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We present a straightforward method to solve gas damping problems for perforated structures in two dimensions (2D) utilising a Perforation Profile Reynolds (PPR) solver. The PPR equation is an extended Reynolds equation that includes additional terms modelling the leakage flow through the perforations, and variable diffusivity and compressibility profiles. The solution method consists of two phases: 1) determination of the specific admittance profile and relative diffusivity (and relative compressibility) profiles due to the perforation, and 2) solution of the PPR equation with a FEM solver in 2D. Rarefied gas corrections in the slip-flow region are also included. Analytic profiles for circular and square holes with slip conditions are presented in the paper. To verify the method, square perforated dampers with 16–64 holes were simulated with a three-dimensional (3D) Navier-Stokes solver, a homogenised extended Reynolds solver, and a 2D PPR solver. Cases for both translational (in normal to the surfaces) and torsional motion were simulated. The presented method extends the region of accurate simulation of perforated structures to cases where the homogenisation method is inaccurate and the full 3D Navier-Stokes simulation is too time-consuming.

Perforations are used in oscillating microelectromechanical (MEM) sensors and actuators to control damping due to gas for desired frequency-domain operation. Practical MEM devices using perforations include microphones [

The perforations reduce damping in a squeeze-film structure considerably: the characteristic dimensions are reduced from the outer dimensions of the surface to the separation of the perforations. Also, the cut-off frequency (when the spring and damping forces are equal) rises considerably due to the perforations. Often this cut-off frequency in perforated structures is much higher than the operating frequencies of interest, which simplifies the design of such devices since only the damping forces need to be considered in the device design.

3D flow simulations are needed in solving perforation problems generally. However, accurate 3D simulation of the Navier-Stokes (N-S) equation is, in practice, impossible due to the huge number of elements needed, especially in cases where the number of perforations is relatively large. The number of perforations may vary from a few holes to a grid of thousands of holes. Moreover, the N-S equations are limited to the region of a slightly rarefied gas (slip-flow region). Accurate modelling of these structures requires the consideration of rarefied gas flow in tiny flow channels and around the structure.

Different methods that reduce the 3D flow problem into 2D are available, depending on the boundary conditions, number of perforations, the relative hole size _{0}/_{0} is the hole diameter and _{0}/_{0}/

If the _{0}/

The models referenced above assume motion normal to the surfaces, and the squeezing force in the air gap makes the gas flow. A different problem where the perforated surface moves tangentially to the surfaces has been discussed in [

We have presented a Perforation Profile Reynolds (PPR) method [

In this paper, we present improvements to this method. Flow admittance profiles that vary across the perforations are used instead of constant profiles. Also, an improved perforation cell model [

The method was verified with 3D FEM simulations using a N-S solver. Simulations of a parallel-plate damper with 16 to 64 square perforations were performed for various perforation ratios. Only incompressible flow is considered here, since time-harmonic reference N-S simulation with sufficient accuracy is not realistic.

Gas lubrication and damping problems in small air gaps are traditionally modelled with the Reynolds equation [_{ch} is the relative flow rate, _{A} is the ambient pressure, _{z}(^{jωt} is the surface velocity in the ^{jωt} is the pressure variation to be solved from the equation. This form allows the static displacement _{ch}(

We extend the Reynolds equation by adding a term that models an additional flow component through the perforations similarly as in [

_{h}(_{h}(_{h}(

The perforation admittance _{h} is specified as an inverse of the specific acoustic impedance _{S}:
_{h} and _{h} are the gas pressure and velocity in the _{h}, _{S}, _{h}, and _{h}.

The approximation for the relative flow rate due to the rarefied gas effects is [_{n,ch}= λ/_{A}). The relative accuracy is valid for _{n,ch} < 880 and its maximum relative error is less than 5%. When the slip correction is considered, as in this paper,
_{n,ch} < 0.1, whereas the approximation in

The PPR method consists of two phases. First, diffusivity, compressibility, and perforation admittance profiles in _{h}(_{h} includes the flow resistance of the perforation cell, that is, the flow in the air gap and the perforations.

