Convective Thermal Flow

Convective Thermal Flow

Background
Benchmark ID	ctfChipCoolingv0_n20082m1q5 ctfChipCoolingv01_n20082m1q5 ctfFlowMeterv0_n9669m1q5 ctfFlowMeterv05_n9669m1q5
Category	oberwolfach
System-Class	LTI-FOS
Parameters
nstates	20082 20082 9669 9669
ninputs	1
noutputs	5
nparameters	0
components	A, B, C
Copyright
License	NA
Creator	Christian Himpe
Editor	Christian Himpe
Location
NA

Description: Convective Thermal Flow Problems

Figure 1: Convective heat flow example: 2D anemometer model

Figure 2: Convective heat flow example: 3D cooling structure

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow. The most elaborate approach to this problem is computational fluid dynamics (CFD). However, CFD is computationally expensive. A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model. However, caution has to be taken to select the film coefficient.

An intermediate level is to include a flow region with a given velocity profile that adds convective transport to the model. Compared to convection boundary conditions this approach has the advantage that the film coefficient does not need to be specified and that information about the heat profile in the flow can be obtained. A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation because the solution accuracy when employing upwind finite element schemes depends on the element size. While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one. So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Two different designs are tested: a 2D model of an anemometer-like structure mainly consisting of a tube and a small heat source (Fig. 1) . The solid model has been generated and meshed in ANSYS. Triangular PLANE55 elements have been used for meshing and discretizing by the finite element method, resulting in 19282 elements and 9710 nodes. The second design is a 3D model of a chip cooled by forced convection (Fig. 2) . In this case, the tetrahedral element type SOLID70 was used, resulting in 107989 elements and 20542 nodes. Since the implementation of the convective term in ANSYS does not allow defining the fluid speed on a per-element basis but on a per-region one, the flow profile has to be approximated by piece-wise step functions. The approximation used for these benchmarks is shown in Fig. 1.

The Dirichlet boundary conditions are applied to the original system. In both models, the reference temperature is set to $300 \, \text{K}$, and Dirichlet boundary conditions as well as initial conditions are set to $0$ with respect to the reference. The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the non-symmetric case (with forced convection). Table 1 shows the output nodes specified for the two benchmarks, Table 2 links the filenames according to the different cases.

Practically, only a few nodes are considered quantities of interest. Hence, a small subset of five nodes is selected as output nodes, which are filtered from the discretized state by a linear transformation.

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 38867, see .

Data

Matrices are in the Matrix Market format. The matrix name is used as an extension of the matrix file. *.C.names contains a list of output names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4fem.

Table 1: Output nodes for the two models.

Model	Number	Code	Comment
Flow Meter	1	out1	outlet position
	2	out2	outlet position
	3	SenL	left sensor position
	4	Heater	within the heater
	5	SenR	right sensor position
Cooling Structure	1	out1	outlet position
	2	out2	outlet position
	3	out3	outlet position
	4	out4	outlet position
	5	Heater	within the heater

Table 2: Provided files.

Model	Fluid Speed (m/s)	Link	Size
Flow Meter	$0$	Convection-dim1e4-flow_meter_model_v0.tgz	649.4 kB
	$0.5$	Convection-dim1e4-flow_meter_model_v0.5.tgz	757.8 kB
Cooling Structure	$0$	Convection-dim1e4-chip_cooling_model_v0.tgz	3.9 MB
	$0.1$	Convection-dim1e4-chip_cooling_model_v0.1.tgz	4.0 MB

Further information on the models can be found in , where model reduction by means of the Arnoldi algorithm is also presented.

Dimensions

System structure:

$\begin{align} E \dot{x}(t) &= A x(t) + B u(t) \\ y(t) &= C x(t) \end{align}$

System dimensions:

$E \in \mathbb{R}^{n \times n}$, $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times 1}$, $C \in \mathbb{R}^{5 \times n}$.

System variants:

flow_v0: $n = 9669$, flow_v0.5: $n = 9669$, chip_v0: $n = 20082$, chip_v0.1: $n = 20082$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Convective Thermal Flow. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Convective_Thermal_Flow

@MISC{morwiki_convection,
  author =       {{The MORwiki Community}},
  title =        {Convective Thermal Flow},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {http://modelreduction.org/index.php/Convection},
  year =         {20XX}
}

• For the background on the benchmark:

@INPROCEEDINGS{morMooRGetal04,
  author =       {C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink},
  title =        {Model Order Reduction for Linear Convective Thermal Flow},
  booktitle =    {Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems},
  year =         {2004},
  url =          {http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf}
}