 ## Numerical Methods

###  Figure 1: Schematics for Problem 1

A square plate with simply supported edges is subject to an areal load q (Figure 1). The deflection in the z dimension can be determined by solving the partial differential equation: (1)

subject to the boundary conditions that, at the edges, the deflection and slope normal to the boundary are zero. The parameter D is the flexural rigidity, (2)

where E = the modulus of elasticity, Δz = the plate’s thickness, and σ = Poisson’s ratio. If a new variable is defined as

(3)

Equation 1 can be rewritten as (4)

Therefore the problem reduces to successively solving two Poisson equations. First, Equation (4) can be solved for u subject to the boundary condition that u = 0 at the edges. Then, the results can be employed in conjunction with (5)

to solve for z subject to the condition that z = 0 at the edges.

### Questions for Problem 1  代写Numerical Methods

1)Determine whether equation (4) is elliptic, parabolic or hyperbolic. Show your reasoning and calculations.[5 marks]

2)Using the Taylor series expansion, show that the central difference method for the second derivative is second-order accurate. [10 marks]  代写Numerical Methods

3)Use a second-order central difference method to derive the finite difference schemes for the discretisation of all derivative terms for Equations (4) and (5) in the interior of the domain.[10 marks]

4)Write a MATLAB programme to determine the deflections z(x, y) for a square plate subject to a constant areal load, using the finite difference method in question 3. Use the followingdata:

• Length of the plate (both x and y directions), L = 2
• Number of grid points (both x and y directions), N = 81
• Areal load, q = 33.6kN/m2.
• Poisson’s ratio, σ =3.
• Plate’s thickness, Δz = 0.01
• Modulus of elasticity, E = 2 × 1011

Plot the solution (contour plot).[20 marks]

Error-free, clean, well commented and well indented Matlab code will get you additional 5 marks. An efficient, vectorised code with sensible and justified use of user-defined functions can get you a bonus of up to 5 marks (provided that the total marks for Problem 1 do not exceed 50).[5 (+5) Marks]

Important: in order for me to assess whether your answers are supported by your Matlab code when required, please write down detailed instructions on how to run your code to obtain the above answers.

### Problem 2 [50 marks]

We have a cylindrical chemical reactor, as depicted in the following figure: Figure 2: Schematics for Problem 2

A chemical substance is injected into the reactor and it travels along the x axis until it exits the reactor. Assuming the dimensions in the y and z directions are negligible, so that we can consider the concentration uniform in y and z (but not x, of course), we can model the evolution of the concentration in time and space (=along x) using the following equation, obtained from the application of the conservation of mass: (6)

Where C is the concentration, t is time, D a diffusion coefficient, x is the distance along the reactor’s longitudinal axis where x = 0 at the reactor’s inlet, U is the velocity in the x direction and k is a reaction rate whereby the chemical decays to another form.

Use the following data:

• D = 1.7m2/s
• k = 0.0025s-1
• U = 0.02m/s
• Inflow concentration: Cin= 100 mg/m3
• Initial concentration: C0= 0 mg/m3

Note that we haven’t set up a boundary value for the concentration at the outlet, because its value is not known. What we can use as a boundary condition is at the outlet boundary, meaning that the concentration does not change around the final point of the reactor.  代写Numerical Methods

### Questions for Problem 2  代写Numerical Methods

1)Determine whether equation (6) is elliptic, parabolic or hyperbolic. Show your reasoning and calculations.[5 marks]

2)Discretise equation (6) in both the interior of the domain and the boundaries. For the interior nodes, use a forward difference scheme in time, a central difference scheme for the diffusion term, and a backward difference scheme for the advective term; for the Neumann condition at the outlet, use the backward difference scheme.[10 marks]

3)Writea Matlab programme to solve the above  In particular, calculate the solution C(x, t) using the following assumptions:

• Δx = 0.25m
• Δt = 0.01s

#### Let your simulation run until t = 2 minutes. Plot the final concentration profile along x.[12 marks]

4)Nowlet the simulation from question 3 run until the system has reached a steady  Consider the system to have reached a steady state when the change between the concentration values in all the nodes from the previous time step to the current one is less than 0.0001%. Report the time when this happens and plot the evolution in time of the concentration value at x = 5 m.[8 marks]

5)Discretiseequation (6) using the same finite difference schemes in space as in question 2, but this time using an implicit method. Briefly describe what are the advantages and disadvantages of implicit methods compared to explicit methods.[10 marks]  代写Numerical Methods

Error-free, clean, well commented and well indented Matlab code will get you additional 5 marks. An efficient, vectorised code with sensible and justified use of user-defined functions can get you a bonus of up to 5 marks (provided that the total marks for Problem 2 do not exceed 50).[5 (+5) Marks]

Important: in order for me to assess whether your answers are supported by your Matlab code when required, please write down detailed instructions on how to run your code to obtain the above answers.