In [_{h} is the velocity profile and _{h} is the pressure.

The diffusivity of the air gap also effectively changes at the perforation. For the flow passing in the air gap below the perforation, the frictional surface area will be reduced, effectively increasing the diffusivity coefficient. On the other hand, the flow profile is changed due to the perforations, effectively decreasing the diffusivity coefficient _{h}. The compressibility coefficient _{h} is zero at the perforations and unity elsewhere.

There are two alternate ways to determine the flow admittance profiles for perforations: 3D FEM (or other discretising methods) simulations and analytic (approximate) equations. Usually, the perforations are relatively short and the analytic model for long channels are not applicable. Here, flow resistances extracted from FEM simulations of the perforation cells are used [

The flow fields above the perforated surface are coupled, especially when the perforation ratio is large. In [_{z}(

Next, analytic expressions for two typical perforations, circular and square, are given (See

For the flow resistance of the perforation we utilise the perforation cell model [_{c}) with a circular cross-section (with a radius of _{0}) in the slip-flow region is [_{IC}, _{C} and _{E} are given in _{C} is the mechanical resistance of a long channel with a circular cross-section. In the slip-flow region, the relative flow rate of a channel with a circular cross-section _{tb} is
_{n,tb} is the Knudsen number of the capillary

The normalised velocity profile (average velocity is 1) in a long circular capillary is a function of

The specific admittance profile resulting from

Mechanical flow resistance of a long square channel is [_{sq} is the flow rate coefficient and _{c}_{sq} using the elongation model for a circular channel [_{0} is needed. It is specified such that the acoustic resistances (hydraulic resistances) of the square and circular long channels are equal. That is
_{0}:
_{tb} is a function of _{0}, the accurate solution of _{0} on the Knudsen number is small, and the simple approximation in _{n,sl} < 0.14.

In the homogenisation method, the flow admittane _{h} caused by the perforations is assumed to be smoothly distributed over the whole surface. In the actual 2D FEM simulation, the holes are excluded from the simulated structure and their flow admittance is included in the extended Reynolds equation. In the case of uniform perforation, this flow admittance is constant over the whole surface. Damping models using the homogenization approach have been published, but not properly verified [

The PPR _{h}. In addition to the flow admittance of the perforations itself, Yh should now include the flow component in the air gap.

In deriving the model for _{h}, the concept of a “perforation cell” is applied here. Relative simple analytic models for a cylindrical and square perforation is derived in [_{x} and _{0} are the outer and inner radii of the cylindrical perforation cell, respectively. _{S} is the flow resistance in the air gap and _{C} is the flow resistance of the perforation. All other resistances model the flow in the intermediate region under and above the perforation. Values for these resistances are given in

Radius _{x} isThe perforation pitch is selected such that the bottom areas of circular and square cells match. This results in

In the case of uniform perforation and translational motion, the specific impedance spread all over the surface is
_{eff} = _{n,ch}) is the effective dimension of the surface.

Several perforated dampers in perpendicular and rotating motion are simulated both with the 3D N-S solver with slip conditions and with the time-harmonic PPR solver. Both solvers are implemented in the multiphysical simulation software Elmer [

The surface is a square, and it consists of _{1}, as is the distance from the edges of the holes to the edges of the damper _{2} = _{1}. The perforation pitch is _{x} = _{0} + _{1}. A comparison is made well below the cut-off frequency, that is, the compressibility of the gas is ignored (_{h} = 0). The simplifies the N-S simulations, since response to a constant velocity is calculated. The dimensions are summarised in

First, full 3D N-S simulations of the structure are performed to generate reference data for the methods presented in this paper. The simulations can be performed relatively reliably, thanks to the small number of holes and the symmetry of the structure. Slip boundary conditions are used for the surfaces. The simulated gas volume is extended around the damper: free space around and above the damper are 6_{0}, respectively. A mesh of 450 000 elements is used. Both translational and torsional motion cases were simulated.

