A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement-pressure) and the three-field (stress-displacement-pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated

Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques

Numerical solution strategies for the Stokes eigenvalue problem based on the use of penalty formulations are investigated in this study. It is shown that the penalty method approach can successfully be adapted for the eigenproblem to rectify the associated problems such as the existence of zero diagonal entries in the resulting algebraic system. Two different schemes, namely, the standard penalisation with a small penalty parameter, and the iterative penalisation that enables relatively large parameters, are implemented. The employment of the latter leads to a so-called inhomogeneous generalised eigenvalue problem which requires a special attention. A feasible solution strategy is presented which is adapted from a procedure based on Newton's method proposed for the corresponding standard (inhomogeneous) eigenvalue problems. Concerning the spatial discretisation, among other possible options, the Chebyshev spectral collocation method based on expanding the unknown fields in tensor product of Chebyshev polynomials is employed. It is shown that the method constitutes a novel way of efficiently examining the approximate eigensolutions of the Stokes operator with the use of Chebyshev spectral collocation method directly, without a decoupling of velocity and pressure

THE EFFECT OF PLYOMETRIC TRAINING ON ATHLETIC PERFORMANCE AND OXYGEN SATURATION IN YOUNG MALE BASKETBALL PLAYERS

The aim of the study is to examine the effect of plyometric training on athletic performance and oxygen saturation in young male basketball players. 22 male basketball players who regularly practice basketball participated in the study. Participants were divided into two different groups as the experimental group (n:11 age: 20.41±3.27) and the control group (n:11 age: 21.78±2.32). A plyometric training program was applied to the experimental group 3 days a week for 6 weeks. Both groups continued their normal basketball training. Exercises known as drop jump, box jump, squat jump, split squat jump and overhead slam were applied to the plyometric training group. Anaerobic power, speed (20 m), flexibility (sit and reach) and oxygen saturation (SpO2) values were measured before and after the plyometric training. SPSS 22.0 package program was used for statistical evaluation. Shapiro-Wilk test was used for the normality of the data. Paired Sample t-test was used for within-group comparisons for statistical analysis of the data. The significance level was applied as p<0.05. Anaerobic power, speed, flexibility, and SpO2 values of the experimental group were found to be significant at the p<0.05 level. The values of the control group were not significant (p>0.05). As a result, it can be said that the 6-week plyometric training program applied to young male basketball players has a positive effect on athletic performance and SpO2 values.  Article visualizations

A DRBEM approximation of the Steklov eigenvalue problem

In this study, we propose a novel approach based on the dual reciprocity boundary element method (DRBEM) to approximate the solutions of various Steklov eigenvalue problems. The method consists in weighting the governing differential equation with the fundamental solutions of the Laplace equation where the definition of interior nodes is not necessary for the solution on the boundary. DRBEM constitutes a promising tool to characterize such problems due to the fact that the boundary conditions on part or all of the boundary of the given flow domain depend on the spectral parameter. The matrices resulting from the discretization are partitioned in a novel way to relate the eigenfunction with its flux on the boundary where the spectral parameter resides. The discretization is carried out with the use of constant boundary elements resulting in a generalized eigenvalue problem of moderate size that can be solved at a smaller expense compared to full domain discretization alternatives. We systematically investigate the convergence of the method by several experiments including cases with selfadjoint and non-selfadjoint operators. We present numerical results which demonstrate that the proposed approach is able to efficiently approximate the solutions of various mixed Steklov eigenvalue problems defined on arbitrary domains

Chebyshev Spectral Collocation Method for Natural Convection Flow of a Micropolar Nanofluid in the Presence of a Magnetic Field

The two-dimensional, laminar, unsteady natural convection flow of a micropolar nanofluid (Al2O3-water) in a square enclosure under the influence of a magnetic field, is solved numerically using the Chebyshev spectral collocation method (CSCM). The nanofluid is considered as Newtonian and incompressible, and the nanoparticles and water are assumed to be in thermal equilibrium. The governing equations in nondimensional form are given in terms of stream function, vorticity, micrototaion and temperature. The coupled and nonlinear equations are solved iteratively in the time direction, and an implicit backward difference scheme is employed for the time integration. The boundary conditions of vorticity are computed within this iterative process using a CSCM discretization of the stream function equation. The main advantages of CSCM, such as the high accuracy and the ease of implementation, aremade used of to obtain solutions for very high values of Ra and Ha, up to 10(7) and 1000, respectively

Hava kirliliğinde reaksiyon-difüzyon-adveksiyon denklemlerinin sonlu elemanlar yöntemi ile çözümü.

We consider the reaction-diffusion-advection (RDA) equations resulting in air pollution mod- eling problems. We employ the finite element method (FEM) for solving the RDA equations in two dimensions. Linear triangular finite elements are used in the discretization of problem domains. The instabilities occuring in the solution when the standard Galerkin finite element method is used, in advection or reaction dominated cases, are eliminated by using an adap- tive stabilized finite element method. In transient problems the unconditionally stable Crank- Nicolson scheme is used for the temporal discretization. The stabilization is also applied for reaction or advection dominant case in the time dependent problems. It is found that the stabilization in FEM makes it possible to solve RDA problems for very small diffusivity constants. However, for transient RDA problems, although the stabilization improves the solution for the case of reaction or advection dominance, it is not that pronounced as in the steady problems. Numerical results are presented in terms of graphics for some test steady and unsteady RDA problems. Solution of an air pollution model problem is also provided.M.S. - Master of Scienc

Chebyshev Spectral Collocation Method Approximation to Thermally Coupled MHD Equations

