Overview

Parameter sweeps are efficiently handled in an high throughput computing environment. To keep things simple, we sweep a single parameter in a simple model. The model is driven-damped harmonic oscillator and is based on ordinary differential equation (ODE). The ODE is solved using MATLAB. For the driven-damped harmonic oscillator, the resonance is set when the applied frequency is equal to the natural frequency. Under the resonance condition, the oscillator vibrates with large amplitude. We sweep the applied frequency and find out when the oscillator has a maximum amplitude.

fig 1

Fig.1. The amplitude of driven-damped oscillator as a function of driving frequency. The amplitude is normalized with respect to the initial displacement and the driving frequency is normalized with respect to the natural frequency of the oscillator.

Tutorial files

It is easiest to start with the tutorial command. In the command prompt, type

 $ tutorial matlab-ResonanceODE # Copies input and script files to the directory tutorial-matlab-ResonanceODE.

This will create a directory tutorial-matlab-ResonanceODE. Inside the directory, you will see the following files

driv_damp_osc.m             # matlab script - solves second order ODE for driven-damped harmonic oscillator
driv_damp_osc               # compiled executable binary of driv_damp_osc.m
driv_damp_osc.submit        # condor job description file
driv_damp_osc.sh            # condor execution script
Log/                        # Directory to copy the standard output, error and log files from condor jobs. 
post-script.bash            # script used to gather the output data after completing condor jobs

MATLAB script - Damped, Driven Harmonic Oscillator

The matlab script takes one argument fnumber. The argument fnumber is used to label the output file. The matlab script solves the ODE using the in-built ODE45 solver. The ODE45 solves non-stiff ODE's based on Runge-Kutta formula.

%  Damped, driven harmonic oscillator: nonstiff ODE

function dd_oscillator(fnumber)

    % Set up the initial condition
    omega = 1;      % natural frequency = sqrt(k/m)
    b = 0.3;        % drag coefficient s
    m = 1.0;        % mass⋅
    x0 = 1.0;       % initial position
    v0 = 1.0;       % initial velocity
    F0 = 1.0;       % strength of applied frequency⋅
    tBegin = 0;     % time begin
    tEnd = 80;      % time end

    % Applied frequency is chosen from random generator⋅
    rng('shuffle');
    omega_app = rand(1)*2.5;   % driving frequency


    % Use Runge-Kutta integrator to solve the ODE
    [t,w] = ode45(@derivatives, [tBegin tEnd], [x0 v0]);
    x = w(:,1);     % extract positions from first column of w matrix
    v = w(:,2);     % extract velocities from second column of w matrix

    xmax_norm = max(x)/x0   %normalize the displacement⋅
    omega_app_norm = omega_app/omega  %normalize the applied frequency


    % Write the outputs on a file⋅
    filenumber = num2str(fnumber);
    outfilename = sprintf ( '%s%s%s', 'XmaxFreq', filenumber, '.dat' );
    fileID = fopen(outfilename,'w');
    fprintf(fileID,'xmax_norm= %9.3f  omega_app_norm= %9.3f\n', xmax_norm, omega_app_norm);
    fclose(fileID);

    % Function to compute the  derivatives of dx/dt and dv/dt
    % The parameters m, b, F0, omega_app are from the main program⋅
    function derivs = derivatives(tf,wf)
        xf = wf(1);            % wf(1) stores x
        vf = wf(2);            % wf(2) stores v
        dxdt = vf;                                     % set dx/dt = velocity
        dvdt = - m*xf - b * vf + F0*cos(omega_app*tf);  % set dv/dt = acceleration
        derivs = [dxdt; dvdt];  % return the derivatives
    end

end

In this MATLAB script, the applied frequency omega_app is a random number that falls between 0 and 2.5. Whenever omega_app = omega, the resonance is set.

MATLAB runtime execution

As mentioned in the lesson on basics of MATLAB compilation, we need to compile the matlab script on a machine with license. At present, we don't have license for matlab on OSG conect. On a machine with matlab license, invoke the compiler mcc. We turn off all graphical options (-nodisplay), disable Java (-nojvm), and instruct MATLAB to run this program as a single-threaded application (-singleCompThread). The flag -m means c language translation during compilation.

mcc -m -R -singleCompThread -R -nodisplay -R -nojvm driv_damp_osc.m

would produce the files: driv_damp_osc, run_driv_damp_osc.sh, mccExcludedFiles.log and readme.txt. The file driv_damp_osc is the compiled binary file that we run on OSG Connect.

Job execution and submission files

Let us take a look at driv_damp_osc.submit file:

Universe = vanilla                          # the job universe is "vanilla"

Executable =  driv_damp_osc.sh    # Job execution file which is transferred to worker machine
Arguments = $(Process)   # list of arguments: process ID used to label the output filename.
transfer_input_files = wigner_distribution  # list of file(s) need be transferred to the remote worker machine

Output = Log/job.$(Process).out⋅            # standard output 
Error =  Log/job.$(Process).err             # standard error
Log =    Log/job.$(Process).log             # log information about job execution

requirements =  Arch == "X86_64" && HAS_MODULES == True  # Check if the worker machine has CVMFS

queue 100                                   # Submit 100  jobs

The above job description instructs condor to submit 100 jobs. Each job would find the response of the oscillator with a random applied frequency. The executable is a wrapper⋅ script driv_damp_osc

#!/bin/bash
source /cvmfs/oasis.opensciencegrid.org/osg/modules/lmod/current/init/bash
module load matlab/2014b
chmod +x driv_damp_osc
./driv_damp_osc $1

that loads the module matlab/2014b and executes the MATLAB compiled binary driv_damp_osc. The only required argument is a numerical⋅ label attached with the name of the output file.⋅

Job submission

We submit the job using condor_submit command as follows

$ condor_submit driv_damp_osc.submit  # Submit the condor job

Now you have submitted 100 MATLAB jobs that solves ODE for randomly generated applied frequency. The present job should be finished quickly (less than two hours). You can check the status of the submitted job by using the condor_q command as follows

$ condor_q username  # The status of the job is printed on the screen. Here, username is your login name.

Each job produce XmaxFreq$(Process).dat file, where $(Process) is the process ID runs from 0 to 99. Each output file contains the amplitude and the applied frequency.

Post process

After all jobs finished, we want to gather the output data. The script post-script.bash gathers the output values in a file all_XmaxFreq.dat and generates the figure all_XmaxFreq.png. The plotting package gnuplot is used to produce the figure. To get the plot from the output data, type

$ post-script.bash

Getting Help

For assistance or questions, please email the OSG User Support team at user-support@opensciencegrid.org or visit the help desk and community forums.

 

This page was updated on Dec 19, 2018 at 08:02 from tutorials/tutorial-matlab-ResonanceODE/README.md.