257 research outputs found
An approximate solution of the MHD Falkner-Skan flow by Hermite functions pseudospectral method
Based on a new approximation method, namely pseudospectral method, a solution
for the three order nonlinear ordinary differential laminar boundary layer
Falkner-Skan equation has been obtained on the semi-infinite domain. The
proposed approach is equipped by the orthogonal Hermite functions that have
perfect properties to achieve this goal. This method solves the problem on the
semi-infinite domain without truncating it to a finite domain and transforming
domain of the problem to a finite domain. In addition, this method reduces
solution of the problem to solution of a system of algebraic equations. We also
present the comparison of this work with numerical results and show that the
present method is applicable.Comment: 15 pages, 4 figures; Published online in the journal of
"Communications in Nonlinear Science and Numerical Simulation
Numerical Study on Wall Temperature and Surface Heat Flux Natural Convection Equations Arising in Porous Media by Rational Legendre Collocation Approach
Abstract: In this paper, a new powerful approach, called rational Legendre collocation method (RLC) is used to obtain the solution for nonlinear ordinary deferential equations that often appear in boundary layers problems arising in heat transfer. These kinds of the equations contain infinity boundary condition. The main objective is to reduce the solution of the problem to a solution of a system of algebraic equations, which do not require linearization and imposing the asymptotic condition transforming and physically unrealistic assumptions. Numerical results are compared with those of other methods, showing that the collocation method leads to more accurate results
Mixed Pseudospectral Method for Heat Transfer
In this paper, we propose a mixed generalized Laguerre-Legendre pseudospectral method for non-isotropic heat transfer with inhomogeneous boundary conditions on an infinite strip. Some properties about the mixed generalized LaguerreLegendre approximation are established. By reformulating the equation with suitable functional transform defined on an infinite strip, a mixed Laguerre-Legendre pseudospectral scheme is constructed. Its convergence is proved. Numerical results are presented to demonstrate the efficiency of this new approach and to validate our theoretical analysis
Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems
We consider the usage of higher order spectral element methods for the solution of
problems in structures and fluid mechanics areas. In structures applications we study
different beam theories, with mixed and displacement based formulations, consider
the analysis of plates subject to external loadings, and large deformation analysis of
beams with continuum based formulations. Higher order methods alleviate the problems
of locking that have plagued finite element method applications to structures, and also
provide for spectral accuracy of the solutions. For applications in computational fluid
dynamics areas we consider the driven cavity problem with least squares based finite element
methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving
problems effectively along with domain decomposition algorithms for fluid problems. In
the context of least squares finite element methods we also explore the usage of Multigrid
techniques to obtain faster convergence of the the solutions for the problems of interest.
Applications of the traditional Lagrange based finite element methods with the Penalty finite
element method are presented for modelling porous media flow problems. Finally, we explore
applications to some CFD problems namely, the flow past a cylinder and forward facing
step
Uncertainty quantification integrated to computational fluid dynamic modeling of synthetic jet actuators
The Point Collocation Non-Intrusive Polynomial Chaos (NIPC) method was applied to a stochastic synthetic jet actuator problem to demonstrate the integration of computationally efficient uncertainty quantification to the high-fidelity CFD modeling of Synthetic Jet Actuators. The uncertainty quantification approach was first implemented in two stochastic model problem cases for the prediction of peak exit plane velocity using a Fluid Dynamic Based analytical model of the Synthetic Jet Actuator, which is computationally less expensive than CFD simulations. The NIPC results were compared with direct Monte Carlo sampling results. To demonstrate the efficient uncertainty quantification in CFD modeling of synthetic jet actuators, a test case, Case 1 (synthetic jet issuing into quiescent air), was selected from the CFDVal2004 workshop. In the stochastic CFD problem, the NIPC method was used to quantify the uncertainty in the long-time averaged u and v-velocities at several locations in the flow field, due to the uncertainty in the amplitude and frequency of the oscillation of the piezo-electric membrane. Fifth order NIPC expansions were used to obtain the uncertainty information which showed that the variation in the v-velocity is high in the region directly above the jet slot and the variation in the u-velocity is maximum in the region immediately adjacent to the slot. Even with a ten percent variation in the amplitude and frequency, the long-time averaged u and v-velocity profiles could not match the experimental measurements at y = 0.1mm above the slot, indicating that the discrepancy may be due to other uncertainty sources in CFD or measurement errors. A global sensitivity analysis using linear regression approach indicated that the frequency had a stronger contribution to the overall uncertainty in the long-time averaged flow field velocity for the range of input uncertainties considered in this study. Overall, the results obtained in this study showed the potential of Non-Intrusive Polynomial Chaos as an effective uncertainty quantification method for computationally expensive high-fidelity CFD simulations applied to the stochastic modeling of synthetic jet flow fields --Abstract, page iii
Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems
We consider the usage of higher order spectral element methods for the solution of
problems in structures and fluid mechanics areas. In structures applications we study
different beam theories, with mixed and displacement based formulations, consider
the analysis of plates subject to external loadings, and large deformation analysis of
beams with continuum based formulations. Higher order methods alleviate the problems
of locking that have plagued finite element method applications to structures, and also
provide for spectral accuracy of the solutions. For applications in computational fluid
dynamics areas we consider the driven cavity problem with least squares based finite element
methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving
problems effectively along with domain decomposition algorithms for fluid problems. In
the context of least squares finite element methods we also explore the usage of Multigrid
techniques to obtain faster convergence of the the solutions for the problems of interest.
