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Convolution finite element method: an alternative approach for time integration and time-marching algorithms

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Abstract

A finite element procedure is proposed for wave propagation in elastic media. The method is based on an alternative formulation for the equations of motion that can systematically be constructed for linear evolutionary partial differential equations. A weak formulation—corresponding to convolutional variational principles—is then defined, which paves the way for introducing a particular type of time-wise shape functions. Next, some mathematical characteristics of the method are investigated, and upon those properties, a new solution procedure for elastodynamics problems is proposed. Subsequently, several numerical examples are considered, including a single degree of freedom mass-spring-damper system as the prototype of structural dynamics along with 1d and 2d elastodynamics problems for the case of the wave motion in elastic solids. The present method can be considered as an alternative approach for time integration and time-marching algorithms, e.g., Newmark’s algorithm, to solve time-domain problems in elastic media.

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References

  1. Surana KS, Reddy JN (2017) The finite element method for initial value problems: mathematics and computations. CRC Press, Boca Raton

    Book  Google Scholar 

  2. Rektorys K (1982) The method of discretization in time and partial differential equations. Equadiff 5:293–296

    MATH  Google Scholar 

  3. Gottlieb S, Shu CW (1998) Total variation diminishing Runge-Kutta schemes. Math Comput 67:73–85

    Article  MathSciNet  Google Scholar 

  4. Gottlieb S, Shu CW, Tadmor E (2001) Strong stability-preserving high-order time discretization methods. SIAM Rev 43:89–112

    Article  MathSciNet  Google Scholar 

  5. Hughes TJ, Hulbert GM (1988) Space-time finite element methods for elastodynamics: formulations and error estimates. Comput Methods Appl Mech Eng 66:339–363

    Article  MathSciNet  Google Scholar 

  6. Bathe K, Wilson E (1972) Stability and accuracy analysis of direct integration methods. Earthq Eng Struct Dyn 1:283–291

    Article  Google Scholar 

  7. Hilber HM, Hughes TJ (1978) Collocation, dissipation and [overshoot] for time integration schemes in structural dynamics. Earthq Eng Struct Dyn 6:99–117

    Article  Google Scholar 

  8. Krenk S (2006) Energy conservation in Newmark based time integration algorithms. Comput Methods Appl Mech Eng 195:6110–6124

    Article  MathSciNet  Google Scholar 

  9. Hilber HM, Hughes TJ, Taylor RL (1977) Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq Eng Struct Dyn 5:283–292

    Article  Google Scholar 

  10. Wood W, Bossak M, Zienkiewicz O (1980) An alpha modification of Newmark’s method. Int J Numer Methods Eng 15:1562–1566

    Article  MathSciNet  Google Scholar 

  11. Hoff C, Pahl P (1988) Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics. Comput Methods Appl Mech Eng 67:367–385

    Article  MathSciNet  Google Scholar 

  12. Hoff C, Pahl P (1988) Practical performance of the \(\theta \)1-method and comparison with other dissipative algorithms in structural dynamics. Comput Methods Appl Mech Eng 67:87–110

    Article  MathSciNet  Google Scholar 

  13. Chung J, Hulbert G (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha \) method. J Appl Mech 60:371–375

    Article  MathSciNet  Google Scholar 

  14. Idesman AV (2007) A new high-order accurate continuous Galerkin method for linear elastodynamics problems. Comput Mech 40:261–279

    Article  MathSciNet  Google Scholar 

  15. Oden JT (1969) A general theory of finite elements. II. Applications. Int J Numer Methods Eng 1:247–259

    Article  Google Scholar 

  16. Masud A, Hughes TJ (1997) A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems. Comput Methods Appl Mech Eng 146:91–126

    Article  MathSciNet  Google Scholar 

  17. Hulbert GM, Hughes TJ (1990) Space-time finite element methods for second-order hyperbolic equations. Comput Methods Appl Mech Eng 84:327–348

    Article  MathSciNet  Google Scholar 

  18. Abedi R, Petracovici B, Haber RB (2006) A space-time discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance. Comput Methods Appl Mech Eng 195:3247–3273

    Article  MathSciNet  Google Scholar 

  19. Idesman AV (2007) Solution of linear elastodynamics problems with space-time finite elements on structured and unstructured meshes. Comput Methods Appl Mech Eng 196:1787–1815

    Article  Google Scholar 

  20. Gurtin ME (1964) Variational principles for linear elastodynamics. Arch Rational Mech Anal 16:34–50

    Article  MathSciNet  Google Scholar 

  21. Nickell RE, Sackman JL (1968) Variational principles for linear coupled thermoelasticity. Q Appl Math 26:11–26

  22. Amiri-Hezaveh A, Karimi P, Ostoja-Starzewski M (2020) IBVP for electromagneto-elastic materials: variational approach. Math Mech Complex Syst 8:47–67

  23. Peng J, Zhang J (1992) A semi-analytical approach to general transient problems and its applications to dynamics. Acta Mech Sin 24:708–716

    Google Scholar 

  24. Peng J, Lewis R, Zhang J (1995) A semi-analytic method for dynamic response analysis based on Gurtin’s variational principle. Commun Numer Methods Eng 11:297–306

    Article  MathSciNet  Google Scholar 

  25. Peng J, Lewis R, Zhang J (1996) A semi-analytical approach for solving forced vibration problems based on a convolution-type variational principle. Comput Struct 59:167–177

    Article  Google Scholar 

  26. Jianshe P, Jingyu Z, Jie Y (1997) Formulation of a semi-analytical approach based on Gurtin variational principle for dynamic response of general thin plates. Appl Math Mech (Engl Ed) 59:167–177

    MATH  Google Scholar 

  27. Js Peng, Yang J, Yq Yuan, Gb Luo (2009) A convolution-type semi-analytic DQ approach to transient response of rectangular plates. Appl Math Mech (Engl Ed) 30:1143–1151

    Article  MathSciNet  Google Scholar 

  28. Hughes TJ (2012) The finite element method: linear static and dynamic finite element analysis. Courier Corporation, North Chelmsford

    Google Scholar 

  29. Ignaczak J (1959) Direct determination of stresses from the stress equations of motion in elasticity. Arch Mech Stos 11:671–678

    MathSciNet  MATH  Google Scholar 

  30. Ignaczak J (1963) A completeness problem for stress equations of motion in the linear elasticity theory. Arch Mech Stos 15:225

    MathSciNet  MATH  Google Scholar 

  31. Gottlieb D, Shu CW (1997) On the Gibbs phenomenon and its resolution. SIAM Rev 39:644–668

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank anonymous reviewers for their constructive comments that improved the quality of the manuscript.

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Authors and Affiliations

  1. Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, USA

    A. Amiri-Hezaveh & M. Ostoja-Starzewski

  2. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, USA

    A. Masud

  3. Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, USA

    M. Ostoja-Starzewski

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  1. A. Amiri-Hezaveh
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  2. A. Masud
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  3. M. Ostoja-Starzewski
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Correspondence to M. Ostoja-Starzewski.

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Amiri-Hezaveh, A., Masud, A. & Ostoja-Starzewski, M. Convolution finite element method: an alternative approach for time integration and time-marching algorithms. Comput Mech 68, 667–696 (2021). https://doi.org/10.1007/s00466-021-02046-w

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