B-spline methods in R-matrix theory for scattering in two-electron systems

The use of B-spline basis sets in R-matrix theory for scattering processes has been investigated. In the present approach a B-spline basis is used for the description of the inner region, which is matched to the physical outgoing wavefunctions by the R-matrix. Using B-splines, continuum basis functions can be determined easily, while pseudostates can be included naturally. The accuracy for low-energy scattering processes is demonstrated by calculating inelastic scattering cross sections for e colliding on H. Very good agreement with other calculations has been obtained. Further extensions of the codes to quasi two-electron systems and general atoms are discussed as well as the application to (multi) photoionization.

Partial and total multiphoton detachment rates for H- in a perturbative B-spline-based approach

An ab initio approach has been applied to study multiphoton detachment rates for the negative hydrogen ion in the lowest nonvanishing order of perturbation theory. The approach is based on the use of B splines allowing an accurate treatment of the electronic repulsion. Total detachment rates have been determined for two- to six-photon processes as well as partial rates for detachment into the different final symmetries. It is shown that B-spline expansions can yield accurate continuum and bound-state wave functions in a very simple manner. The calculated total rates for two- and three-photon detachment are in good agreement with other perturbative calculations. For more than three-photon detachment little information has been available before now. While the total cross sections show little structure, a fair amount of structure is predicted in the partial cross sections. In the two-photon process, it is shown that the detached electrons mainly have s character. For four- and six-photon processes, the contribution from the d channel is the most important. For three- and five-photon processes p electrons dominate the electron emission spectrum. Detachment rates for s and p electrons show minima as a function of photon energy. © 1994 The American Physical Society.

Genetic and bacterial programming for B-Spline neural networks design

The design phase of B-spline neural networks is a highly computationally complex task. Existent heuristics have been found to be highly dependent on the initial conditions employed. Increasing interest in biologically inspired learning algorithms for control techniques such as Artificial Neural Networks and Fuzzy Systems is in progress. In this paper, the Bacterial Programming approach is presented, which is based on the replication of the microbial evolution phenomenon. This technique produces an efficient topology search, obtaining additionally more consistent solutions.

A hybrid training method for B-spline neural networks

Current and past research has brought up new views related to the optimization of neural networks. For a fixed structure, second order methods are seen as the most promising. From previous works we have shown how second order methods are of easy applicability to a neural network. Namely, we have proved how the Levenberg-Marquard possesses not only better convergence but how it can assure the convergence to a local minima. However, as any gradient-based method, the results obtained depend on the startup point. In this work, a reformulated Evolutionary algorithm - the Bacterial Programming for Levenberg-Marquardt is proposed, as an heuristic which can be used to determine the most suitable starting points, therefore achieving, in most cases, the global optimum.

B-spline and neuro-fuzzy models design with function and derivative equalities

The design of neuro-fuzzy models is still a complex problem, as it involves not only the determination of the model parameters, but also its structure. Of special importance is the incorporation of a priori information in the design process. In this paper two known design algorithms for B-spline models will be updated to account for function and derivatives equality restrictions, which are important when the neural model is used for performing single or multi-objective optimization on-line.

Design of B-spline neural networks using a bacterial programming approach

The design phase of B-spline neural networks represents a very high computational task. For this purpose, heuristics have been developed, but have been shown to be dependent on the initial conditions employed. In this paper a new technique, Bacterial Programming, is proposed, whose principles are based on the replication of the microbial evolution phenomenon. The performance of this approach is illustrated and compared with existing alternatives.

Completely supervised training algorithms for B-spline neural networks and neuro-fuzzy systems

Complete supervised training algorithms for B-spline neural networks and fuzzy rule-based systems are discussed. By interducing the relationship between B-spline neural networks and certain types of fuzzy models, training algorithms developed initially for neural networks can be adapted by fuzzy systems.

