Green’s Function Partitioning in Galerkin-Based Integral Solution of the Diffusion Equation

1990 ◽  
Vol 112 (1) ◽  
pp. 28-34 ◽  
Author(s):  
A. Haji-Sheikh ◽  
J. V. Beck
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Large Time ◽  
Small Time ◽  
Solution Technique ◽  
Heterogeneous Solids ◽  
Function Solution ◽  
Nonhomogeneous Boundary ◽  
Time Partitioning ◽  
Time Accuracy

A procedure to obtain accurate solutions for many transient conduction problems in complex geometries using a Galerkin-based integral (GBI) method is presented. The nonhomogeneous boundary conditions are accommodated by the Green’s function solution technique. A Green’s function obtained by the GBI method exhibits excellent large-time accuracy. It is shown that the time partitioning of the Green’s function yields accurate small-time and large-time solutions. In one example, a hollow cylinder with convective inner surface and prescribed heat flux at the outer surface is considered. Only a few terms for both large-time and small-time solutions are sufficient to produce results with excellent accuracy. The methodology used for homogeneous solids is modified for application to complex heterogeneous solids.

The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary

2004 ◽  
Vol 56 (3) ◽  
pp. 590-611
Author(s):  
Yilong Ni
Keyword(s):  
Riemannian Manifold ◽  
Heisenberg Group ◽  
Green’S Function ◽  
Heat Kernel ◽  
Green's Function ◽  
Beltrami Operator ◽  
Small Time ◽  
Time Estimates

AbstractWe study the Riemannian Laplace-Beltrami operator L on a Riemannian manifold with Heisenberg group H1 as boundary. We calculate the heat kernel and Green's function for L, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of H1. We also restrict L to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary.

A method for the coupled field of an orthotropic piezoelectric plate subjected to an arbitrary electro-mechanical load

2020 ◽  
Vol 25 (11) ◽  
pp. 2132-2152
Author(s):  
ShouMing Shang ◽  
PengFei Hou ◽  
J Tong
Keyword(s):  
General Solution ◽  
Green’S Function ◽  
Green's Function ◽  
Mechanical Load ◽  
Piezoelectric Plate ◽  
Numerical Results ◽  
Function Solution ◽  
Coupled Field ◽  
Piezoelectric Devices ◽  
Plate Type

There are a number of plate-type piezoelectric devices in engineering, hence it is crucial to search for a method that can accurately acquire the electro-mechanical coupled field of a piezoelectric plate. A method for calculating the coupled field of an orthotropic piezoelectric plate with arbitrary thickness under an arbitrary electro-mechanical load is put forward in this article. First, the Green’s function solution of an orthotropic piezoelectric plate subjected to a line charge and a normal line force is derived based on the general solution of the orthotropic piezoelectric material. All stress and electric components of the orthotropic piezoelectric plate are derived when the general solution is substituted into suitable harmonic functions containing undetermined constants. Once the boundary conditions and electro-mechanical equilibrium conditions are satisfied, those constants can be solved. In addition, according to the obtained Green’s function solution and superposition principle, the coupled field of the orthotropic piezoelectric plate subjected to an arbitrary electro-mechanical load can be solved. Numerical results indicate that the convergence and precision of the method are quite good. A concise skill without repeated calculations is also presented for acquiring the coupled fields in the orthotropic piezoelectric plates with various thickness, which facilitates the effective design of plate thickness in plate-type piezoelectric devices. Finally, some valuable conclusions for the fine design of plate-type piezoelectric sensors, energy harvesters and actuators are presented based on the numerical results.

Green’s Functions for Two-Phase Transversely Isotropic Materials

1979 ◽  
Vol 46 (3) ◽  
pp. 551-556 ◽  
Author(s):  
Y.-C. Pan ◽  
T.-W. Chou
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Elastic Constants ◽  
Transversely Isotropic ◽  
Two Phase ◽  
Closed Form Solutions ◽  
Isotropic Materials ◽  
Function Solution ◽  
Present Solution ◽  
Plane Of Isotropy

Closed-form solutions are obtained for the Green’s function problems of point forces applied in the interior of a two-phase material consisting of two semi-infinite transversely isotropic elastic media bonded along a plane interface. The interface is parallel to the plane of isotropy of both media. The solutions are applicable to all combinations of elastic constants. The present solution reduces to Sueklo’s expression when the point force is normal to the plane of isotropy and (C11C33)1/2 ≠ C13 + 2C44 for both phases. When the elastic constants of one of the phases are set to zero, the solution can be reduced to the Green’s function for semi-infinite media obtained by Michell, Lekhnitzki, Hu, Shield, and Pan and Chou. The Green’s function solution of Pan and Chou for an infinite transversely isotropic solid can be reproduced from the present expression by setting the elastic constants of both phases to be equal. Finally, the Green’s function for isotropic materials can also be obtained from the present solution by suitable substitution of elastic constants.

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