# Application of the ps−Version of the Finite Element Method to the Analysis of Laminated Shells

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

#### 2.1. Geometry Description

#### 2.2. Curvilinear Fiber Path Description

#### 2.3. Shell Description

## 3. The $\mathit{ps}$−Version of the Finite Element Method

#### 3.1. p−Refinement

#### 3.2. s−Refinement

#### 3.3. $ps$−Refinement

## 4. Results

#### 4.1. Validation

#### 4.1.1. Test Case 1: Vibrations of Elliptical Shells

#### 4.1.2. Test Case 2: Vibration of Variable Stiffness Plate

#### 4.1.3. Test Case 3: Static Analysis of Plate with Cutout

#### 4.2. Applications

#### 4.2.1. Example 1: Vibration and Buckling of a Highly Anisotropic Laminated Plate

#### 4.2.2. Example 2: Stress Analysis of Variable Stiffness Elliptical Shell

#### 4.2.3. Example 3: Snap−Back of Cylindrical Panel

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Reddy, J. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis; CRC Press: Boca Raton, CA, USA, 2004. [Google Scholar]
- Gürdal, Z.; Olmedo, R. Composite laminates with spatially varying fiber orientations: Variable stiffness panel concept. In Proceedings of the 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Dallas, TX, USA, 13–15 April 1992. [Google Scholar]
- Gürdal, Z.; Tatting, B.; Wu, C. Variable stiffness composite panels: Effects of stiffness variation on the in-plane and buckling response. Compos. Part A Appl. Sci. Manuf.
**2008**, 39, 911–922. [Google Scholar] [CrossRef] - Wu, Z.; Weaver, P.; Raju, G.; Kim, B. Buckling analysis and optimisation of variable angle tow composite plates. Thin-Walled Struct.
**2012**, 60, 163–172. [Google Scholar] [CrossRef] - Raju, G.; Wu, Z.; Weaver, P. Postbuckling analysis of variable angle tow plates using differential quadrature method. Compos. Struct.
**2013**, 106, 74–84. [Google Scholar] [CrossRef] - Szabó, B.; Babuška, I.B. Introduction to Finite Element Analysis: Formulation, Verification and Validation; John Wiley & Sons Ltd.: Chichester, UK, 2011. [Google Scholar]
- Bathe, K. Finite Element Procedures; Prentice Hall: Hoboken, NJ, USA, 2006. [Google Scholar]
- Babuška, I.; Szabó, B.; Katz, I. The p-version of the finite element method. SIAM J. Numer. Anal.
**1981**, 18, 515–545. [Google Scholar] [CrossRef] - Bacciocchi, M.; Luciano, R.; Majorana, C.; Tarantino, A. Free vibrations of sandwich plates with damaged soft-core and non-uniform mechanical properties: Modeling and finite element analysis. Materials
**2019**, 12, 2444. [Google Scholar] [CrossRef] - Bacciocchi, M.; Tarantino, A. Natural frequency analysis of functionally graded orthotropic cross-ply plates based on the finite element method. Math. Comput. Appl.
**2019**, 24, 52. [Google Scholar] [CrossRef] - Duc, N.; Trinh, T.; Do, T.V.; Doan, D. On the buckling behavior of multi-cracked FGM plates. In Proceedings of the International Conference on Advances in Computational Mechanics 2017: ACOME 2017, Phu Quoc, Vietnam, 2–4 August 2018. [Google Scholar]
- Dong, H.; Zheng, X.; Cui, J.; Nie, Y.; Yang, Z.; Ma, Q. Multi-scale computational method for dynamic thermo-mechanical performance of heterogeneous shell structures with orthogonal periodic configurations. Comput. Methods Appl. Mech. Eng.
