# Presentation: 284395

## Steady Heat Conduction in Cartesian Coordinates and a Library of Green's Functions Kevin D. Cole Dept. of Mechanical EngineeringPaul E. CrittendenDept. of MathematicsUniversity of Nebraska--Lincoln

## Motivation

- Verification of fully-numeric codes
- Sponsor: Sandia National Laboratory
- Personnel: K. Dowding, D. Amos (Sandia);    J. Beck , D. Yen, R. McMasters, (MI State); K. D. Cole, P. E. Crittenden (Nebraska)
- Geometry:  Parallelepiped

## Outline

- Temperature problem, Cartesian domains
- Green’s function solution
- Green’s function in 1D, 2D and 3D
- Web-based Library of Green’s Functions
- Summary

## Temperature Problem 

- Domain **R** includes the slab, rectangle, and parallelepiped.

- The boundary condition represents one of three types : Type 1. *k**i*=0, *h**i*=1, and *f**i* a specified temperature; Type 2. *k**i*=*k*, *h**i*=0, and *f**i* a specified heat flux [*W*/*m*2]; Type 3. *k**i*=*k* and *h**i* a heat transfer coefficient [*W*/*m*2/o*K*].

## What is a Green's Function?

- Green's function (GF) is the response of a body (with homogeneous boundary conditions) to a concentrated energy source.  The GF depends on the differential equation, the body shape, and the *type*  of boundary conditions present.
- Given the GF for a geometry, *any* temperature problem can be solved by integration.
- Green's functions are named in honor of English mathematician George Green (1793-1841).

## Green’s function solution

**T ( r ) =**

- Green’s function ***G *** is the response at location ***r****** *** to an
- infinitessimal heat source located at coordinate ***r’.***

## Green’s function for 1D Slab

- Boundary conditions are homogeneous, and of the same *type*  (1, 2, or 3) as the temperature problem.  There are 32 = 9 combinations of boundary types for the 1D slab.

## 1D Example  

- G=0 at y=0 and at y=W.
- Y11 case.  Two forms:
-   Series.
-   Polynomial.

## Y11 case, continued

- Plot of ***G(y,y’)*** versus ***y***                 Y11 Geometry.
-   for several ***y’***  values.

## GF for the 2D Rectangle

- Here G is dimensionless.
- There are 34 = 81 different combinations of boundary
-   conditions (different GF) in the rectangle.

## 2D Example

- Case X21Y11. *G*=0 at edges,
- except insulated at *x*=0.

- Double sum form:

- where

## 2D Example, case X21Y11

- Single sum form:

- where kernel function *P**n* for this case is:

## Case X21Y11 heated at (0.4,0.4)

## GF for the 3D Parallelepiped

- There are 36=729 combinations of boundary types.

## 3D ExampleCase X21Y11Z12Triple sum form:

## 3D Example, X21Y11Z12Alternate double-sum forms:

## Web Publication:  Promise

- Material can be presented in multiple digital formats, may be cut and pasted into other digital documents.
- Immediate world-wide distribution.
- Retain control of content, easily updated.
- Hyperlinks to related sites.

## Web Publication:  Pitfalls

- No editorial support, no royalties.
- Unclear copyright protection.
- Continued operating costs (service provider, computer maintenance, etc.)
- Little academic reward; doesn’t “count” as a publication.

## NIST Digital Library of Mathematical Functions

- Web-based revision of handbook by Abramowitz and Stegun (1964).
- Emphasis on text, graphics with few colors, photos used sparingly.
- Navigational tools on every page.
- No proprietary file formats (HTML only).
- Source code developed in AMS-TeX.

## Green’s Function Library

- Source code is LateX, converted to HTML with shareware code latex2html run on a Linux PC
- GF are organized by equation, coordinate system, body shape, and type of boundary conditions
- Each GF also has an identifying number

## Contents of the GF Library

**Heat Equation. Transient Heat Conduction**
- 		Rectangular Coordinates.  Transient 1-D
- 		Cylindrical Coordinates.  Transient 1-D
- 		Radial-Spherical Coordinates.Transient 1-D
**Laplace Equation. Steady Heat Conduction**
- 		Rectangular Coordinates.  Steady 1-D
- 		Rectangular Coordinates. Finite Bodies, Steady.
- 		Cylindrical Coordinates.  Steady 1-D
- 		Radial-Spherical Coordinates.Steady 1-D
**Helmholtz Equation.  Steady with Side Losses**
- 		Rectangular Coordinates.  Steady 1-D

## Slide 22

## Slide 23

## Slide 24

## Summary

- GF in slabs, rectangle, and parallelepiped for 3 types of boundary conditions
- These GF have components in common: 9 eigenfunctions and 18 kernel functions
- Alternate forms of each GF allow efficient numerical evaluation

## Summary, continued.

- Web Publishing:  wide dissemination, local control, updatable; continuing expense, little academic reward.
- Green’s Function Library:  source code developed in LateX (runs on any computer) and converted to HTML with latex2html (runs on Linux).

## Work in progress: Dynamic Math

- Currently GF web page is static, book-like
- Temperature solutions are too numerous for pre-determined display
- Working to create and display temperature solutions on demand, in response to user input.
- Code with open standards Perl, latex2html

## Acknowledgments

- Green’s Function work supported by Sandia National Laboratories,    University of Nebraska-Lincoln,           and by J. V. Beck
- Web-page development assisted by undergraduate student researchers Christine Lam, Lloyd Lim, Sean Dugan, and Chootep Teppratuangtip.