# Effects of Internal Boundary Layers and Sensitivity on Frequency Response of Shells of Revolution

## Abstract

**:**

## 1. Introduction

#### 1.1. Novelty of This Work

#### 1.2. Brief Review of Literature

#### 1.3. Structure of Discussion

## 2. Preliminaries on Shells

#### 2.1. Shell Geometries

#### 2.2. Perforations

#### 2.3. Shell Models

**Remark 1.**

#### 2.4. Shallow Shell Model

#### 2.5. Sensitive Shells

## 3. Frequency Response

#### 3.1. Damping Model

#### 3.2. Low-Rank Approximation

## 4. Layers

**Definition 1**

**.**The subset of the domain from which the boundary layer decays exponentially is called the layer generator. Formally, the layer generator is of measure zero.

#### Propagation of Singularities

## 5. Numerical Simulations

#### 5.1. What to Expect

#### 5.2. Analysis

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Auvinen, V.V.; Virtanen, J.; Merivaara, A.; Virtanen, V.; Laurén, P.; Tuukkanen, S.; Laaksonen, T. Modulating sustained drug release from nanocellulose hydrogel by adjusting the inner geometry of implantable capsules. J. Drug Deliv. Sci. Technol.
**2020**, 57, 101625. [Google Scholar] [CrossRef] - Vicente, W.; Picelli, R.; Pavanello, R.; Xie, Y. Topology optimization of frequency responses of fluid–structure interaction systems. Finite Elem. Anal. Des.
**2015**, 98, 1–13. [Google Scholar] [CrossRef] - Chapelle, D.; Bathe, K.J. The Finite Element Analysis of Shells; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Sanchez-Palencia, E.; Millet, O.; Béchet, F. Singular Problems in Shell Theory; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Giani, S.; Hakula, H. On effects of perforated domains on parameter-dependent free vibration. J. Comput. Appl. Math.
**2021**, 394, 113526. [Google Scholar] [CrossRef] - Giani, S.; Hakula, H. Free Vibration of Perforated Cylindrical Shells of Revolution: Asymptotics and Effective Material Parameters. Comput. Methods Appl. Mech. Eng.
**2023**, 403, 115700. [Google Scholar] [CrossRef] - Giani, S.; Hakula, H. On effects of concentrated loads on perforated sensitive shells of revolution. J. Comput. Appl. Math.
**2023**, 408, 115165. [Google Scholar] [CrossRef] - Hakula, H.; Laaksonen, M. Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage. Appl. Sci.
**2019**, 9, 5299. [Google Scholar] [CrossRef] [Green Version] - Hakula, H.; Laaksonen, M. Low-Rank Approximation of Frequency Response Analysis of Perforated Cylinders under Uncertainty. Appl. Sci.
**2022**, 12, 3559. [Google Scholar] [CrossRef] - Pitkäranta, J.; Sanchez-Palencia, E. On the asymptotic behaviour of sensitive shells with small thickness. C. R. L’AcadÉmie Des Sci. Ser. Iib Mech.-Phys.-Chem.-Astron.
**1997**, 325, 127–134. [Google Scholar] [CrossRef] - Bathe, K.J.; Chapelle, D.; Lee, P.S. A shell problem ‘highly sensitive’ to thickness changes. Int. J. Numer. Methods Eng.
**2003**, 57, 1039–1052. [Google Scholar] [CrossRef] - Artioli, E.; da Veiga, L.B.; Hakula, H.; Lovadina, C. On the asymptotic behaviour of shells of revolution in free vibration. Comput. Mech.
**2009**, 44, 45–60. [Google Scholar] [CrossRef] - Chaussade-Beaudouin, M.; Dauge, M.; Faou, E.; Yosibash, Z. High Frequency Oscillations of First Eigenmodes in Axisymmetric Shells as the Thickness Tends to Zero. In Recent Trends in Operator Theory and Partial Differential Equations; Springer International Publishing: Berlin/Heidelberg, Germany, 2017; pp. 89–110. [Google Scholar] [CrossRef] [Green Version]
- Schenk, C.A.; Schuëller, G.I. Uncertainty Assessment of Large Finite Element Systems. In Lecture Notes in Applied and Computational Mathematics; Springer: Berlin/Heidelberg, Germany, 2005; Volume 24. [Google Scholar]
- Benson, D.; Bazilevs, Y.; Hsu, M.; Hughes, T. Isogeometric shell analysis: The Reissner–Mindlin shell. Comput. Methods Appl. Mech. Eng.
**2010**, 199, 276–289. [Google Scholar] [CrossRef] [Green Version] - Naghdi, P. Foundations of elastic shell theory. In Progress in Solid Mechanics; North-Holland Publishing Company: Amsterdam, The Netherlands, 1963; Volume 4, pp. 1–90. [Google Scholar]
- Malinen, M. On the classical shell model underlying bilinear degenerated shell finite elements: General shell geometry. Int. J. Numer. Methods Eng.
