Alternative Numerical Procedure Research Articles

Natural convection in a sinusoidally heated cavity filled with ferrofluid in the presence of partial variable magnetic field

PurposeThe purpose of this study is to investigate partial magnetic source (MS) effect on natural convection (NC) flow of a ferrofluid flow in a cavity with sinusoidally heated vertical walls. The combination of ferrohydrodynamics and magnetohydrodynamics due to the variable magnetic field (MF) and magnetite nanoparticles in one part of the cavity, and the classical NC in the other part of the cavity are concerned.Design/methodology/approachThe dimensionless equations in stream function-vorticity form are numerically solved by radial basis functions (RBF) based collocation method.FindingsA remarkable change in fluid flow and heat transfer is noted if the MS location is close to the left sinusoidally heated wall. In particular, the average Nusselt number is the smallest for the middle centered partial MF through the left wall at a large Hartmann number.Research limitations/implicationsRBF collocation approach is limited to small geometries due to the obtained solution globally in the entire domain of the problem.Practical implicationsIf the partial restriction of the effect of MF is done in real life, it would be a control parameter at some required/requested areas of the concerned problem.Social implicationsThis is a physical problem.Originality/valueIf the proposed idea of partial variable MF is able to be applied to a system in real life, it would be a good controller on fluid flow and heat transfer. RBF-based methods are also alternative numerical procedures to solve heat transfer and fluid flow problems.

Numerical solutions for tunnels excavated in strain-softening rock masses considering a combined support system

In this article, an alternative numerical procedure to calculate displacements and stresses of supported circular tunnels is proposed, considering the whole process of tunnel advancement, and sequential installation of the primary and secondary support systems. In the derivation, the plastic area of the rock mass is divided into a large number of annuli around the tunnel, and then the Finite Difference Method is employed. First, the strain-softening behaviour model is taken to simulate the post-failure behaviour of the rock mass. Furthermore, the Mohr-Coulomb or the Hoek-Brown failure criteria can be chosen, a non-associated plastic flow rule is assumed and the dilatancy of the rock mass is considered. After that, the fictitious support forces concept is used to simulate the process of tunnel advancement, and thus, the three-dimensional effect of the tunnel face is considered. Finally, the solutions of displacements and stresses for the rock mass and the supports can be obtained, by using the compatibility conditions of stresses and displacements at both rock-support and support-support interfaces. The results obtained from these solutions agree well with those of the self-similar solutions for circular openings, and the compatibility conditions of supported tunnels were verified. The proposed method has been compared with the convergence-confinement method. Parametric analyses are then carried out to investigate the sensitivity of support forces and displacements to the rock mass behaviour model selection. Then, the application of the proposed solutions in the design of tunnels is presented. The proposed method provides a convenient alternative method for the preliminary design of tunnels.

Two-layer geostrophic tripoles comprised by patches of uniform potential vorticity

The so-called carousel tripoles are constructed and characterized in the framework of two-layer quasi-geostrophic contour dynamics, and their stability is examined. Such a tripole is a steadily rotating doubly symmetric ensemble of three collinear vortices, or more specifically, uniform-potential-vorticity patches, with the central, core vortex, located in the upper layer, and the two remaining, satellite vortices, in the lower layer, or vice versa. The carousel tripole solutions are obtained with the use of a numerical iterative procedure. A tripole with zero total potential vorticity can be generally identified by a point in the plane spanned by two parameters, namely, the typical size of the patches relative to the Rossby deformation radius, and some shape parameter. We consider two kinds of the parameter plane by taking as the second parameter either the distance d between the centroids of the core and one of the satellites (termed also separation) or, alternatively, the minimal distance h between the core centroid and the satellite contour, measured along the symmetry axis that passes through the centroids of the core and satellites. Accordingly, to capture the stationary tripoles, we use two alternative numerical procedures, which are based on fixing the first or the second pair of parameters. This is done because the areas of convergence of the two procedures differ somewhat from each other. The areas of convergence are plotted in the parameter planes, and in each of the planes, two branches of solutions are found bifurcating from some segments of the lines bounding the convergence areas. Stability is tested in numerical simulations with the numerical noise taken as a perturbation factor. Stability/instability of a tripole is determined by examining the oscillations in the perimeter of one of the vortex satellites. For each tripole size, both stable and unstable solutions exist. The stability bounds coincide with the bifurcation lines, so that one branch of the solutions is stable while the other is not. As a whole, tripoles with considerable separation behave stably.

