Fractal geometry provides a new insight to the approximation and modelling of scientific data.This paper presents the construction of coalescence cubic spline fractal interpolation surfaces over a rectangular grid D through the corresponding univariate basis of coalescence cubic fractal splines of Type-I or Type-II. Coalescence cubic spline fractal surfaces are self-affine or nonself- affine in nature depending on the iterated function systems parameters of these univariate fractal splines. Upper bounds of L_∞-norm of the errors between between a coalescence cubic spline fractal surface and an original function f ∈ C^4[D], and their derivatives are deduced. Finally, the effects of free variables, constrained free variables and hidden variables are discussed for coalescence cubic spline fractal interpolation surfaces through suitably chosen examples.

We consider the possibility of passing high-frequency signals past power transformers forming part of an electrical grid. We first model a transformer, including its laminated core, to obtain asymptotic behaviour of currents and voltages in the secondary circuit. Having got this we are able to determine the effects of different by-pass mechanisms which might be tried to get the high-frequency signal from the primary to the secondary circuit.

The livelihood of humanity depends crucially on the growing and harvesting of crops, the processing of crops to produce the various foods that are eaten and the distribution of the resulting products and produce to the various consumers. The underlying biological foundation on which the success of this complex industrial hierarchy of activity rests is the success of the ongoing process of plant breeding. Not only must plants be bred to ensure that the planned end-products, such as bread, cakes, pasta and noodles, are of acceptable quality, they must, in order to minimize crop failure and thereby ensure food security and supply, also be insect and/or disease resistant. The success of such endeavours rests on the quality of the underlying science, which has become highly sophisticated in recent years. Its utilization, in terms of the modern understanding of the genetics of plant growth and the increasing sophistication of experimentation and instrumentation, has greatly improved the speed and quality of plant breeding. The associated implementation of these new plant breeding protocols is generating a need for improved quantication through the utilization of mathematical modelling. In order to illustrate the diverse range of mathematics required to support such quantification, this paper discusses some illustrative aspects connected with the recent modelling of the flow and deformation of wheat-flour dough, information recovery from spectroscopic data (e.g. such as the determination of the protein content in wheat), antiviral resistance in plants and pattern formation in plants. Various aspects of the mathematics involved are highlighted from a mathematical modelling perspective, with a key secondary goal, using the discussion about these examples, of illustrating how applications impact on mathematics with the resulting mathematical developments in turn contributing to the solution of other applications with the process starting all over again.

Accelerator Driven Sub-critical nuclear reactor System (ADS) are envisaged to enhance neutronics of reactors as well as safety physics. The spallation target module or target system is the most innovative and key component for an ADS. In the target module, a high energy proton beam from the accelerator irradiates a heavy metal target like Lead Bismuth Eutectic (LBE) to produce spallation neutrons, which initiate the fission reaction in the sub-critical core. The removal of the spallation heat by the same LBE is a challenging thermal-hydraulic issue. Also the presence of any recirculation or stagnation zones of LBE in the flow path may lead to local hot spots either in the window or in the flowing liquid metal which is detrimental to the performance of the target. The beam window, a physical barrier separating the liquid metal (LBE) from the proton beam, is a critical component as it is subject to high heat fluxes as well as thermal and mechanical stresses. In addition to heat deposited in the bulk of LBE in the spallation region, large amount of heat also gets deposited on the window. To incorporate the physical situation in a more realistic way, a conjugate heat transfer problem (solving the conduction equation of the beam window in conjunction with the energy equation) is accomplished. As the conjugate heat transfer problem is found to be computationally very demanding, the energy equation module is parallelized following the paradigm of domain decomposition method using MPI (Message passing Interface) library. In this study, the equations governing the axisymmetric flow and thermal energy are solved numerically using a Streamline Upwind Petrov-Galerkin (SUPG) Finite Element (FE) method. The turbulent kinetic energy and its dissipation rate are modeled using k-$\in$ model with standard wall function approach. The interface temperature as a result of conjugate heat transfer and Nusselt number distribution at the interface with a reasonable speedup is computed and quantified.

