Research Article
BibTex RIS Cite

Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability

Year 2017, Volume: 9 Issue: 2, 1 - 21, 14.04.2017

Francesco Tornabene Nicholas Fantuzzi Michele Bacciocchi

https://doi.org/10.24107/ijeas.304376
Cited By: 3

Abstract

The authors are presenting a novel
formulation based on the Differential Quadrature (DQ) method which is used to
approximate derivatives and integrals. The resulting scheme has been termed
strong and weak form finite elements (SFEM or WFEM), according to the numerical
scheme employed in the computation. Such numerical methods are applied to solve
some structural problems related to the mechanical behavior of plates and
shells, made of isotropic or composite materials.

The main differences
between these two approaches rely on the initial formulation – which is strong
or weak (variational) – and the implementation of the boundary conditions, that
for the former include the continuity of stresses and displacements, whereas in
the latter can consider the continuity of the displacements or both.





The two methodologies
consider also a mapping technique to transform an element of general shape
described in Cartesian coordinates into the same element in the computational
space. Such technique can be implemented by employing the classic
Lagrangian-shaped elements with a fixed number of nodes along the element edges
or blending functions which allow an “exact mapping” of the element. In
particular, the authors are employing NURBS (Not-Uniform Rational B-Splines)
for such nonlinear mapping in order to use the “exact” shape of CAD designs.

Keywords

Structural analysis Numerical methods Strong formulation finite element method Weak formulation finite element method Differential and integral quadrature

