Exact series solutions of reactive transport models with general initial conditions

Exact solutions of partial differential equation models describing the transport and decay of single and coupled multispecies problems can provide insight into the fate and transport of solutes in saturated aquifers. Most previous analytical solutions are based on integral transform techniques, meaning that the initial condition is restricted in the sense that the choice of initial condition has an important impact on whether or not the inverse transform can be calculated exactly. In this work we describe and implement a technique that produces exact solutions for single and multispecies reactive transport problems with more general, smooth initial conditions. We achieve this by using a different method to invert a Laplace transform which produces a power series solution. To demonstrate the utility of this technique, we apply it to two example problems with initial conditions that cannot be solved exactly using traditional transform techniques.

Task constraints

Newell (1985, 1986) identified the importance of interacting constraints on the emergent behaviours of learners or performers in sport as they assemble functional states of movement organisation in achieving task goals. Constraints, related to the person, task and environment, were defined as ‘boundaries or features that limit motion of the entity under consideration at any moment in time’ (Newell, 1986, p.347). Personal (or organismic) constraints include factors such as individual anthropometrics (height, weight, and limb lengths), fitness (e.g., strength, speed, aerobic capacity, and flexibility), mental skills (e.g. concentration, confidence, emotional control and motivation), perceptual and decisionmaking skills (e.g., recognising patterns of play, anticipation by reading the movements of opponents) and personality factors (e.g., risk taking or conservative behaviours). Newell (1986, p.350) distinguished between general environmental constraints, such as gravity, ambient temperature, natural light and altitude and task constraints, which are task specific and concerned with the goals of a specific activity. More recently, socio-cultural constraints (e.g., family support, cultural expectations and access to facilities) have also been considered as environmental constraints. Application of the constraints framework to the study of sport performance has led to task constraints being defined to include factors such as rules of games, equipment used, boundary playing areas and markings, nets and goals, the number of...

Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton-Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

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.

GNSS ambiguity resolution with controllable failure rate for long baseline network RTK

Many large-scale GNSS CORS networks have been deployed around the world to support various commercial and scientific applications. To make use of these networks for real-time kinematic positioning services, one of the major challenges is the ambiguity resolution (AR) over long inter-station baselines in the presence of considerable atmosphere biases. Usually, the widelane ambiguities are fixed first, followed by the procedure of determination of the narrowlane ambiguity integers based on the ionosphere-free model in which the widelane integers are introduced as known quantities. This paper seeks to improve the AR performance over long baseline through efficient procedures for improved float solutions and ambiguity fixing. The contribution is threefold: (1) instead of using the ionosphere-free measurements, the absolute and/or relative ionospheric constraints are introduced in the ionosphere-constrained model to enhance the model strength, thus resulting in the better float solutions; (2) the realistic widelane ambiguity precision is estimated by capturing the multipath effects due to the observation complexity, leading to improvement of reliability of widelane AR; (3) for the narrowlane AR, the partial AR for a subset of ambiguities selected according to the successively increased elevation is applied. For fixing the scalar ambiguity, an error probability controllable rounding method is proposed. The established ionosphere-constrained model can be efficiently solved based on the sequential Kalman filter. It can be either reduced to some special models simply by adjusting the variances of ionospheric constraints, or extended with more parameters and constraints. The presented methodology is tested over seven baselines of around 100 km from USA CORS network. The results show that the new widelane AR scheme can obtain the 99.4 % successful fixing rate with 0.6 % failure rate; while the new rounding method of narrowlane AR can obtain the fix rate of 89 % with failure rate of 0.8 %. In summary, the AR reliability can be efficiently improved with rigorous controllable probability of incorrectly fixed ambiguities.

Retrospective evaluation of dosimetric quality for prostate carcinomas treated with 3D conformal, intensity-modulated and volumetric-modulated arc radiotherapy

Introduction This study examines and compares the dosimetric quality of radiotherapy treatment plans for prostate carcinoma across a cohort of 163 patients treated across 5 centres: 83 treated with three-dimensional conformal radiotherapy (3DCRT), 33 treated with intensity-modulated radiotherapy (IMRT) and 47 treated with volumetric-modulated arc therapy (VMAT). Methods Treatment plan quality was evaluated in terms of target dose homogeneity and organ-at-risk sparing, through the use of a set of dose metrics. These included the mean, maximum and minimum doses; the homogeneity and conformity indices for the target volumes; and a selection of dose coverage values that were relevant to each organ-at-risk. Statistical significance was evaluated using two-tailed Welch’s T-tests. The Monte Carlo DICOM ToolKit software was adapted to permit the evaluation of dose metrics from DICOM data exported from a commercial radiotherapy treatment planning system. Results The 3DCRT treatment plans offered greater planning target volume dose homogeneity than the other two treatment modalities. The IMRT and VMAT plans offered greater dose reduction in the organs-at-risk: with increased compliance with recommended organ-at-risk dose constraints, compared to conventional 3DCRT treatments. When compared to each other, IMRT and VMAT did not provide significantly different treatment plan quality for like-sized tumour volumes. Conclusions This study indicates that IMRT and VMAT have provided similar dosimetric quality, which is superior to the dosimetric quality achieved with 3DCRT.

