od of Finite Elements II Assignment #1 Method of Finite Elements II Consider a steady state heat transfer problem (i.e. 𝜃 is the temperature) in a one dimensional channel (of a unit cross-­‐sectional are) as a fluid flows through it with prescribed velocity 𝑣 under a temperature gradient 𝑑𝜃
𝑑! 𝜃
𝑣
= 𝑎 ! 0 ≤ 𝑥 ≤ 𝐿 𝑑𝑥
𝑑𝑥
𝜃 𝑥 = 0 = 𝜃! 𝜃 𝑥 = 𝐿 = 𝜃! where 𝛼 is the diffusivity of the fluid. The analytical solution of the problem is given by 𝑃!
𝜃 𝑥 − 𝜃! 𝑒𝑥𝑝 𝐿 𝑥 − 1
=
0 ≤ 𝑥 ≤ 𝐿 𝜃! − 𝜃!
𝑒𝑥𝑝 𝑃! − 1
where P! =
!"
!
is known as the Péclet number. Answer the following questions using the parameters: 𝐿 = 1, 𝜃! = 1, 𝜃! = 0. 1. Plot the behavior of the analytical solution for increasing values of P! and explain what you observe. Tip: For different values of P! , what are the insulating properties of the pipe? 2. Write the Weak form of the problem. 3a. Using Galerkin’s method, formulate the corresponding problem 𝐾𝑑 = 𝑓 and describe the properties of 𝐾 . Reminder: In Galerkin's method the same shape functions are used for the trial and weight function approximation). Extra Credit: Explain how the properties and structure of 𝐾 impact the solution process from a computational aspect. 3b. Given: 𝛼=0.01, 𝑣=1, linear shape functions, and a uniform discretization of the domain, modify the MATLAB code given, to solve this problem and obtain the nodal temperature solutions with a number of nodes 𝑁np=20. Plot the FE and the analytical solution on the same figure and comment on the quality of the solution. Extra Credit: a) Can you think of one way to obtain a better solution (other than the one in 4.). !
b) Assuming the exact shape functions are given as 𝐻 = 1
𝑒 !! , is Gaussian integration still applicable? Explain your reasoning. Note: The main routine is FEMaxialbar.m. Make sure to properly modify the number of used nodes (nnp), the essential (e_bc) and natural (n_bc) boundary conditions in input_file.m and of course the setup of the element stiffness and force matrices based on your derivation for the Galerkin method (question 3a) in elem.m) 4. Using a Petrov-­‐Galerkin method, repeat question 3a and 3b. In a Petrov-­‐Galerkin method, the shape functions used for the trial and weight functions are different and are written given by: 𝑁! 𝑢! 𝑢 𝑥 =
𝑤 𝑥 =
𝑤! 𝑁! + 𝛾
ℎ! 𝑑𝑁!
, 𝑓𝑜𝑟 𝑣 > 0 2 𝑑𝑥
where 𝛾 = 𝑐𝑜𝑡ℎ
!!
!
!
− ! !