Random regression analyses using B-spline functions to model growth of Nellore cattle

The objective of this study was to estimate (co)variance components using random regression on B-spline functions to weight records obtained from birth to adulthood. A total of 82 064 weight records of 8145 females obtained from the data bank of the Nellore Breeding Program (PMGRN/Nellore Brazil) which started in 1987, were used. The models included direct additive and maternal genetic effects and animal and maternal permanent environmental effects as random. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of age (cubic regression) were considered as random covariate. The random effects were modeled using B-spline functions considering linear, quadratic and cubic polynomials for each individual segment. Residual variances were grouped in five age classes. Direct additive genetic and animal permanent environmental effects were modeled using up to seven knots (six segments). A single segment with two knots at the end points of the curve was used for the estimation of maternal genetic and maternal permanent environmental effects. A total of 15 models were studied, with the number of parameters ranging from 17 to 81. The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most appropriate and parsimonious model to describe the covariance structure of the data. Selection for higher weight, such as at young ages, should be performed taking into account an increase in mature cow weight. Particularly, this is important in most of Nellore beef cattle production systems, where the cow herd is maintained on range conditions. There is limited modification of the growth curve of Nellore cattle with respect to the aim of selecting them for rapid growth at young ages while maintaining constant adult weight.

Random regression analyses using B-spline functions to model growth of Nellore cattle

The objective of this study was to estimate (co)variance components using random regression on B-spline functions to weight records obtained from birth to adulthood. A total of 82 064 weight records of 8145 females obtained from the data bank of the Nellore Breeding Program (PMGRN/Nellore Brazil) which started in 1987, were used. The models included direct additive and maternal genetic effects and animal and maternal permanent environmental effects as random. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of age (cubic regression) were considered as random covariate. The random effects were modeled using B-spline functions considering linear, quadratic and cubic polynomials for each individual segment. Residual variances were grouped in five age classes. Direct additive genetic and animal permanent environmental effects were modeled using up to seven knots (six segments). A single segment with two knots at the end points of the curve was used for the estimation of maternal genetic and maternal permanent environmental effects. A total of 15 models were studied, with the number of parameters ranging from 17 to 81. The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most appropriate and parsimonious model to describe the covariance structure of the data. Selection for higher weight, such as at young ages, should be performed taking into account an increase in mature cow weight. Particularly, this is important in most of Nellore beef cattle production systems, where the cow herd is maintained on range conditions. There is limited modification of the growth curve of Nellore cattle with respect to the aim of selecting them for rapid growth at young ages while maintaining constant adult weight.

Hardware accelerated ray cast of volume data and volume gradient for an optimized splines-based multi-resolution 2D-3D registration

This paper describes a method for DRR generation as well as for volume gradients projection using hardware accelerated 2D texture mapping and accumulation buffering and demonstrates its application in 2D-3D registration of X-ray fluoroscopy to CT images. The robustness of the present registration scheme are guaranteed by taking advantage of a coarse-to-fine processing of the volume/image pyramids based on cubic B-splines. A human cadaveric spine specimen together with its ground truth was used to compare the present scheme with a purely software-based scheme in three aspects: accuracy, speed, and capture ranges. Our experiments revealed an equivalent accuracy and capture ranges but with much shorter registration time with the present scheme. More specifically, the results showed 0.8 mm average target registration error, 55 second average execution time per registration, and 10 mm and 10° capture ranges for the present scheme when tested on a 3.0 GHz Pentium 4 computer.

A subject-specific model of human buttocks and thighs in a seated posture

Finite element analyses of the human body in seated postures requires digital models capable of providing accurate and precise prediction of the tissue-level response of the body in the seated posture. To achieve such models, the human anatomy must be represented with high fidelity. This information can readily be defined using medical imaging techniques such as Magnetic Resonance Imaging (MRI) or Computed Tomography (CT). Current practices for constructing digital human models, based on the magnetic resonance (MR) images, in a lying down (supine) posture have reduced the error in the geometric representation of human anatomy relative to reconstructions based on data from cadaveric studies. Nonetheless, the significant differences between seated and supine postures in segment orientation, soft-tissue deformation and soft tissue strain create a need for data obtained in postures more similar to the application posture. In this study, we present a novel method for creating digital human models based on seated MR data. An adult-male volunteer was scanned in a simulated driving posture using a FONAR 0.6T upright MRI scanner with a T1 scanning protocol. To compensate for unavoidable image distortion near the edges of the study, images of the same anatomical structures were obtained in transverse and sagittal planes. Combinations of transverse and sagittal images were used to reconstruct the major anatomical features from the buttocks through the knees, including bone, muscle and fat tissue perimeters, using Solidworks® software. For each MR image, B-splines were created as contours for the anatomical structures of interest, and LOFT commands were used to interpolate between the generated Bsplines. The reconstruction of the pelvis, from MR data, was enhanced by the use of a template model generated in previous work CT images. A non-rigid registration algorithm was used to fit the pelvis template into the MR data. Additionally, MR image processing was conducted to both the left and the right sides of the model due to the intended asymmetric posture of the volunteer during the MR measurements. The presented subject-specific, three-dimensional model of the buttocks and thighs will add value to optimisation cycles in automotive seat development when used in simulating human interaction with automotive seats.

