Q.1. Derive the central difference approximation for a 2nd derivative : ∂ 2w ∂z2 ' wj+1 − 2wj + wj−1 ∆z 2 using a Taylor expansion as shown in the lectures. Keep track of the truncation error in your calculation and use this to prove that the expression is 2nd order accurate. Q.2. A steel bar, 70 mm long is struck at one end by a heavy mass travelling at 20 m/s. The impact causes a compression wave to travel along the bar until it reaches the other end, which is kept fixed. The bar has a density of 7900 kg/m3 and a Youngs modulus of 200 GPa. a. What are the boundary conditions at the two ends? You may assume that the heavy mass is a rigid body that does not decelerate when it hits the bar, i.e. the initial velocity of the impacted end is the same as the heavy mass. b. Construct a spreadsheet to solve this problem, using an explicit FD scheme with 6 computational nodes (8 nodes including boundary conditions) and a timestep ∆t = 1 × 10−6 s. Plot the results for the displacement w at times t = 1 × 10−5 s and t = 2 × 10−5 s. Please submit your spreadsheet and graphs (on paper!) – and please use scientific notation and appropriate decimal places where necessary to make it easy to read. c. Replot the results from part b for a timestep of ∆t = 3 × 10−6 s. What is the critical value of CL∆t/∆z for stability? d. Revise the spreadsheet to use 13 computational points (15 including boundaries). Plot the stress at the mid-point of the bar as a function of time. e. Using the 15 node simulation again, plot the velocity of the mid-point of the bar as a function of time. At what time does the bar lose contact with the impacting mass? What boundary condition should you use at the impacted end once this occurs? Q.3. Use the von Neumann stability analysis to find an expression for the error growth factor G for this algorithm (you can leave this as a quadratic). Show that the scheme is unstable for a value of CL∆t/∆z = √ 2

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