Series Truncation Method Research Articles

FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-Homogeneous Circular Region

Objective: This work presents an implementation of a stable algorithm that recovers sources located at the boundary separating two homogeneous media in field-programmable gate arrays. Two loop unrolling architectures were developed and analyzed for this purpose. This inverse source problem is ill-posed due to numerical instability, i.e., small errors in the measurement can produce significant changes in the source location. Methodology: To handle the numerical instability when recovering these sources, the Tikhonov regularization method in combination with the Fourier series truncation method are applied in the stable algorithm. This stable algorithm is implemented in two different architectures developed in this work: The first architecture (Mode 1) allows for different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (Mode 2) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. Results: The architectures were tested on four devices of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two hardware implementations of the stable algorithm were validated using synthetic examples implemented in MATLAB, which shows the advantages of each architecture. Contributions: We developed two efficient architectures based on a loop unrolling design for source identification problems. These are effective strategies to divide and assign tasks to the configurable hardware, and they appear as an appropriate technique for implementing the algorithm. The first one is simple and allows for different operating speeds. The second one uses a control system based on multiplexors that reduce resource consumption and complexity of the design and can be used for multichannel problems. From the numerical test, we found the regularization parameters. The synthetic examples developed here can be considered for similar problems and can be extended to concentric spheres.

Correlated Queues with Inter-Arrival Times Depending on Service Times

We consider a type of correlated queue in which the inter-arrival time between two consecutive customers linearly depends on the service time of the first customer. We first derive infinite systems of linear equations for the moments of the waiting time in two special cases, based on which we then develop several methods to calculate the moments of the waiting time by using MacLaurin series approximation, Padé approximation and truncation method. In addition, we show how the moments and covariances of the inter-departure times of the correlated queue can be calculated based on the moments of the waiting time. Finally, extensive numerical examples are provided to validate our methods.

Free Surface Flow Between Two Semi-Infinite Inclined Plate Without Gravity Effect

In this paper, we study the problem of two-dimensional free surface flows past an inclined plate of an infinite length. The Riabouchinsky model is considered. The fluid is assumed to be inviscid and incompressible and the flow to be steady and irrotational. For this study, the effects of gravity and surface tension are negligible. This problem is solved by using two methods. The first method allow us to compute analytical solution by employing Schwarz-Christoffel mapping. The other is numerically using the series truncation method. The obtained results showed a good agreement between the numerical solution and the exact solution.

Improved calculations of waterfalls and weir flows

Many works have considered two-dimensional free-surface flow over the edge of a plate, forming a waterfall, and with uniform horizontal flow far upstream. The flow is assumed to be steady and irrotational, whilst the fluid is assumed to be inviscid and incompressible. Gravity is also taken into account. In particular, amongst these works, numerical solutions for supercritical flows have been computed, utilising conformal mappings as well as a series truncation and collocation method. We present an extension to this work where a more appropriate expression is taken for the assumed form of the complex velocity. The justification of this lies in the behaviour of the waterfall flow far downstream and the wish to better encapsulate the parabolic nature of such a free-falling jet. New numerical results will be presented, demonstrating the improved shape of the new free-surface profiles. These numerical solutions will also be validated through comparisons with asymptotic solutions, in particular for flows with larger Froude numbers. For flows with Froude numbers closer to 1, we demonstrate that the revised complex velocity ansatz should be employed in place of the asymptotic solution. We present further adjustments to the method that lead to enhanced coefficient decay. The aforementioned adjustments are also applied to supercritical weir flows and similar improvements to the jet shape can be observed.

