Complicated Boundary Shape Research Articles

Unsteady thermal field in bars of doubly connected cross section with heat generation

The title problem is approximately solved by means of a variational formulation. It is shown that in the case of complicated boundary shapes of the cross section, the technique of conformal mapping is, sometimes, quite advantageous. The analytical predictions are compared with the results obtained by means of the finite element method and, in general, good engineering agreement is shown to exist.

A modification of the Galerkin method and the solution of the Helmholtz equation in regions of complicated boundary shape

An approximate approach proposed by R. Schmidt [J. Appl. Mech. 49, 639–640 (1982)] for optimizing the value of fundamental frequency coefficients when determined by variational methods in the case of one-dimensional problems is extended in the present letter in order to deal with domains of arbitrary shape, provided that the mapping function which maps the given shape onto a unit circle is known.

Three-Dimensional Analysis of RF Electromagnetic Field by Finite Element Method

We have developed a program which computes the resonant frequencies and fields in an arbitrary shaped three dimensional cavity by finite element method. We take electric field E = (Ex, Ey, Ez) as unknown variables and solve the eigenvalue problem for the Maxwell equations in time-harmonic case imposing boundary conditions. Divergence-free condition is treated approximately by penalty method. We show the formulation for three dimensional analysis of RF electromagnetic fields and some test calculations. Satisfactory results are obtained for rectangular cavities, cylindrical cavities and those with smoothly deformed parts. In the present calculations, number of the nodal points is limited to less than 1000 because of the memory size of the computer, which means about ten nodes in each direction. For very complicated boundary shape, errors become large because mesh size is coarse and therefore the condition div E = 0 is not fully satisfied. But this difficulty will be overcome with increasing available memory size.

Solution of a nonlinear diffusion type problem in doubly-connected domains of complicated boundary shape

Variable transformation and the conformal mapping methods are invoked when solving a nonlinear diffusion type problem in doubly-connected domains of complicated boundary shape. Results are presented for prisms with concentric, circular perforations. The results are in good agreement with numerical predictions obtained with a finite element code.

Temperature distributions in cladded configurations of complicated boundary shape

This paper deals with an approximate analytical solution of certain cladded shapes of basic interest in nuclear engineering technology. In one case a solution of Laplace's equation is obtained. In the other, a solution to a system of partial differential equations (Poisson's and Laplace's) is attained. An approximate conformal mapping approach is also used in both situations, and it is shown that the analytical predictions are in good agreement with the values obtained by means of a finite element code.

Experiments on the solution of the helmholtz equation using the finite element method and a variational approach in the case of domains of complicated boundary shape

Domains of ‘exotic’ boundary shape are of interest in several technological applications—acoustic and electromagnetic waveguides, solid propellant rocket cross-sections, printed circuit boards, etc. The present paper describes some experiments performed to determine the eigenvalues of a vibrating, ideal membrane. It is shown that the finite element method yields results which are in very good agreement with values determined by means of an analytical approach in the case of a membrane of a cardioidal shape.

Construction of a type of unsteady two-dimensional temperature profile in the case of domains of complicated boundary shape

This paper presents a simple, analytical solution to an unsteady state temperature distribution of an orthotropic two-dimensional configuration of complicated boundary shape. It is assumed that the initial, non-uniform temperature distribution can be expressed, approximately, by a functional relation which is generated multiplying the equations which define each portion of the boundary of the configuration under study. The results are in good engineering agreement with the values obtained by means of a finite element solution.

Unsteady heat conduction in orthotropic plates of arbitrary shape with geometrically complicated initial conditions

A complex variable approach, coupled with a variational method, is used in order to obtain an approximate analytical description of the unsteady temperature distribution in orthotropic plates of complicated boundary shape which possess, initially, a ‘hot spot’ in a concentric, circular region. It is also shown that the analytical predictions are in reasonably good agreement with the calculated values obtained by means of a finite element code.

Comparison of variational and finite element solutions of Helmholtz’s equation

This paper deals with some numerical experiments on the determination of natural frequencies of vibration of doubly connected membranes of complicated boundary shape. Finite element results are compared with values obtained using an approximate method which consists of constructing appropriate coordinate functions which identically satisfy the boundary conditions and use of a variational approach to generate a simple frequency equation. The results are in good agreement with values previously published in the open literature which have been calculated using a conformal mapping-variational approach.

On the numerical analysis of compressible flow problems by the “modified FLIC method”

This paper presents a new computational technique for transonic flow problem analysis. This method, named Modified FLIC Method, is based on a time-marching technique of FLIC (fluid in cell) method and employs triangular elements conventionally used in finite element method. This technique can be applied to transonic flows with any complicated boundary shapes. Three problems were solved in this paper, the first was a supersonic flow around a circular cylinder, the second was a transonic flow between tubrine blade cascades and the last was an unsteady flow in a duct with a junction. The calculated results showed a good agreement with the experimental data.

Numerical experiments on the solution of the Helmholtz equation in the case of domains of complicated boundary shape

Domains of complicated boundary shape are of great practical importance in several fields of technology and applied science; e.g. solid propellant rocket grains, electromagnetic and acoustic waveguides, and certain elements used in nuclear engineering. The technical literature contains very few comparative studies of analytical and numerical solutions when dealing with such rather complex geometries. The present study constitutes an effort in that direction.

