FEM Coding Assignment: LST, Q4, Q9 Shape Functions

This assignment is similar to assignment 2: the objetive is to implement shape functions for linear strain triangle (LST), bilinear quadrilateral (Q4), and Larange quadrilateral (Q9) elements.

Initial setup

1. Check out the problem repository

     git clone https://github.com/solidsgroup/EM525-PS03.git
   

2. Change into the problem directory

     cd EM525-PS03
   

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
   

   Note that the code will not compile initially. You should see the following error message:

     src/main.cpp:4:10: fatal error: Element/CST.H: No such file or directory
   

   To fix this, you must copy your completed src/Element/CST.H from PS02 to the src/Element/ directory.

5. Compile the code again (using make) and run with ./bin/main. The output should indicate that the DiracTest, SumToUnityTest, and EtaDerivativeTest passes for CST (which you implemented before), and that it fails for LST, Q4, and Q9.

Implementation

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

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

For each of these, you must compute the shape functions (eta) and their derivatives (deta), just as you did for the CST element. Note that the shape functions are all given to you in the notes; you only need to implement them and compute their derivatives.

Check result

You will know that the implementation is successful if all of the tests pass. For your reference:

• The "Weak Dirac" test checks that each shape function returns 1 for its associated node and zero for all other nodes. The error message will tell you which shape function / node is problematic.

• The "SumToUnity" test checks that all the shape functions sum to one for points sampled randomly inside the element.

• The "EtaDerivative" checks the exact derivative (deta) against a numerically calculated derivative to ensure consistency.