User:Eml5526.s11.team5.srv/hw6

Example 8.3
Consider the heat conduction problem given in example 8.1 modeled with 16 quadrilateral elements. Solve this problem using finite element coding given in Fish and Belyschko section 12.5.

Given
(i) The conductivity of the plate is isotropic and $$ k = 5 W^0 C^{-1} \quad $$

(ii) Essential boundary condition T = 0 along the bottom edge and left edge.

(iii) Natural boundary condition $$ \bar{q} = 0 \ W m^{-1} $$ and  $$ \bar{q} = 20 \ W m^{-1} $$ along right edge and top edge respectively.

(iv) A constant heat source $$ s = 6 \ W m^{-2} $$

Find
Reproduce the post processing results obtained from the finite element code goven in section 12.5 in Fish and Belyschko book.

Solution
Using the matlab codes provided in section 12.5 of the book, we obtain the following results.

The Matlab code to analyze the problem 8.3 can be downloaded from here:Matlab code for Problem 8.3

This code can be altered to solve 2D heat conduction problems. We need to modify the 'input_file_xxxx.m' according to the boundary conditions, mesh size and the 'mesh2d.m' code also needs to be changed according to the given geometry.

For each problem, mesh configuration we create a separate 'input_file' which is called from 'preprocessor.m'.

The input files for each case are given below.

Given


(i) The conductivity of the concrete (inner material) is $$ k = 2 W^0 C^{-1} \quad $$ and that of the brick (outer material) is $$ k = 0.9 W^0 C^{-1} \quad $$

(ii) Essential boundary condition T = 10 along the top edge and T = 140 along the bottom edge.

(iii) Natural boundary condition $$ \bar{q} = 0 \ W m^{-1} $$ and  $$ \bar{q} = 20 \ W m^{-1} $$ along right edge and left edge respectively.

Find
Find the temperature distribution for different mesh densities:

(i) 2 x 2 quadrilateral element

(ii)4 x 4 quadrilateral element

(iii)8 x 8 quadrilateral element

Solution
Since the geometry has changed in this problem, we need to edit the mesh2d code to create a proper mesh. The code used for mesh creation for this geometry is shown below: % Mesh2D for 1 element