Circular Piston

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Description: Axi-Symmetric Infinite Element Model for Circular Piston

This example is a model of the form

$M\ddot{x}(t) + E\dot{x}(t) + Kx(t) = f,$

with $M$, $C$, and $K$ non-symmetric matrices and $M$ singular. This is thus a differential algebraic equation. It is shown that it has index . The input of the system is $f$, the output is the state vector $x$. The motivation for using model reduction for this type of problems is the reduction of the computation time of a simulation.

This is an example from an acoustic radiation problem discussed in . Consider a circular piston subtending a polar angle $0<\theta<\theta_p$ on a submerged massless and rigid sphere of radius $\delta$. The piston vibrates harmonically with a uniform radial acceleration. The surrounding acoustic domain is unbounded and is characterized by its density $\rho$ and sound speed $c$.

We denote by $p$ and $a_r$ the prescribed pressure and normal acceleration respectively. In order to have a steady state solution $\tilde{p}(r,\theta,t)$ verifying

$\tilde{p}(r,\theta,t) = \mathcal{R} e (p(r,\theta) e^{i\omega t})$

the transient boundary condition is chosen as:

$a_r = \frac{-1}{\rho} \frac{\partial p(r,\theta)}{\partial r} \big|_{r=a} = \begin{cases} a_0 \sin(\omega t) & 0<\theta<\theta_p \\ 0 & \theta>\theta_p \end{cases}$

The axi-symmetric discrete finite-infinite element model relies on a mesh of linear quadrangle finite elements for the inner domain (region between spherical surfaces $r=\delta$ and $r=1.5\delta$). The numbers of divisions along radial and circumferential directions are $5$ and $80$, respectively. The outer domain relies on conjugated infinite elements of order $5$. For this example we used $\delta=1 [m]$, $\rho=1.225 [kg/m^3]$, $c=340 [m/s]$, $a_0 = 0.001 [m/s^2]$ and $\omega = 1000 [rad/s]$.

The matrices $K$, $C$, $M$ and the right-hand side $f$ are computed by Free Field Technologies. The dimension of the second-order system is $N=2025$.

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 38890.

Data

Download matrices in the Matrix Market format piston.tar.gz (1.9 MB).