Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Computational Mechanics
Published in: Computational Mechanics 2/2020

12-10-2019 | Original Paper

A new conservative/dissipative time integration scheme for nonlinear mechanical systems

Authors: Cristian Guillermo Gebhardt, Ignacio Romero, Raimund Rolfes

Published in: Computational Mechanics | Issue 2/2020

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.
previous article Anatomically realistic lumen motion representation in patient-specific space–time isogeometric flow analysis of coronary arteries with time-dependent medical-image data
next article Selective enrichment of moment fitting and application to cut finite elements and cells

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Appendix
Available only for authorised users
Literature
1.
Arnold VI (1989) Mathematical methods of classical mechanics. Springer, Berlin
2.
Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration, 2nd edn. Springer, BerlinMATH
3.
Stuart AM, Humphries AR (1996) Dynamical systems and numerical analysis. Cambridge University Press, CambridgeMATH
4.
Simo JC, Tarnow N (1992) The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z Angew Math Phys 43:757–792MathSciNetMATH
5.
Simo JC, Tarnow N (1994) A new energy and momentum conserving algorithm for the non-linear dynamics of shells. Int J Numer Methods Eng 37:2527–2549MathSciNetMATH
6.
7.
Kane C, Marsden JE, Ortiz M (1999) Symplectic-energy-momentum preserving variational integrators. J Math Phys 40:3353–3371MathSciNetMATH
8.
McLachlan RI, Quispel GRW, Robideux N (1999) Geometric integration using discrete gradients. Philos Trans Math Phys Eng Sci 357:1021–1045MathSciNetMATH
9.
Armero F, Romero I (2001) On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics. Comput Methods Appl Mech Eng 190:2603–2649MATH
10.
11.
Simo JC, Tarnow N, Doblaré M (1995) Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. Int J Numer Methods Eng 38:1431–1473MathSciNetMATH
12.
Romero I, Armero F (2002) An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics. Int J Numer Methods Eng 54:1683–1716MathSciNetMATH
13.
Armero F, Petocz E (1998) Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems. Comput Methods Appl Mech Eng 158:269–300MathSciNetMATH
14.
Goicolea JM, García Orden JC (2000) Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy-momentum schemes. Comput Methods Appl Mech Eng 188:789–804MathSciNetMATH
15.
Betsch P, Hesch C, Sänger N, Uhlar S (2010) Variational integrators and energy-momentum schemes for flexible multibody dynamics. J Comput Nonlinear Dyn 5:031001-1–031001-11
16.
Gonzalez O (2000) Exact energy-momentum conserving algorithms for general models in nonlinear elasticity. Comput Methods Appl Mech Eng 190:1763–1783MathSciNetMATH
17.
Laursen TA, Meng XN (2001) A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elastodynamics. Comput Methods Appl Mech Eng 190:6300–6309MathSciNetMATH
18.
Gotusso L (1985) On the energy theorem for the Lagrange equations in the discrete case. Appl Math Comput 17:129–136MathSciNetMATH
19.
Itoh T, Abe K (1988) Hamiltonian-conserving discrete canonical equations based on variational difference quotients. J Comput Phys 76:85–102MathSciNetMATH
20.
Romero I (2012) An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics. Comput Mech 50:603–610MathSciNetMATH
21.
Harten A, Lax B, Leer P (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev 25:35–61MathSciNetMATH
22.
French DA, Schaeffer JW (1990) Continuous finite element methods which preserve energy properties for nonlinear problems. Appl Math Comput 39:271–295MathSciNetMATH
23.
Groß M, Betsch P, Steinmann P (2005) Conservation properties of a time fe method. Part IV: higher order energy and momentum conserving schemes. Int J Numer Methods Eng 63:1849–1897MATH
24.
Betsch P, Janz A, Hesch C (2018) A mixed variational framework for the design of energy-momentum schemes inspired by the structure of polyconvex stored energy functions. Comput Methods Appl Mech Eng 335:660–696MathSciNet
25.
García Orden JC (2018) Energy and symmetry-preserving formulation of nonlinear constraints and potential forces in multibody dynamics. Nonlinear Dyn 95:823–837
26.