In this paper we study a finite volume approximation for the conservative formulation of multiple fragmentation models. We investigate the convergence of the numerical solutions towards a weak solution of the continuous problem by considering locally bounded kernels. The proof is based on the Dunford-Pettis theorem by using the weak L^1 compactness method. The analysis of the method allows us to prove the convergence of the discretized approximated solution towards a weak solution to the continuous problem in a weighted L^1 space.

A thin liquid film subject to a temperature gradient undergoes thermocapillary convection because of the non-uniform surface tension at the free surface. This induced flow perturbs the film free surface and generate a free surface velocity field. These observable consequences can be thought of as the “signature” of the imposed temperature field and this work investigates whether the temperature field can be reconstructed from this signature for general three-dimensional flows. Using a model based on the lubrication approximation, we show that one can explicitly formulate the partial differential equation which governs this inverse problem. This equation is solved using finite differences. We illustrates the feasibility of this reconstruction exercise on a set of “artificial” experimental data obtained by first solving the direct problem which consists in computing the free surface deformation and free surface velocity field for a given applied temperature field.

We present two versions of third order accurate jet schemes, which achieve high order accuracy by tracking derivative information of the solution along characteristic curves. For a benchmark linear advection problem, the effciency of jet schemes is compared with WENO and Discontinuous Galerkin methods of the same order. Moreover, the performance of various schemes in tracking solution contours is investigated. It is demonstrated that jet schemes possess the simplicity and speed of WENO schemes, while showing several of the advantages as well as the accuracy of DG methods.

Computational mathematics is constantly evolving to develop novel techniques for solving coupled processes that arise in multi-disciplinary applications. Often such analysis may be accomplished by efficient techniques which involve partitioning the global domain (on which the coupled process evolves) into several sub-domains on each of which local problems are solved. The solution to the global problem is then constructed by suitably piecing together solutions obtained locally from independently modeled sub-domains. In this paper we develop a multilevel computational approach for coupled fluid-structure interaction problems. The method relies on computing coupled solutions over different sub-domains with different multigrid levels. Numerical results for the reliability of the schemes introduced are also presented.

The reliability of computational models of physical processes has received much attention and involves issues such as the validity of the mathematical models being used, the error in any data that the models need, and the accuracy of the numerical schemes being used. These issues are considered in the context of elastic, viscoelastic and hyperelastic deformation, when finite element approximations are applied. Goal oriented techniques using specific quantities of interest (QoI) are described for estimating discretisation and modelling errors in the hyperelastic case. The computational modelling of the rapid large inflation of hyperelastic circular sheets modelled as axisymmetric membranes is then treated, with the aim of estimating engineering QoI and their errors. Fine (involving inertia terms) and coarse (quasi-static) models of the inflation are considered. The techniques are applied to thermoforming processes where sheets are inflated into moulds to form thin-walled structures.

Glass wool manufacturing is a multiphysics problem which requires the understanding of the rotational melt-spinning of ten thousands of viscous thermal slender jets by fast air streams. Due to its high complexity an uniform numerical treatment is impossible. In this work we present a multimethods approach that is based on an asymptotic modeling framework of slender-body theory, homogenization and surrogate models. The algorithm weakly couples melting and spinning phases via iterations. The possibility of combining commercial software and self-implemented code yields satisfying efficiency off-the-shelf. The simulation results are very promising and demonstrate the applicability and practical relevance of our approach for ongoing optimization strategies of the production processes.

The model for pricing of American option gives rise to a parabolic variational inequality. We first use penalty function approach to reformulate it as an equality problem. Since the problem is defined on an unbounded domain, we truncate it to a bounded domain and discuss error due to truncation and penalization. Finite element method is then applied to the penalized problem on the truncated domain. By coupling the penalty parameter and the discretization parameters, error estimates are established when the initial data in H^1_0 . Finally, some numerical experiments are conducted to confirm the theoretical findings.