A shallow water model for the propagation of tsunami via Lattice Boltzmann method

An efficient implementation of the lattice Boltzmann method (LBM) for the numerical simulation of the propagation of long ocean waves (e.g. tsunami), based on the nonlinear shallow water (NSW) wave equation is presented. The LBM is an alternative numerical procedure for the description of incompressible hydrodynamics and has the potential to serve as an efficient solver for incompressible flows in complex geometries. This work proposes the NSW equations for the irrotational surface waves in the case of complex bottom elevation. In recent time, equation involving shallow water is the current norm in modelling tsunami operations which include the propagation zone estimation. Several test-cases are presented to verify our model. Some implications to tsunami wave modelling are also discussed. Numerical results are found to be in excellent agreement with theory.

Convected field analysis of flat panels response to turbulent boundary layer induced excitation

Abstract The main objective of the work was the development of a procedure allowing the study of the effect of fluid loading over the vibration response of a plate driven by a TBL excitation field. The reasons for such analysis are concerned with the limitations of the main numerical codes available on the market when dealing with the subject, and the need for the extension of the existing predictive capabilities of deterministic methodologies, when facing with the stochastic pressure distribution over a flat plate. The availability of such specific tools is judged as mandatory for the improvement and/or better classification of the design criteria for the enhancement of modern aircraft acoustic comfort. After recalling and rearranging the theoretical concepts applicable to the aeroelastic behaviour of flat plates, to be used as a reference for the validation process, two different numerical approaches are herein presented. A first procedure was elaborated by using a vibro-acoustic coupling formulation, in which an appropriate convected field Green function was employed, and by relying on the modal information provided by FEM based numerical tools. An alternative numerical procedure was developed by employing the aeroelastic module of NASTRAN commercial code, in which the baffle condition was sought and obtained throughout proper adaptation and tuning of the input parameters. Both procedures were built up to be applicable to any baffled flat plate, including sandwich and composite panels. A two-ways validation process, based on the assumption of point forces loading conditions, was carried out by introducing the fluid back reaction contribution associated to the convected field in the frequency response solution of the first procedure. The evidence of the convected field effect on the panel response was highlighted, mainly associated to a reduction of the investigated panel stiffness properties due to aeroelastic interactions. A preliminary study on stochastic loading conditions response was performed by considering a “rain on the roof” excitation, while the final investigation, focused on the TBL induced excitation field, confirmed the potentiality of the proposed procedures in dealing with the subject. The relevance of a correct estimation of the convected field contribution on the plate response in terms of both qualitative and quantitative aspects is pointed out, specially for what concerns the attenuation of a plate vibration energy.

An alternative procedure for approximating hyperbolic systems of conservation laws

This work presents an alternative numerical procedure for simulating a class of nonlinear hyperbolic systems, using Glimm's method for advancing in time. The standard procedure to implement this methodology suffers from the disadvantage of requiring a complete solution of the associated Riemann problem—a task, in general, not easily reached. The alternative procedure introduced in this article consists in approximating the solution of the associated Riemann problem by piecewise constant functions always satisfying the jump condition—thus circumventing the difficulty of solving the Riemann problem and giving rise to an approximation easier to implement with lower computational cost. In order to illustrate the good performance of the alternative methodology proposed, two problems are considered—namely the transport of a pollutant in the atmosphere and the dynamics of the filling up of a rigid porous medium, modeled under a mixture theory viewpoint. Comparison with the standard procedure, employing the complete solution of the associated Riemann problem for implementing Glimm's scheme, has shown good agreement.

A METHOD FOR QUASI-STATIC ANALYSIS OF TOPOLOGICALLY VARIABLE LATTICE STRUCTURES

The proposed time-independent quasi-static approach for simulations of lattice structures with imperfections is based on integration of the Inverse Broyden's Method suitable for finding the equilibrium state for a large system of atoms interacting through strongly nonlinear potentials and the Recursive Inverse Matrix Algorithm (RIMA) capable of updating the inverse matrix when topological changes (broken or new bonds between the atoms) take place. In this approach, the crystal structure is treated as a truss system while the forces between the atoms situated at the nodes are defined by the inter-atomic potentials. Since both the Broyden's and the RIMA algorithms deal with the inverse matrices of the structure their coupling makes the procedure computationally efficient. In addition, the method allows analysis of lattices subjected to mixed boundary conditions. The developed code was verified by the comparison with an alternative numerical procedure based on energy minimization technique. The model and the code developed were applied to the case of a 2D hexagonal lattice with the mode I crack embedded into the structure. For the cases considered, it was observed that the crack nucleation and growth were accompanied by the dislocation emission.