The number of elements (450 000) in the 3D N-S simulations was in practice the maximum that could be performed considering the computing time and the memory consumption. The computer used was Sun Fire 25K. Simulating each (of 432) topology took between one and two hours and 6 Gb of memory.

To estimate the accuracy of the results, simulations with 250 000 elements were made, too. Comparative results are discussed later in this paper.

The PPR _{0} is used. The PPR solver also includes the compressibility of the gas in the air gap. The compressibility of the perforations can also be included specifying a complex-valued specific admittance. This is essential in modelling, e.g., closed perforations. Compressibility profile is not needed here, since steady flow calculations are made.

We use _{h} = 1 since there is no model available for the diffusivity profile. The friction surface for the flow under the hole is reduced by half, and this could justify setting _{h} = 2. However, the changes in the flow profile will introduce additional losses that justify decreasing the diffusivity.

Considering the edge effects at the outer borders of the structure is essential for accurate results. Each border of the structure is extended by an amount 0.65_{n,ch}), as suggested in [

Both translational and torsional motion cases are simulated. Each of the simulations with the Reynolds solver with 20 000 elements took about 10 seconds.

First, the structure is set into translational motion with a velocity of _{z} = 1 m/s. The mechanical resistance, or the damping coefficient is _{m} = _{z}, where

The damping coefficients are compared with the ones from the N-S simulations as a function of the perforation ratio

Next, the structure is set into torsional motion about the _{z}_{t} = _{z}, where _{z} = _{z}/(

The extended Reynolds equation is solved for perpendicularly moving rectangular surfaces. The perforation cell admittance calculated from _{h} = 1 is used. Each damper border is extended by 0.65_{n,ch}). The quarter of the rectangular surface is meshed having 5000 elements.

_{c}

In the case of torsional motion,

To analyse the importance of the varying pressure profile at the perforations, a set of PPR simulations was performed using a constant specific admittance _{ci} of the perforations. The results are summarised in

The maximum relative errors of the homogenisation method (

The verification in this paper is limited to the slip-flow region just to make the comparison with 3D solutions possible. The PPR method itself is valid for arbitrary gas rarefaction as long as the model for _{h} is valid. The comparison is made in the region where the validity of the slip conditions is questionable (_{n,sl} < 0.14, _{n,ch} < 0.14). In spite of the small additional error in the results, the comparison is justified since slip flow models are used in both cases. The consideration of the slip correction is essential for accurate results for the micromechanical structures simulated. If the rarefied gas effects are ignored (_{n,sl} = _{n,ch} = 0), the error in the damping force would increase by about 40 % and 15 % for air gaps of 1

In squeeze-film dampers, the compressibility of the gas will change the damping coefficient at higher frequencies. The cut-off frequency specifies a frequency when the viscous and compressibility (spring-) forces are equal. For perforated dampers, the small-frequency assumption made in this paper is generally a valid, since the perforation increases the cut-off frequency considerably.

The cut-off frequency due to compressibility can be estimated by calculating the ratio between the viscous and compressibility forces in a perforation cell [

There is no limitation in the method that prevents compressibility from being accounted for. In this case, a complex-valued specific admittance profile and compressibility profile _{h} are both needed for the PPR solver. For verifying this case a 3D linearised time-harmonic N-S solver may be used.

The 3D simulations were very challenging. A lot of computing resources were used to have the data, still the results are not perfect. To estimate the error in the simulations with 450 000 elements, a comparison is made to simulations with 250 000 elements. The results are shown in

When the perforation ratio is large, the relative difference is the largest. This indicates that the selection of the meshing was not very good for large perforations.

