A bridging transition technique for the combination of meshfree method with finite element method in 2D solids and structures

For certain continuum problems, it is desirable and beneficial to combine two different methods together in order to exploit their advantages while evading their disadvantages. In this paper, a bridging transition algorithm is developed for the combination of the meshfree method (MM) with the finite element method (FEM). In this coupled method, the meshfree method is used in the sub-domain where the MM is required to obtain high accuracy, and the finite element method is employed in other sub-domains where FEM is required to improve the computational efficiency. The MM domain and the FEM domain are connected by a transition (bridging) region. A modified variational formulation and the Lagrange multiplier method are used to ensure the compatibility of displacements and their gradients. To improve the computational efficiency and reduce the meshing cost in the transition region, regularly distributed transition particles, which are independent of either the meshfree nodes or the FE nodes, can be inserted into the transition region. The newly developed coupled method is applied to the stress analysis of 2D solids and structures in order to investigate its’ performance and study parameters. Numerical results show that the present coupled method is convergent, accurate and stable. The coupled method has a promising potential for practical applications, because it can take advantages of both the meshfree method and FEM when overcome their shortcomings.

I-cobb - An innovative technique in spinal radiographic measurement

The measurement of Cobb angles on radiographs of patients with spinal deformities is routine practice in spinal clinics. The technique relies on the use and availability of specialist equipment such as a goniometer, cobbometer or protractor. The aim of this study was to validate the use of i-Phone (Apple Inc) combined with Tilt Meter Pro software as compared to a protractor in the measurement of Cobb angles. The i-Phone combined with Tilt Meter Pro software offers a faster alternative to the traditional method of Cobb angle measurement. The use of i-Phone offers a more convenient way of measuring Cobb angles in the outpatient setting. The intra-observer repeatability of the iPhone is equivalent to the protractor in the measurement of Cobb angles.

Effect of inaccuracies in anthropometric data and linked-segment inverse dynamic modeling on kinetics of gait in persons with partial foot amputation

The accuracy of data derived from linked-segment models depends on how well the system has been represented. Previous investigations describing the gait of persons with partial foot amputation did not account for the unique anthropometry of the residuum or the inclusion of a prosthesis and footwear in the model and, as such, are likely to have underestimated the magnitude of the peak joint moments and powers. This investigation determined the effect of inaccuracies in the anthropometric input data on the kinetics of gait. Toward this end, a geometric model was developed and validated to estimate body segment parameters of various intact and partial feet. These data were then incorporated into customized linked-segment models, and the kinetic data were compared with that obtained from conventional models. Results indicate that accurate modeling increased the magnitude of the peak hip and knee joint moments and powers during terminal swing. Conventional inverse dynamic models are sufficiently accurate for research questions relating to stance phase. More accurate models that account for the anthropometry of the residuum, prosthesis, and footwear better reflect the work of the hip extensors and knee flexors to decelerate the limb during terminal swing phase.

Social adaptation of children with mild intellectual disability : effects of partial integration within primary school classes

Examined the social adaptation of 32 children in grades 3–6 with mild intellectual disability: 13 Ss were partially integrated into regular primary school classes and 19 Ss were full-time in separate classes. Sociometric status was assessed using best friend and play rating measures. Consistent with previous research, children with intellectual disability were less socially accepted than were a matched group of 32 children with no learning disabilities. Children in partially integrated classes received more play nominations than those in separate classes, but had no greater acceptance as a best friend. On teachers' reports, disabled children had higher levels of inappropriate social behaviours, but there was no significant difference in appropriate behaviours. Self-assessments by integrated children were more negative than those by children in separate classes, and their peer-relationship satisfaction was lower. Ratings by disabled children of their satisfaction with peer relationships were associated with ratings of appropriate social skills by themselves and their teachers, and with self-ratings of negative behaviour. The study confirmed that partial integration can have negative consequences for children with an intellectual disability.

Application of bag sampling technique for particle size distribution measurements

Bag sampling techniques can be used to temporarily store an aerosol and therefore provide sufficient time to utilize sensitive but slow instrumental techniques for recording detailed particle size distributions. Laboratory based assessment of the method were conducted to examine size dependant deposition loss coefficients for aerosols held in VelostatTM bags conforming to a horizontal cylindrical geometry. Deposition losses of NaCl particles in the range of 10 nm to 160 nm were analysed in relation to the bag size, storage time, and sampling flow rate. Results of this study suggest that the bag sampling method is most useful for moderately short sampling periods of about 5 minutes.