Two-scale computational modelling of unsaturated flow in soils exhibiting small scale heterogeneities

Unsaturated water flow in soil is commonly modelled using Richards’ equation, which requires the hydraulic properties of the soil (e.g., porosity, hydraulic conductivity, etc.) to be characterised. Naturally occurring soils, however, are heterogeneous in nature, that is, they are composed of a number of interwoven homogeneous soils each with their own set of hydraulic properties. When the length scale of these soil heterogeneities is small, numerical solution of Richards’ equation is computationally impractical due to the immense effort and refinement required to mesh the actual heterogeneous geometry. A classic way forward is to use a macroscopic model, where the heterogeneous medium is replaced with a fictitious homogeneous medium, which attempts to give the average flow behaviour at the macroscopic scale (i.e., at a scale much larger than the scale of the heterogeneities). Using the homogenisation theory, a macroscopic equation can be derived that takes the form of Richards’ equation with effective parameters. A disadvantage of the macroscopic approach, however, is that it fails in cases when the assumption of local equilibrium does not hold. This limitation has seen the introduction of two-scale models that include at each point in the macroscopic domain an additional flow equation at the scale of the heterogeneities (microscopic scale). This report outlines a well-known two-scale model and contributes to the literature a number of important advances in its numerical implementation. These include the use of an unstructured control volume finite element method and image-based meshing techniques, that allow for irregular micro-scale geometries to be treated, and the use of an exponential time integration scheme that permits both scales to be resolved simultaneously in a completely coupled manner. Numerical comparisons against a classical macroscopic model confirm that only the two-scale model correctly captures the important features of the flow for a range of parameter values.

Output feedback optimal control with contraints

In Chapters 1 through 9 of the book (with the exception of a brief discussion on observers and integral action in Section 5.5 of Chapter 5) we considered constrained optimal control problems for systems without uncertainty, that is, with no unmodelled dynamics or disturbances, and where the full state was available for measurement. More realistically, however, it is necessary to consider control problems for systems with uncertainty. This chapter addresses some of the issues that arise in this situation. As in Chapter 9, we adopt a stochastic description of uncertainty, which associates probability distributions to the uncertain elements, that is, disturbances and initial conditions. (See Section 12.6 for references to alternative approaches to model uncertainty.) When incomplete state information exists, a popular observer-based control strategy in the presence of stochastic disturbances is to use the certainty equivalence [CE] principle, introduced in Section 5.5 of Chapter 5 for deterministic systems. In the stochastic framework, CE consists of estimating the state and then using these estimates as if they were the true state in the control law that results if the problem were formulated as a deterministic problem (that is, without uncertainty). This strategy is motivated by the unconstrained problem with a quadratic objective function, for which CE is indeed the optimal solution (˚Astr¨om 1970, Bertsekas 1976). One of the aims of this chapter is to explore the issues that arise from the use of CE in RHC in the presence of constraints. We then turn to the obvious question about the optimality of the CE principle. We show that CE is, indeed, not optimal in general. We also analyse the possibility of obtaining truly optimal solutions for single input linear systems with input constraints and uncertainty related to output feedback and stochastic disturbances.We first find the optimal solution for the case of horizon N = 1, and then we indicate the complications that arise in the case of horizon N = 2. Our conclusion is that, for the case of linear constrained systems, the extra effort involved in the optimal feedback policy is probably not justified in practice. Indeed, we show by example that CE can give near optimal performance. We thus advocate this approach in real applications.

Weak conformance of process models with respect to data objects

Process models specify behavioral aspects by describing ordering constraints between tasks which must be accomplished to achieve envisioned goals. Tasks usually exchange information by means of data objects, i.e., by writing information to and reading information from data objects. A data object can be characterized by its states and allowed state transitions. In this paper, we propose a notion which checks conformance of a process model with respect to data objects that its tasks access. This new notion can be used to tell whether in every execution of a process model each time a task needs to access a data object in a particular state, it is ensured that the data object is in the expected state or can reach the expected state and, hence, the process model can achieve its goals.

