A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Infinite Dimensional Slow Modulations In A Delayed Model For Orthogonal Cutting Vibrations

We apply the method of multiple scales (MMS) to a well known model of regenerative cutting vibrations in the large delay regime. By ``large'' we mean the delay is much larger than the time scale of typical cutting tool oscillations. The MMS upto second order for such systems has been developed recently, and is applied here to study tool dynamics in the large delay regime. The second order analysis is found to be much more accurate than first order analysis. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy. The main advantage of the present analysis is that infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space. Lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS. Finally, the strong sensitivity of the dynamics to small changes in parameter values is seen clearly.

A three-dimensional hybrid smoothed finite element method (H-SFEM) for nonlinear solid mechanics problems

This paper presents a novel three-dimensional hybrid smoothed finite element method (H-SFEM) for solid mechanics problems. In 3D H-SFEM, the strain field is assumed to be the weighted average between compatible strains from the finite element method (FEM) and smoothed strains from the node-based smoothed FEM with a parameter α equipped into H-SFEM. By adjusting α, the upper and lower bound solutions in the strain energy norm and eigenfrequencies can always be obtained. The optimized α value in 3D H-SFEM using a tetrahedron mesh possesses a close-to-exact stiffness of the continuous system, and produces ultra-accurate solutions in terms of displacement, strain energy and eigenfrequencies in the linear and nonlinear problems. The novel domain-based selective scheme is proposed leading to a combined selective H-SFEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed 3D H-SFEM is an innovative and unique numerical method with its distinct features, which has great potential in the successful application for solid mechanics problems.

A Method of Characteristics for Solving Two-dimensional Unsteady Flow Problems

The aim of this investigation is to evolve a method of solving two-dimensional unsteady flow problems by the method of characteristics. This involves the reduction of the given system of equations to an equivalent system where only interior derivatives occur on a characteristic surface. From this system, four special bicharacteristic directional derivatives are chosen. A finite difference scheme is prescribed for solving the equations. General rectangular lattices are also considered. As an example, we investigate the propagation of an initial pressure distribution in a medium at rest.

New nondestructive method for three-dimensional photoelasticity

A new set of equations describing completely the optical phenomena in a model involving continuous rotation of secondary axes and secondary principal-stress differences are obtained. These are solved by Peano-Baker method using experimentally determined characteristic parameters for several wavelengths of light. Experimental verifications are obtained for a rectangular bar subjected to combined torsion and tension. Paper was presented at Third SESA International Congress on Experimental Mechanics held in Los Angeles, CA on May 13–18, 1973.

A new nondestructive method for three-dimensional photoelasticity

A new set of equations describing completely the optical phenomena in a model involving continuous rotation of secondary axes and secondary principal-stress differences are obtained. These are solved by Peano-Baker method using experimentally determined characteristic parameters for several wavelengths of light. Experimental verifications are obtained for a rectangular bar subjected to combined torsion and tension.

Regularized numerical integration of multibody dynamics with the generalized alpha method*

This paper discusses the consistent regularization property of the generalized α method when applied as an integrator to an initial value high index and singular differential-algebraic equation model of a multibody system. The regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take. The regularization involves increase of the smallest singular values of the ill-conditioned Jacobian of the discretization and is different from Baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter. This regularization also helps where pre-conditioning the Jacobian by scaling is of limited effect, for example, when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present. The feed-forward control in Kane's equation models is additionally considered in the numerical examples to illustrate the effect of regularization. The discretization presented in this work is adopted to the first order DAE system (unlike the original method which is intended for second order systems) for its A-stability and same order of accuracy for positions and velocities.

A method for separation of stresses in Two- and three-dimensional photoelasticity

A method for separation of stresses in two and three-dimensional photo elasticity using the harmonisation ofjrst stress invariant along a straight section is deve- ,dped. For two-dimensions, the equations of equilibrium are reformulated in terms ojsum and difference of normal stresses and relations are obtained which can be used for harmonisation of the first invariant of stress along a straight section. A similar procedure is adopted for three-dimensions by making use of the Beltrmi-MicheN equations. The new relations are used in finite d~yerencefo rm to evaluate the sum of normal stresses along straight sections in a three-dimensional body. The method requires photoelastic data along the section as well ~rra djacent sections. This method could be used as an alternative to the shear d@erence method for separation of stresses in photoelasticity. 7he accuracy and reliability of the method is ver$ed by applying the method to problems whose solutions are known.

Simulation of laminar flow in a three-dimensional lid-driven cavity by lattice Boltzmann method

Purpose - The purpose of this paper is to apply lattice Boltzmann equation method (LBM) with multiple relaxation time (MRT) model, to investigate lid-driven flow in a three-dimensional (3D), rectangular cavity, and compare the results with flow in an equivalent two-dimensional (2D) cavity. Design/methodology/approach - The second-order MRT model is implemented in a 3D LBM code. The flow structure in cavities of different aspect ratios (0.25-4) and Reynolds numbers (0.01-1000) is investigated. The LBM simulation results are compared with those from numerical solution of Navier-Stokes (NS) equations and with available experimental data. Findings - The 3D simulations demonstrate that 2D models may predict the flow structure reasonably well at low Reynolds numbers, but significant differences with experimental data appear at high Reynolds numbers. Such discrepancy between 2D and 3D results are attributed to the effect of boundary layers near the side-walls in transverse direction (in 3D), due to which the vorticity in the core-region is weakened in general. Secondly, owing to the vortex stretching effect present in 3D flow, the vorticity in the transverse plane intensifies whereas that in the lateral plane decays, with increase in Reynolds number. However, on the symmetry-plane, the flow structure variation with respect to cavity aspect ratio is found to be qualitatively consistent with results of 2D simulations. Secondary flow vortices whose axis is in the direction of the lid-motion are observed; these are weak at low. Reynolds numbers, but become quite strong at high Reynolds numbers. Originality/value - The findings will be useful in the study of variety of enclosed fluid flows.

