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High Performance Computing

Continuous Assessment 1

Date Set: 9 February 2023

Date Due: 27 February 2023

Return Date: 17 March 2023

This CA comprises 40% of the overall module assessment.

This is an individual exercise and your attention is drawn to the College and University guidelines

on collaboration and plagiarism, which are available from the College website.

This is the coursework for this module and tests your understanding of how to find numerical

solutions to partial di?erential equations, and parallel programming.

ECMM461: CA1

This continuous assessment is for the ECMM461 version of the High Performance

Computing module. If you are a student on an MSc programme you should follow

these instructions. If you are an undergraduate student please refer to the “ECM3446:

Continuous assessment” instructions instead.

1 The Atmospheric Boundary Layer

The part of the Earth’s atmosphere nearest the surface is known as the “Atmospheric Boundary

Layer”, and in this part of the atmosphere the wind speed increases with height. This causes

material emitted from a chimney to be advected at di?erent horizontal speeds depending on its

height. Figure 1 shows material from a chimney being advected by the wind.

Figure 1: Material emitted from a chimney being advected by a horizontal wind

The wind in the Atmospheric Boundary Layer can be approximated as a horizontal velocity

which depends only on height (z) and zero vertical velocity. The horizontal velocity as a function

of height vx(z) can be represented by a logarithmic profile

(1)

where u is a parameter called the “friction velocity”, z0 is a parameter called the “roughness

length”, and = 0.41 is a constant called Von Ka′rma′n’s constant.

2 The assignment

In this assignment you will calculate a numerical solution to the advection equation to simulate

the movement of a cloud of material in the atmospheric boundary layer. There are four parts to

this assignment:

Task 1: For the first part of this assignment you will parallelise a program which calculates a

numerical solution to the 2D advection equation. You have been provided with the program

1

ECMM461: CA1

and you will need to parallelise it using OpenMP. This part of the assignment is described

in section 2.1.

Task 2: For the second part of this assignment you will modify the program to change

the parameters of the test problem into a more realistic configuration. This is described in

section 2.2.

Task 3: For the third part of this assignment you will add a logarithmic velocity profile to

the program. This is described in section 2.3.

Task 4: For the fourth part of the assignment you will calculate the horizontal profile of the

advected material by calculating a vertical average. This is described in section 2.4.

This assignment assesses the following Intended Learning Outcomes:

Demonstrate skills in parallel processing algorithm design and the practical implementation

of such algorithms

Demonstrate an awareness of numerical e?ects and the influence of floating point number

representation in high-performance computing applications.

Interpret an informal requirement specification

Systematically analyse information and make appropriate design choices

2.1 Task 1: Parallelising the program

The advection equation for a two-dimensional scalar field u(x, y) is

(2)

where vx and vy are the velocities in the x and y direction respectively. You have been provided

with a C program which calculates a numerical solution to the advection equation for a two-

dimensional scalar field u(x, y) using finite di?erence approximations. The program uses one-sided

di?erences to calculate the spatial derivatives

(4)

where ui,j is the value of the scalar field u(x, y) at grid point (i, j), dx is the spacing of grid points

in the x direction and dy is the spacing of grid points in the y direction. The solution starts

from specified initial conditions at time t = 0 and is updated from time t to time t + t using

forward-Euler time steps

u(t+t) ? u(t) + @u

t (5)

The program uses 1000 ? 1000 grid points covering a unit square computational domain 0 ?

x ? 1.0 and 0 ? y ? 1.0. The initial conditions are a Gaussian given by the expression

(6)

The Gaussian is centred at x0 = 0.1, y0 = 0.1 and has a width x = y = 0.03. The boundary

conditions are u = 0 on all boundaries. The program calculates 1500 time steps, and the time step

size is calculated from the Courant condition with a CFL number of 0.9. The velocities in the x

and y directions are vx = 0.01 and vy = 0.01 respectively.

The first part of the assignment is to parallelise the program provided using OpenMP. There

are 10 loops or loop nests (a loop over j inside a loop over i) in the program and these have been

identified with a comment of the form

2

ECMM461: CA1

/* LOOP N */

where N is an integer between 1 and 10 inclusive. For each loop or loop nest you should decide

whether it can be correctly parallelised by adding an OpenMP directive:

If the loop can be correctly parallelised then you should parallelise it by adding an OpenMP

directive, ensuring that all variables are correctly scoped. You do not need to scope variables

which have the const qualifier.

If the loop cannot be parallelised then you should add a comment in the program to say that

the loop cannot be parallelised and explain why.

To build the serial program on a Lovelace lab PC use the command

gcc -o advection2D -std=c99 advection2D.c -lm

To build the program on a Lovelace lab PC with OpenMP enabled use the command

gcc -fopenmp -o advection2D -std=c99 advection2D.c -lm

A gnuplot script which plot the final values of u(x, y) has been provided.

The parallelised version of the program must produce exactly the same output as

the original program.

