Pressure Dynamics in 1D Ducts - Solving Wave Equations by Discretized PDEs

Currently Modelica cannot handle partial differential equations directly since there is only the notion of differentiation with respect to time built into the language. However, in many cases derivatives with respect to other variables such as for example spatial dimensions can be handled by simple discretizations schemes represented using the array capabilities in Modelica. Here we will give an example of how the one dimensional wave equation can be represented in Modelica and how OpenModelica can be used for simulation and display of the results, as well as representing the result as a 2D interpolating function.

1 WaveEquationSample Model

loadModel(Modelica)
model WaveEquationSample import Modelica.SIunits; parameter SIunits.Length L = 10 "Length of duct"; parameter Integer n = 30 "Number of sections"; parameter SIunits.Length dL = L/n "Section length"; parameter SIunits.Velocity c = 1; SIunits.Pressure[n] p(start = initialPressure(n), fixed = {i > 1 and i < n for i in 1:n}); Real[n] dp(start = fill(0,n), fixed = {i > 1 and i < n for i in 1:n}); equation p[1] = exp(-(-L/2)^2); p[n] = exp(-(L/2)^2); dp = der(p); for i in 2:n-1 loop der(dp[i]) = c^2 * (p[i+1] - 2 * p[i] + p[i-1]) / dL^2; end for; end WaveEquationSample;

Here we are using a Modelica function initialPressure (defined further below) to specify the initial value of the pressure along the duct. The initial pressure profile of the duct.The function initialPressure returns the initial pressure profile function in discretized form as an array.

function initialPressure input Integer n; output Real p[n]; protected parameter Modelica.SIunits.Length L = 10; algorithm for i in 1:n loop p[i] := exp(-(-L/2 + (i - 1) / (n - 1) * L)^2); end for; end initialPressure;

2 Simulation of WaveEquationSample

We simulate the WaveEquationSample model...

simulate( WaveEquationSample, stopTime=10 )
plot({p[2], p[1],p[3]})