User:Eml5526.s11.team04.premchand/HW6

= Problem 6.6 = Result obtained by ABAQUS





= Problem 6.7 - Matrix Algebra =

Problem Statement
Recall matrix algebra: (1) $$(\mathbf{A}\mathbf{B})^{T}=\boldsymbol{B}^{T}\boldsymbol{A}^{T}$$ (2) $$(\boldsymbol{A}^{-1})^{T}=(\boldsymbol{A}^{T})^{-1}=\boldsymbol{A}^{-T}$$ Given the matrices: $$ \boldsymbol{A}=\begin{bmatrix} 1 & 1 & 1\\ 2 & -1 & 3\\ 3 & 2 & 6 \end{bmatrix} $$ $$ \boldsymbol{B}=\begin{bmatrix} 1 & 3 & 5\\ 1 & -4 & 1\\ 2 & 5 & 8 \end{bmatrix} $$ Verify the matrix algebra properties shown above (properties 1 and 2). Find and explain the syntax of Wolframalpha

Solution
Condition 1

(1a) is equal to (1b), this proves that $$(\mathbf{A}\mathbf{B})^{T}=\boldsymbol{B}^{T}\boldsymbol{A}^{T}$$

The wolfram alpha calculation for $$(\mathbf{A}\mathbf{B})^{T}$$ is given in the following pdf file

Click on the pdf file to see the second page.

The wolfram alpha calculation for $$\boldsymbol{B}^{T}\boldsymbol{A}^{T}$$ is given in the following pdf file

Click on the pdf file to see the second page.

The Wolfram alpha results match with the manually calculated results of condition 1 of problem statement.This validates the proof.

Condition 2

(2a) is equal to (2b).This proves that $$(\boldsymbol{A}^{-1})^{T}=(\boldsymbol{A}^{T})^{-1}=\boldsymbol{A}^{-T}$$

The wolfram alpha calculation for $$(\boldsymbol{A}^{-1})^{T}$$ is given in the following pdf file

The wolfram alpha calculation for $$(\boldsymbol{A}^{T})^{-1}$$ is given in the following pdf file

Click on the pdf file to see the second page.

The Wolfram alpha results match with the manually calculated results of condition 2 of problem statement.This validates the proof.

= Problem 6.9 - F&B Ex 8.3, Ex 8.4, and Problem 8.6 =

Problem Statement
Problem 1: The heat conduction problem in Example 8.1 ( Fish and Belytschko ) is modeled with 16 quadrilateral finite elements.The problem is solved manually using the finite element code given in Section 12.5

Solution
1.) The problem is solved using the MATLAB code given in section 12.5 of companion site of the text book Fish and Belytschko.On running the code we receive the following post process result plots for temperature and flux figs (a),(b). Fluxes are calculated by looping over the number of elements.For the four node quadrilateral there are four Gauss points as shown in fig (c).







2.) Reproduce Example 8.4 (page 205) from Fish and Belytschko. 3.) Complete problem 8.6 (page 212) from Fish Belytschko.