**2002**, 55, 629–652. [Google Scholar] [CrossRef] - Pitkäranta, J. The problem of membrane locking in finite element analysis of cylindrical shells. Numer. Math.
**1992**, 61, 523–542. [Google Scholar] [CrossRef] - Hakula, H.; Leino, Y.; Pitkäranta, J. Scale resolution, locking, and high-order finite element modelling of shells. Comput. Methods Appl. Mech. Eng.
**1996**, 133, 157–182. [Google Scholar] [CrossRef] - Bieber, S.; Oesterle, B.; Ramm, E.; Bischoff, M. A variational method to avoid locking—Independent of the discretization scheme. Numer. Methods Eng.
**2018**, 114, 801–827. [Google Scholar] [CrossRef] - Szabo, B.; Babuska, I. Finite Element Analysis; Wiley: Hoboken, NJ, USA, 1991. [Google Scholar]
- Schwab, C. p- and hp-Finite Element Methods; Oxford University Press: Oxford, UK, 1998. [Google Scholar]
- Ko, Y.; Lee, P.S.; Bathe, K.J. A new MITC4+ shell element. Comput. Struct.
**2017**, 182, 404–418. [Google Scholar] [CrossRef] - Ho-Nguyen-Tan, T.; Kim, H.G. Polygonal shell elements with assumed transverse shear and membrane strains. Comput. Methods Appl. Mech. Eng.
**2019**, 349, 595–627. [Google Scholar] [CrossRef] - Jung, J.; Hyun, J.; Goo, S.; Wang, S. An efficient design sensitivity analysis using element energies for topology optimization of a frequency response problem. Comput. Methods Appl. Mech. Eng.
**2015**, 296, 196–210. [Google Scholar] [CrossRef] - Liu, Q.; Chan, R.; Huang, X. Concurrent topology optimization of macrostructures and material microstructures for natural frequency. Mater. Des.
**2016**, 106, 380–390. [Google Scholar] [CrossRef] - Venini, P. Dynamic compliance optimization: Time vs. frequency domain strategies. Comput. Struct.
**2016**, 177, 12–22. [Google Scholar] [CrossRef] - Zhao, J.; Wang, C. Topology optimization for minimizing the maximum dynamic response in the time domain using aggregation functional method. Comput. Struct.
**2017**, 190, 41–60. [Google Scholar] [CrossRef] - Takezawa, A.; Daifuku, M.; Nakano, Y.; Nakagawa, K.; Yamamoto, T.; Kitamura, M. Topology optimization of damping material for reducing resonance response based on complex dynamic compliance. J. Sound Vib.
**2016**, 365, 230–243. [Google Scholar] [CrossRef] [Green Version] - Do Carmo, M. Differential Geometry of Curves and Surfaces; Prentice Hall: Hoboken, NJ, USA, 1976. [Google Scholar]
- Forskitt, M.; Moon, J.R.; Brook, P.A. Elastic properties of plates perforated by elliptical holes. Appl. Math. Model.
**1991**, 15, 182–190. [Google Scholar] [CrossRef] - Burgemeister, K.; Hansen, C. Calculating Resonance Frequencies of Perforated Panels. J. Sound Vib.
**1996**, 196, 387–399. [Google Scholar] [CrossRef] - Jhung, M.J.; Yu, S.O. Study on modal characteristics of perforated shell using effective Young’s modulus. Nucl. Eng. Des.
**2011**, 241, 2026–2033. [Google Scholar] [CrossRef] - Pitkäranta, J.; Matache, A.M.; Schwab, C. Fourier mode analysis of layers in shallow shell deformations. Comput. Methods Appl. Mech. Eng.
**2001**, 190, 2943–2975. [Google Scholar] [CrossRef] [Green Version] - Niemi, A.H.; Pitkäranta, J. Bilinear finite elements for shells: Isoparametric quadrilaterals. Int. J. Numer. Methods Eng.
**2008**, 75, 212–240. [Google Scholar] [CrossRef] - Malinen, M.; Pitkäranta, J. A benchmark study of reduced-strain shell finite elements: Quadratic schemes. Int. J. Numer. Methods Eng.
**2000**, 48, 1637–1671. [Google Scholar] [CrossRef] - Inman, D.J. Engineering Vibration; Pearson: New York, NY, USA, 2008. [Google Scholar]
- Eid, R.; Salimbahrami, B.; Lohmann, B.; Rudnyi, E.B.; Korvink, J.G. Parametric Order Reduction of Proportionally Damped Second-Order Systems. Sens. Mater.
**2007**, 19, 149–164. [Google Scholar] - Freund, R.W. Krylov-subspace methods for reduced-order modeling in circuit simulation. J. Comput. Appl. Math.
**2000**, 123, 395–421. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Three realistic capsule designs with the limit payload regions indicated. The first design (

