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. 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.
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
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
#!/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.⋅
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.
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
This page was updated on Oct 23, 2019 at 11:00 from tutorials/tutorial-matlab-ResonanceODE/README.md.