References

  • [1] Oden, J.T., and Reddy, J.N. (1976) An introduction to the mathematical theory of finite elements, John Wiley & Sons.
  • [2] Ochoa, O.O. and Reddy, J.N. (1992) Finite Element Analysis of Composite Laminates, Springer.
  • [3] Zienkiewicz, O.C. and Taylor R.L. (2005) The Finite Element Method for Solid and Structural Mechanics, 6th edition, Elsevier.
  • [4] Reddy, J.N. (2006) An introduction to the finite element method, 3rd edition, McGraw-Hill.
  • [5] Tornabene, F., Fantuzzi, N., Ubertini, F. and Viola, E. (2015) Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey, Applied Mechanics Reviews 67, 020801-1-55.
  • [6] Tornabene, F., Fantuzzi, N., and Bacciocchi, M. (2014) The Strong Formulation Finite Element Method: Stability and Accuracy, Fracture and Structural Integrity 29, 251-265.
  • [7] Gottlieb, D., and Orszag, S.A. (1977) Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics., SIAM.
  • [8] Boyd, J.P. (2001) Chebyshev and Fourier Spectral Methods, Dover Publications.
  • [9] Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A., (2006) Spectral Method. Fundamentals in Single Domains, Springer.
  • [10] Bellman, R., and Casti, J., (1971) Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications 34, 235-238.
  • [11] Bert, C.W., and Malik, M. (1996) Differential quadrature method in computational mechanics, Applied Mechanics Reviews 49, 1-27.
  • [12] Quan, J.R., and Chang, C.T. (1989) New insights in solving distributed system equations by the quadrature method - I. Analysis, Computers & Chemical Engineering 13, 779-788.
  • [13] Striz, A.G., Chen, W.L., and Bert, C.W. (1994) Static analysis of structures by the quadrature element method (QEM), International Journal of Solids and Structures 31, 2807-2818.
  • [14] Shu, C., (2000) Differential Quadrature and Its Application in Engineering, Springer.
  • [15] Reddy, J.N. (2004) Mechanics of Laminated Composites Plates and Shells, 2nd edition, CRC Press, New York.
  • [16] Chen, C.-N. (2006) Discrete Element Analysis Methods of Generic Differential Quadratures, Springer.
  • [17] Zong, Z., and Zhang, Y.Y. (2009) Advanced Differential Quadrature Methods, CRC Press.
  • [18] Cottrell, J.A., Hughes, T.J.R. and Bazilevs, Y. (2009) Isogeometric Analysis. Toward Integration of CAD and FEA, John Wiley & Sons.
  • [19] Reali, A. (2006) An isogeometric analysis approach for the study of structural vibrations, Journal of Earthquake Engineering 10, 1-30.
  • [20] Tornabene, F., Fantuzzi, N. and Bacciocchi M. (2016) The GDQ Method for the Free Vibration Analysis of Arbitrarily Shaped Laminated Composite Shells Using a NURBS-Based Isogeometric Approach, Composite Structures 154, 190-218.
  • [21] Fantuzzi, N. and Tornabene F. (2016) Strong Formulation Isogeometric Analysis (SFIGA) for Laminated Composite Arbitrarily Shaped Plates, Composites Part B Engineering 96, 173-203.
  • [22] Viola, E., Tornabene, F. and Fantuzzi N. (2013) Generalized Differential Quadrature Finite Element Method for Cracked Composite Structures of Arbitrary Shape, Composite Structures 106, 815-834.
  • [23] Fantuzzi, N., Bacciocchi, M., Tornabene, F., Viola, E. and Ferreira, A.J.M. (2015) Radial Basis Functions Based on Differential Quadrature Method for the Free Vibration of Laminated Composite Arbitrary Shaped Plates, Composites Part B Engineering 78, 65-78.
  • [24] Tornabene, F., Fantuzzi, N., Bacciocchi, M., Neves, A.M.A. and Ferreira, A.J.M. (2016) MLSDQ Based on RBFs for the Free Vibrations of Laminated Composite Doubly-Curved Shells, Composites Part B: Engineering 99, 30-47.
  • [25] Fantuzzi, N., Tornabene, F., Bacciocchi, M., Neves, A.M.A. and A.J.M. Ferreira (2017) Stability and Accuracy of Three Fourier Expansion-Based Strong Form Finite Elements for the Free Vibration Analysis of Laminated Composite Plates, International Journal for Numerical Methods in Engineering (in press).
  • [26] Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2014) The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free vibration analysis, Composite Structures 116, 637-660.
  • [27] Tornabene, F., Brischetto, S., Fantuzzi, N., and Bacciocchi M. (2016) Boundary Conditions in 2D Numerical and 3D Exact Models for Cylindrical Bending Analysis of Functionally Graded Structures, Shock and Vibration Vol. 2016, Article ID 2373862, 1-17.
  • [28] Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Reddy J.N. (2017) An Equivalent Layer-Wise Approach for the Free Vibration Analysis of Thick and Thin Laminated Sandwich Shells, Applied Sciences 7, 17, 1-34.
  • [29] Brischetto, S., Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2017) Interpretation of Boundary Conditions in the Analytical and Numerical Shell Solutions for Mode Analysis of Multilayered Structures, International Journal of Mechanical Sciences 122, 18-28.
  • [30] Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E. and Reddy J.N. (2017) A Numerical Investigation on the Natural Frequencies of FGM Sandwich Shells with Variable Thickness by the Local Generalized Differential Quadrature Method, Applied Sciences 7, 131, 1-39.
  • [31] Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016) Laminated Composite Doubly-Curved Shell Structures. Differential Geometry. Higher-Order Structural Theories, Esculapio.
  • [32] Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016) Laminated Composite Doubly-Curved Shell Structures. Differential and Integral Quadrature. Strong Formulation Finite Element Method, Esculapio.

