Analytical Solution Technique Research Articles

Stability of orthotropic rectangular plates

An analytical method of solution is presented for the problem of elastic buckling of orthotropic rectangular plates. Uniaxially compressed plates are treated for different boundary conditions along the loaded and unloaded edges. The advantage of the proposed method lies in the introduction of a unified analytical solution technique for a multitude of combinations of boundary conditions. For some combinations, only numerical solution techniques were available in the literature.

A bidomain model for ring stimulation of a cardiac strand.

A bidomain model is developed to represent a cardiac muscle strand undergoing stimulation from a ring electrode. Expressions are derived for the distribution of the potentials in the strand as well as in the bath perfusing the tissue, under the condition of equal anisotropy ratios in the intracellular and interstitial spaces. The present derivation provides an example of an analytical solution technique that might also prove useful for other problems in theoretical electrophysiology.

Convective heat transfer from a sphere due to acoustic streaming

Convective heat transfer from a rigid sphere due to acoustic streaming is discussed for large streaming Reynolds numbers. The work is motivated by a need for the thermal analysis of the experiments on containerless materials processing using acoustic levitation. The acoustic signals used in these experiments give rise to a strong steady fluid motion around levitated alloy samples. The associated convective heat transfer rates are significant enough to give rise to heating/cooling rates in the samples which are crucial in determining their material structure and hence their thermophysical properties. The steady heat transport effect due to the streaming motion is studied with the levitated sample modeled as an isolated solid sphere in a pure plane standing sound field. Strong acoustic signals in the range of 150–170 dB and 1–2 kHz are considered. Analytical and numerical solution techniques are used to obtain a Nusselt number correlation for the heat transfer when the surrounding fluid is a gas with Prandtl number ∼O(1). A simple experiment performed to confirm some of the predictions is found to give very good agreement with the analysis. A more detailed report of the results of this study may be found elsewhere [J. Heat Transfer 115, 332–341 (1993)]. [Work supported by NSF Grant No. CTS-8918777 and the NRC.]

Flow dynamics across end-to-end vascular bypass graft anastomoses.

In this article, a numerical simulation of steady flow across an end-to-end vascular bypass graft anastomosis is presented. In vitro experiments were performed to determine the variations in the conduit cross section at the anastomosis. Penrose surgical drainage tubing was used to simulate an artery and was anastomosed with Polytetrafluoroethylene (PTFE) vascular grafts using a continuous suturing technique. Artery to artery anastomosis was simulated by suturing two Penrose tubing segments. The anastomotic specimens were subject to static transmural pressure in the physiologic range to determine the instantaneous diameter and compliance as a function of the distance from the anastomotic site. The experimentally determined geometries were used to simulate steady flow through an end-to-end anastomosis using the finite analytic (FA) numerical solution technique. The results demonstrated a region of flow separation 2 mm distal to the Penrose tubing-Penrose tubing anastomosis (simulating an artery-artery anastomosis) at higher transmural pressures. Moreover, wall shear stresses increased proximal to the anastomosis in flow from the Penrose tubing to the graft. In flow from the graft to the Penrose tubing, low wall shear stresses were observed distal to the anastomosis. Flow separation was observed distal to the anastomosis at higher transmural pressures with uniform inlet velocity condition. The region of low shear stress in flow from PTFE graft to the Penrose tubing was located nearer to the anastomosis with thin wall grafts than that with standard wall thickness grafts. Our steady flow model studies suggest a correlation between regions of low wall shear stress and the development of anastomotic neointimal fibrous hyperplasia in end-to-end anastomoses.

Convective Heat Transfer From a Sphere Due to Acoustic Streaming

Convective heat transfer from a sphere due to acoustic streaming is examined for large streaming Reynolds numbers. Analytical and numerical solution techniques are used to obtain Nusselt number correlations for a wide range of Prandtl numbers with particular emphasis on the case of Pr~1. A simple experiment performed to confirm some of the predictions is described. The results obtained can be used for the thermal analysis of containerless materials processing in space using acoustic levitation.