**a**) is a cylinder, whereas the other two (

**b**,

**c**) have non-uniform curvature inner shells. Images courtesy of V. Auvinen/University of Helsinki.

**Figure 2.**Reference configurations. (

**a**) Parabolic shell of revolution. (

**b**) Perforated computational domain. Boundaries at $y=0$ and $y=2\pi $ are periodic. Clamped case: All displacements are inhibited at $x\pm \pi $. Sensitive case: All displacements are inhibited at $x=-\pi $ only. Regular $20\times 20$ grid with all holes free where the hole coverage percentage is 25%.

**Figure 4.**Layer charts for different geometries. (

**a**) Parabolic. (

**b**) Hyperbolic. The arrows indicate the direction in which the width of the layer varies.

**Figure 5.**Most energetic response: Parabolic (Naghdi). (

**a**) Surface. (

**b**) Contours. Clamped case, symmetric point load at $\omega =90\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$.

**Figure 6.**Most energetic response: Hyperbolic (Naghdi). (

**a**) Surface. (

**b**) Contours. Clamped case, symmetric point load at $\omega =130\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$.

**Figure 7.**Reference configurations. (

**a**) Hyperbolic shell of revolution. (

**b**) Shell of revolution with nonuniform curvature type.

**Figure 8.**Most energetic response: Parabolic (Shallow Shell Model). (

**a**) Surface. (

**b**) Contours. Clamped case, symmetric point load at $\omega =90\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$.

**Figure 9.**Most energetic response: Parabolic (Naghdi). (

**a**) Surface. (

**b**) Comparison of energy and displacement. Clamped case.

**Figure 10.**Most energetic response: Hyperbolic (Shallow Shell Model). (

**a**) Surface. (

**b**) Contours. Clamped case, symmetric point load at $\omega =200\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$.

**Figure 11.**Most energetic response: Hyperbolic (Naghdi). (

**a**) Surface. (

**b**) Contours. Sensitive case, symmetric point load at $\omega =60\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$.