The maximum likelihood estimate in reduced‐rank regression

AbstractIn previous work by Stoica and Viberg the reduced‐rank regression problem is solved in a maximum likelihood sense. The present paper proposes an alternative numerical procedure. The solution is written in terms of the principal angles between subspaces spanned by the data matrices. It is demonstrated that the solution is meaningful also in the case when the maximum likelihood criterion is not valid. A numerical example is given. Copyright © 2005 John Wiley & Sons, Ltd.

Least squares finite element methods for fluid-structure interaction problems

The first order least squares finite element method is an alternative numerical procedure to standard Galerkin methods for the solution of partial differential equations. Its advantages are mainly the resulting symmetric positive definite matrices and the inherent numerical stability. Several first order formulations were proposed for the incompressible Navier–Stokes equations and for the equations of linear elasticity. Here we analyse which of these methods could provide an efficient and accurate tool for the solution of fluid-structure interaction problems. It seems that the combination of inherent least squares stability and high order ansatz functions is very promising.

Discussion: An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour

D. GALLIPOLI, A. GENS, R. SHARMA and J. VAUNAT (2003).Ge´otechnique53, No. 1, 123–135J. Robles, Technical University of MadridL. R. Hoyos, University of Texas at ArlingtonThe authors should be commended for presenting such aninnovative, elaborate elasto-plastic framework for predictingconstitutive unsaturated soil response under isotropic stressstates. One of the key aspects of the work is the authors’deep knowledge of the physical interactions that lie behindthe mechanical behaviour of unsaturated soils. The proposedconstitutive model and their associated conceptual frame-work provide an important improved understanding of soilbehaviour. The present review is intended to offer somecomments that further highlight the relevance of the workand its potential for further refinement and tuning.The objective of this review is twofold:(a) to propose an alternative numerical procedure forobtaining the analytical expression for the assumedfundamental assumption(b) to discuss the effectiveness of the proposed model withregard to the ‘direct’ relationship between the mech-anical and hydraulic responses of unsaturated soils forirreversible stress paths.Inspection of Fig. 6 suggests that, for a suction state ofs

Vibration of Plates with Constrained V-Notches or Cracks

This paper reports the first known free vibration solutions for thin circular plates with V-notches having various edge conditions. The classical Ritz method is employed with 2 sets of admissible functions assumed for the transverse vibratory displacements. These sets include: 1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained; and 2) corner functions to account for the bending moment singularities at the sharp re-entrant corner of the V-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondimensional frequencies for circular plates having a free circumferential edge and various combinations edge conditions of the V-notch. Accurate (5 significant figures) frequencies are presented for clamped-free, clamped-hinged, and hinged-free notches for the spectra of notch angles, causing a re-entrant vertex corner of the radial edges. For very small notch angles, a clamped-free, clamped-hinged, or hinged-free radial crack ensues. One general observation is that, for the range of notch angles considered, there is a substantial increase in the first 6 frequencies as the notch depth increases. The frequency increase with increasing notch depth is more pronounced in the higher modes than the lower ones, and is quite substantial for segmental plates with notch angles equal to 180 deg. A large reduction in frequencies is also observed as the notch angle decreases at a constant notch depth. A new database of accurate frequencies and mode shapes for sectorial, semicircular and segmental plates is presented with which future solutions drawn from alternative numerical procedures and finite element and boundary element techniques may be compared. Normalized contours of the transverse vibratory displacement are shown for plates with various notch depths and notch angles of 5, 30, 60, 90, and 180 deg.

