Circular Piston

Description: Axi-Symmetric Infinite Element Model for Circular Piston

This example is a model of the form

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

with $M$, $E$, and $K$ non-symmetric matrices and $M$ singular. This is thus a differential algebraic equation. It is shown that it has index 1. The input of the system is $B$, 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 $K$, $E$, $M$ 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, see .

Data

Download matrices in the Matrix Market format CircularPiston-dim1e3-piston.tar.gz (1.9 MB).

Dimensions

System structure:

$\begin{align} M\ddot{x}(t) + E\dot{x}(t) + Kx(t) &= B \\ y(t) &= x(t) \end{align}$

System dimensions:

$M \in \mathbb{R}^{2025 \times 2025}$, $E \in \mathbb{R}^{2025 \times 2025}$, $K \in \mathbb{R}^{2025 \times 2025}$, $B \in \mathbb{R}^{2025 \times 1}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Circular Piston. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Circular_Piston

@MISC{morwiki_piston,
  author =       {{The MORwiki Community}},
  title =        {Circular Piston},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {http://modelreduction.org/index.php/Circular_Piston},
  year =         {20XX}
}

• For the background on the benchmark:

@ARTICLE{PinA91,
  author =       {P.M. Pinsky and N.N. Abboud},
  title =        {Finite element solution of the transient exterior structural acoustics problem based on the use of radially asymptotic boundary conditions},
  journal =      {Computer Methods in Applied Mechanics and Engineering},
  volume =       {85},
  pages =        {311--348},
  year =         {1991},
  doi =          {10.1016/0045-7825(91)90101-B}
}