**2019**, 354, 143–180. [Google Scholar] [CrossRef] - Tsapetis, D.; Sotiropoulos, G.; Stavroulakis, G.; Papadopoulos, V.; Papadrakakis, M. A stochastic multiscale formulation for isogeometric composite Kirchhoff-Love shells. Comput. Methods Appl. Mech. Eng.
**2021**, 373, 113541. [Google Scholar] [CrossRef] - Cao, Z.; Guo, D.; Fu, H.; Han, Z. Mechanical simulation of thermoplastic composite fiber variable-angle laminates. Materials
**2020**, 13, 3374. [Google Scholar] [CrossRef] - Sanchez-Majano, A.; Pagani, A.; Petrolo, M.; Zhang, C. Buckling sensitivity of tow-steered plates subjected to multiscale defects by high-order finite elements and polynomial chaos expansion. Materials
**2021**, 14, 2706. [Google Scholar] [CrossRef] - Akhavan, H.; Ribeiro, P.; Moura, M.D. Large deflection and stresses in variable stiffness composite laminates with curvilinear fibres. Int. J. Mech. Sci.
**2013**, 73, 14–26. [Google Scholar] [CrossRef] - Yazdani, S.; Ribeiro, P. A layerwise p-version finite element formulation for free vibration analysis of thick composite laminates with curvilinear fibres. Compos. Struct.
**2015**, 120, 531–542. [Google Scholar] [CrossRef] - Bank, R.; Sherman, A.; Weiser, A. Some refinement algorithms and data structures for regular local mesh refinement. Sci. Comput. Appl. Math. Comput. Phys. Sci.
**1983**, 1, 3–17. [Google Scholar] - Zhu, J.; Zienkiewicz, O. Adaptive techniques in the finite element method. Commun. Appl. Numer. Methods
**1988**, 4, 197–204. [Google Scholar] [CrossRef] - Zienkiewicz, O.; Zhu, J. A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Methods Eng.
**1987**, 24, 337–357. [Google Scholar] [CrossRef] - BabuÅ¡ka, I.; Dorr, M. Error estimates for the combined h and p versions of the finite element method. Numer. Math.
**1981**, 37, 257–277. [Google Scholar] - Guo, B.; Babuška, I. The hp version of the finite element method. Comput. Mech.
**1986**, 1, 21–41. [Google Scholar] [CrossRef] - Gui, W.; Babuška, I. The h, p and hp versions of the finite element method in 1 dimension. Part 1. In The Error Analysis of the p-Version; TN BN-1036; Laboratory for Numerical Analysis, University of Maryland: College Park, MD, USA, 1985. [Google Scholar]
- Babuška, I.; Rank, E. An expert-system-like feedback approach in the hp-version of the finite element method. Finite Elem. Anal. Des.
**1987**, 3, 127–147. [Google Scholar] [CrossRef] - Verfürth, R. A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Eng.
**1999**, 176, 419–440. [Google Scholar] [CrossRef] - Eibner, T.; Melenk, J. An adaptive strategy for hp-FEM based on testing for analyticity. Comput. Mech.
**2007**, 39, 575–595. [Google Scholar] [CrossRef] - Bern, M.; Flaherty, J.; Luskin, M. Grid generation and adaptive algorithms. In The IMA Volumes in Mathematics and its Applications; Springer: New York, NY, USA, 2012. [Google Scholar]
- Mote, C. Global-local finite element. Int. J. Numer. Methods Eng.
**1971**, 3, 565–574. [Google Scholar] [CrossRef] - Belytschko, T.; Fish, J.; Bayliss, A. The spectral overlay on finite elements for problems with high gradients. Comput. Methods Appl. Mech. Eng.
**1990**, 81, 71–89. [Google Scholar] [CrossRef] - Fish, J. The s-version of the finite element method. Comput. Struct.
**1992**, 43, 539–547. [Google Scholar] [CrossRef] - Fish, J.; Markolefas, S. The s-version of the finite element method for multilayer laminates. Int. J. Numer. Methods Eng.
**1992**, 33, 1081–1105. [Google Scholar] [CrossRef] - Fish, J.; Guttal, R. The s-version of finite element method for laminated composites. Int. J. Numer. Methods Eng.
**1996**, 39, 3641–3662. [Google Scholar] [CrossRef] - Fish, J.; Suvorov, A.; Belsky, V. Hierarchical composite grid method for global-local analysis of laminated composite shells. Appl. Numer. Math.
**1997**, 23, 241–258. [Google Scholar] [CrossRef] - Park, J.; Hwang, J.; Kim, Y. Efficient finite element analysis using mesh superposition technique. Finite Elem. Anal. Des.
**2003**, 39, 619–638. [Google Scholar] [CrossRef] - Reddy, J.; Robbins, D. Theories and Computational Models for Composite Laminates. Appl. Mech. Rev.
