Nsemf Examination 2017 PDF

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Summary

Numerical Simulation of Electromagnetic Fields...

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Tas k 1: Gen era wle dge on F EM Task Genera erall kno knowle wledge FEM EM.. 1. Explain one advantage and one disadvantage of FEM. 2. Given linear PDE: 𝐿[𝑓(𝒙)] = 𝑔(𝒙). How is the residual 𝑟(𝒙) defined? Which value does it have for the exact solution? Why is it not possible to set it to zero directly with the FEM approach? 3. What is Galerkin’s Method? Name one property of the system matrix arising from it!

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Tas k 2: First Ord er F EM on a 1D Me sh. Task Order FEM Mesh. Goal is to solve ODE

𝑑2

𝑑𝑥2

𝑓(𝑥) = −1 with boundary conditions 𝑓 ′ (𝑥0 ) = −1, 𝑓(𝑥2 ) = 1

on this 1D mesh.

*expertly drawn 1. Solve the problem analytically! 2. Transform the ODE into the weak form. How many testing function are required? __ ≤ 𝑗 ≤ __ 3. How are the tent ansatz functions 𝜑𝑖 (𝑥) and their derivatives 𝜑′𝑖 (𝑥) from the sketch defined? 4. How is the approximated solution 𝑓󰆻 defined? 𝑥 5. Create a table with all relevant values of 𝑎𝑖𝑗 = ∫ 𝑥 1𝜑′𝑖 (𝑥)𝜑′𝑗 (𝑥) 𝑑𝑥 ! 𝑥

0

6. Create a table with all relevant values of 𝑏𝑗 = ∫ 𝑥 1𝜑𝑗 (𝑥) 𝑑𝑥 ! 0

7. Assemble the system of linear equations 𝑨𝒖 = 𝒃 ! (Do not solve it!) 8. In a general way: What is the advantage in solving this problem with the FEM? Why use it on a simple problem like this?

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Task Triangu ngular Mesh esh.. Tas k 3: FEM on a 2D Tria ngu lar M esh

left: reference element with local coordinates right: arbitrary element with global coordinates 1. How do the global coordinates (𝑥, 𝑦) depend on the local coordinates (𝑢, 𝑣) of the reference element in general? 2. What are the coefficients of the coordinate transformation / mapping for the given element with coordinates (𝑥1 , 𝑦1 ) = (1, 2), (𝑥2 , 𝑦2 ) = (4, −1) and (𝑥3 , 𝑦3 ) = (4, 4) ? 3. Why is the coordinate transformation useful regarding the 2D-FEM? (Explain explicitly where the simplification comes into play!) 4. How many tent ansatz functions 𝜑𝑖𝑅 (𝑢, 𝑣) exist on the reference element? How are they defined? What are their values at the reference element’s nodes? 5. The 𝜑𝑖𝑅 (𝑢, 𝑣) are scalar functions. What is the problem in discretising a vector valued equation (i.e. component-wise) with them? Which ansatz functions are used to solve a (electromagnetic) vector value PDE with the FEM?

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