A new analytic technique for 3-D heat transfer from a cylinder with two or more axially interacting eccentrically embedded vessels with application to countercurrent blood flow

A new approximate analytic solution technique is developed for treating the heat exchange between two or more axially interacting vessels which are eccentrically embedded in a cylinder. The cylinder exchanges heat by convection with the environment and has a non-uniform surface temperature due to the vessel eccentricity. The flow in the vessels can be either countercurrent or unidirectional. The solution is constructed by superposition of a new solution for a single embedded vessel which exactly satisfies the boundary conditions when the thermal conductivity of the fluid is equal to that of the cylinder material and is a very good approximation when these conductivities are not equal. Comparison with an exact solution for a single embedded vessel, when the conductivities are not equal, shows that the approximate solution results in very minor errors for a wide range of the governing parameters. As an application of the new technique the two-vessel solution is used to examine the countercurrent heat exchange in a human arm.

An absorbing boundary condition for wave propagation in saturated poroelastic media — Part II: Finite element formulation

In a finite element formulation for dynamic soil-structure interaction, an absorbing boundary condition is needed to model wave propagation towards infinity. When the soil is saturated, its dynamic behaviour can be modelled by means of Biot's poroelastic theory. In Part I (Degrande, G. & De Roeck, G., Soil Dynamics & Earthquake Eng., 1993, 12(7), 411-21), a local absorbing boundary condition for wave propagation in saturated poroelastic media has been developed. In the present paper, this boundary condition is implemented in an irreducible finite element formulation for a compressible pore fluid. Spurious reflections for oblique incident waves on the absorbing boundary contribute to the solution errors. Therefore, a spectral element method, based on classical analytical solution techniques, is used to assess the accuracy of the finite element formulation.

Flexural‐Torsional Stability of Thin‐Walled Columns

This paper discusses the problem of flexural‐torsional stability of columns with open thin‐walled cross sections. It introduces an analytical solution technique applicable to the differential equation system describing the current problem. The approach derived is more comprehensive than those presented traditionally in the literature and is applicable in determining the value of critical load due to combined buckling modes. The system considered consists of three separate differential equations that are coupled in a general case. Various symmetries of the cross section release these couplings. This presentation is limited, for simplicity, only to columns having singly symmetric cross sections, or at least cross sections of point symmetry. Thus, there are couplings between two of the governing equations, The elastic restraints having varying stiffnesses with respect to translation, rotation, torsion and warping, separately, have been studied as boundary conditions through numerical examples. The stiffnesses of the restraints applied are chosen to cover the range from zero to infinity.

Partial Differential Equations: Analytical Solution Techniques (J. Kevorkian)

Previous article Next article Partial Differential Equations: Analytical Solution Techniques (J. Kevorkian)Susan ColeSusan Colehttps://doi.org/10.1137/1033026PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Partial Differential Equations: Analytical Solution Techniques (J. Kevorkian)." SIAM Review, 33(1), pp. 135–136 Previous article Next article FiguresRelatedReferencesCited byDetails Volume 33, Issue 1| 1991SIAM Review History Published online:18 July 2006 InformationCopyright © 1991 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1033026Article page range:pp. 135-136ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Non-linear dynamics of a geared rotor-bearing system with multiple clearances

Non-linear frequency response characteristics of a geared rotor-bearing system are examined in this paper. A three-degree-of-freedom dynamic model is developed which includes non-linearities associated with radial clearances in the radial rolling element bearings and backlash between a spur gear pair; linear time-invariant gear meshing stiffness is assumed. The corresponding linear system problem is also solved, and predicted natural frequencies and modes match with finite element method results. The bearing non-linear stiffness function is approximated for the sake of convenience by a simple model which is identical to that used for the gear mesh. This approximate bearing model has been verified by comparing steady state frequency spectra. Applicability of both analytical and numerical solution techniques to the multi-degree-of-freedom non-linear problem is investigated. Satisfactory agreement has been found between our theory and available experimental data. Several key issues such as non-linear modal interactions and differences between internal static transmission error excitation and external torque excitation are discussed. Additionally, parametric studies are performed to understand the effect of system parameters such as bearing stiffness to gear mesh stiffness ratio, alternating to mean force ratio and radial bearing preload to mean force ratio on the non-linear dynamic behavior. A criterion used to classify the steady state solutions is presented, and the conditions for chaotic, quasi-periodic and subharmonic steady state solutions are determined. Two typical routes to chaos observed in this geared system are also identified.