Regularization of a final value problem for a linear and nonlinear biharmonic equation with observed data in $ L^{q} $ space

<abstract><p>In this work, we focus on the final value problem of an inverse problem for both linear and nonlinear biharmonic equations. The aim of this study is to provide a regularized method for the bi-harmonic equation, once the observed data are obtained at a terminal time in $ L^{q}(\Omega) $. We obtain an approximated solution using the Fourier series truncation method and the terminal input data in $ L^{q}(\Omega) $ for $ q \ne 2 $. In comparision with previous studies, the most highlight of this study is the error between the exact and regularized solutions to be estimated in $ L^{q}(\Omega) $; wherein an embedding between $ L^{q}(\Omega) $ and Hilbert scale spaces $ \mathcal{H}^{\rho}(\Omega) $ is applied.</p></abstract>

On some inverse problem for bi-parabolic equation with observed data in L^{p} spaces

The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in \(L^p\). We are interested in looking at three types of inverse problems. Regularization results in the \(L^2\) space appears in many related papers, but the survey results are rare in \(L^p\), \(p \neq 2\). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in \(L^p\) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in \(L^p\), we obtain the approximated solution also in the space \(L^p\). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space \(L^p\). This paper seems to generalize to previous results for bi-parabolic equation on this direction.

Dynamic Stress Concentrations in Infinite Plates with a Circular Cutout Based on Refined Theory

In this paper, based on the refined dynamic equations of plate bending vibration, the wave function expansion method is employed to investigate the elastic wave scattering and dynamic stress concentrations in infinite plates with a circular cutout. By applying the orthogonal function expansion method, the problem is reduced into solving a set of infinite algebraic equations. Then, by using the series truncation method, the above infinite algebraic equations reduce to the finite algebraic equations. As examples, the numerical results of dynamic moment concentration factor (DMCF) in infinite plates with one circular cutout under free boundary conditions subjected to an incident flexural wave are presented, and then the influences of the parameters including incident wavenumber, thickness of plates and frequency on the dynamic moment distributions are also analysed. The analyses indicate that the wave type of scattering waves around the circular cutout converts into the propagating wave from the attenuating wave when the cutoff frequency calculated by this study is 0.861 and the DMCFs vary remarkably nearby the cutoff frequency. This phenomenon is more prominent in thick plates.

The Two-Dimensional Heat Transfer Analysis in Arrayed Fins with the Thermal Dissipation Substrate

The aim of the present study is to investigate the two-dimensional heat transfer analysis in arrayed fins with thermal dissipation substrate. The governing equations for the fins and the substrate are expressed with Laplace equations, and the boundary conditions around the fins and substrate are Robin conditions. The present investigation first aims to provide a solution with regard to the geometry models by a series truncation method. Then the research will compare the results of the series truncation method with the point-matching method. Furthermore, the present study will also discuss the effects of dimension and Biot number of the fins on local dimensionless temperature, mean temperature, and heat transfer rate.

Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves

Hydroelastic waves propagating at a constant velocity at the surface of a fluid are considered. The flow is assumed to be two-dimensional and potential. Gravity is included in the dynamic boundary condition. The fluid is bounded above by an elastic sheet which is described by the Plotnikov-Toland model. Fully nonlinear solutions are computed by a series truncation method. The findings generalised previous results where the sheet was described by a simplified model known as the Kirchhoff-Love model. Periodic and generalised solitary waves are computed. The results strongly suggest that there are no true solitary waves (i.e., solitary waves characterised by a flat free surface in the far field). Detailed comparisons with results obtained with the Kirchhoff-Love model and for the related problem of gravity capillary waves are also presented.

Forced Convection Past an Oblate Spheroid at Low to Moderate Reynolds Numbers

AbstractForced convection past a heated oblate spheroid is studied in an attempt to investigate the effect of the axis ratio on the heat transfer rate. The time-dependent full Navier–Stokes and energy equations are solved using a series truncation method. The axis ratios considered range from 1∕2 to 1 (a perfect sphere). The results for the flow and thermal fields are satisfactorily compared with relevant published research. The results are presented in the form of streamlines, isotherms, and the local and averaged Nusselt number distributions.