Year 2017, Volume: 9 Issue: 2, 1 - 21, 14.04.2017

Francesco Tornabene Nicholas Fantuzzi Michele Bacciocchi

https://doi.org/10.24107/ijeas.304376
Cited By: 3

Abstract

References

  • [1] Oden, J.T., and Reddy, J.N. (1976) An introduction to the mathematical theory of finite elements, John Wiley & Sons.
  • [2] Ochoa, O.O. and Reddy, J.N. (1992) Finite Element Analysis of Composite Laminates, Springer.
  • [3] Zienkiewicz, O.C. and Taylor R.L. (2005) The Finite Element Method for Solid and Structural Mechanics, 6th edition, Elsevier.
  • [4] Reddy, J.N. (2006) An introduction to the finite element method, 3rd edition, McGraw-Hill.
  • [5] Tornabene, F., Fantuzzi, N., Ubertini, F. and Viola, E. (2015) Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey, Applied Mechanics Reviews 67, 020801-1-55.
  • [6] Tornabene, F., Fantuzzi, N., and Bacciocchi, M. (2014) The Strong Formulation Finite Element Method: Stability and Accuracy, Fracture and Structural Integrity 29, 251-265.
  • [7] Gottlieb, D., and Orszag, S.A. (1977) Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics., SIAM.
  • [8] Boyd, J.P. (2001) Chebyshev and Fourier Spectral Methods, Dover Publications.
  • [9] Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A., (2006) Spectral Method. Fundamentals in Single Domains, Springer.
  • [10] Bellman, R., and Casti, J., (1971) Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications 34, 235-238.
  • [11] Bert, C.W., and Malik, M. (1996) Differential quadrature method in computational mechanics, Applied Mechanics Reviews 49, 1-27.
  • [12] Quan, J.R., and Chang, C.T. (1989) New insights in solving distributed system equations by the quadrature method - I. Analysis, Computers & Chemical Engineering 13, 779-788.
  • [13] Striz, A.G., Chen, W.L., and Bert, C.W. (1994) Static analysis of structures by the quadrature element method (QEM), International Journal of Solids and Structures 31, 2807-2818.
  • [14] Shu, C., (2000) Differential Quadrature and Its Application in Engineering, Springer.
  • [15] Reddy, J.N. (2004) Mechanics of Laminated Composites Plates and Shells, 2nd edition, CRC Press, New York.
  • [16] Chen, C.-N. (2006) Discrete Element Analysis Methods of Generic Differential Quadratures, Springer.
  • [17] Zong, Z., and Zhang, Y.Y. (2009) Advanced Differential Quadrature Methods, CRC Press.
  • [18] Cottrell, J.A., Hughes, T.J.R. and Bazilevs, Y. (2009) Isogeometric Analysis. Toward Integration of CAD and FEA, John Wiley & Sons.
  • [19] Reali, A. (2006) An isogeometric analysis approach for the study of structural vibrations, Journal of Earthquake Engineering 10, 1-30.
  • [20] Tornabene, F., Fantuzzi, N. and Bacciocchi M. (2016) The GDQ Method for the Free Vibration Analysis of Arbitrarily Shaped Laminated Composite Shells Using a NURBS-Based Isogeometric Approach, Composite Structures 154, 190-218.
  • [21] Fantuzzi, N. and Tornabene F. (2016) Strong Formulation Isogeometric Analysis (SFIGA) for Laminated Composite Arbitrarily Shaped Plates, Composites Part B Engineering 96, 173-203.
  • [22] Viola, E., Tornabene, F. and Fantuzzi N. (2013) Generalized Differential Quadrature Finite Element Method for Cracked Composite Structures of Arbitrary Shape, Composite Structures 106, 815-834.
  • [23] Fantuzzi, N., Bacciocchi, M., Tornabene, F., Viola, E. and Ferreira, A.J.M. (2015) Radial Basis Functions Based on Differential Quadrature Method for the Free Vibration of Laminated Composite Arbitrary Shaped Plates, Composites Part B Engineering 78, 65-78.
  • [24] Tornabene, F., Fantuzzi, N., Bacciocchi, M., Neves, A.M.A. and Ferreira, A.J.M. (2016) MLSDQ Based on RBFs for the Free Vibrations of Laminated Composite Doubly-Curved Shells, Composites Part B: Engineering 99, 30-47.
  • [25] Fantuzzi, N., Tornabene, F., Bacciocchi, M., Neves, A.M.A. and A.J.M. Ferreira (2017) Stability and Accuracy of Three Fourier Expansion-Based Strong Form Finite Elements for the Free Vibration Analysis of Laminated Composite Plates, International Journal for Numerical Methods in Engineering (in press).
  • [26] Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2014) The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free vibration analysis, Composite Structures 116, 637-660.
  • [27] Tornabene, F., Brischetto, S., Fantuzzi, N., and Bacciocchi M. (2016) Boundary Conditions in 2D Numerical and 3D Exact Models for Cylindrical Bending Analysis of Functionally Graded Structures, Shock and Vibration Vol. 2016, Article ID 2373862, 1-17.
  • [28] Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Reddy J.N. (2017) An Equivalent Layer-Wise Approach for the Free Vibration Analysis of Thick and Thin Laminated Sandwich Shells, Applied Sciences 7, 17, 1-34.
  • [29] Brischetto, S., Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2017) Interpretation of Boundary Conditions in the Analytical and Numerical Shell Solutions for Mode Analysis of Multilayered Structures, International Journal of Mechanical Sciences 122, 18-28.
  • [30] Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E. and Reddy J.N. (2017) A Numerical Investigation on the Natural Frequencies of FGM Sandwich Shells with Variable Thickness by the Local Generalized Differential Quadrature Method, Applied Sciences 7, 131, 1-39.
  • [31] Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016) Laminated Composite Doubly-Curved Shell Structures. Differential Geometry. Higher-Order Structural Theories, Esculapio.
  • [32] Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016) Laminated Composite Doubly-Curved Shell Structures. Differential and Integral Quadrature. Strong Formulation Finite Element Method, Esculapio.
There are 32 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Francesco Tornabene 0000-0002-5968-3382

