research in optimising the behaviour of the PWFB. The effect of the amount of water on its energy absorption capacity is investigated and the results have practical applications in the design of PWFBs.

Minimization of railhead edge stresses through shape optimization

The railhead is severely stressed under the localized wheel contact patch close to the gaps in insulated rail joints. A modified railhead profile in the vicinity of the gapped joint, through a shape optimization model based on a coupled genetic algorithm and finite element method, effectively alters the contact zone and reduces the railhead edge stress concentration significantly. Two optimization methods, a grid search method and a genetic algorithm, were employed for this optimization problem. The optimal results from these two methods are discussed and, in particular, their suitability for the rail end stress minimization problem is studied. Through several numerical examples, the optimal profile is shown to be unaffected by either the magnitude or the contact position of the loaded wheel. The numerical results are validated through a large-scale experimental study.

Accuracy of FEM and BEM for electromagnetic flux calculations in high power applications

The development and design of electric high power devices with electromagnetic computer-aided engineering (EM-CAE) software such as the Finite Element Method (FEM) and Boundary Element Method (BEM) has been widely adopted. This paper presents the analysis of a Fault Current Limiter (FCL), which acts as a high-voltage surge protector for power grids. A prototype FCL was built. The magnetic flux in the core and the resulting electromagnetic forces in the winding of the FCL were analyzed using both FEM and BEM. An experiment on the prototype was conducted in a laboratory. The data obtained from the experiment is compared to the numerical solutions to determine the suitability and accuracy of the two methods.

Damage detection in slab-on-girder bridges using vibration characteristics

This paper develops and applies a multi-criteria procedure, incorporating changes in natural frequencies, modal flexibility and the modal strain energy, for damage detection in slab-on-girder bridges. The proposed procedure is first validated through experimental testing of a model bridge. Numerically simulated modal data obtained through finite element analyses are then used to evaluate the vibration parameters before and after damage and used as the indices for assessment of the state of structural health. The procedure is illustrated by its application to full scale slab-on-girder bridges under different damage scenarios involving single and multiple damages on the deck and girders.

A non-linear dynamic analysis of heavy haul track line sleepers under wheel flat

Significant wheel-rail dynamic forces occur because of imperfections in the wheels and/or rail. One of the key responses to the transmission of these forces down through the track is impact force on the sleepers. Dynamic analysis of nonlinear systems is very complicated and does not lend itself easily to a classical solution of multiple equations. Trying to deduce the behaviour of track components from experimental data is very difficult because such data is hard to obtain and applies to only the particular conditions of the track being tested. The finite element method can be the best solution to this dilemma. This paper describes a finite element model using the software package ANSYS for various sized flat defects in the tread of a wheel rolling at a typical speed on heavy haul track. The paper explores the dynamic response of a prestressed concrete sleeper to these defects.

Mixed convection along a vertical flat plate in a non-absorbing medium

Here mixed convection boundary layer flow of a viscous fluid along a heated vertical semi-infinite plate is investigated in a non-absorbing medium. The relationship between convection and thermal radiation is established via boundary condition of second kind on the thermally radiating vertical surface. The governing boundary layer equations are transformed into dimensionless parabolic partial differential equations with the help of appropriate transformations and the resultant system is solved numerically by applying straightforward finite difference method along with Gaussian elimination technique. It is worthy to note that Prandlt number, Pr, is taken to be small (<< 1) which is appropriate for liquid metals. Moreover, the numerical results are demonstrated graphically by showing the effects of important physical parameters, namely, the modified Richardson number (or mixed convection parameter), Ri*, and surface radiation parameter, R, in terms of local skin friction and local Nusselt number coefficients.

Natural convection flow in a strong cross magnetic field with radiation

The problem of MHD natural convection boundary layer flow of an electrically conducting and optically dense gray viscous fluid along a heated vertical plate is analyzed in the presence of strong cross magnetic field with radiative heat transfer. In the analysis radiative heat flux is considered by adopting optically thick radiation limit. Attempt is made to obtain the solutions valid for liquid metals by taking Pr≪1. Boundary layer equations are transformed in to a convenient dimensionless form by using stream function formulation (SFF) and primitive variable formulation (PVF). Non-similar equations obtained from SFF are then simulated by implicit finite difference (Keller-box) method whereas parabolic partial differential equations obtained from PVF are integrated numerically by hiring direct finite difference method over the entire range of local Hartmann parameter, $xi$ . Further, asymptotic solutions are also obtained for large and small values of local Hartmann parameter $xi$ . A favorable agreement is found between the results for small, large and all values of $xi$ . Numerical results are also demonstrated graphically by showing the effect of various physical parameters on shear stress, rate of heat transfer, velocity and temperature.

Magnetic flux analysis in superconducting Fault Current Limiters under short-circuit conditions

In order to obtain a more compact Superconducting Fault Current limiter (SFCL), a special geometry of core and AC coil is required. This results in a unique magnetic flux pattern which differs from those associated with conventional round core arrangements. In this paper the magnetic flux density within a Fault Current Limiter (FCL) is described. Both experimental and analytical approaches are considered. A small scale prototype of an FCL was constructed in order to conduct the experiments. This prototype comprises a single phase. The analysis covers both the steady state and the short-circuit condition. Simulation results were obtained using commercial software based on the Finite Element Method (FEM). The magnetic flux saturating the cores, leakage magnetic flux giving rise to electromagnetic forces and leakage magnetic flux flowing in the enclosing tank are computed.

Patient-specific simulations of spinal deformity surgery

Adolescent idiopathic scoliosis (AIS) is a three-dimensional spinal deformity involving the side-to-side curvature of the spine in the coronal plane and axial rotation of the vertebrae in the transverse plane. For patients with a severe or rapidly progressing deformity, corrective instrumented fusion surgery is performed. The wide choice of implants and large variability between patients make it difficult for surgeons to choose optimal treatment strategies. This paper describes the patient specific finite element modelling techniques employed and the results of preliminary analyses predicting the surgical outcomes for a series of AIS patients. This report highlights the importance of not only patient-specific anatomy and material parameters, but also patient-specific data for the clinical and physiological loading conditions experienced by the patient who has corrective scoliosis surgery.