Application of conformal mapping to the determination of unsteady temperature distribution in thermally orthotropic plates

Very few papers and reports are available on the solution of heat conduction problems in anisotropic solids. This situation is of particular interest in some nuclear engineering applications (for instance, fuel elements possess, in general, anisotropic characteristics). The present study deals with the solution of an unsteady heat conduction problem in domains of complicated boundary shape, considering a particular case of anisotropy: a thermally orthotropic material. It is shown that the conformal mapping technique coupled with the Ritz method leads to a unified solution of the title problem for an arbitrary orientation for the axes of orthotropy with respect to the directions of the sides of the plates.

Numerical experiments on the determination of unsteady state temperature distribution in domains of complicated boundary shape

This paper compares analytical and finite element results for an unsteady heat-conduction problem in simply and doubly connected plates of regular polygonal shape. A numerical solution is obtained by means of the powerful finite element method and the results are shown to agree with an approximate conformal mapping-variational technique previously developed by the first author and coworkers.

A comparison of analytical and numerical solutions in heat conduction problems

This paper deals with a comparison of analytical and finite element results in two types of steady state diffusion problems: (a) determination of solutions in domains of complicated boundary shape, and (b) analysis diffusion-type problems in nonhomogeneous media.

Waveguides of Arbitrary Cross Section by Solution of a Nonlinear Integral Eigenvalue Equation

The problem of electromagnetic wave propagation in hollow conducting waveguides of arbitrary cross section is formulated as an integro-differential equation in terms of fields at the waveguide boundary. Cutoff wave numbers and wall currents appear as eigenvalues and eigenfunctions of a nonlinear eigenvalue problem involving an integro-differential operator. A variational solution is effected by reducing the problem to matrix form using the method of moments. A specific solution of the problem is developed using triangle expansion functions in the method of moments. The solution is simplified by symmetry considerations and is implemented by two digital computer programs. Listings and full documentation of these programs are available. This solution yields accurate determinations of cutoff wave numbers, wall currents, and distributions of both longitudinal and transverse modal field components for the first several modes. Illustrative computations are presented for the single-ridge waveguide, which has a complicated boundary shape that does not lend itself to exact solution.

On the approximate solution of flow and heat transfer through non-circular conduits with uniform wall temperature and heat generation

The analytical solution for flow and heat transfer through conduits with heat sources in the fluid stream is studied. The flow is assumed to have a steady uniform velocity profile defined as slug flow. A variety of conduits of different cross-sectional shapes is analyzed in a unified manner by mapping the cross section of the duct onto a unit circle and then transforming the governing equations. The transformed equations are solved using a weighted residuals approach. Solutions for the dimensionless mixing temperatures and Nusselt numbers are obtained for conduits of different cross-sectional shapes (square, hexagon, circle, corrugated, cardioid). In the case of a square duct the results are in excellent agreement with the exact values. The method is applicable to conduits of arbitrary shape and it may prove useful to design engineers since conduits of complicated boundary shape are commonly used in aerospace vehicles, nuclear reactors, etc.

Solution of Eigenvalue Problems Using Conformal Mapping Techniques

The study of many problems of importance in modern technology involves the determination of eigenvalues in domains of complicated boundary shape. Some examples can be cited—solid-propellant rocket motors: A simplified mathematical model of a rocket motor is an infinitely long, circular cylinder with a star-shaped perforation; electromagnetic waveguides: Within the last 15 yr the desire for more suitable waveguide shapes has often appeared. Investigation of different modes in waveguides of arbitrary cross section has been conducted by several authors. Flow and heat transfer through noncircular conduits and unsteady flow problems in ducts of arbitrary shape are also problems of great interest in aerospace engineering and biophysics. Conformal mapping techniques have been used by several authors when solving for the eigenvalues of the problem. A brief review of these applications is presented, and fundamental frequencies of vibrating parallelogram plates are computed as application of the method.

Vibrations of Rib-Stiffened Thin Elastic Plates Carrying Concentrated Masses

The present study deals with the calculation of the fundamental frequency of vibration of clamped and simply supported plates stiffened in two perpendicular directions and carrying concentrated masses. It is assumed that the stiffeners are closely spaced and that their height is moderate when compared with the thickness of the plate. Use of polynomial approximations and a variational technique simplifies the calculations considerably. The approach is also applicable in the case of plates of complicated boundary shape. Numerical calculations are presented for a clamped square plate with ribs in one direction, and the results are in good agreement with those obtained following the “orthotropic-plate” approach.

An Application of Conformal Mapping to a Three-Dimensional Unsteady Heat Conduction Problem

SummaryThis paper deals with an unsteady-state three-dimensional heat conduction problem in a prismatic body which has a complicated boundary shape. Numerical techniques such as a relaxation method must be used to solve a problem of this type; this method, however, requires unusually large amounts of labour. The method used in this paper consists of the application of conformal mapping and numerical solution of a partial differential equation and is applied first to a prismatic body of square cross section for which the exact solution is well known. As a second application, the approximate solution for a problem involving a star-shaped cross section is presented.