Ctrl+K

Exercises

2.4. Exercises#

helpful files for the exercises 2d geometries 3d geometries wave equation

Exercise 0#

Draw a unit_cube using the webgui of NGSolve utilizing one of the methods to run NGSolve described here

Exercise 1#

Read Section 2.2 and Section 2.3 (more information can be found in the iFEM tutorial (especially the basic example )) and answer the following:

  • For a given strong formulation of a time-domain wave problem (e.g., (2.1), describe in at least 6 steps how one approximates the solution of said problem using a method of lines approach with finite elements (assuming you have a finite element code like NGSolve readily available).

  • Which of these steps are inherent to the fact that we treat a time-domain problem using the method of lines (compared to finite elements for stationary problems)?

  • Which of these steps are inherent to the fact that we specifically use a finite element method (compared to a general Galerkin method)?

  • Which choices do you have to make in each step?

Exercise 2#

Add a parameter order = n for \(n\in\mathbb N\) in the initialization of the H1 finite element space and look at the resulting basis functions (default order = 1 if the parameter is ommited). Write down at least 4 characterizing properties of the basis functions. Indicate which of this properties characterize the space \(V\) and which specifically the choice of the basis.

Exercise 3#

Create a geometry (two or three-dimensional) and mesh where you would expect interesting acoustic effects (e.g., some (very simplified) musical instrument, …). Simulate the wave problem on your geometry/mesh.

Exercise 4#

Verify that the implementation of the Newmark time-stepping scheme in Section 2.3 is actually consistent with the formulae presented above. How does it have to be modified if \(B\neq 0\) or \(r\neq 0\)?

Change your code to do the following

  • Add time-harmonic source term of the form \(f(t,x)=\cos(\omega t) f_0(x)\) for some \(\omega>0\) and \(f_0:\Omega\to \mathbb R\) with small support.

  • Add a first order absorbing boundary condition to (at least) one of the boundaries.

Exercise 5#

Derive a weak formulation for the 2nd order time domain Maxwell system (1.14). What are the natural, homogeneous boundary conditions for this system? Look at the basis functions of the space HCurl (instead of H1). Use the option vectors = True in the Draw command. How do you think these basis functions are constructed? Simulate a time-domain electromagnetic wave on the unit_cube with initial data of your choice and zero right hand side and homogeneous natural boundary conditions.

Contents