Anemometer

Anemometer

Background
Benchmark ID	anemometer1Param_n29008m1q1 anemometer3Param_n29008m1q1
Category	misc
System-Class	AP-LTI-FOS
Parameters
nstates	29008
ninputs	1
noutputs	1
nparameters	2 5
components	A, B, C, E
Copyright
License	NA
Creator	Lihong Feng
Editor	Lihong Feng Christian Himpe Norman Lang Ulrike Baur Judith Schneider Kathryn Lund
Location
NA

Description

Figure 1: Schematic 2D-Model-Anemometer

Figure 2: Calculated temperature profile for the Anemometer function

An Anemometer (see thermal mass flow meter) is a flow sensing device, consisting of a heater and temperature sensors before and after the heater, placed either directly in the flow or in its vicinity Fig. 1. They are located on a membrane to minimize heat dissipation through the structure. Without any flow, the heat dissipates symmetrically into the fluid. This symmetry is disturbed if a flow is applied to the fluid, which leads to a convection on the temperature field and therefore to a difference between the temperature sensors (see Fig. 2) from which the fluid velocity can be determined.

The physical model can be expressed by the convection-diffusion partial differential equation :

$\rho c \frac{\partial T}{\partial t} = \nabla \cdot (\kappa \nabla T) - \rho c v \nabla T + \dot{q},$

where $\rho$ denotes the mass density, $c$ is the specific heat capacity, $\kappa$ is the thermal conductivity, $v$ is the fluid velocity, $T$ is the temperature, and $\dot q$ is the heat flow into the system caused by the heater.

The solid model has been generated and meshed in ANSYS. Triangular PLANE55 elements have been used for the finite element discretization. The order of the system is $n = 29008$.

Example with one parameter:

The $n$ dimensional ODE system has the following transfer function

$G(s, p) = C (s E - A_1 - p (A_2 - A_1))^{-1} B$

with the fluid velocity $p(=v)$ as single parameter. Here $E$ is the heat capacitance matrix, $B$ is the load vector which is derived from separating the spatial and temporal variables in $\dot{q}$ and the FEM discretization w.r.t. the spatial variables. $A_i$ are the stiffness matrices with $i=1$ for pure diffusion and $i=2$ for diffusion and convection. Thus, for obtaining pure convection you have to compute $A_2 - A_1$.

Example with three parameters:

Here, all fluid properties are identified as parameters. Thus, we consider the following transfer function

$G(s, p_0, p_1, p_2) = C ( s \underbrace{(E_s + p_0 E_f)}_{E(p_0)} - \underbrace{(A_{d,s} + p_1 A_{d,f} + p_2 A_c)}_{A(p_1,p_2)} )^{-1} B$

with parameters $p_0$, $p_1$, $p_2$ which are combinations of the original fluid parameters $\rho$, $c$, $\kappa$, $v$: $p_0 = \rho c$, $p_1 = \kappa$, and $p_2 = \rho c v$, see . So far, we have considered the mass density as fixed, i.e. $\rho=1$.

Origin

• IMTEK Freiburg, Simulation group, Prof Dr Jan G. Korvink has taken on a position as Director of the Institute of Microstructure Technology (IMT) at the Karlsruhe Institute of Technology (KIT).

Data

Matrices are in the Matrix Market format. All matrices (for the one parameter system and for the three parameter case) can be found and uploaded in Anemometer.tar.gz. The matrix name is used as an extension of the matrix file. The system matrices have been extracted from ANSYS models by means of mor4fem.

For more information about computing the system matrices, the choice of the output, applying the permutation, please look into the readme file.

Example with one parameter:

• .B: load vector
• .E: heat capacitance matrix
• .P: permutation matrix
• .A: stiffness matrices (2)

Example with three parameters:

• .B: load vector
• .E: heat capacitance matrices (2)
• .A: stiffness matrices (5)

To test the quality of the reduced order systems, harmonic simulations as well as transient step responses could be computed, see .

The output matrix $C \in \mathbb{R}^{1 \times 29008}$ is a vector with non-zero elements $C_{173} = 1$ and $C_{133} = -1$.

Dimensions

System structure (1 parameter):

$\begin{align} E \dot{x}(t) &= (A_1 + p (A_2 - A_1)) x(t) + B u(t) \\ y(t) &= Cx(t) \end{align}$

System dimensions:

$E \in \mathbb{R}^{29008 \times 29008}$, $A_{1,2} \in \mathbb{R}^{29008 \times 29008}$, $B \in \mathbb{R}^{29008 \times 1}$, $C \in \mathbb{R}^{1 \times 29008}$.

System structure (3 parameter):

$\begin{align} (E_1 + p_0 (E_2 - E_1)) \dot{x}(t) &= (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1)) x(t) + B u(t) \\ y(t) &= Cx(t) \end{align}$

System dimensions:

$E_{1,2} \in \mathbb{R}^{29008 \times 29008}$, $A_{1,2,3,4,5} \in \mathbb{R}^{29008 \times 29008}$, $B \in \mathbb{R}^{29008 \times 1}$, $C \in \mathbb{R}^{1 \times 29008}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

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

@MISC{morwiki_anemom,
  author =       {{The MORwiki Community}},
  title =        {Anemometer},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/index.php/Anemometer},
  year =         {2018}
}

• For the background on the benchmark:

References

a) About the Anemometer

b) MOR for non-parametrized Anemometer

c) MOR for parametrized Anemometer

Contact

Christian Himpe