The comparison shows that damping coefficients with 250 000 elements are generally larger than with 450 000 elements. It is then expected that the damping coefficients become smaller when the element count is increased. When considering such fictitious simulations and studying the maximum relative errors of the PPR method in

A straightforward method solving perforation problems with a 2D PPR solver was presented. Damping coefficients of a large number of perforated dampers were solved with the PPR method and they were compared with 3D N-S simulation results with very good agreement. Compared to direct N-S simulations, the PPR method offers orders of magnitudes faster simulation times and less memory consumption. The maximum relative errors for translational motion for

Perforations were modelled with their specific admittance profile. Both, constant and spatially varying admittance profiles at the perforations were used. It was shown that the errors were smaller when a varying admittance profile was used. Uniform diffusivity profile (_{h} = 1) was assumed. A further study is needed to resolve if a non-trivial profile could be extracted form FEM simulations, and if the utilization of this profile would improve the accuracy of the model.

The PPR method was also compared with the homogenisation method. The results were almost as good (maximum relative error about 10%) for

The square perforations were approximated with a model for a cylindrical perforation cell. It is expected that a model for a square cell [

The fringe flow effects due to the sidewalls and the upper surface clearly contributed the damping coefficient for small _{0}/

Incompressible flow was assumed. For a squeeze-film damper this sounds like a very limiting assumption. But due to the perforations, the cut-off frequency is significantly higher than in the non-perforated case. Thus the valid frequency range of the incompressible damper model is relatively wide. Further increase of the frequency range is not trivial, since, besides the compressibility effects, the inertia of the gas should be considered. Also, the thermal aspects should be considered since isothermal assumptions will not be sufficient due to the high cut-off frequencies.

The method needs to be verified in cases where the gas compressibility and inertia are considered. However, this is not easy, since the time-harmonic analysis of the 3D structure with a sufficient accuracy will be very challenging. Extraction and usage of non-uniform perforation profiles and coupling between perforations will be studied in the future.

_{p} of the perforation cell consists of lumped flow resistances and their effective elongations. The equations have been derived partly analytically and partly by fitting the model to FEM simulations by varying the coefficients in heuristic equations [

The mechanical resistance of the perforation cell is [_{x} and _{0} are the outer and inner radii of the cylindrical perforation cell, respectively.

The lumped resistances are

The effective elongations in the previous equations are:

General geometry of a perforated gas damper.

a) Topology of a single perforation in 2D and b) reduced profiles in 1D for

Cross-sections and dimensions of a) circular and b) square perforations.

Structure of simulated dampers. Topology with _{2} is set equal to the separation of holes _{1}.

Damping coefficients of the perforated dampers in translational motion simulated with the full 3D N-S solver (□) and with the PPR solver (——) as a function of the perforation ratio. _{c}

Pressure profiles on the bottom surface simulated with the full 3D N-S solver (left) and with the PPR solver (right) for translational movement. The hole diameters are: a) 1 _{c}

Damping coefficients of the perforated dampers in torsional motion simulated with the full 3D N-S solver (□) and with the PPR solver (——) as a function of the perforation ratio. _{c}

Topology and dimensions of the axisymmetric perforation cell.

Mechanical resistances used in modelling the flow in different regions of a perforation cell.

Parameters for the simulated dampers. Altogether, 504 different topologies were generated and simulated.

Description | Values | Unit |
---|---|---|

Number of holes |
16, 36, 64 | |

Surface length |
20, 30, 40 | 10^{−6} m |

Hole diameter _{0} |
0.5, 1.0,…, 4.5 | 10^{−6} m |

Thickness _{c} |
0.5, 1, 2, 5 | 10^{−6} m |

Air gap height |
1, 2 | 10^{−6} m |

Viscosity coefficient |
20 | 10^{−6} Ns/m^{2} |

Mean free path λ | 69 | 10^{−9} m |

Maximum relative errors in the translational motion damping coefficient _{m} simulated with the PPR method. Each row includes cases for _{c}