Magnetic Shaping of a Liquid Metal Column and Deformation of a Bubble in a Vortex Flow

Two different physical problems are considered: the magnetic shaping of a liquid metal column and the distortion of a bubble in a corner vortex flow. It is shown that the two problems can be modeled with a virtually identical set of equations. These equations are solved numerically using a conformal mapping and a series truncation method, which permits fast and efficient computation of the bubble or column shapes. It is found that the two problems exhibit different limiting configurations. For the bubble problem, the deformation becomes more severe as the vortex moves further into the corner until eventually the free surface makes contact with the walls. For the magnetic shaping problem, columns approach a limiting configuration featuring either a finite number of cusps or a fixed number of trapped bubbles along the perimeter. The division between these two different behaviors is explained by means of an exact solution for zero surface tension.

Numerical simulation of the unsteady flow over an elliptic cylinder at different orientations

AbstractA numerical method is developed for investigating the two‐dimensional unsteady viscous flow over an inclined elliptic cylinder placed in a uniform stream of infinite extent. The direction of the free stream is normal to the cylinder axis and the flow field unsteadiness arises from two effects, the first is due to the flow field development following the start of the motion and the second is due to vortex shedding in the wake region. The time‐dependent flow is governed by the full conservation equations of mass and momentum with no boundary layer approximations. The parameters involved are the cylinder axis ratio, Reynolds number and the angle of attack. The investigation covers a Reynolds number range up to 5000. The minor–major axis ratio of the elliptic cylinder ranges between 0.5 and 0.6, and the angle of attack ranges between 0° and 90°. A series truncation method based on Fourier series is used to reduce the governing Navier–Stokes equations to two coupled infinite sets of second‐order differential equations. These equations are approximated by retaining only a finite number of terms and are then solved by approximating the derivatives using central differences. The results reveal an unusual phenomenon of negative lift occurring shortly after the start of motion. Various comparisons are made with previous theoretical and experimental results, including flow visualizations, to validate the solution methodology. Copyright © 2001 John Wiley & Sons, Ltd.

Viscous flow over a sphere with fluctuations in the free-stream velocity

The problem of viscous flow over a sphere with fluctuations in the free-stream velocity is considered. The governing conservation equations are expressed terms of the stream function and vorticity and solved using the series truncation method where the stream function and vorticity are approximated using finite series of Legendre and first associated Legendre functions. The effects of the Reynolds number, Strouhal number, and the amplitude of the fluctuations on the flow characteristics are studied. Results are presented for periodic variation of the drag coefficient, surface vorticity and pressure distributions for Reynolds numbers ranging from 5 to 100, Strouhal numbers ranging from π/4 to π, and amplitude of fluctuations from 0.0 to 0.5. The time variation of the velocity field during one complete cycle is presented in the form of streamline and equi-vorticity patterns. The periodic variation of the angle of separation as well as the length of the separation region are also presented.

Oscillating flow over oblate spheroids

This paper deals with the problem of oscillating viscous flow over an oblate spheriod. The flow is assumed incompressible and axisymmetric and the motion is governed by the Navier-Stokes equations. The method of solution is based on the series truncation method where the stream function and vorticity are expressed in terms of a finite series of Legendre functions. The effects of the Reynolds and Strouhal numbers on the flow characteristics are studied and compared with previous available solutions. Results are presented for the periodic variation of the drag coefficient, surface vorticity and pressure distributions for Reynolds numbers ranging from 5 to 100 and Strouhal numbers ranging from π/4 to π while keeping the spheroid axis ratio unchanged. The time variation of the velocity field during one complete oscillation is presented in the form of streamline and equi-vorticity patterns. The periodic variation of the angle of separation as well as the length of the separation bubble are also presented. The double boundary layer structure previously observed in the case of a sphere is also detected.

The impulsively started flow over oblate spheroids

The problem of the impulsively started flow over an oblate spheroid is solved using the series truncation method in which the stream function and vorticity are expanded in terms of series of Legendre functions. The resulting time-dependent differential equations are solved using the Crank-Nicolson finite difference scheme. The parameters involved are the Reynolds number and the axis ratio. The range of Reynolds numbers considered is from 5 to 100 while the axis ratio is considered at values of A r = 0.6 and 0.76. The time variation of the flow field is presented in the form of streamline and equi-vorticity patterns as well as pressure and surface vorticity distributions. The time development of the separation angle, wake length, and the frictional and total drag coefficients are also presented. The large time (steady) values of the drag coefficient for the special case of a sphere are compared with previously known data and the agreement is satisfactory.