Applications of the traditional Lagrange based finite element methods with the Penalty finite
element method are presented for modelling porous media flow problems. Finally, we explore
applications to some CFD problems namely, the flow past a cylinder and forward facing
step
Computational and numerical analysis of differential equations using spectral based collocation method.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally eļ¬cient spectral collocation-based methods,
both modiļ¬ed and new, and apply them to solve diļ¬erential equations. Spectral collocation-based
methods are the most commonly used methods for approximating smooth solutions of diļ¬erential
equations deļ¬ned over simple geometries. Procedurally, these methods entail transforming the gov
erning diļ¬erential equation(s) into a system of linear algebraic equations that can be solved directly.
Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported
in the literature, researchers often transform their models to reduce the number of variables or
narrow them down to problems with fewer dimensions. Such a process is accomplished by making
a series of assumptions that limit the scope of the study. To address this deļ¬ciency, the present
study explores the development of numerical algorithms for solving ordinary and partial diļ¬erential
equations deļ¬ned over simple geometries. The solutions of the diļ¬erential equations considered are
approximated using interpolating polynomials that satisfy the given diļ¬erential equation at se
lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the
computational domain is particularly emphasized as it plays a key role in determining the number
of grid points that are used; a feature that dictates the accuracy and the computational expense of
the spectral method. To solve diļ¬erential equations deļ¬ned on large computational domains much
eļ¬ort is devoted to the development and application of new multidomain approaches, based on
decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time
interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con
ļ¬rms the superiority of these multiple domain techniques in terms of accuracy and computational
eļ¬ciency over the single domain approach when applied to problems deļ¬ned over large domains.
The structure of the thesis indicates a smooth sequence of constructing spectral collocation method
algorithms for problems across diļ¬erent dimensions. The process of switching between dimensions
is explained by presenting the work in chronological order from a simple one-dimensional problem
to more complex higher-dimensional problems. The preliminary chapter explores solutions of or
dinary diļ¬erential equations. Subsequent chapters then build on solutions to partial diļ¬erential
i
equations in order of increasing computational complexity. The transition between intermediate
dimensions is demonstrated and reinforced while highlighting the computational complexities in
volved. Discussions of the numerical methods terminate with development and application of a
new method namely; the trivariate spectral collocation method for solving two-dimensional initial
boundary value problems. Finally, the new error bound theorems on polynomial interpolation are
presented with rigorous proofs in each chapter to benchmark the adoption of the diļ¬erent numerical
algorithms. The numerical results of the study conļ¬rm that incorporating domain decomposition
techniques in spectral collocation methods work eļ¬ectively for all dimensions, as we report highly
accurate results obtained in a computationally eļ¬cient manner for problems deļ¬ned on large do
mains. The ļ¬ndings of this study thus lay a solid foundation to overcome major challenges that
numerical analysts might encounter
On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.
Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations.
Most of these equations are highly nonlinear and exact solutions are not always possible.
Exact solutions always give a good account of the physical nature of the phenomena modeled.
However, existing analytical methods can only handle a limited range of these equations.
Semi-numerical and numerical methods give approximate solutions where exact solutions are
impossible to find. However, some common numerical methods give low accuracy and may lack
stability. In general, the character and qualitative behaviour of the solutions may not always
be fully revealed by numerical approximations, hence the need for improved semi-numerical
methods that are accurate, computational efficient and robust.
In this study we introduce innovative techniques for finding solutions of highly nonlinear
coupled boundary value problems. These techniques aim to combine the strengths of both
analytical and numerical methods to produce efficient hybrid algorithms. In this work, the
homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral
methods are well known for their high levels of accuracy. The new spectral homotopy analysis
method is further improved by using a more accurate initial approximation to accelerate
convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral
methods are used to solve the linearised equations. The new techniques were used to solve
mathematical models in fluid dynamics.
The thesis comprises of an introductory Chapter that gives an overview of common numerical
methods currently in use. In Chapter 2 we give an overview of the methods used in this
work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional
squeezing flow of a viscous fluid between two approaching parallel plates and the
steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter
4 the methods were used to find solutions of the laminar heat transfer problem in a rotating
disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and
the classical von KĪ¬rmĪ¬n equations for boundary layer flow induced by a rotating disk. In
Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a
rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem
due to a shrinking sheet with a chemical reaction, were solved using the new methods
- ā¦