In this study, a Chebyshev spectral collocation method (CSCM) approximation is proposed for solving the full magnetohydrodynamics (MHD) equations coupled with energy equation. The MHD flow is two-dimensional, unsteady, laminar and incompressible, and the heat transfer is considered using the Boussinesq approximation for thermal coupling. The flow takes place in a square cavity which is subjected to a vertically applied external magnetic field, and the presence of the induced magnetic field is also taken into account due to the electrical conductivity of the fluid. The governing equations given in terms of stream function, vorticity, temperature, magnetic stream function, and current density, are solved iteratively using CSCM for the spatial discretisation, and an unconditionally stable backward difference scheme for the time integration. The induced magnetic field is obtained by means of its relation to the magnetic stream function. The behaviours of the flow and the heat transfer are investigated for varying values of Reynolds ($Re$), magnetic Reynolds ($Rem$), Rayleigh ($Ra$) and Hartmann ($Ha$) numbers

Chebyshev Spectral Collocation Method Approximation to Thermally Coupled MHD Equations

In this study, a Chebyshev spectral collocation method (CSCM) approximation is proposed for solving the full magnetohydrodynamics (MHD) equations coupled with energy equation. The MHD flow is two-dimensional, unsteady, laminar and incompressible, and the heat transfer is considered using the Boussinesq approximation for thermal coupling. The flow takes place in a square cavity which is subjected to a vertically applied external magnetic field, and the presence of the induced magnetic field is also taken into account due to the electrical conductivity of the fluid. The governing equations given in terms of stream function, vorticity, temperature, magnetic stream function, and current density, are solved iteratively using CSCM for the spatial discretisation, and an unconditionally stable backward difference scheme for the time integration. The induced magnetic field is obtained by means of its relation to the magnetic stream function. The behaviours of the flow and the heat transfer are investigated for varying values of Reynolds ($Re$), magnetic Reynolds ($Rem$), Rayleigh ($Ra$) and Hartmann ($Ha$) numbers

An MHD Stokes eigenvalue problem and its approximation by a spectral collocation method

An eigenvalue problem is introduced for the magnetohydrodynamic (MHD) Stokes equations describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. The solution of the eigenproblem is approximated by using a spectral collocation method that is based on vanishing the residual equation at the collocation points on the physical domain which are chosen to be the Chebyshev–Gauss–Lobatto points. As the solutions are sought in the physical space, the approximations to the derivatives of the unknowns are directly evaluated. The equations are formulated in the primitive variables, and hence with inclusion of the continuity equation, the discretization of the operator results in a generalized eigenproblem with zero diagonal entries. Therefore, a penalty method is applied to circumvent the degeneracy where a perturbed form of the problem is considered, and a zero mean pressure value is introduced. The numerical prospects of the algorithm are investigated and demonstrated by a number of characteristic tests. The key features of interest are the effects of introducing a magnetic field on the eigenspectrum focusing mainly on the change of the fundamental eigenpairs, and the consequential variation of the eigenstructure with the magnetic field. The mechanisms that underlie these effects are examined by the numerical model proposed, the implications of these effects are presented, and it is shown that the flow field is considerably affected with the introduction of a magnetic field into the physical model

Magnetohidrodinamik ve biyomanyetik akışkan kanal akımlarının sonlu elemanlar yöntemi ile çözümü.

In this thesis, solutions to steady and unsteady flow problems of incompressible viscous fluids are obtained numerically. In computational aspects, the primary focus is on the finite element analysis, however, spectral collocation and boundary element methods are also employed. The two-dimensional Navier-Stokes (N-S) equations in stream function-vorticity form are solved by using both finite element method (FEM) and Chebyshev spectral collocation method (CSCM). The accuracy of the FEM and CSCM methodologies is investigated by solving some benchmark fluid flow problems such as lid-driven cavity flow, and natural convection flow in enclosures. The natural convection flow problem is also considered under the effect of an externally applied magnetic field. The magnetohydrodynamic (MHD) system is coupled with the temperature effects through the gravitational force by means of the Boussinesq approximation. Different flow configurations with various boundary conditions are examined on both inclined and non-inclined enclosures, and the solutions are obtained by using FEM and CSCM for the case of small magnetic Reynolds number. The problem of unsteady, one-dimensional MHD flow and heat transfer between parallel plates, is solved with CSCM due to its simplicity in computations. For the time discretization, an implicit backward finite difference scheme is presented. The effect of the movement of the upper plate on the flow, and the convection action in terms of inflow/outflow through plates are examined. The MHD flow between parallel plates is extended to the case of dusty fluid by including differential equations for the dust particles. The Navier-slip conditions for both the fluid and dust particle velocities are introduced. The Hartmann number, viscosity parameter, and Navier-slip parameter influences on the flow and temperature are visualized in terms of graphics together with discussions. The biomagnetic fluid flow (blood flow) and heat transfer in channels between plates with various physical configurations are simulated. A blood model consistent with biomagnetic fluid dynamics (BFD), which includes the principles of MHD and ferrohydrodynamics (FHD), is considered. The fluid is assumed to be Newtonian, and both electrically conducting and nonconducting fluid flows are separately considered. The FEM and DRBEM applications are introduced for the steady biomagnetic fluid flow model where the fluid is considered as electrically non-conducting. The effects of the externally applied magnetic field on the flow and heat distribution are analyzed in details. FEM applications are also presented for the solution of biomagnetic fluid flow through channels between plates with differing constriction profiles. Alterations in the behaviors of the flow and temperature of the biomagnetic fluid due to the stenoses in the channel and location and intensity of the magnetic source are analyzed.Ph.D. - Doctoral Progra