An Iterative Boundary Integral Numerical Solution for General Steady Heat Conduction Problems

1981 ◽  
Vol 103 (1) ◽  
pp. 26-31 ◽  
Author(s):  
M. S. Khader ◽  
M. C. Hanna
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Three Dimensional ◽  
Computational Procedure ◽  
Boundary Integral ◽  
Temperature Dependent ◽  
Convective Boundary ◽  
Arbitrary Boundary ◽  
Steady Heat Conduction ◽  
Steady Heat

An iterative boundary integral numerical method for solving the steady conduction of heat is developed. The method is general for two- and three-dimensional regions with arbitrary boundary shapes. The development is generalized to include the first, second, and third kind of boundary conditions and also radiative boundary and temperature-space dependent convective coefficient cases. With Kirchhoff’s transformation, cases of temperature-dependent thermal conductivity with general boundary conditions are also accounted for by the present method. A variety of problems are analyzed with this method and their solutions are compared to those obtained analytically. A comparison between the present method and the finite difference predictions is also investigated for a case of mixed temperature and convective boundary conditions. Moreover, two-dimensional regions with three kinds of boundary conditions and irregular-shaped boundaries are used to illustrate the versatility of the technique as a computational procedure.

scholarly journals Three-Dimensional Vibration Analysis of Rectangular Thick Plates on Pasternak Foundation with Arbitrary Boundary Conditions

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Huimin Liu ◽  
Fanming Liu ◽  
Xin Jing ◽  
Zhenpeng Wang ◽  
Linlin Xia
Keyword(s):  
Boundary Conditions ◽  
Admissible Function ◽  
Three Dimensional ◽  
Future Research ◽  
Pasternak Foundation ◽  
Arbitrary Boundary ◽  
Thick Plates ◽  
Arbitrary Boundary Conditions ◽  
Ritz Procedure ◽  
Three Dimensional Elasticity

This paper presents the first known vibration characteristic of rectangular thick plates on Pasternak foundation with arbitrary boundary conditions on the basis of the three-dimensional elasticity theory. The arbitrary boundary conditions are obtained by laying out three types of linear springs on all edges. The modified Fourier series are chosen as the basis functions of the admissible function of the thick plates to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges. The exact solution is obtained based on the Rayleigh–Ritz procedure by the energy functions of the thick plate. The excellent accuracy and reliability of current solutions are demonstrated by numerical examples and comparisons with the results available in the literature. In addition, the influence of the foundation coefficients as well as the boundary restraint parameters is also analyzed, which can serve as the benchmark data for the future research technique.

Entrance, Viscous Dissipation and Temperature Dependent Viscosity Effects in Microchannel Flows With Convective Boundary Conditions

2006 ◽  
Author(s):  
C. Nonino ◽  
S. Del Giudice ◽  
S. Savino
Keyword(s):  
Boundary Conditions ◽  
Viscous Dissipation ◽  
Circular Cross Section ◽  
Laminar Flows ◽  
Projection Algorithm ◽  
Convective Boundary Conditions ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Convective Boundary ◽  
Dependent Viscosity

The effects of viscous dissipation and temperature dependent viscosity in simultaneously developing laminar flows of liquids in straight microchannels of circular cross-section are studied with reference to convective boundary conditions. Viscosity is assumed to vary linearly with temperature, in order to allow a parametric investigation, while the other fluid properties are held constant. A finite element procedure, based on a projection algorithm, is employed for the step-by-step solution of the parabolized momentum and energy equations. Axial distributions of the local overall Nusselt number and of the apparent Fanning friction factor are presented with reference to both heating and cooling conditions for three different values of the Biot number. Examples of temperature profiles at different axial locations are also shown.

Three-Dimensional Vibrations of Truncated Hollow Cones

1995 ◽  
Vol 1 (2) ◽  
pp. 145-158 ◽  
Author(s):  
Arthur W. Leissa ◽  
Jinyoung So
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Arbitrary Boundary ◽  
Conical Shells ◽  
Hole Radius ◽  
Geometric Boundary

This work presents a three-dimensional (3-D) method of analysis for determining the free vibration frequencies and corresponding mode shapes of truncated hollow cones of arbitrary thickness and having arbitrary boundary conditions. It also supplies the first known numerical results from 3-D analysis for such problems. The analysis is based upon the Ritz method. The vibration modes are separated into their Fourier components in terms of the circumferential coordinate. For each Fourier component, displacements are expressed as algebraic polynomials in the thickness and slant length coordinates. These polynomials satisfy the geometric boundary conditions exactly. Because the displacement functions are mathematically complete, upper bound values of the vibration frequencies are obtained that are as close to the exact values as desired. This convergence is demonstrated for a representative truncated hollow cone configuration where six-digit exactitude in the frequencies is achieved. The method is then used to obtain accurate and extensive frequencies for two sets of completely free, truncated hollow cones, one set consisting of thick conical shells and the other being tori having square-generating cross sections. Frequencies are presented for combinations of two values of apex angles and two values of inner hole radius ratios for each set of problems.

Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media

2017 ◽  
Vol 29 (6) ◽  
pp. 1255-1271 ◽  
Author(s):  
MingHao Zhao ◽  
Yuan Li ◽  
CuiYing Fan
Keyword(s):  
Stress Intensity ◽  
Boundary Conditions ◽  
Stress Intensity Factors ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Boundary Integral ◽  
Displacement Discontinuity ◽  
Green Functions ◽  
Intensity Factors

An arbitrarily shaped planar crack under different thermal and electric boundary conditions on the crack surfaces is studied in three-dimensional transversely isotropic thermopiezoelectric media subjected to thermal–mechanical–electric coupling fields. Using Hankel transformations, Green functions are derived for unit point extended displacement discontinuities in three-dimensional transversely isotropic thermopiezoelectric media, where the extended displacement discontinuities include the conventional displacement discontinuities, electric potential discontinuity, as well as the temperature discontinuity. On the basis of these Green functions, the extended displacement discontinuity boundary integral equations for arbitrarily shaped planar cracks in the isotropic plane of three-dimensional transversely isotropic thermopiezoelectric media are established under different thermal and electric boundary conditions on the crack surfaces, namely, the thermally and electrically impermeable, permeable, and semi-permeable boundary conditions. The singularities of near-crack border fields are analyzed and the extended stress intensity factors are expressed in terms of the extended displacement discontinuities. The effect of different thermal and electric boundary conditions on the extended stress intensity factors is studied via the extended displacement discontinuity boundary element method. Subsequent numerical results of elliptical cracks subjected to combined thermal–mechanical–electric loadings are obtained.

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