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

The neuro-muscular and musculo-skeletal characterization of children with joint hypermobility

In children, joint hypermobility (typified by structural instability of joints) manifests clinically as neuro-muscular and musculo-skeletal conditions and conditions associated with development and organization of control of posture and gait (Finkelstein, 1916; Jahss, 1919; Sobel, 1926; Larsson, Mudholkar, Baum and Srivastava, 1995; Murray and Woo, 2001; Hakim and Grahame, 2003; Adib, Davies, Grahame, Woo and Murray, 2005:). The process of control of the relative proportions of joint mobility and stability, whilst maintaining equilibrium in standing posture and gait, is dependent upon the complex interrelationship between skeletal, muscular and neurological function (Massion, 1998; Gurfinkel, Ivanenko, Levik and Babakova, 1995; Shumway-Cook and Woollacott, 1995). The efficiency of this relies upon the integrity of neuro-muscular and musculo-skeletal components (ligaments, muscles, nerves), and the Central Nervous System’s capacity to interpret, process and integrate sensory information from visual, vestibular and proprioceptive sources (Crotts, Thompson, Nahom, Ryan and Newton, 1996; Riemann, Guskiewicz and Shields, 1999; Schmitz and Arnold, 1998) and development and incorporation of this into a representational scheme (postural reference frame) of body orientation with respect to internal and external environments (Gurfinkel et al., 1995; Roll and Roll, 1988). Sensory information from the base of support (feet) makes significant contribution to the development of reference frameworks (Kavounoudias, Roll and Roll, 1998). Problems with the structure and/ or function of any one, or combination of these components or systems, may result in partial loss of equilibrium and, therefore ineffectiveness or significant reduction in the capacity to interact with the environment, which may result in disability and/ or injury (Crotts et al., 1996; Rozzi, Lephart, Sterner and Kuligowski, 1999b). Whilst literature focusing upon clinical associations between joint hypermobility and conditions requiring therapeutic intervention has been abundant (Crego and Ford, 1952; Powell and Cantab, 1983; Dockery, in Jay, 1999; Grahame, 1971; Childs, 1986; Barton, Bird, Lindsay, Newton and Wright, 1995a; Rozzi, et al., 1999b; Kerr, Macmillan, Uttley and Luqmani, 2000; Grahame, 2001), there has been a deficit in controlled studies in which the neuro-muscular and musculo-skeletal characteristics of children with joint hypermobility have been quantified and considered within the context of organization of postural control in standing balance and gait. This was the aim of this project, undertaken as three studies. The major study (Study One) compared the fundamental neuro-muscular and musculo-skeletal characteristics of 15 children with joint hypermobility, and 15 age (8 and 9 years), gender, height and weight matched non-hypermobile controls. Significant differences were identified between previously undiagnosed hypermobile (n=15) and non-hypermobile children (n=15) in passive joint ranges of motion of the lower limbs and lumbar spine, muscle tone of the lower leg and foot, barefoot CoP displacement and in parameters of barefoot gait. Clinically relevant differences were also noted in barefoot single leg balance time. There were no differences between groups in isometric muscle strength in ankle dorsiflexion, knee flexion or extension. The second comparative study investigated foot morphology in non-weight bearing and weight bearing load conditions of the same children with and without joint hypermobility using three dimensional images (plaster casts) of their feet. The preliminary phase of this study evaluated the casting technique against direct measures of foot length, forefoot width, RCSP and forefoot to rearfoot angle. Results indicated accurate representation of elementary foot morphology within the plaster images. The comparative study examined the between and within group differences in measures of foot length and width, and in measures above the support surface (heel inclination angle, forefoot to rearfoot angle, normalized arch height, height of the widest point of the heel) in the two load conditions. Results of measures from plaster images identified that hypermobile children have different barefoot weight bearing foot morphology above the support surface than non-hypermobile children, despite no differences in measures of foot length or width. Based upon the differences in components of control of posture and gait in the hypermobile group, identified in Study One and Study Two, the final study (Study Three), using the same subjects, tested the immediate effect of specifically designed custom-made foot orthoses upon balance and gait of hypermobile children. The design of the orthoses was evaluated against the direct measures and the measures from plaster images of the feet. This ascertained the differences in morphology of the modified casts used to mould the orthoses and the original image of the foot. The orthoses were fitted into standardized running shoes. The effect of the shoe alone was tested upon the non-hypermobile children as the non-therapeutic equivalent condition. Immediate improvement in balance was noted in single leg stance and CoP displacement in the hypermobile group together with significant immediate improvement in the percentage of gait phases and in the percentage of the gait cycle at which maximum plantar flexion of the ankle occurred in gait. The neuro-muscular and musculo-skeletal characteristics of children with joint hypermobility are different from those of non-hypermobile children. The Beighton, Solomon and Soskolne (1973) screening criteria successfully classified joint hypermobility in children. As a result of this study joint hypermobility has been identified as a variable which must be controlled in studies of foot morphology and function in children. The outcomes of this study provide a basis upon which to further explore the association between joint hypermobility and neuro-muscular and musculo-skeletal conditions, and, have relevance for the physical education of children with joint hypermobility, for footwear and orthotic design processes, and, in particular, for clinical identification and treatment of children with joint hypermobility.