Partial differential equations: analytical solution techniques

1. The Diffusion Equation.- 2. Laplace's Equation.- 3. The Wave Equation.- 4. Linear Second-Order Equations with Two Independent Variables.- 5. The Scalar Quasilinear First-Order Equation.- 6. Nonlinear First-Order Equations.- 7. Quasilinear Hyperbolic Systems.- 8. Approximate Solutions by Perturbation Methods.- A.1. Review of Green's Function for ODEs Using the Dirac Delta Function.- A.2. Review of Fourier and Laplace Transforms.- A.3. Review of Asymptotic Expansions.- References.

Partial differential equations—analytical solution techniques, by J. Kevorkian. Pp 547. 1989. ISBN 0-534-12216-7 (Wadsworth and Brooks/Cole)

Partial differential equations—analytical solution techniques, by J. Kevorkian. Pp 547. 1989. ISBN 0-534-12216-7 (Wadsworth and Brooks/Cole)

A triangularization algorithm which determines the Lie symmetry algebra of any system of PDEs

The author presents several algorithms which have been automated in the symbolic language MACSYMA. Algorithm STANDARD FORM reduces any system of linear PDEs to a simplified triangular form which has its integrability conditions identically satisfied. Generally a system's standard form is more amenable to numerical or analytical solution techniques than the system itself. The dimension of the solution space and the consistency or inconsistency of a system are directly determinable from its standard form. Algorithm TAYLOR uses a system's standard form to compute its Taylor series solution to any prescribed finite degree. The author presents an algorithm STRUCTURE CONSTANT based on STANDARD FORM and TAYLOR which, unlike existing symbolic algorithms for determining symmetries, always computes the dimension and structure constants of the Lie symmetry algebra of any system of PDEs.

Mathematical Modeling of Landfill Gas Extraction

A model, incorporating Darcy’s law, is developed for single-species gas flow inside a nonisotropic porous medium, within which gas is continuously generated. A linear parabolic equation is derived to express the internal pressure variations in terms of the permeability, gas production rate, and boundary conditions of a site when a series of extraction wells are present. The model is used to determine the gas fluxes within a site of rectangular cross section, into which an arbitrary number of horizontal extraction pipes have been installed. This solution is used to predict the consequences of cumulative failures within a complex pumping system, and hence estimate the circumstances under which gas is first able to escape from a site. Using the results of the model some brief guidelines are offered for designing such installations. The analytical solution technique is contrasted with purely numerical methods for similar problems, and the speed and accuracy of each are compared. An efficient computer implementation for preliminary design studies is suggested.

Solutions approximating solute transport in a leaky aquifer receiving wastewater injection

A mathematical model amenable to analytical solution techniques is developed for the investigation of contaminant transport from an injection well into a leaky aquifer system, which comprises a pumped and an unpumped aquifer connected to each other by an aquitard. A steady state groundwater flow field is assumed, where the injected fluids move horizontally in the pumped aquifer and vertically in the aquitard. The unpumped aquifer is assumed to remain in hydrostatic condition due to its large transmissivity. Descriptions of groundwater velocities are based on appropriate leaky aquifer well hydraulics. The model assumes that contaminants are transmitted in the pumped aquifer by radial advection, and in the aquitard by vertical one‐dimensional advection and longitudinal dispersion. Two coupled linear differential equations are formulated to deal with this transport problem; they involve variable coefficients dependent on Bessel functions. The unpumped aquifer is incorporated into the model as a boundary condition. Semianalytical transient solutions for concentration distributions in the pumped aquifer and the aquitard can be obtained by numerically inverting the Laplace domain solutions of the model with the Crump [1976] method. However, the steady state solutions of the model are determined in closed forms. In general, the modeling approach neglecting longitudinal dispersion in the pumped aquifer and lateral dispersion in the aquitard does not introduce significant error to the modelling results, as shown by the good agreement between numerical solutions including these two transport mechanisms and the solutions of the proposed model. Within the context of the assumed hydrogeological conditions, the proposed model appears to be a valid approximation to the problem of interest. Type curves for concentration evaluated at the well bore and the top of the aquitard are given, which can be employed to estimate whether the unpumped aquifer would be polluted as the consequence of the injection. Neglecting the leakage effects in a leaky aquifer can cause significant error in studying the transport problem.

