A finite volume method with linearisation in time for the solution of advection-reaction-diffusion systems

The numerical solution in one space dimension of advection--reaction--diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature. Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models. This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest. The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non- classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick’s law. We consider a space-fractional advection-dispersion equation based on a fractional Fick’s law. Zhang et al. [Water Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Gru ̈nwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach.

Geometric sensitivity of patient-specific finite element models of the spine to variability in user-selected anatomical landmarks

Software to create individualised finite element (FE) models of the osseoligamentous spine using pre-operative computed tomography (CT) data-sets for spinal surgery patients has recently been developed. This study presents a geometric sensitivity analysis of this software to assess the effect of intra-observer variability in user-selected anatomical landmarks. User-selected landmarks on the osseous anatomy were defined from CT data-sets for three scoliosis patients and these landmarks were used to reconstruct patient-specific anatomy of the spine and ribcage using parametric descriptions. The intra-observer errors in landmark co-ordinates for these anatomical landmarks were calculated. FE models of the spine and ribcage were created using the reconstructed anatomy for each patient and these models were analysed for a loadcase simulating clinical flexibility assessment. The intra-observer error in the anatomical measurements was low in comparison to the initial dimensions, with the exception of the angular measurements for disc wedge and zygapophyseal joint (z-joint) orientation and disc height. This variability suggested that CT resolution may influence such angular measurements, particularly for small anatomical features, such as the z-joints, and may also affect disc height. The results of the FE analysis showed low variation in the model predictions for spinal curvature with the mean intra-observer variability substantially less than the accepted error in clinical measurement. These findings demonstrate that intra-observer variability in landmark point selection has minimal effect on the subsequent FE predictions for a clinical loadcase.

Analysis of strain-rate dependent mechanical behavior of single chondrocyte : a finite element study

Various studies have been conducted to investigate the effects of impact loading on cartilage damage and chondrocyte death. These have shown that the rate and magnitude of the applied strain significantly influence chondrocyte death, and that cell death occurred mostly in the superficial zone of cartilage suggesting the need to further understand the fundamental mechanisms underlying the chondrocytes death induced at certain levels of strain-rate. To date there is no comprehensive study providing insight on this phenomenon. The aim of this study is to examine the strain-rate dependent behavior of a single chondrocyte using a computational approach based on Finite Element Method (FEM). An FEM model was developed using various mechanical models, which were Standard Neo-Hookean Solid (SnHS), porohyperelastic (PHE) and poroviscohyperelastic (PVHE) to simulate Atomic Force Microscopy (AFM) experiments of chondrocyte. The PVHE showed, it can capture both relaxation and loading rate dependent behaviors of chondrocytes, accurately compared to other models.

Methanesulfonic acid-catalyzed conversion of glucose and xylose mixtures to levulinic acid and furfural

Methanesulfonic acid (MSA) was compared with sulfuric acid for the conversion of glucose and xylose mixtures to produce levulinic acid and furfural. The interactions of glucose and xylose, the predominant sugars found in biomass, were found to influence product yields with furfural degradation reactions enhanced under higher reactant loadings. Fast heating rates allowed maximal yields (>60 mol%) of levulinic acid and furfural to be achieved under short reaction times. Under the range of conditions examined, sulfuric acid produced a slight increase in levulinic acid yield by 6% (P = 0.02), although there was no significant difference (P = 0.11) between MSA and sulfuric acid in levulinic acid formed from glucose alone. The amount and type of the solid residue is similar between MSA and sulfuric acid. As such, MSA is a suitable alternative because its use minimizes corrosion and disposal issues associated with mineral acid catalysts. The heating value of the residue was 22 MJ/kg implying that it is a suitable source of fuel. On the basis of these results, a two-stage processing strategy is proposed to target high levulinic acid and furfural yields, and other chemical products (e.g., lactic acid, xylitol, acetic acid and formic acid). This will result in full utilization of bagasse components.

Second-order elastic finite element analysis of structures using a single element per member

Finite element frame analysis programs targeted for design office application necessitate algorithms which can deliver reliable numerical convergence in a practical timeframe with comparable degrees of accuracy, and a highly desirable attribute is the use of a single element per member to reduce computational storage, as well as data preparation and the interpretation of the results. To this end, a higher-order finite element method including geometric non-linearity is addressed in the paper for the analysis of elastic frames for which a single element is used to model each member. The geometric non-linearity in the structure is handled using an updated Lagrangian formulation, which takes the effects of the large translations and rotations that occur at the joints into consideration by accumulating their nodal coordinates. Rigid body movements are eliminated from the local member load-displacement relationship for which the total secant stiffness is formulated for evaluating the large member deformations of an element. The influences of the axial force on the member stiffness and the changes in the member chord length are taken into account using a modified bowing function which is formulated in the total secant stiffness relationship, for which the coupling of the axial strain and flexural bowing is included.