A multistep transversal linearization (MTL) method in non-linear dynamics through a Magnus characterization

A new form of a multi-step transversal linearization (MTL) method is developed and numerically explored in this study for a numeric-analytical integration of non-linear dynamical systems under deterministic excitations. As with other transversal linearization methods, the present version also requires that the linearized solution manifold transversally intersects the non-linear solution manifold at a chosen set of points or cross-section in the state space. However, a major point of departure of the present method is that it has the flexibility of treating non-linear damping and stiffness terms of the original system as damping and stiffness terms in the transversally linearized system, even though these linearized terms become explicit functions of time. From this perspective, the present development is closely related to the popular practice of tangent-space linearization adopted in finite element (FE) based solutions of non-linear problems in structural dynamics. The only difference is that the MTL method would require construction of transversal system matrices in lieu of the tangent system matrices needed within an FE framework. The resulting time-varying linearized system matrix is then treated as a Lie element using Magnus’ characterization [W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., VII (1954) 649–673] and the associated fundamental solution matrix (FSM) is obtained through repeated Lie-bracket operations (or nested commutators). An advantage of this approach is that the underlying exponential transformation could preserve certain intrinsic structural properties of the solution of the non-linear problem. Yet another advantage of the transversal linearization lies in the non-unique representation of the linearized vector field – an aspect that has been specifically exploited in this study to enhance the spectral stability of the proposed family of methods and thus contain the temporal propagation of local errors. A simple analysis of the formal orders of accuracy is provided within a finite dimensional framework. Only a limited numerical exploration of the method is presently provided for a couple of popularly known non-linear oscillators, viz. a hardening Duffing oscillator, which has a non-linear stiffness term, and the van der Pol oscillator, which is self-excited and has a non-linear damping term.

Optical Studies of the Antarctic Glacio-Oceanic System

The Antarctic system comprises of the continent itself, Antarctica, and the ocean surrounding it, the Southern Ocean. The system has an important part in the global climate due to its size, its high latitude location and the negative radiation balance of its large ice sheets. Antarctica has also been in focus for several decades due to increased ultraviolet (UV) levels caused by stratospheric ozone depletion, and the disintegration of its ice shelves. In this study, measurements were made during three Austral summers to study the optical properties of the Antarctic system and to produce radiation information for additional modeling studies. These are related to specific phenomena found in the system. During the summer of 1997-1998, measurements of beam absorption and beam attenuation coefficients, and downwelling and upwelling irradiance were made in the Southern Ocean along a S-N transect at 6°E. The attenuation of photosynthetically active radiation (PAR) was calculated and used together with hydrographic measurements to judge whether the phytoplankton in the investigated areas of the Southern Ocean are light limited. By using the Kirk formula the diffuse attenuation coefficient was linked to the absorption and scattering coefficients. The diffuse attenuation coefficients (Kpar) for PAR were found to vary between 0.03 and 0.09 1/m. Using the values for KPAR and the definition of the Sverdrup critical depth, the studied Southern Ocean plankton systems were found not to be light limited. Variabilities in the spectral and total albedo of snow were studied in the Queen Maud Land region of Antarctica during the summers of 1999-2000 and 2000-2001. The measurement areas were the vicinity of the South African Antarctic research station SANAE 4, and a traverse near the Finnish Antarctic research station Aboa. The midday mean total albedos for snow were between 0.83, for clear skies, and 0.86, for overcast skies, at Aboa and between 0.81 and 0.83 for SANAE 4. The mean spectral albedo levels at Aboa and SANAE 4 were very close to each other. The variations in the spectral albedos were due more to differences in ambient conditions than variations in snow properties. A Monte-Carlo model was developed to study the spectral albedo and to develop a novel nondestructive method to measure the diffuse attenuation coefficient of snow. The method was based on the decay of upwelling radiation moving horizontally away from a source of downwelling light. This was assumed to have a relation to the diffuse attenuation coefficient. In the model, the attenuation coefficient obtained from the upwelling irradiance was higher than that obtained using vertical profiles of downwelling irradiance. The model results were compared to field measurements made on dry snow in Finnish Lapland and they correlated reasonably well. Low-elevation (below 1000 m) blue-ice areas may experience substantial melt-freeze cycles due to absorbed solar radiation and the small heat conductivity in the ice. A two-dimensional (x-z) model has been developed to simulate the formation and water circulation in the subsurface ponds. The model results show that for a physically reasonable parameter set the formation of liquid water within the ice can be reproduced. The results however are sensitive to the chosen parameter values, and their exact values are not well known. Vertical convection and a weak overturning circulation is generated stratifying the fluid and transporting warmer water downward, thereby causing additional melting at the base of the pond. In a 50-year integration, a global warming scenario mimicked by a decadal scale increase of 3 degrees per 100 years in air temperature, leads to a general increase in subsurface water volume. The ice did not disintegrate due to the air temperature increase after the 50 year integration.