FEM Coding Assignment: Elastic strain energy density,and element energy, and derivatives

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-PS05.git
   

2. Change into the problem directory

     cd EM525-PS05
   

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. These will disappear once you have implemented the assignment correctly.

Notational comment

You will notice that matrices here are called Set::Matrix instead of Eigen::Matrix2d. These two forms are equivalent and interchangeable, but I am going to try to stick to consistent use of Set::Matrix rather than Eigen::Matrix2d in the future.

Similarly, Set::Scalar is identical to double and can be used interchangeably.

We also introduce here Set::Tensor4 which is a fourth-order tensor object. You can access elements similarly to how you access elements for Set::Matrix.

Part 1: Copying previous implementation of LST, CST, Q4, Q9

Copy and paste your previous implementation for the following functions where indicated by "TODO"

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

(For future assignments, you will be able to simply copy/paste the files.)

If this passes successfully, you will see the usual "test.element..." tests pass (except for the "energyderivative" tests)

Part 2: Implement isotropic elasticity

You will find TODOs in the following file

• src/Model/Isotropic.H

In this element, implement the calculation of the isotropic elastic energy density and its derivatives. See Problem Set 4 Problem 2 for the equations that you need to implement. Once complete, you should see the "model.isotropic.derivative" test pass.

Part 3: Implement element energy and derivatives

You will find TODOs in the following file

• src/Element/Element.H

In this element, implement the calculation of the element's energy (W) and its derivatives (DW, DDW). See Problem 1 of this problem set for the relevant equations.

Note: for this assignment, you can assume that the element is not an isoparametric element. This means that X = Y, and you do not need to include Jacobian corrections. We will introduce these corrections in a subsequent assignment.