Nicholas Fantuzzi 0000-0002-8406-4882

Michele Bacciocchi This is me 0000-0002-1152-2336

Publication Date April 14, 2017
Published in Issue Year 2017 Volume: 9 Issue: 2

Cite

APA Tornabene, F., Fantuzzi, N., & Bacciocchi, M. (2017). Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability. International Journal of Engineering and Applied Sciences, 9(2), 1-21. https://doi.org/10.24107/ijeas.304376
AMA Tornabene F, Fantuzzi N, Bacciocchi M. Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability. IJEAS. July 2017;9(2):1-21. doi:10.24107/ijeas.304376
Chicago Tornabene, Francesco, Nicholas Fantuzzi, and Michele Bacciocchi. “Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability”. International Journal of Engineering and Applied Sciences 9, no. 2 (July 2017): 1-21. https://doi.org/10.24107/ijeas.304376.
EndNote Tornabene F, Fantuzzi N, Bacciocchi M (July 1, 2017) Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability. International Journal of Engineering and Applied Sciences 9 2 1–21.
IEEE F. Tornabene, N. Fantuzzi, and M. Bacciocchi, “Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability”, IJEAS, vol. 9, no. 2, pp. 1–21, 2017, doi: 10.24107/ijeas.304376.
ISNAD Tornabene, Francesco et al. “Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability”. International Journal of Engineering and Applied Sciences 9/2 (July 2017), 1-21. https://doi.org/10.24107/ijeas.304376.
JAMA Tornabene F, Fantuzzi N, Bacciocchi M. Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability. IJEAS. 2017;9:1–21.
MLA Tornabene, Francesco et al. “Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability”. International Journal of Engineering and Applied Sciences, vol. 9, no. 2, 2017, pp. 1-21, doi:10.24107/ijeas.304376.
Vancouver Tornabene F, Fantuzzi N, Bacciocchi M. Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability. IJEAS. 2017;9(2):1-21.

Cited By

Refined shear deformation theories for laminated composite arches and beams with variable thickness: Natural frequency analysis
Engineering Analysis with Boundary Elements
Francesco Tornabene
https://doi.org/10.1016/j.enganabound.2017.07.029
Finite Element Vibration Analysis of Variable Thickness Thin Cylindrical FGM Shells under Various Boundary Conditions
Materials Performance and Characterization
Iman Fakhari Golpayegani
https://doi.org/10.1520/MPC20180148
Free vibration analysis of a rotary smart two directional functionally graded piezoelectric material in axial symmetry circular nanoplate
Mechanical Systems and Signal Processing
Mohammad Mahinzare
https://doi.org/10.1016/j.ymssp.2017.07.041
 Download Cover Image

21357download