A Second Order Finite Volume Technique for Simulating Transport in Anisotropic Media

An existing two-dimensional finite volume technique is modified by introducing a correction term to increase the accuracy of the method to second order. It is well known that the accuracy of the finite volume method strongly depends on the order of the approximation of the flux term at the control volume (CV) faces. For highly orthotropic and anisotropic media, first order approximations produce inaccurate simulation results, which motivates the need for better estimates of the flux expression. In this article, a new approach to approximate the flux term at the CV face is presented. The discretisation involves a decomposition of the flux and an improved least squares approximation technique to calculate the derivatives of the dependent function on the CV faces for estimating both the cross diffusion term and a correction for the primary flux term. The advantage of this method is that any arbitrary unstructured mesh can be used to implement the technique without considering the shapes of the mesh elements. It was found that the numerical results well matched the available exact solution for a representative transport equation in highly orthotropic media and the benchmark solutions obtained on a fine mesh for anisotropic media. Previously proposed CV techniques are compared with the new method to highlight its accuracy for different unstructured meshes

Modeling anisotropic diffusion using a departure from isotropy approach

There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method

Generalised Finite Volume Strategies for Simulating Transport in Strongly Orthotropic Porous Media

In this work two different finite volume computational strategies for solving a representative two-dimensional diffusion equation in an orthotropic medium are considered. When the diffusivity tensor is treated as linear, this problem admits an analytic solution used for analysing the accuracy of the proposed numerical methods. In the first method, the gradient approximation techniques discussed by Jayantha and Turner [Numerical Heat Transfer, Part B: Fundamentals, 40, pp.367--390, 2001] are applied directly to the diffusion equation. In the second method, the diffusion equation is transformed via scaling parameters to an isotropic model and then the control volume techniques discussed by Jayantha and Turner are used to obtain the numerical results on the transformed domain. Both methods are shown to produce reasonable results in comparison with the exact solution for a range of anisotropy ratios typical of wood. However, only the first method is appropriate for use in non-linear coupled transport systems. This work highlights the necessity of determining a higher order gradient approximation to improve the numerical results on the untransformed domain

On the Use of Surface Interpolation Techniques in Generalised Finite Volume Strategies for Simulating Transport in Highly Anisotropic Porous Media

A control volume technique for solving a representative di3usion equation in an orthotropic medium is considered.The approximation of the cross-di3usion 6ux term is of utmost important for the accuracy of the solution.A preliminary investigation that used exact function values from an available analytical solution to approximate this term during the numerical simulation provided excellent agreement with the exact solution. This &nding motivated the need for accurate surface interpolation techniques for estimating the cross-di3usion term.The use of radial basis functions is a well-known interpolation technique for &tting scattered data, which can be considered as a global interpolating method because function values in the whole solution domain contribute towards the interpolation.A number of radial basis functions (RBF) was used to approximate the gradients in the cross-di3usion 6ux term and it was found that the accuracy of the &nite volume solution was generally poor.It was concluded that the RBF estimated function does not re6ect local variation of the solution, particularly for the gradients.Another strategy for local function estimation concerns the weighted least-squares method.Di3erent variants of this method were analysed here for approximating the cross-di3usion term and it was found that the numerical results well matched the exact solution.The results highlight that the development of an accurate, generalised &nite volume strategy requires a highly accurate 6ux approximation to enable second-order spatial accuracy to be achieved

A Second Order Control-Volume Finite-Element Least-Squares Strategy for Simulating Diffusion in Strongly Anisotropic Media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality

A Comparison of Gradient Approximations for Use in Finite-Volume Computational Models for Two-Dimensional Diffusion Equations

Finite-volume methods (FVMs) are now a popular choice among practitioners in scientific computation and engineering. This article focuses on generalized FVMs that can be implemented on any mesh structure. The accuracy of FVMs is primarily influenced by the numerical approximation of the flux term at the control-volume face. Here, different flux approximations are compared to identify which approximation is the most accurate, independent of the mesh structure. The accuracy of the classical two-node approximation can be improved significantly by using a local gradient reconstruction to capture the crossdiffusion term of the flux at the control-volume face. A simple two-dimensional isotropic diffusion equation for which an analytical solution is available is chosen for benchmarking purposes

A finite volume method with linearisation in time for the solution of advection-reaction-diffusion systems

The numerical solution in one space dimension of advection--reaction--diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature. Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models. This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest. The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations

Nonlinear spin waves in magnetic thin films - foldover, dispersive shock waves, and spin pumping

Includes bibliographical references.2016 Fall.Three nonlinear phenomena of spin waves and the spin Seebeck effect in yttrium iron garnet (YIG)/Pt bi-layer structures are studied in this thesis and are reported in detail in Chapters 4-7. In the fourth chapter, the first observation of foldover effect of nonlinear eigenmodes in feedback ring systems is reported. The experiments made use of a system that consisted of a YIG thin film strip, which supported the propagation of forward volume spin waves, and a microwave amplifier, which amplified the signal from the output of the YIG strip and then fed it back to the input of the strip. The signal amplitude vs. frequency response in this ring system showed resonant peaks which resulted from ring eigenmodes. With an increase in the resonance amplitude, those resonant peaks evolved from symmetric peaks to asymmetric ones and then folded over to higher frequencies. The experimental observations were reproduced by theoretical calculations that took into account the nonlinearity-produced frequency shift of the traveling spin waves. The fifth chapter presents the first experimental observation of the formation of envelope dispersive shock wave (DSW) excitations from repulsive nonlinear spin waves. The experiments used a microwave step pulse to excite a spin-wave step pulse in a YIG thin film strip, in which the spin-wave amplitude increases rapidly. Under certain conditions, the spin-wave pulse evolved into a DSW excitation that consisted of a train of dark soliton-like dips with both the dip width and depth increasing from the front to the back and was terminated by a black soliton that had an almost zero intensity and a nearly 180 degree phase jump at its center. The sixth chapter reports on the spin pumping due to traveling spin waves. The experiment used a micron-thick YIG strip capped by a nanometer-thick Pt layer. The YIG film was biased by an in-plane magnetic field. The spin waves pumped spin currents into the Pt layer, and the later produced electrical voltages across the length of the Pt strip through the inverse spin Hall effect (ISHE). Several distinct pumping regimes were observed and were interpreted in the frame work of the nonlinear three-wave splitting processes of the spin waves. The seventh chapter presents the first experimental work on the roles of damping in the spin Seebeck effect (SSE). The experiments used YIG/Pt bi-layered structures where the YIG films exhibited very similar structural and static magnetic properties but very different damping. The data indicate that a decrease in the damping of the YIG film gives rise to an increase in the SSE coefficient, and this response shows quasi-linear behavior. The data also indicate that the SSE coefficient shows no notable dependences on the enhanced damping due to spin pumping