30.10: Questions


Quick questions

You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!

Which is true? For second order elements:

 a The interpolation between the element nodes is linear b The shape function derivatives sum to 1 c The solution between the nodes can be approximated using a second order polynomial d The solution between the nodes has no dependence on the solution at the nodes

C

Which is true? For first order elements:

 a The shape functions sum to 0 b The shape function derivatives sum to 1 c Field variables vary linearly between the element nodes d The elements contain midside nodes

C

Which is true? The error between an exact and finite element solution will always be reduced by:

 a Decreasing the element density b Increasing the element density c Increasing the element order d Decreasing the element order

B

If your finite element mesh contains 16 nodes, how many degrees of freedom are there in a coupled thermomechanical simulation:

 a 64 b 48 c 24 d 16

A

In a structural mechanics analysis, what type of boundary condition is an applied pressure:

 a Dirichlet b Neumann c Robin d Dirichlet + Neumann

B

Discretise the following function using three equal length elements between$$0 \leq x \leq 6$$. Assume the elements are linear (first order), and calculate $$\phi(x=3.2)$$ using the finite element method. Compare your answer to the exact solution.

$\phi = x(x-3.5)(x+3) + 30$

hint:

FEM Solution (3 first order elements) = 32.4, Exact Solution = 24.05, Error = 34%

Discretise the same function using six equal length elements and find $$\phi(x=3.2)$$ using the finite element method. Compare your answer to the exact solution and to the answer obtained using a three element discretisation.

FEM Solution (6 first order elements) = ,25.6, Exact Solution = 24.05, Error = 6.4%

Discretise the same function using three equal length but QUADRATIC elements. Calculate $$\phi(x=3.2)$$ and compare your answer to the ones obtained previously.

FEM Solution (3 second order elements) = 24.24, Exact Solution = 24.05, Error = 0.8%

Using an equal length, 4-element discretisation of $$f(x)=10 - x^2$$, calculate $$f(x = 0.6)$$ and the error between the finite element and exact solutions.

hint:

FEM Solution (4 first order elements) = 9.525, Exact Solution = 9.64, Error = 1.2%

For the system of springs below, determine the global stiffness matrix.

$K = \begin{pmatrix} k1 & -k1 & 0 & 0\\ -k1 & k1+k2+k3 & -k2 & -k3\\ 0 & -k2 & k2 & 0\\ 0 & -k3 & 0 & k3 \end{pmatrix}$
$\begin{pmatrix} k1 & -k1 & 0 & 0 & 0 & 0\\ -k1 & k1+k2+k3+k4+k5 & -k2 & -k3 & -(k4+k5) & 0\\ 0 & -k2 & k2 & 0 & 0 & 0\\ 0 & -k3 & 0 & k3 & 0 & 0\\ 0 & -(k4+k5) & 0 & 0 & k4+k5+k6 & -k6\\ 0 & 0 & 0 & 0 & -k6 & k6 \end{pmatrix}$