2.2 Task 2: Modifying the calculation

In this section you will modify the example program to change the test problem so that it more

closely resembles a cloud of material emitted from a chimney. The calculation will represent ma-

terial advecting into the computational domain from the left hand side by specifying a boundary

condition which varies with height and varies over time. Modifying the calculation requires chang-

ing the size of the computational domain, the initial conditions, one of the boundary conditions,

the velocities and the number of time steps. In the following instructions distances are measured

in metres (m), time in seconds (s) and velocities in metres per second (m/s).

The test problem uses a computational domain which is a unit square (it covers the range. Change the computational domain so that it covers the range.

The calculation will start with an empty computational domain. Change the initial conditions

so that u(x, y) = 0 everywhere.

Modify the boundary condition at x = 0 (the left boundary) so that it varies with height (y)

and time (t) according to the following equation:

(7)

Set the values of the parameters to y0 = 15.0m, y = 5.0m, t0 = 3.0 s, t = 1.0 s

The material in the atmospheric boundary layer advects horizontally but not vertically. In

this part of the assignment you will test the calculation with a constant horizontal velocity

of 1.0m/s. Change change the horizontal velocity to vx = 1.0m/s and change the vertical

velocity to vy = 0.

Change the maximum number of time steps to 1000 so that the material does not advect out

of the computational domain.

After you have made these changes re-compile and re-run the program. If necessary update

the OpenMP directives to ensure that the program produces identical results with and without

OpenMP.

Now plot the final values of u(x, y). The final values are in the file final.dat. If you

are using the example gnuplot script you will need to change the axis limits to match the new

computational domain size. A plots of the final values of u(x, y) is shown in figure 2 for comparison.

3

ECMM461: CA1

Figure 2: Final values of u(x, y) from the calculation in section 2.2

2.3 Task 3: Adding vertical shear

In this section you will modify the program so that the horizontal velocity varies with height

according to the logarithmic profile given in equation 1. The logarithmic profile should only be

used if z is greater than the roughness length z0. If z > z0 calculate the horizontal velocity using

equation 1, and if z ? z0 set the horizontal velocity to zero. The parameters of the profile should

be set to u? = 0.1m/s, z0 = 1.0m and ? = 0.41.

In the program the variable y represents the height z in equation 1. The computational domain,

the initial conditions and the number of time steps should be kept at the values specified in the

previous section (section 2.2).

Modify the horizontal velocity as described above and re-run the program. If necessary update

the OpenMP directives to ensure that the program produces identical results with and without

OpenMP.

Now plot the final values of u(x, y). Plots of the final values of u(x, y) are shown in figure 3.

Figure 3: Final values of u(x, y) from the calculation in section 2.3

4

ECMM461: CA1

2.4 Task 4: Calculating the vertically averaged distribution

In this last section you should calculate the vertically averaged distribution of u(x, y) at the end

of the calculation in section 2.3. The vertical average should cover the whole vertical domain but

should not include the boundary values. The code you add to the program does not need to be

parallelised. Plot the vertically averaged values of u as a function of x using a line

graph.

3 Deliverables

The deliverables for this assignment are the source code for the program with your modifications,

and the three plots showing your results:

1. Source code for the final version of the program. The program should include the OpenMP

directives added in section 2.1 with any changes required to ensure the modified program

continues to work correctly. The computational domain, the initial conditions, the velocities

and the number of time steps should be set to the values used in section 2.3.

2. Three plots showing the output from running the program. The required plots are:

A plot of the final values of u(x, y) from section 2.2

A plot of the final values of u(x, y) from section 2.3

A plot showing the vertically averaged profile of u from section 2.4

The plots may be generated using the gnuplot script provided, or alternative plotting software

of your choice provided the data values in the plot are clearly represented. Your plots should

be delivered as PNG or PDF files.

The deliverables should uploaded to BART a single zip or tar file containing the

source code for the final version of the program, and the four plots. The deadline for

submission is 12 noon 27 February 2023.

4 Mark scheme

A total of 100 marks are available for this assignment:

1. Task 1: Parallelising the program (30 marks)

(a) For each of the 10 loops or loop nests identified in the program there are two marks to

be awarded according to the following criteria:

The loop has been correctly identified as either one that can be parallelised or one

that cannot be parallelised (1 mark)

If the loop can be parallelised then an appropriate OpenMP directive has been

added which unambiguously and correctly scopes all variables. Variables with the

const qualifier do not need to be scoped (1 mark)

OR

If the loop cannot be parallelised then a comment has been added with a correct

explanation of why the loop cannot be parallelised (1 mark).

(b) An additional 10 marks are awarded if the parallelised program produces results which

exactly match the original serial program.

2. Task 2: Modifying the calculation (20 marks)

15 marks are awarded for correctly modifying the computational domain, the initial con-

ditions, the boundary condition at x = 0 and the number of time steps, as specified in

section 2.2. 5 marks are awarded for the plot of the final results.

5

ECMM461: CA1

3. Task 3: Adding vertical shear (25 marks)

20 marks are awarded for correctly modifying the velocity profile as specified in section 2.3.

5 marks are awarded for the plot of the final results.

4. Task 4: Calculating the vertically averaged distribution (25 marks)

20 marks are awarded for correctly modifying the program to calculate the vertically averaged

distribution of u, as specified in section 2.4. 5 marks are awarded for the plot of the final

results.

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