**1994**, 47, 147. [Google Scholar] [CrossRef] - Sakata, S.; Chan, Y.; Arai, Y. On accuracy improvement of microscopic stress/stress sensitivity analysis with the mesh superposition method for heterogeneous materials considering geometrical variation of inclusions. Int. J. Numer. Methods Eng.
**2020**, 121, 534–559. [Google Scholar] [CrossRef] - Kishi, K.; Takeoka, Y.; Fukui, T.; Matsumoto, T.; Suzuki, K.; Shibanuma, K. Dynamic crack propagation analysis based on the s-version of the finite element method. Comput. Methods Appl. Mech. Eng.
**2020**, 366, 113091. [Google Scholar] [CrossRef] - Rank, E. Adaptive remeshing and hp domain decomposition. Comput. Methods Appl. Mech. Eng.
**1992**, 101, 299–313. [Google Scholar] [CrossRef] - Schillinger, D.; Düster, A.; Rank, E. The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int. J. Numer. Methods Eng.
**2012**, 89, 1171–1202. [Google Scholar] [CrossRef] - Krause, R.; Rank, E. Multiscale computations with a combination of the h-and p-versions of the finite-element method. Comput. Methods Appl. Mech. Eng.
**2003**, 192, 3959–3983. [Google Scholar] [CrossRef] - Zander, N.; Bog, T.; Kollmannsberger, S.; Schillinger, D.; Rank, E. Multi-level hp-adaptivity: High-order mesh adaptivity without the difficulties of constraining hanging nodes. Comput. Mech.
**2015**, 55, 499–517. [Google Scholar] [CrossRef] - Zander, N.; Bog, T.; Elhaddad, M.; Frischmann, F.; Kollmannsberger, S.; Rank, E. The multi-level hp-method for three-dimensional problems: Dynamically changing high-order mesh refinement with arbitrary hanging nodes. Comput. Methods Appl. Mech. Eng.
**2016**, 310, 252–277. [Google Scholar] [CrossRef] - Zander, N.; Ruess, M.; Bog, T.; Kollmannsberger, S.; Rank, E. Multi-level hp-adaptivity for cohesive fracture modeling. Int. J. Numer. Methods Eng.
**2017**, 109, 1723–1755. [Google Scholar] [CrossRef] - Tornabene, F.; Fantuzzi, N.; Bacciocchi, M.; Dimitri, R. Dynamic analysis of thick and thin elliptic shell structures made of laminated composite materials. Compos. Struct.
**2015**, 133, 278–299. [Google Scholar] [CrossRef] - Amabili, M. Nonlinear Vibrations and Stability of Shells and Plates; Cambridge University Press: Cambrige, UK, 2008. [Google Scholar]
- Gordon, W.; Hall, C. Construction of curvilinear coordinate systems and applications to mesh generation. Int. J. Numer. Methods Eng.
**1973**, 7, 461–477. [Google Scholar] [CrossRef] - Gordon, W.; Hall, C. Transfinite element methods: Blending-function interpolation over arbitrary curved element domains. Numer. Math.
**1973**, 21, 109–129. [Google Scholar] [CrossRef] - Vescovini, R.; Spigarolo, E.; Dozio, L. Efficient post-buckling analysis of variable-stiffness plates using a perturbation approach. Thin-Walled Struct.
**2019**, 143, 106211. [Google Scholar] [CrossRef] - Tornabene, F.; Fantuzzi, N.; Bacciocchi, M.; Dimitri, R. Free vibrations of composite oval and elliptic cylinders by the generalized differential quadrature method. Thin-Walled Struct.
**2015**, 97, 114–129. [Google Scholar] [CrossRef] - Houmat, A. Nonlinear free vibration of laminated composite rectangular plates with curvilinear fibers. Compos. Struct.
**2013**, 106, 211–224. [Google Scholar] [CrossRef] - Vescovini, R.; Dozio, L.; d’Ottavio, M.; Polit, O. On the application of the Ritz method to free vibration and buckling analysis of highly anisotropic plates. Compos. Struct.
**2018**, 192, 460–474. [Google Scholar] [CrossRef] - Kriegesmann, B.; Jansen, E.; Rolfes, R. Design of cylindrical shells using the single perturbation load approach potentials and application limits. Thin-Walled Struct.
**2016**, 108, 369–380. [Google Scholar] [CrossRef] - Wisniewski, K. Finite rotation shells. In Basic Equations and Finite Elements for Reissner Kinematics; Springer: Berlin, Germany, 2010. [Google Scholar]

