FEM Coding Assignment: Quadrature Points and the Isoparametric Element Jacobian

This assignment extends the Q4, Q9, CST, and LST elements that you implemented in PS03 to include quadrature weights and points, as well as to add the calculation of the Jacobian.

Initial setup (same as before)

1. Check out the problem repository

     git clone https://github.com/solidsgroup/EM525-PS04.git
   

2. Change into the problem directory

     cd EM525-PS04
   

3. Use this command to install eigen (optional: you can skip this if you have eigen installed)

     make eigen
   

4. Now, compile the code

     make
   

   You should see a number of warnings indicating "no return statement". These should disappear once you have implemented the assignment.

Part 1: Adding quadrature points and weights

You will find two TODOs in each of the following files:

• src/Element/LST.H
• src/Element/CST.H
• src/Element/Q4.H
• src/Element/Q9.H

To address the todo items:

1. For the eta and deta functions, simply copy/paste your implementations from PS03.
2. For the getArea function, you can simply return a value (like return 1.0; if the area of the template element is 1.0).
3. For Qp, add vectors corresponding to the quadrature points. (This should be a comma-separated list of Eigen::Vector2ds, like for Y0.
4. For Qw, add a comma-separated list of weights corresponding to the quadrature points.

Part 2: Implement the Jacobian matrix calculation

You will find a TODO in the following file:

• src/Element/Element.H

(Note that this is an abstract parent element, meaning that everything you imlement here is automatically inherited by all of the other elements.)

Follow the instructions to implement the Jacobian matrix.

Check result

You will find a TODO in the following file:

• src/main.cpp

Follow the instructions in this file to create a CST element with embedded coordinates (recommended that you use the coordinates in the CST example in the notes.) When you have completed the example, running ./bin/main will run the tests, followed by the output from your test.

Play around with the inputs X0 to ensure that this implmentation makes sense. Provide a brief discussion in your submission.