Free surface problems for static Coulomb-Mohr granular solids

This paper is concerned with some plane strain and axially symmetric free surface problems which arise in the study of static granular solids that satisfy the Coulomb-Mohr yield condition. Such problems are inherently nonlinear, and hence difficult to attack analytically. Given a Coulomb friction condition holds on a solid boundary, it is shown that the angle a free surface is allowed to attach to the boundary is dependent only on the angle of wall friction, assuming the stresses are all continuous at the attachment point, and assuming also that the coefficient of cohesion is nonzero. As a model problem, the formation of stable cohesive arches in hoppers is considered. This undesirable phenomena is an obstacle to flow, and occurs when the hopper outlet is too small. Typically, engineers are concerned with predicting the critical outlet size for a given hopper and granular solid, so that for hoppers with outlets larger than this critical value, arching cannot occur. This is a topic of considerable practical interest, with most accepted engineering methods being conservative in nature. Here, the governing equations in two limiting cases (small cohesion and high angle of internal friction) are considered directly. No information on the critical outlet size is found; however solutions for the shape of the free boundary (the arch) are presented, for both plane and axially symmetric geometries.

A geometrical approach to the motion planning problem for a submerged rigid body

The main focus of this paper is the motion planning problem for a deeply submerged rigid body. The equations of motion are formulated and presented by use of the framework of differential geometry and these equations incorporate external dissipative and restoring forces. We consider a kinematic reduction of the affine connection control system for the rigid body submerged in an ideal fluid, and present an extension of this reduction to the forced affine connection control system for the rigid body submerged in a viscous fluid. The motion planning strategy is based on kinematic motions; the integral curves of rank one kinematic reductions. This method is of particular interest to autonomous underwater vehicles which can not directly control all six degrees of freedom (such as torpedo shaped AUVs) or in case of actuator failure (i.e., under-actuated scenario). A practical example is included to illustrate our technique.

Decoupled trajectory planning for a submerged rigid body subject to dissipative and potential forces

This paper studies the practical but challenging problem of motion planning for a deeply submerged rigid body. Here, we formulate the dynamic equations of motion of a submerged rigid body under the architecture of differential geometric mechanics and include external dissipative and potential forces. The mechanical system is represented as a forced affine-connection control system on the configuration space SE(3). Solutions to the motion planning problem are computed by concatenating and reparameterizing the integral curves of decoupling vector fields. We provide an extension to this inverse kinematic method to compensate for external potential forces caused by buoyancy and gravity. We present a mission scenario and implement the theoretically computed control strategy onto a test-bed autonomous underwater vehicle. This scenario emphasizes the use of this motion planning technique in the under-actuated situation; the vehicle loses direct control on one or more degrees of freedom. We include experimental results to illustrate our technique and validate our method.

Controlling a submerged rigid body : a geometric analysis

In this paper we analyze the equations of motion of a submerged rigid body. Our motivation is based on recent developments done in trajectory design for this problem. Our goal is to relate some properties of singular extremals to the existence of decoupling vector fields. The ideas displayed in this paper can be viewed as a starting point to a geometric formulation of the trajectory design problem for mechanical systems with potential and external forces.

Processing load and the use of concrete representations and strategies for solving linear equations

This paper is a report of students' responses to instruction which was based on the use of concrete representations to solve linear equations. The sample consisted of 21 Grade 8 students from a middle-class suburban state secondary school with a reputation for high academic standards and innovative mathematics teaching. The students were interviewed before and after instruction. Interviews and classroom interactions were observed and videotaped. A qualitative analysis of the responses revealed that students did not use the materials in solving problems. The increased processing load caused by concrete representations is hypothesised as a reason.

Sequential nonlinear manifold learning

The computation of compact and meaningful representations of high dimensional sensor data has recently been addressed through the development of Nonlinear Dimensional Reduction (NLDR) algorithms. The numerical implementation of spectral NLDR techniques typically leads to a symmetric eigenvalue problem that is solved by traditional batch eigensolution algorithms. The application of such algorithms in real-time systems necessitates the development of sequential algorithms that perform feature extraction online. This paper presents an efficient online NLDR scheme, Sequential-Isomap, based on incremental singular value decomposition (SVD) and the Isomap method. Example simulations demonstrate the validity and significant potential of this technique in real-time applications such as autonomous systems.