**Figure 6.**Conditions to satisfy when performing s−refinements: (

**a**) compatibility and (

**b**) linear indipendency.

**Figure 14.**Computed values of ${N}_{xx}$ at $({\xi}_{1},{\xi}_{2})=(0,r)$ for different levels and types of $ps$−refinement: (

**a**) linear, (

**b**) uniform, and (

**c**) graded.

**Figure 15.**Highlyanisotropic plate – vibration and buckling analysis: (

**a**) Base mesh, (

**b**) $s-$refined mesh.

**Figure 16.**Highly anisotropic plate: mode shapes for free vibration analysis: (

**a**) deflection shape and (

**b**) twisting moment.

**Figure 17.**Highly anisotropic plate: mode shapes for buckling analysis: (

**a**) deflection shape and (

**b**) twisting moment.

**Figure 18.**Highly anisotropic plate: convergence for different refinement strategies: (

**a**) nondimensional fundamental frequency and (

**b**) nondimensional buckling load.

**Figure 19.**Variable Stiffness cylindrical elliptical shell – static analysis: (

**a**) Base mesh, (

**b**) $s-$refined mesh.

**Figure 20.**Variable stiffness cylindrical elliptical shell: comparison between p− and $ps$−U−models: (

**a**) displacement and (

**b**) normal stress field.

**Figure 21.**Variable stiffness cylindrical elliptical shell: stress field ${\sigma}_{22}$: (

**a**) p−model and (

**b**) $ps$−U−model.

**Figure 24.**Cylindrical panel: load−deflection curves for different levels of (

**a**) h−, (

**b**) p− and (

**c**) $ps$−refinements.

L (mm) | ${\mathit{R}}_{\mathit{A}}$ (mm) | ${\mathit{R}}_{\mathit{B}}$ (mm) | e (-) | |
---|---|---|---|---|

Shell 1 | 594.9188 | 304.8000 | 304.8000 | 0.0000 |

Shell 2 | 595.3250 | 328.9300 | 279.6540 | 0.5265 |

Shell 3 | 595.3760 | 365.5060 | 237.4900 | 0.7601 |

**Table 2.**Natural frequencies $\omega $ (rad/s) for elliptical shells with different eccentricities e. Subscript: percent difference against ref. [49].

Mode N° | $\mathit{e}=0.0000$ | $\mathit{e}=0.5265$ | $\mathit{e}=0.7601$ |
---|---|---|---|

1 | 611.2500${}^{\left(0.0031\right)}$ | 597.7019${}^{\left(0.0033\right)}$ | 539.5629${}^{\left(0.0041\right)}$ |

2 | 611.2500${}^{\left(0.0031\right)}$ | 597.7020${}^{\left(0.0034\right)}$ | 539.5683${}^{\left(0.0059\right)}$ |

3 | 643.1510${}^{\left(0.0003\right)}$ | 618.4085${}^{\left(0.0016\right)}$ | 542.8651${}^{\left(0.0025\right)}$ |

4 | 643.1510${}^{\left(0.0003\right)}$ | 618.4086${}^{\left(0.0015\right)}$ | 542.8681${}^{\left(0.0023\right)}$ |

5 | 701.4954${}^{\left(0.0042\right)}$ | 712.9803${}^{\left(0.0025\right)}$ | 715.4793${}^{\left(0.0009\right)}$ |

6 | 701.4954${}^{\left(0.0042\right)}$ | 712.9810${}^{\left(0.0025\right)}$ | 715.5105${}^{\left(0.0049\right)}$ |

7 | 857.9875${}^{\left(0.0014\right)}$ | 847.2782${}^{\left(0.0010\right)}$ | 785.3787${}^{\left(0.0042\right)}$ |

8 | 857.9875${}^{\left(0.0014\right)}$ | 847.2840${}^{\left(0.0014\right)}$ | 785.5405${}^{\left(0.0080\right)}$ |