Impact response and damage in laminated composite cylindrical shells

Impact response of a laminated composite cylindrical shell is determined both by the classical Fourier series and the finite element methods. Impact response determined by the finite element method also includes a prediction of the impact-induced damage deploying the semi-empirical damage prediction model of Choi–Chang. In the analytical method, the solution is obtained by means of an alternative numerical procedure incorporating the non-linear Hertz’s contact law which enables consideration of local indentation produced by the indentor on the impacted surface, instead of the Laplace Transforms technique used in an earlier work. The above procedure is also extended to the problem of impact on a cylindrical shell panel. The analytical procedure also provided information such as the natural frequencies of vibration of the impacted shell which helped in selecting appropriate mesh and time step sizes for the finite element method. A parametric study was carried out by the finite element method to determine the effect of varying the controlling parameters such as impactor mass, its approach velocity, curvature of the shell, on both the impact response and on the impact-induced damage. A reduction of the stiffnesses of the failed laminas on the impact response concurrently as the solution proceeded has also been incorporated.

The 3-D BEM implementation of a numerical Green's function for fracture mechanics applications

The use of Green's functions has been considered a powerful technique in the solution of fracture mechanics problems by the boundary element method (BEM). Closed-form expressions for Green's function components, however, have only been available for few simple 2-D crack geometry applications and require complex variable theory. The present authors have recently introduced an alternative numerical procedure to compute the Green's function components that produced BEM results for 2-D general geometry multiple crack problems, including static and dynamic applications. This technique is not restricted to 2-D problems and the computational aspects of the 3-D implementation of the numerical Green's function approach are now discussed, including examples. Copyright © 2000 John Wiley & Sons, Ltd.

Computational electrochemistry: three-dimensional boundary element simulations of double electrode geometries

The boundary element method (BEM) is employed to simulate the current response of double microelectrode geometries operating in generator–collector mode under transport conditions of diffusion control. Specifically, results are presented which quantify influence of electrode shape and position on the current flowing during electrolysis. The nature of the problems presented requires solution of the three-dimensional form of the Laplace equation, which for objects of unusual topography have proved numerically demanding using the more traditionally employed finite difference method. The reduction in dimensionality brought about by the BEM formulation renders the class of problem tackled as two-dimensional and, thus, offers substantial benefits in both simulation time and grid generation procedures over alternative numerical procedures.

Numerical methods for differential equations: fundamental concepts for scientific and engineering applications

Part 1 Introduction to partial differential equations: behaviour physical systems definitions and equation properties conclusion problems. Part 2 Finite difference methods: discrete approximations in one dimension a generalized formulation of differences example calculations in one dimension solution of initial value problems finite differences multiple dimension two-dimensional example calculations conclusion problems. Part 3 Finite element methods: the methods of weighted residuals nomenclature computational procedures in one dimension mathematical requirements method of weighted residuals in two dimensions method of weighted residuals in three dimensions mathematical properties of the Galerkin finite element method conclusion. Part 4 Discretization considerations and design of approximations: spatial discretization temporal discretization space-time discretization alternative numerical procedures conclusion problems. Part 5 Accuracy and error reduction: improved accuracy through Mesh refinement improved accuracy through higher order approximation considerations of space-time problems conclusion problems.

An alternative numerical procedure for simulating the dynamical response of non‐linear elastic rods

AbstractIn this paper there is presented an alternative numerical procedure for obtaining approximations to non‐linear conservation laws like those that describe the dynamical behaviour of elastic rods (composed of materials whose stress–strain relation is non‐linear). The above‐mentioned procedure consists of approximating the solution of the Riemann problem (associated with the considered conservation law) by a piecewise constant function (satisfying the jump conditions) and using Glimm's scheme for advancing in time, step by step. The proposed numerical approach eliminates the necessity of solving (in a complete way) the associated Riemann problem, easing and cheapening its computational implementation. This procedure is employed for simulating the dynamical response of an elastic‐non‐linear rod, fixed at its edges, that is left in a non‐equilibrium state. There is presented a comparison between results obtained through a classical procedure and through the procedure proposed in this work.

Linear methods of structural analysis

The foundations of the finite element methods and their connection to alternative numerical procedures for linear structural problems are outlined. The state-of-the-art of the finite element displacement method in structural mechanics is surveyed. Extension of the method to dynamic structural problems, to initial instability problems of thin plates and shells, to linear fracture mechanics, and to linear thermoelasticity is indicated. Examples representative of structural problems in reactor technology are included.