**Figure 12.**General geometry: $f\left(x\right)=4/5+(1/5)tanh(-2(x-\pi /2\left)\right)$. (

**a**) Profile function. (

**b**) Second derivative. All geometry types are present. The load acts in the parabolic region at $x=-\pi /10$.

**Figure 13.**Most energetic response: General geometry (Naghdi). (

**a**) Surface. (

**b**) Comparison of energy and displacement. Clamped case.

**Figure 14.**Most energetic responses: General geometry (Naghdi). (

**a**) Surface at $\omega =100\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$. (

**b**) Contours at $\omega =100\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$. (

**c**) Surface at $\omega =40\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$. (

**d**) Contours at $\omega =40\phantom{\rule{3.33333pt}{0ex}}\pi $ $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$. (

**e**) Energy vs. frequency. (

**f**) Maximal transverse deflection vs frequency. Sensitive case.

Geometry | Boundary Layer | Internal Layer | Boundary Oscillation |
---|---|---|---|

Parabolic | t, $\sqrt{t}$ | $\sqrt[4]{t}$ | - |

Hyperbolic | t, $\sqrt{t}$ | $\sqrt[3]{t}$ | - |

Elliptic | t, $\sqrt{t}$ | - | $K\sim -logt$ |

**Table 2.**Simulation configuration data. Computational domain $\mathsf{\Omega}=[-\pi ,\pi ]\times [0,2\pi ]$.

Geometry | Shell Model | Profile Data |
---|---|---|

Parabolic | Naghdi | $f\left(x\right)=1$ |

Shallow | $a=0,b=1,c=0$ | |

Hyperbolic | Naghdi | $f\left(x\right)=1+(1/2){(x/\pi )}^{2}$ |

Shallow | $a=1,b=-1,c=0$ | |

General | Naghdi | $f\left(x\right)=4/5+(1/5)tanh(-2(x-\pi /2\left)\right)$ |

**Table 3.**Simulation results. Observed angular frequency of the most energetic response and the correlation between energies and transverse displacements.

Geometry | Shell Model | Mesh | BC | $\mathit{\omega}$ rad s${}^{-1}$ | Correlation |
---|---|---|---|---|---|

Parabolic | Naghdi | Standard | Clamped | 130 $\pi $ | 0.82 |

Sensitive | 80 $\pi $ | 0.68 | |||

Perforated | Clamped | 80 $\pi $ | 0.88 | ||

Sensitive | 30 $\pi $ | 0.94 | |||

Parabolic | Shallow | Standard | Clamped | 90 $\pi $ | 0.89 |

Sensitive | 90 $\pi $ | 0.62 | |||

Perforated | Clamped | 80 $\pi $ | 0.87 | ||

Sensitive | 30 $\pi $ | 0.99 | |||

Hyperbolic | Naghdi | Standard | Clamped | 130 $\pi $ | 0.69 |

Sensitive | 70$\phantom{\rule{3.33333pt}{0ex}}\pi $ | 0.69 | |||

Perforated | Clamped | 110 $\pi $ | 0.84 | ||

Sensitive | 60 $\pi $ | 0.90 | |||

Hyperbolic | Shallow | Standard | Clamped | 200 $\pi $ | 0.99 |

Sensitive | 160 $\pi $ | 0.83 | |||

Perforated | Clamped | 190 $\pi $ | 0.99 | ||

Sensitive | 120 $\pi $ | 0.72 | |||

General | Naghdi | Standard | Clamped | 100 $\pi $ | 0.77 |

Sensitive | 40 $\pi $ | 0.95 | |||

Perforated | Clamped | 90$\phantom{\rule{3.33333pt}{0ex}}\mathit{\pi}$ | 0.91 | ||

Sensitive | 100$\phantom{\rule{3.33333pt}{0ex}}\mathit{\pi}$ | 0.60 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hakula, H.
Effects of Internal Boundary Layers and Sensitivity on Frequency Response of Shells of Revolution. *Vibration* **2023**, *6*, 566-583.
https://doi.org/10.3390/vibration6030035

**AMA Style**

Hakula H.
Effects of Internal Boundary Layers and Sensitivity on Frequency Response of Shells of Revolution. *Vibration*. 2023; 6(3):566-583.
https://doi.org/10.3390/vibration6030035

**Chicago/Turabian Style**

Hakula, Harri.
2023. "Effects of Internal Boundary Layers and Sensitivity on Frequency Response of Shells of Revolution" *Vibration* 6, no. 3: 566-583.
https://doi.org/10.3390/vibration6030035