9 | 864.4224${}^{\left(0.0025\right)}$ | 870.6317${}^{\left(0.0001\right)}$ | 892.8773${}^{\left(0.0072\right)}$ |

10 | 864.4224${}^{\left(0.0025\right)}$ | 870.6408${}^{\left(0.0003\right)}$ | 893.0180${}^{\left(0.0147\right)}$ |

**Table 3.**Natural frequencies (rad/s) of rectangular plate with different variable stiffness layups. Subscript: percent difference against ref. [50].

p | $\u2329-45|45\u232a$ | $\u2329-45|30\u232a$ | $\u2329-45|15\u232a$ | $\u2329-45|0\u232a$ | $\u2329-45|-15\u232a$ | $\u2329-45|-30\u232a$ | $\u2329-45|-45\u232a$ |
---|---|---|---|---|---|---|---|

4 | 0.0894${}^{\left(3.95\right)}$ | 0.0966${}^{\left(1.68\right)}$ | 0.1028${}^{\left(1.78\right)}$ | 0.1058${}^{\left(2.72\right)}$ | 0.1049${}^{\left(0.87\right)}$ | 0.1010${}^{\left(1.00\right)}$ | 0.0962${}^{\left(0.21\right)}$ |

6 | 0.0882${}^{\left(2.56\right)}$ | 0.0957${}^{\left(0.74\right)}$ | 0.1016${}^{\left(0.59\right)}$ | 0.1044${}^{\left(1.36\right)}$ | 0.1037${}^{(-0.29)}$ | 0.1005${}^{\left(0.50\right)}$ | 0.0960${}^{\left(0.00\right)}$ |

8 | 0.0868${}^{\left(0.93\right)}$ | 0.0948${}^{(-0.21)}$ | 0.1011${}^{\left(0.10\right)}$ | 0.1040${}^{\left(0.97\right)}$ | 0.1035${}^{(-0.48)}$ | 0.1004${}^{\left(0.40\right)}$ | 0.0959${}^{(-0.10)}$ |

10 | 0.0866${}^{\left(0.70\right)}$ | 0.0946${}^{(-0.42)}$ | 0.1009${}^{(-0.10)}$ | 0.1039${}^{\left(0.87\right)}$ | 0.1035${}^{(-0.48)}$ | 0.1004${}^{\left(0.40\right)}$ | 0.0959${}^{(-0.10)}$ |

Linear $\mathit{ps}$-Refinement | Uniform $\mathit{ps}$-Refinement | Graded $\mathit{ps}$-Refinement ($\mathit{m}=1$) | ||
---|---|---|---|---|

${\mathit{p}}_{\mathit{s}}=1$ | ${\mathit{p}}_{\mathit{s}}=\mathit{p}$ | ${\mathit{p}}_{\mathit{s}}=\mathit{p}$ − Floor$\left(\frac{\mathit{s}}{\mathit{m}}\right)$ | ||

$p=1$ | 52.4124 | 52.4570 | 52.3428 | |

$p=2$ | 52.5307 | 52.5348 | 52.5254 | |

$p=3$ | 52.5433 | 52.5439 | 52.5433 | |

$p=4$ | 52.5445 | 52.5446 | 52.5445 | |

$p=5$ | 52.5446 | 52.5446 | 52.5446 | |

$p=6$ | 52.5446 | 52.5446 | 52.5446 | |

Reference | 52.5446 |

Refinement Strategy | Mesh Resolution (h) | Polynomial Order (p) | Superposition Levels (s) | |
---|---|---|---|---|

h−model | h | Increased | Fixed | - |

p−model | p | Fixed | Increased | - |

$ps$−L−model | Linear $ps$ | Fixed | Increased | Increased, ${p}_{s}=1$ |

$ps$−U−model | Uniform $ps$ | Fixed | Increased | Increased, ${p}_{s}=p$ |

$ps$−G−model | Graded $ps$ | Fixed | Increased | Increased, ${p}_{s}$ set with Equation (44) |

**Table 6.**Nondimensional frequency $\widehat{\omega}$ and buckling load ${\widehat{N}}_{xx}$ for a SSSS anisotropic plate using a uniform $ps$−refinement strategy.