Locally analytical prediction of the steady inviscid incompressible flow through an airfoil cascade

A mathematical model and locally analytical solution technique are developed to analyze the steady, inviscid, incompressible, flow through a two-dimensional turbomachinery cascade comprised of cambered, thick, airfoils. The cascade is analyzed by considering a single flow channel. The boundary conditions require a zero velocity normal to the airfoil surfaces, the Kutta condition to be satisfied, and the inlet and exit flow from the cascade to be periodic. The flow field velocity potential is separated into circulatory and noncirculatory components, each individually described by a Laplace equation. A body fitted computational grid is utilized which permits grid points to be specified along the entire computational boundary and results in a smoothly spaced, nonoverlapping grid. General analytic solutions to the Laplace equations in the transformed computational plane are determined by separation of variables. Locally analytical solutions are then developed by applying these solutions to individual grid elements, i.e. the integration and separation constants are determined from the boundary conditions in each grid element. The complete flow field solution is then obtained by assembling these locally analytical solutions. The validity of this locally analytical solution and the flow modelling are then demonstrated by correlation predictions with data from various cascade experiments.

Locally analytical prediction of the steady inviscid incompressible flow through an airfoil cascade

A mathematical model and locally analytical solution technique are developed to analyze the steady, inviscid, incompressible, flow through a two-dimensional turbomachinery cascade. The velocity potential is separated into circulatory and non-circulatory components, each individually described by a Laplace euation. A body fitted computational grid is utilized. General analytical solutions in the transformed computational plane are determined by separation of variables. The validity of this locally analytical solution and the flow modeling are then demonstrated by correlating predictions with cascade data.

Electrochemistry of Iodide in Propylene Carbonate: II . Theoretical Model

An algorithm based on Duhamel's superposition integral is used to simulate cyclic voltammograms for iodide electrochemistry on platinum in propylene carbonate. The electrochemistry is represented by the reaction sequencewhere reaction [ I ] is rate limited (Butler‐Volmer kinetics) and reaction [ II ] is reversible (Nernst equilibrium behavior). Since the nonunity stoichiometries and Butler‐Volmer kinetics in the reaction sequence prohibit the use of analytical solution techniques, the superposition‐integral algorithm is used as a fast and accurate method for the numerical simulation. The simulated voltammograms for the iodide system agree very well with experimental observations and support the proposed reaction sequence.

Quadratic analytical solution of general systems through formal hamiltonization

A technique for the quadratic analytical solution of general nonlinearly perturbed periodic systems is presented. It relies on a device recognized as early as Birkhoff (1927), through which any system of ordinary differential equations can be cast in Hamiltonian form through the introduction of a set of auxiliary ‘conjugate’ variables. The particular implementation applies the author's quadratic Hamiltonian approach, utilizing Lie transforms (so admitting of easy inversion), and featuring the ability to determine the frequencies of the system to twice the order of the solution at the last step. The method is exemplified through an analysis of the van der Pol equation to find the solution to second order, and frequencies to fourth, of the limit cycle of the system. Finally, the relationship of the approach to other perturbation techniques, particularly the vector/matrix Lie transform method, is discussed.

First- and second-order sensitivity analysis of linear and nonlinearstructures

This paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches. The computations required are based on additional load conditions characterized by imposed initial strains, body forces, or surface tractions. As such, they are equally applicable to numerical or analytical solution techniques. The relative efficiency of various approaches for calculating first and second derivatives is assessed. It is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first-order sensitivities and adjoint vectors. Two example problems are used for demonstrating the various approaches.