Assessment and application of clustering techniques to atmospheric particle number size distribution for the purpose of source apportionment

Long-term measurements of particle number size distribution (PNSD) produce a very large number of observations and their analysis requires an efficient approach in order to produce results in the least possible time and with maximum accuracy. Clustering techniques are a family of sophisticated methods which have been recently employed to analyse PNSD data, however, very little information is available comparing the performance of different clustering techniques on PNSD data. This study aims to apply several clustering techniques (i.e. K-means, PAM, CLARA and SOM) to PNSD data, in order to identify and apply the optimum technique to PNSD data measured at 25 sites across Brisbane, Australia. A new method, based on the Generalised Additive Model (GAM) with a basis of penalised B-splines, was proposed to parameterise the PNSD data and the temporal weight of each cluster was also estimated using the GAM. In addition, each cluster was associated with its possible source based on the results of this parameterisation, together with the characteristics of each cluster. The performances of four clustering techniques were compared using the Dunn index and Silhouette width validation values and the K-means technique was found to have the highest performance, with five clusters being the optimum. Therefore, five clusters were found within the data using the K-means technique. The diurnal occurrence of each cluster was used together with other air quality parameters, temporal trends and the physical properties of each cluster, in order to attribute each cluster to its source and origin. The five clusters were attributed to three major sources and origins, including regional background particles, photochemically induced nucleated particles and vehicle generated particles. Overall, clustering was found to be an effective technique for attributing each particle size spectra to its source and the GAM was suitable to parameterise the PNSD data. These two techniques can help researchers immensely in analysing PNSD data for characterisation and source apportionment purposes.

MRI-based strain and strain rate analysis of left ventricle: A modified hierarchical transformation model

Background Different from other indicators of cardiac function, such as ejection fraction and transmitral early diastolic velocity, myocardial strain is promising to capture subtle alterations that result from early diseases of the myocardium. In order to extract the left ventricle (LV) myocardial strain and strain rate from cardiac cine-MRI, a modified hierarchical transformation model was proposed. Methods A hierarchical transformation model including the global and local LV deformations was employed to analyze the strain and strain rate of the left ventricle by cine-MRI image registration. The endocardial and epicardial contour information was introduced to enhance the registration accuracy by combining the original hierarchical algorithm with an Iterative Closest Points using Invariant Features algorithm. The hierarchical model was validated by a normal volunteer first and then applied to two clinical cases (i.e., the normal volunteer and a diabetic patient) to evaluate their respective function. Results Based on the two clinical cases, by comparing the displacement fields of two selected landmarks in the normal volunteer, the proposed method showed a better performance than the original or unmodified model. Meanwhile, the comparison of the radial strain between the volunteer and patient demonstrated their apparent functional difference. Conclusions The present method could be used to estimate the LV myocardial strain and strain rate during a cardiac cycle and thus to quantify the analysis of the LV motion function.

Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.

The Reproducing Kernel DMS-FEM: 3D Shape Functions and Applications to Linear Solid Mechanics

We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and ID NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested by enforcing the condition of polynomial reproduction whilst deriving the shape functions. Nevertheless, given the higher complexity in forming the knotclouds for tetrahedral elements especially when higher demand is placed on the order of continuity of the shape functions across inter-element boundaries, we presently emphasize an exploration of the triangular prism based formulation in the context of several benchmark problems of interest in linear solid mechanics. In the absence of a more rigorous study on the convergence analyses, the numerical exercise, reported herein, helps establish the method as one of remarkable accuracy and robust performance against numerical ill-conditioning (such as locking of different kinds) vis-a-vis the conventional FEM.