Non-linear refined finite element elastic analysis of steel frames with generalized transverse member loads

In the finite element modelling of steel frames, external loads usually act along the members rather than at the nodes only. Conventionally, when a member is subjected to these transverse loads, they are converted to nodal forces which act at the ends of the elements into which the member is discretised by either lumping or consistent nodal load approaches. For a contemporary geometrically non-linear analysis in which the axial force in the member is large, accurate solutions are achieved by discretising the member into many elements, which can produce unfavourable consequences on the efficacy of the method for analysing large steel frames. Herein, a numerical technique to include the transverse loading in the non-linear stiffness formulation for a single element is proposed, and which is able to predict the structural responses of steel frames involving the effects of first-order member loads as well as the second-order coupling effect between the transverse load and the axial force in the member. This allows for a minimal discretisation of a frame for second-order analysis. For those conventional analyses which do include transverse member loading, prescribed stiffness matrices must be used for the plethora of specific loading patterns encountered. This paper shows, however, that the principle of superposition can be applied to the equilibrium condition, so that the form of the stiffness matrix remains unchanged with only the magnitude of the loading being needed to be changed in the stiffness formulation. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. The results are verified using analytical stability function studies, as well as with numerical results reported by independent researchers on several simple structural frames.

Experimental and finite element analysis to identify the source of vibration of a coach

This paper presents an investigation on the cause of severe vibration problem of a coach with four-cylinder engine running at an idle state using vibration and impact hammer modal experiments to obtain the main vibration frequency components and the natural characteristics of the coach. The vibration results indicate that the main vibration component comes from the vibration transmitted from the engine to the chassis frame, which is closely related with the engine idle speed. Based on structural simulation analysis of the coach’s chassis frame and comparison with modal testing, the coach severe vibration problem was due to coupling resonance between the engine idle frequency and the fourth natural frequency of the chassis frame. The solution to eliminate the vibration problem is provided by changing the local structure stiffness of the chassis frame. The contribution of this paper lies in providing a solution to solve similar problems.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Finite element analysis of the in-plane shear behaviour of masonry panels confined with reinforced grouted cores

A combined experimental and numerical program was conducted to study the in-plane shear behaviour of hollow concrete masonry panels containing reinforced grout cores. This paper is focused on the numerical program. A two dimensional macromodelling strategy was used to simulate the behaviour of the confined masonry (CM) shear panels. Both the unreinforced masonry and the confining element were modelled using macromasonry properties and the steel reinforcement was modelled as an embedded truss element located within the grout using perfectly bonded constraint. The FE model reproduced key behaviours observed in the experiments, including the shear strength, the deformation and the crack patterns of the unconfined and confined masonry panels. The predictions of the validated model were used to evaluate the existing in-plane shear expressions available in the national masonry standards and research publications.

Swelling induced finite strain flexure in a rectangular block of an isotropic elastic material

The deformation of a rectangular block into an annular wedge is studied with respect to the state of swelling interior to the block. Nonuniform swelling fields are shown to generate these flexure deformations in the absence of resultant forces and bending moments. Analytical expressions for the deformation fields demonstrate these effects for both incompressible and compressible generalizations of conventional hyperelastic materials. Existing results in the absence of a swelling agent are recovered as special cases.

Using patient specific finite element models to understand the biomechanics of paediatric spinal deformity surgery

Adolescent idiopathic scoliosis (AIS) is a spinal deformity, which may require surgical correction by attaching rods to the patient’s spine using screws inserted into the vertebrae. Complication rates for deformity correction surgery are unacceptably high. Determining an achievable correction without overloading the adjacent spinal tissues or implants requires an understanding of the mechanical interaction between these components. We have developed novel patient specific modelling software to create individualized finite element models (FEM) representing the thoracolumbar spine and ribcage of scoliosis patients. We are using these models to better understand the biomechanics of spinal deformity correction.

Finite-SNR diversity-multiplexing trade-off of inter-vehicular cooperative diversity protocols

The finite-signal-to-noise ratio (SNR) diversity-multiplexing trade-off (DMT) of cooperative diversity protocols are investigated in vehicular networks based on cascaded Rayleigh fading. Lower bounds of DMT at finite SNR for orthogonal and non-orthogonal protocols are derived. The results showcase the first look into the achievable DMT trade-off of cooperative diversity in volatile vehicular environments. It is shown that the diversity gains are significantly suboptimal at realistic SNRs.

Determination of strain-rate-dependent mechanical behavior of living and fixed osteocytes and chondrocytes using atomic force microscopy and inverse finite element analysis

The aim of this paper is to determine the strain-rate-dependent mechanical behavior of living and fixed osteocytes and chondrocytes, in vitro. Firstly, Atomic Force Microscopy (AFM) was used to obtain the force-indentation curves of these single cells at four different strain-rates. These results were then employed in inverse finite element analysis (FEA) using Modified Standard neo-Hookean Solid (MSnHS) idealization of these cells to determine their mechanical properties. In addition, a FEA model with a newly developed spring element was employed to accurately simulate AFM evaluation in this study. We report that both cytoskeleton (CSK) and intracellular fluid govern the strain-rate-dependent mechanical property of living cells whereas intracellular fluid plays a predominant role on fixed cells’ behavior. In addition, through the comparisons, it can be concluded that osteocytes are stiffer than chondrocytes at all strain-rates tested indicating that the cells could be the biomarker of their tissue origin. Finally, we report that MSnHS is able to capture the strain-rate-dependent mechanical behavior of osteocyte and chondrocyte for both living and fixed cells. Therefore, we concluded that the MSnHS is a good model for exploration of mechanical deformation responses of single osteocytes and chondrocytes. This study could open a new avenue for analysis of mechanical behavior of osteocytes and chondrocytes as well as other similar types of cells.