Stationary Transverse Diffusion in a Horizontal Porous Medium Boundary-Layer Flow with Concentration-Dependent Fluid Density and Viscosity

Stable transport of high-concentrated solute is considered in horizontal boundary-layer flows above a wall of constant concentration. Mixing is accomplished by advection and molecular diffusion only. The utilized boundary-layer approximation allows to investigate the exclusive influence of gravity on vertical diffusion. The hydrodynamic dispersion mechanism was disregarded in the present study which confines its applicabilty to flows with small molecular Peclet numbers. A linear variability of both the fluid's density and viscosity with changing concentration is taken into account as well as the complete set of mass-fraction based balance equations. Steady-state concentration and velocity distributions above the horizontal wall have been obtained using the series truncation method which recently had proven successful to solve the corresponding problem using the Boussinesq assumption. The impact of the latter on these distributions is discussed by what has been additionally-facilitated by the existence of an exact analytical solution for the simpler Boussinesq case. Whereas no density variability influence exists with use of the Boussinesq assumption the complete system of mass-fraction based equations predicts opposing effects of density and viscosity differences between oncoming and near-wall fluids on concentration distributions. Larger density differences narrow the transition zone between both fluids, larger viscosity differences widen it. Thus, a compensation of both effects can be observed for individual fluids and for certain regions of the flow field.

A model for the free-surface flow due to a submerged source in water of infinite depth

AbstractWe consider a free-surface flow due to a source submerged in a fluid of infinite depth. It is assumed that there is a stagnation point on the free surface just above the source. The free-surface condition is linearized around the rigid-lid solution, and the resulting equations are solved numerically by a series truncation method with a nonuniform distribution of collocation points. Solutions are presented for various values of the Froude number. It is shown that for sufficiently large values of the Froude number, there is a train of waves on the free surface. The wavelength of these waves decreases as the distance from the source increases.

Oscillating viscous flow over a sphere

This paper deals with the problem of oscillating viscous flow over a sphere. The flow is assumed axisymmetric and governed by the Navier-Stokes equations for incompressible fluids. The method of solution is based on the series truncation method where the stream function and vorticity are expressed in terms of a finite series of Legendre and first associated Legendre functions. The effects of the Reynolds and Strouhal numbers on the flow characteristics are studied and compared with available data from previous work and also from potential flow solutions. Results are presented for periodic variation of the drag coefficient, surface vorticity and pressure distributions for Reynolds numbers ranging from 5 to 200. The time variation of the velocity field during one complete oscillation is presented in the form of streamline and equivorticity patterns. The periodic variation of the angle of separation as well as the length of the separation bubble are also presented. Calculations are performed for the time-averaged stream function and vorticity through which the double boundary layer structure is confirmed for the range of Reynolds and Strouhal numbers considered.

Withdrawal of fluid through a line sink beneath a free surface above a sloping boundary

A series truncation method is used to compute the flow into a line sink from a region of fluid with a free surface and a sloping wall beneath the sink. The method admits a well-known exact solution for a particular value of the slope. Solutions with a cusp above the sink, and with a stagnation point above the sink are computed for all values of the slope, and compared with results at both ends of the range, i.e. with results for both a vertical wall and a horizontal bottom, with good agreement.

Nonlinear bow flows with spray

The steady flow past the bow of a two-dimensional ship in water of infinite depth is considered. The ship is assumed to be a semi-infinite flat-bottomed body terminated by a face inclined at an angle β with the horizontal. The spray is modelled by a layer of water rising along the bow and falling back as a jet. A series truncation method is used to solve the fully nonlinear problem numerically. It is shown that for a prescribed value of β, there is a one-parameter family of solutions. Values of the drag and of the jet thickness are presented for different values of β.