Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.

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.

Numerical analysis of shape transition in graphene nanoribbons

Graphene nanoribbon (GNR) with free edges can exhibit non-flat morphologies due to pre-existing edge stress. Using molecular dynamics (MD) simulations, we investigate the free-edge effect on the shape transition in GNRs with different edge types, including regular (armchair and zigzag), armchair terminated with hydrogen and reconstructed armchair. The results show that initial edge stress and energy are dependent on the edge configurations. It is confirmed that pre-strain on the free edges is a possible way to limit the random shape transition of GNRs. In addition, the influence of surface attachment on the shape transition is also investigated in this work. It is found that surface attachment can lead to periodic ripples in GNRs, dependent on the initial edge configurations.

Experimental and numerical analysis of the relationship between indoor and outdoor airborne particles in an operating room

This work investigated the impact of the HVAC filtration system and indoor particle sources on the relationship between indoor and outdoor airborne particle size and concentrations in an operating room. Filters with efficiency between 65% and 99.97% were used in the investigation and indoor and outdoor particle size and concentrations were measured. A balance mass model was used for the simulation of the impact of the surgical team, deposition rate, HVAC exhaust and air change rates on indoor particle concentration. The experimental results showed that high efficiency filters would not be expected to decrease the risk associated with indoor particles larger than approximately 1 µm in size because normal filters are relatively efficient for these large particles. A good fraction of outdoor particles were removed by deposition on the HVAC system surfaces and this deposition increased with particle size. For particles of 0.3-0.5 µm in diameter, particle reduction was about 23%, while for particles >10 µm the loss was about 78%. The modelling results showed that depending on the type of filter used, the surgical team generated between 93-99% of total particles, while the outdoor air contributed only 1-6%.

Three-dimensional off-design numerical analysis of an organic Rankine cycle radial-inflow turbine

Optimisation of organic Rankine cycles(ORCs for binary cycle applications could play a major role in determining the competitiveness of low to moderate renewable sources. An important aspect of the optimisation is to maximise the turbine output power for a given resource. This requires careful attention to the turbine design notably through numerical simulations. Challenges in the numerical modelling of radial-inflow turbines using high-density working fluids still need to be addressed in order to improve the turbine design and better optimise ORCs. Thispaper presents preliminary 3D numerical simulations of a high-density radial-inflow ORC turbine in sensible geothermal conditions. Following extensive investigation of the operating conditions and thermodynamic cycle analysis, therefrigerant R143a is chosen as the high-density working fluid. The 1D design of the candidate radial-inflow turbine is presented in details. Furthermore, commercially-available software Ansys-CFX is used to perform preliminary steady-state 3D CFD simulations of the candidate R143a radial-inflow turbine for a number of operating conditions including off-design conditions. The real-gas properties are obtained using the Peng–Robinson equations of state.The thermodynamic ORC cycle is presented. The preliminary design created using dedicated radial-inflow turbine software Concepts-Rital is discussed and the 3D CFD results are presented and compared against the meanline analysis.

A poroviscohyperelastic model for numerical analysis of mechanical behavior of single chondrocyte

The aim of this paper is to utilize a poroviscohyperelastic (PVHE) model which is developed based on the porohyperelastic (PHE) model to explore the mechanical deformation properties of single chondrocytes. Both creep and relaxation responses are investigated by using FEM models of micropipette aspiration and AFM experiments, respectively. The newly developed PVHE model is compared thoroughly with the SnHS and PHE models. It has been found that the PVHE can accurately capture both creep and stress relaxation behaviors of chondrocytes better than other two models. Hence, the PVHE is a promising model to investigate mechanical properties of single chondrocytes.

Porous versus porthole fuel injection in a radical farming scramjet: A numerical analysis

Numerically computed engine performance of a nominally two-dimensional radical farming scramjet with porous (permeable C/C ceramic) and porthole fuel injection is presented. Inflow conditions with Mach number, stagnation pressure, and enthalpy of 6.44, 40.2MPa, and 4.31 MJ/kg respectively, and fuel/air equivalence ratio of 0.44 were maintained, along with engine geometry. Hydrogen fuel was injected at an axial location of 92.33mm downstream of the leading edge for each investigated injection method. Results from this study show that porous fuel injection results in enhanced mixing and combustion compared to porthole fuel injection. This is particularly evident within the first half of the combustion chamber where porous fuel injection resulted in mixing and combustion efficiencies of 76% and 63% respectively. At the same location, porthole fuel injection resulted in efficiencies respectively of 58% and 46%. Key mechanisms contributing to the observed improved performance were the formation of an attached oblique fuel injection shock and associated stronger shock-expansion train ingested by the engine, enhanced spreading of the fuel in all directions and a more rapidly growing mixing layer.

Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Numerical analysis of a new space-time variable fractional order advection-dispersion equation

Many physical processes appear to exhibit fractional order behavior that may vary with time and/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 a new space–time variable fractional order advection–dispersion equation on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by Coimbra’s variable fractional derivative of order α(x)∈(0,1]α(x)∈(0,1], and the first-order and second-order space derivatives by the Riemann–Liouville derivatives of order γ(x,t)∈(0,1]γ(x,t)∈(0,1] and β(x,t)∈(1,2]β(x,t)∈(1,2], respectively. We propose an implicit Euler approximation for the equation and investigate the stability and convergence of the approximation. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical analysis for the interlayer toughening of piezoelectric smart composite plate

The piezoelectric composite material could engender stress concentration resulting from small cracks during layers easily, as the cracks growth will lead to the failure of the whole structure. In this paper, a finite element model for piezoelectric composite materials by ABAQUS including interlayer crack was established, and the J integral and crack tip stress of different types PZT patches were calculated by using the equivalent integral method. Then, the J integral for adhesive layers with different thickness, elastic modulus considering and not considering piezoelectricity was investigated. The results show that the J integral of mode I, II reduces with thicker adhesive layer and lower elastic modules, and the J integral of mode II decreases more sharply than that of mode I.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.