Number of Unknowns | $\widehat{\mathit{\omega}}$ | ${\widehat{\mathit{N}}}_{\mathit{xx}}$ | |
---|---|---|---|

$p=1$ | 1119 | 24.9319 | 52.5977 |

$p=2$ | 2714 | 22.0819 | 31.6995 |

$p=3$ | 4999 | 21.9341 | 30.2107 |

$p=4$ | 7974 | 21.9289 | 30.1487 |

$p=5$ | 11,639 | 21.9267 | 30.1416 |

$p=6$ | 15,994 | 21.9264 | 30.1407 |

$p=7$ | 21,039 | 21.9263 | 30.1402 |

**Table 7.**Nondimensional frequencies $\widehat{\omega}$ and buckling load ${\widehat{N}}_{xx}$ for SSSS plates with different degrees of anisotropy.

${\mathit{E}}_{11}/{\mathit{E}}_{22}$ | $\widehat{\mathit{\omega}}$ | ${\widehat{\mathit{N}}}_{\mathit{xx}}$ | |||||
---|---|---|---|---|---|---|---|

$\mathit{ps}$-FEM | Ritz Method | %diff | $\mathit{ps}$-FEM | Ritz Method | %diff | ||

$\theta =30$ | 73.64 | 22.6796 | 22.6929 | 0.0584 | 38.7882 | 38.8328 | 0.1148 |

40 | 17.9948 | 17.9913 | 0.0195 | 26.7974 | 26.7858 | 0.0432 | |

20 | 14.0520 | 14.0524 | 0.0032 | 17.7050 | 17.7064 | 0.0078 | |

10 | 11.1987 | 11.1987 | 0.0004 | 11.8977 | 11.8978 | 0.0007 | |

$\theta =45$ | 73.64 | 21.9263 | 21.9674 | 0.1873 | 30.1402 | 30.2538 | 0.3754 |

40 | 17.6666 | 17.6557 | 0.0619 | 22.9941 | 22.9633 | 0.1339 | |

20 | 13.9480 | 13.9496 | 0.0115 | 16.4120 | 16.4163 | 0.0262 | |

10 | 11.1537 | 11.1538 | 0.0010 | 11.4625 | 11.4628 | 0.0024 | |

$\theta =60$ | 73.64 | 22.6796 | 22.6929 | 0.0584 | 24.1328 | 24.1571 | 0.1005 |

40 | 17.9948 | 17.9913 | 0.0195 | 19.3018 | 19.2959 | 0.0307 | |

20 | 14.0520 | 14.0524 | 0.0032 | 15.0556 | 15.0563 | 0.0050 | |

10 | 11.1987 | 11.1987 | 0.0004 | 11.7175 | 11.7176 | 0.0010 |

**Table 8.**Summary of the refinement parameters of the FE models used for solving the application example 3.

Refinement | Number of Unknowns | h | p | s | |
---|---|---|---|---|---|

h−model | 91 | 2 × 2 | 1 | - | |

1 | 343 | 4 × 4 | 1 | - | |

2 | 1327 | 8 × 8 | 1 | - | |

p−model | 91 | 2 × 2 | 1 | - | |

1 | 343 | 2 × 2 | 3 | - | |

2 | 1021 | 2 × 2 | 6 | - | |

$ps$−L−model | 91 | 2 × 2 | 1 | 0 | |

1 | 252 | 2 × 2 | 2 | 5 | |

2 | 453 | 2 × 2 | 3 | 10 |

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 authors. 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**

Yan, C.A.; Vescovini, R.
Application of the *ps*−Version of the Finite Element Method to the Analysis of Laminated Shells. *Materials* **2023**, *16*, 1395.
https://doi.org/10.3390/ma16041395

**AMA Style**

Yan CA, Vescovini R.
Application of the *ps*−Version of the Finite Element Method to the Analysis of Laminated Shells. *Materials*. 2023; 16(4):1395.
https://doi.org/10.3390/ma16041395

**Chicago/Turabian Style**

Yan, Cheng Angelo, and Riccardo Vescovini.
2023. "Application of the *ps*−Version of the Finite Element Method to the Analysis of Laminated Shells" *Materials* 16, no. 4: 1395.
https://doi.org/10.3390/ma16041395