On optimal control of constrained linear systems with imperfect state information and stochastic disturbances

We address the problem of finite horizon optimal control of discrete-time linear systems with input constraints and uncertainty. The uncertainty for the problem analysed is related to incomplete state information (output feedback) and stochastic disturbances. We analyse the complexities associated with finding optimal solutions. We also consider two suboptimal strategies that could be employed for larger optimization horizons.

A low-complexity flight controller for Unmanned Aircraft Systems with constrained control allocation

In this paper, we propose a framework for joint allocation and constrained control design of flight controllers for Unmanned Aircraft Systems (UAS). The actuator configuration is used to map actuator constraint set into the space of the aircraft generalised forces. By constraining the demanded generalised forces, we ensure that the allocation problem is always feasible; and therefore, it can be solved without constraints. This leads to an allocation problem that does not require on-line numerical optimisation. Furthermore, since the controller handles the constraints, and there is no need to implement heuristics to inform the controller about actuator saturation. The latter is fundamental for avoiding Pilot Induced Oscillations (PIO) in remotely operated UAS due to the rate limit on the aircraft control surfaces.

Characterising transiting exoplanets by way of differential photometry

This paper describes a simple activity for plotting and characterising the light curve from an exoplanet transit event by way of differential photometry analysis. Using free digital imaging software, participants analyse a series of telescope images with the goal of calculating various exoplanet parameters, including its size, orbital radius and habitability. The activity has been designed for a high-school or undergraduate university level and introduces fundamental concepts in astrophysics and an understanding of the basis for exoplanetary science, the transit method and digital photometry.

High and low mammographic density human breast tissues maintain histological differential in murine tissue engineering chambers

Mammographic density (MD) is the area of breast tissue that appears radiologically white on mammography. Although high MD is a strong risk factor for breast cancer, independent of BRCA1/2 mutation status, the molecular basis of high MD and its associated breast cancer risk is poorly understood. MD studies will benefit from an animal model, where hormonal, gene and drug perturbations on MD can be measured in a preclinical context. High and low MD tissues were selectively sampled by stereotactic biopsy from operative specimens of high-risk women undergoing prophylactic mastectomy. The high and low MD tissues were transferred into separate vascularised biochambers in the groins of SCID mice. Chamber material was harvested after 6 weeks for histological analyses and immunohistochemistry for cytokeratins, vimentin and a human-specific mitochondrial antigen. Within-individual analysis was performed in replicate mice, eliminating confounding by age, body mass index and process-related factors, and comparisons were made to the parental human tissue. Maintenance of differential MD post-propagation was assessed radiographically. Immunohistochemical staining confirmed the preservation of human glandular and stromal components in the murine biochambers, with maintenance of radiographic MD differential. Propagated high MD regions had higher stromal (p = 0.0002) and lower adipose (p = 0.0006) composition, reflecting the findings in the original human breast tissue, although glands appeared small and non-complex in both high and low MD groups. No significant differences were observed in glandular area (p = 0.4) or count (p = 0.4) between high and low MD biochamber tissues. Human mammary glandular and stromal tissues were viably maintained in murine biochambers, with preservation of differential radiographic density and histological features. Our study provides a murine model for future studies into the biomolecular basis of MD as a risk factor for breast cancer.

Control design for positioning of underwater vehicles with control allocation

In this paper, we address the control design problem of positioning of over-actuated underwater vehicles. The proposed design is based on a control architecture with combined position and velocity loops and a control tuning method based on the decoupled models. We derive analytical tuning rules based on requirements of closed-loop stability, positioning performance, and the vehicle velocity dynamic characteristics. The vehicle modelling is considered from force to motion with appropriate simplifications related to low-speed manoeuvring hydrodynamics and vehicle symmetry. The control design is considered together with a control allocation mapping. This approach makes the control tuning independent of the characteristics of the force actuators and provides the basis for control reconfiguration in the presence of actuator failure. We propose an anti-wind-up implementation of the controller, which ensures that the constraints related to actuation capacity are not violated. This approach simplifies the control allocation problem since the actuator constraints are mapped into generalised force constraints.

Anti-wind-up designs for dynamic positioning of marine vehicles with control allocation

This paper presents a framework for the design of a joint motion controller and a control allocation strategy for dynamic positioning of marine vehicles. The key aspects of the proposed designs are a systematic approach to deal with actuator saturation and to inform the motion controller about saturation. The proposed system uses a mapping that translates the actuator constraint sets into constraint sets at the motion controller level. Hence, while the motion controller addresses the constraints, the control allocation algorithm can solve an unconstrained optimisation problem. The constrained control design is approached using a multivariable anti-wind-up strategy for strictly proper controllers. This is applicable to the implementation of PI and PID type of motion controllers.