An analytical method to solve a general class of nonlinear reactive transport models

Three recent papers published in Chemical Engineering Journal studied the solution of a model of diffusion and nonlinear reaction using three different methods. Two of these studies obtained series solutions using specialized mathematical methods, known as the Adomian decomposition method and the homotopy analysis method. Subsequently it was shown that the solution of the same particular model could be written in terms of a transcendental function called Gauss’ hypergeometric function. These three previous approaches focused on one particular reactive transport model. This particular model ignored advective transport and considered one specific reaction term only. Here we generalize these previous approaches and develop an exact analytical solution for a general class of steady state reactive transport models that incorporate (i) combined advective and diffusive transport, and (ii) any sufficiently differentiable reaction term R(C). The new solution is a convergent Maclaurin series. The Maclaurin series solution can be derived without any specialized mathematical methods nor does it necessarily involve the computation of any transcendental function. Applying the Maclaurin series solution to certain case studies shows that the previously published solutions are particular cases of the more general solution outlined here. We also demonstrate the accuracy of the Maclaurin series solution by comparing with numerical solutions for particular cases.

Electron diffusion and back reaction in dye-sensitized solar cells : the effect of nonlinear recombination kinetics

The electron collection efficiency in dye-sensitized solar cells (DSCs) is usually related to the electron diffusion length, L = (Dτ)^1/2, where D is the diffusion coefficient of mobile electrons and τ is their lifetime, which is determined by electron transfer to the redox electrolyte. Analysis of incident photon-to-current efficiency (IPCE) spectra for front and rear illumination consistently gives smaller values of L than those derived from small amplitude methods. We show that the IPCE analysis is incorrect if recombination is not first-order in free electron concentration, and we demonstrate that the intensity dependence of the apparent L derived by first-order analysis of IPCE measurements and the voltage dependence of L derived from perturbation experiments can be fitted using the same reaction order, γ ≈ 0.8. The new analysis presented in this letter resolves the controversy over why L values derived from small amplitude methods are larger than those obtained from IPCE data.

Stability boundary analysis of the dynamic voltage restorer in weak systems with dynamic loads

In this contribution, a stability analysis for a dynamic voltage restorer (DVR) connected to a weak ac system containing a dynamic load is presented using continuation techniques and bifurcation theory. The system dynamics are explored through the continuation of periodic solutions of the associated dynamic equations. The switching process in the DVR converter is taken into account to trace the stability regions through a suitable mathematical representation of the DVR converter. The stability regions in the Thevenin equivalent plane are computed. In addition, the stability regions in the control gains space, as well as the contour lines for different Floquet multipliers, are computed. Besides, the DVR converter model employed in this contribution avoids the necessity of developing very complicated iterative map approaches as in the conventional bifurcation analysis of converters. The continuation method and the DVR model can take into account dynamics and nonlinear loads and any network topology since the analysis is carried out directly from the state space equations. The bifurcation approach is shown to be both computationally efficient and robust, since it eliminates the need for numerically critical and long-lasting transient simulations.

Communicating assessment criteria is not sufficient for influencing students' approaches to assessment tasks – perspectives from a differential equations class

This report presents the findings of an exploratory study into the perceptions held by students regarding the use of criterion-referenced assessment in an undergraduate differential equations class. Students in the class were largely unaware of the concept of criterion referencing and of the various interpretations that this concept has among mathematics educators. Our primary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. Quantitative data and qualitative feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, the manner in which they actually approached the assessment activity was not altered as a result of the use of explicitly communicated grading criteria.

Minimising wave drag for free surface flow past a two-dimensional stern

Free surface flow past a two-dimensional semi-infinite curved plate is considered, with emphasis given to solving for the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the Wiener-Hopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results found using a forced KdV equation are also presented, as are numerical solutions to the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semi-infinite plate, it is shown how the amplitude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely.

Drug diffusion from polymeric delivery devices : a problem with two moving boundaries

An existing model for solvent penetration and drug release from a spherically-shaped polymeric drug delivery device is revisited. The model has two moving boundaries, one that describes the interface between the glassy and rubbery states of polymer, and another that defines the interface between the polymer ball and the pool of solvent. The model is extended so that the nonlinear diffusion coefficient of drug explicitly depends on the concentration of solvent, and the resulting equations are solved numerically using a front-fixing transformation together with a finite difference spatial discretisation and the method of lines. We present evidence that our scheme is much more accurate than a previous scheme. Asymptotic results in the small-time limit are presented, which show how the use of a kinetic law as a boundary condition on the innermost moving boundary dictates qualitative behaviour, the scalings being very different to the similar moving boundary problem that arises from modelling the melting of an ice ball. The implication is that the model considered here exhibits what is referred to as ``non-Fickian'' or Case II diffusion which, together with the initially constant rate of drug release, has certain appeal from a pharmaceutical perspective.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy