Georgia Institute of Technology An attempt at the classification of energy-decaying schemes for structural and multibody dynamics

Transcription

1 Politecnico di Milano Georgia Institute of Technology An attempt at the classification of energy-decaying schemes for structural and multibody dynamics Carlo L. Bottasso Georgia Institute of Technology, GA, USA Lorenzo Trainelli Politecnico di Milano, Italy Multibody Dynamics 2003 Lisbon, July 1 4, 2003

2 Presentation Outline Background on energy-decaying schemes Motivation and characteristics Methodology Underlying time discretization schemes ED scheme A ED scheme B ED scheme C Discussion of their relations Analysis of discretization patterns Std linear indicators: spectral radius & rel. period error Conclusions

3 Motivation for Energy-Decaying Schemes Structural and multibody dynamics lead to stiff, non-linear FE problems Linearly unconditionally stable, high-frequency dissipative integrators are designed for very general problems, but loose their properties in the non-linear regime Non-linear unconditionally stable, high-frequency dissipative integrators: energy-decaying (ED) schemes

4 Energy Decay ED schemes are built to yield a discrete bound on the total mechanical energy in a typical time step for vanishing loads This implies two highly desirable features: Non-linear unconditional stability (energy method) Damping of unresolved and spurious frequencies Minor drawback: need specialization the integrator must know which equations it is solving

5 Preservation vs. Decay energy-preserving (EP) scheme Typically, although still non-linearly unconditionally stable, EP schemes are not well suited for stiff FE problems: high frequency oscillations corrupt the response (see velocity and stress fields ) Energy manifold Energy preserving solution T D Energy decaying solution

6 Structural dynamics EP/ED Construction An EP/ED scheme is designed for each body model: rigid body, cable, beam, shell, Multibody dynamics An energy-consistent integration is developed for each joint model: revolute, prismatic, spherical, Reaction forces at an ideal, time-independent joint perform null algorithmic work within each time step Result: the multibody system inherits the EP/ED properties of the component bodies

7 EP/ED Methodology Basically, two ingredients enter the formulation of an EP/ED method: An underlying time discretization scheme A number of accompanying details These details are typically model-dependent dependent and crucial to the rigorous proof of the energy bound Parameterization of rotations Spatial semi-discretization Specific interpolations, averages, approximations

8 EP/ED Methodology (cont cont.) The details can impact considerably on other conservation properties: Preservation of momenta Symplecticity Preservation of hidden constraints Frame-indifference Example: Solutions that satisfy the invariants Solutions that satisfy the constraints Drifting solution System manifold Constraint manifold Manifold of the invariants

9 The Underlying Scheme We restrict the analysis to the sole underlying time discretization schemes of viable methods (2nd order) Proposed EP methods are all based on modified versions of mid-point/trapezoidal rule: differences lie in the details Proposed ED methods are based on a few discretization patterns We skip the details that incorporate the specific knowledge of the governing equations to be solved Therefore, the linear oscillator provides a simple and meaningful model problem

10 The Linear Oscillator Governing equations Total energy where

11 ED Scheme A Start by applying the TDG method Weak form over a time step Jump discontinuity: Both trial and test functions chosen as linear polynomials

12 ED Scheme A A (cont.)... end up with ED scheme A The discretized equations imply the energy bound where

13 ED Scheme A A (concl.) The scheme can also be interpreted as a L-stable, stiffly accurate, 2nd / 3rd order RK method First presented in 1996 (Bauchau et al.) and 1997 (Bottasso and Borri) Formulated in many versions with different details for geometrically exact beam and shell dynamics, 3-D elastodynamics, MBS dynamics Demonstrated on a number of complex, real-life engineering applications (e.g. full-scale helicopter and tilt-rotor dynamics) Example: tilt-rotor whirl flutter analysis

14 ED Scheme B A finite difference method of the form Conservative approximations chosen to satisfy Dissipative approximations chosen to satisfy where is non-negative

15 ED Scheme B Conservative discretizations Dissipative discretizations B (cont.) obtained through intermediate stage values introduced by

16 ED Scheme B B (cont.) Rearranging the equations: ED scheme B The discretized equations imply the energy bound with exactly the same dissipation function as ED scheme A

17 ED Scheme B B (concl.) The scheme can also be seen as a family of L-stable, stiffly accurate, 2nd order RK methods Presented in 2001 (Armero and Romero) When the EP trapezoidal rule is recovered 3rd order accuracy only when When the scheme coincides with ED scheme A

18 ED Scheme B B (concl.) The scheme can also be seen as a family of L-stable, stiffly accurate, 2nd order RK methods Presented in 2001 (Armero and Romero) When the EP trapezoidal rule is recovered 3rd order accuracy only when When the scheme coincides with ED scheme A Parameter does not affect the amount of dissipated energy In fact, it is not a tuning parameter in the classical sense: It does not control the asymptotic value of the spectral radius (always null) It affects the cut-off frequency at the price of degraded relative period errors Again, the best performance are obtained when

19 ED Schemes A and B : Spectral Radius

20 ED Schemes A and B : Relative Period Error

21 ED Scheme C Motivation: both A and B have no control on the asymptotic value of the spectral radius Solution: modify A with a tuning parameter: ED scheme C (Borri, Bauchau, Bottasso and Trainelli, 2001)

22 ED Scheme C C (concl.) The discretized equations imply the energy bound where now with The asymptotic value of the spectral radius is (same as HHT-α method) When a 2nd / 4th order EP method is obtained When ED scheme A is recovered Hence, the best performances of B with respect to the relative period error correspond to the worst case of C

23 ED Scheme C : Spectral Radius

24 ED Scheme C : Relative Period Error

25 ED Scheme C vs. Generalized-α: Spectral Radius

26 ED scheme C vs. Generalized-α: Relative Period Error

27 Conclusions An attempt to clarify the relations between indipendently proposed approaches to ED schemes for structural and multibody dynamics was made by looking at the underlying time discretization scheme Accompanying details albeit indispensable for guaranteeing non-linear high-frequency energy decay prevail in typical expositions, hiding the similarities of the schemes Scheme A coincides with the best scheme of family B wrt the relative period error Scheme C has the added advantage on A of a tunable high-frequency numerical dissipation