_{0} |
Max. relative error [%] | |||
---|---|---|---|---|

| ||||

0.5 | 1 | -4.8 | -2.3 | -1.4 |

1.0 | 4 | -4.9 | -2.3 | -3.4 |

1.5 | 9 | -4.8 | -2.3 | -1.8 |

2.0 | 16 | -4.8 | -2.8 | -3.4 |

2.5 | 25 | -5.7 | -4.9 | -5.0 |

3.0 | 36 | -8.3 | -7.7 | -7.7 |

3.5 | 49 | -11.6 | -11.2 | -10.9 |

4.0 | 64 | -15.6 | -15.1 | -15.1 |

4.5 | 81 | -19.8 | -19.3 | -19.3 |

Maximum relative errors in the torsional motion damping coefficient Z_{t} simulated with the PPR method. Each row includes cases for _{c} = 0.5

_{0} |
Max. relative error [%] | |||
---|---|---|---|---|

| ||||

0.5 | 1 | -18.0 | -8.0 | -4.4 |

1.0 | 4 | -18.4 | -8.3 | -4.6 |

1.5 | 9 | -18.7 | -8.4 | -4.6 |

2.0 | 16 | -19.0 | -8.5 | -4.8 |

2.5 | 25 | -19.6 | -8.8 | -7.2 |

3.0 | 36 | -20.4 | -11.3 | -9.7 |

3.5 | 49 | -21.9 | -15.1 | -13.7 |

4.0 | 64 | -24.3 | -19.5 | -18.0 |

4.5 | 81 | -28.5 | -24.3 | -22.9 |

Maximum relative errors in the translational motion damping coefficient _{m} simulated with the homogenisation method. Each row includes cases for _{c}

_{0} |
Max. relative error [%] | |||
---|---|---|---|---|

| ||||

0.5 | 1 | -4.9 | -2.3 | 1.9 |

1.0 | 4 | -5.0 | 6.4 | 6.0 |

1.5 | 9 | 8.8 | 10.6 | 9.7 |

2.0 | 16 | 9.4 | 9.6 | 8.7 |

2.5 | 25 | 7.6 | 7.5 | 6.6 |

3.0 | 36 | -8.2 | 5.2 | 5.0 |

3.5 | 49 | -11.3 | -6.6 | -4.3 |

4.0 | 64 | -15.1 | -9.4 | -6.6 |

4.5 | 81 | -19.6 | -13.0 | -9.3 |

Maximum relative errors in the translational motion damping coefficient Z_{m} simulated with the PPR method. Average specific admittances are used for the perforation and _{h} = 1. All other simulation parameters are the same as in the simulations in

_{0} |
Max. relative error [%] | |||
---|---|---|---|---|

| ||||

0.5 | 1 | -4.8 | -2.3 | 1.6 |

1.0 | 4 | -4.9 | -2.3 | -7.8 |

1.5 | 9 | -4.7 | -2.5 | -4.2 |

2.0 | 16 | -6.9 | -5.5 | -7.0 |

2.5 | 25 | -10.8 | -10.5 | -12.6 |

3.0 | 36 | -16.0 | -15.8 | -15.2 |

3.5 | 49 | -21.2 | -20.8 | -21.0 |

4.0 | 64 | -25.7 | -25.2 | -24.8 |

4.5 | 81 | -29.5 | -27.2 | -28.6 |

Maximum relative differences in the translational motion damping coefficient _{m} simulated with 3D N-S solver with 250 000 and 450 000 elements.

_{0} |
Maximum difference [%] | |||
---|---|---|---|---|

| ||||

0.5 | 1 | 0.2 | 1.0 | 1.0 |

1.0 | 4 | 0.2 | 0.5 | 1.9 |

1.5 | 9 | 0.5 | 0.8 | 1.8 |

2.0 | 16 | 0.3 | 1.0 | 1.1 |

2.5 | 25 | 0.5 | 0.6 | 2.4 |

3.0 | 36 | 0.4 | 0.4 | 1.8 |

3.5 | 49 | 0.6 | 0.7 | 1.5 |

4.0 | 64 | 0.4 | -1.8 | 1.2 |

4.5 | 81 | 0.4 | 2.1 | 2.0 |