Kuhl D, Ramm E (1996) Constraint energy momentum algorithm and its application to nonlinear dynamics of shells. Comput Methods Appl Mech Eng 136:293–315MATH
27.
Kuhl D, Crisfield MA (1999) Energy-conserving and decaying algorithms in non-linear structural dynamics. Int J Numer Methods Eng 45:569–599MathSciNetMATH
28.
Bottasso CL, Borri M (1997) Energy preserving/decaying schemes for non-linear beam dynamics using the helicoidal approximation. Comput Methods Appl Mech Eng 143:393–415MathSciNetMATH
29.
Bottasso CL, Borri M, Trainelli L (2001) Integration of elastic multibody systems by invariant conserving/dissipating algorithms. II. Numerical schemes and applications. Comput Methods Appl Mech Eng 190:3701–3733MathSciNetMATH
30.
Romero I (2016) High frequency dissipative integration schemes for linear and nonlinear elastodynamics. In: Betsch P (ed) Structure-preserving integrators in nonlinear structural dynamics and flexible multibody dynamics. Springer, Berlin, pp 1–30
31.
Armero F, Romero I (2001) On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods. Comput Methods Appl Mech Eng 190:6783–6824MATH
32.
Romero I, Armero F (2002) Numerical integration of the stiff dynamics of geometrically exact shells: an energy-dissipative momentum-conserving scheme. Int J Numer Methods Eng 54:1043–1086MathSciNetMATH
33.
Armero F, Romero I (2003) Energy-dissipative momentum-conserving time-stepping algorithms for the dynamics of nonlinear cosserat rods. Comput Mech 31:3–26MathSciNetMATH
34.
Gebhardt CG, Hofmeister B, Hente C, Rolfes R (2019) Nonlinear dynamics of slender structures: a new object-oriented framework. Comput Mech 63:219–252MathSciNetMATH
35.
Gebhardt CG, Steinbach MC, Rolfes R (2019) Understanding the nonlinear dynamics of beam structures: a principal geodesic analysis approach. Thin-Walled Struct 140:357–372
36.
Kerschen G, Golinval J-C, Vakakis AF, Bergman LA (2005) The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn 41:147–169MathSciNetMATH
37.
Jansen EL (2005) Dynamic stability problems of anisotropic cylindrical shells via simplified analysis. Nonlinear Dyn 39:349–367MATH
38.
Kreiss H-O, Ortiz OE (2014) Introduction to numerical methods for time dependent differential equations. Wiley, LondonMATH
39.
Kopmaz O, Gündoğdu O (2003) On the curvature of an Euler–Bernoulli beam. Int J Mech Eng Educ 31:132–142
40.
Nayfeh AH, Pai PF (2008) Linear and nonlinear structural mechanics. Wiley, LondonMATH
41.
Ye F, Zi-Xiong G, Yi-Chao G (2018) An unconditionally stable explicit algorithm for nonlinear structural dynamics. J Eng Mech 144:04018034-1–04018034-8
42.
Gebhardt CG, Rolfes R (2017) On the nonlinear dynamics of shell structures: combining a mixed finite element formulation and a robust integration scheme. Thin-Walled Struct 118:56–72
43.
Betsch P, Sänger N (2009) On the use of geometrically exact shells in a conserving framework for flexible multibody dynamic. Comput Methods Appl Mech Eng 198:1609–1630MathSciNetMATH
44.
Vu-Quoc L, Tan XG (2003) Optimal solid shells for non-linear analyses of multilayer composites. II. Dynamics. Comput Methods Appl Mech Eng 192:1017–1059MATH
Metadata
Title
A new conservative/dissipative time integration scheme for nonlinear mechanical systems
Authors
Cristian Guillermo Gebhardt
Ignacio Romero
Raimund Rolfes
Publication date
12-10-2019
Publisher
Springer Berlin Heidelberg
Published in
Computational Mechanics / Issue 2/2020
Print ISSN: 0178-7675
Electronic ISSN: 1432-0924
DOI
https://doi.org/10.1007/s00466-019-01775-3

Other articles of this Issue 2/2020

Computational Mechanics 2/2020 Go to the issue

Original Paper

Fracture simulation of viscoelastic polymers by the phase-field method

Original Paper

A constrained spline dynamics (CSD) method for interactive simulation of elastic rods

Original Paper

Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

Original Paper

An investigation of stress inaccuracies and proposed solution in the material point method

Original Paper

Strain localization analysis of Hill’s orthotropic elastoplasticity: analytical results and numerical verification

Original Paper

Geometrically nonlinear analysis of solids using an isogeometric formulation in boundary representation