Nonlinear Model Reduction for Rubber Components in Vehicle Engineering

Transcription

1 Nonlinear Model Reduction for Rubber Components in Vehicle Engineering Dr. Sabrina Herkt, Dr. Klaus Dreßler Fraunhofer Institut für Techno- und Wirtschaftsmathematik Kaiserslautern Prof. Rene Pinnau Universität Kaiserslautern Modred2010, Berlin,

2 Multibody Simulation for Vehicles Vehicles are assessed according to handling, comfort and durability Simulation of long time spans Rigid bodies, connected by joints and force elements T M ( q) qɺɺ = f ( q, qɺ, t) G ( q) λ 0 = g( q) System of Differential Algebraic Equations (DAEs) Few degrees of freedom: O(10 2 ) 2

3 Flexible Bodies in Vehicle Simulation Initial situation: Nonlinear dynamical FE-model e.g. tyre, rubber bushing,... Muɺɺ + R( u, uɺ ) = f ext Muɺɺ + Duɺ + R( u) = f ext Large number of DOFs Only suitable for the computation of short time spans Can not be used directly for problems of durability 3

4 Methods of Model Reduction Reduction of structural mechanics e.g. beams, shells,... Projection of a large system of equations onto a low-dimensional subspace Methods for linear systems: Systems of first order: balanced truncation, Krylov subspace methods,... Systems of second order: Craig-Bampton and relatives (frequency response modes,...) Methods for nonlinear systems: Extensions of linear methods, POD P: 4

5 Linear Model Reduction Craig-Bampton Method Classical approach: Assume small deformations ODE system of flexible body can be linearised: Muɺɺ+ Ku = f ( t) constant matrices! Modal representation: M k = 1 2 ( ) ( ) ( ) u( x, t) = p t ϕ x with K λ M ϕ = 0 Mp ˆɺɺ + Kp ˆ = fˆ ( t) k k k k ˆ T ˆ T ˆ T with M = Φ M Φ = diag, K = Φ KΦ = diag, f ( t) = Φ f ( t) Reduction: Use only modes in relevant frequency range! 5

6 Nonlinear Model Reduction Proper Orthogonal Decomposition Given a data set find a subspace which minimises the projection error { } u H, i = 1 m, dim u,..., u = d i { ϕ ϕ } S = span 1,..., l 1 m of given dimension l (snapshots) E( ϕ,..., ϕ ) = u P ( u ) min! 1 m l j S j j= 1 2 ϕ, ϕ = i j δ ij with projection P ( u S j ) = l i = 1 u j, ϕ i ϕ i Optimality conditions lead to eigenvalue problem of correlation matrix m j =1 u j u j, ϕ = λ ϕ i i i T = u, u = U U ij i j KM v = λ v, ϕ = U v j j j j j Eigenvectors of largest eigenvalues yield optimal basis (Karhunen-Loève) KM 6

7 Nonlinear Model Reduction: Subspace POD Proper Orthogonal Decomposition 1. Acquire data set for unknown variable (solution of full system) Take snapshots 2. Construct a subspace that optimally approximates the given data Build correlation matrix and solve eigenvalue problem 3. Solve nonlinear equations on POD subspace Projected system of equations 7

8 Reduction of Nonlinear Models in Practice Linear case: Projection at beginning of computation yields reduced system Nonlinear case: some terms of full system required no reduction if composition of equation system takes longer than solving it! In practice: nonlinearity not explicitly known! use of commercial tools necessary black box! Decouple reduced from full system 8

9 Lookup Table Approach Store lookup data from full model: for each snapshot or chosen time instances { ui, R ( ui ), K ( ui )} Surrogate model: ( ) ˆ Mu ˆɺɺ + Du ˆɺ + Rˆ u = f ext Newton iteration in each time step: for current state use ( ) ( ) ( )( ) ( 2 ) τ = i + i τ i + Ο τ i R u R u K u u u u u u τ ( τ ) = ( i ) + Ο( τ i ) K u K u u u Lookup method: First order Taylor expansion 9

10 Nonlinear Example Using Abaqus: Model Setup 2880 Degrees of freedom Nonlinear material (Neo-Hooke) Geometrical nonlinearities ( ) ext Muɺɺ + Duɺ + R u = f D = αm 10

11 Nonlinear Example Using Abaqus: Procedure Static step: application of inner pressure Dynamical step: Compressive / tensile load at upper elements Reduction methods: Craig-Bampton: Eigenvalue problem using matrices K and M at beginning of dynamical step 15 eigenmodes POD Lookup: Training input yields both snapshots for POD basis and lookup table entries 15 POD modes, 151 lookup entries 11

12 Nonlinear Example Using Abaqus: Amplitude of External Load over Time Training Reference Computation: Full run Amplitude of vertical force Time 12

13 Nonlinear Example Using Abaqus: Training and Full Run Deformations Training Full run 13

14 Nonlinear Example Using Abaqus: Comparison of Displacements (Upper Ring) Craig-Bampton Abaqus Vertical displacement of upper ring POD Lookup Time 14

15 Nonlinear Example Using Abaqus: Comparison of Deformations Result and Training Abaqus result, t=0.18 Training deformation 15

16 Nonlinear Example Using Abaqus : Additional Training Amplitude of vertical force Time 16

17 Nonlinear Example Using Abaqus: Comparison of Displacements (Upper Ring) Vertical displacement of upper ring Abaqus POD Lookup POD Lookup, Extra training Time 17

18 Nonlinear Example Using Abaqus: CPU Time Solve full system (training) Take snapshots: u, K and R Build POD-basis and lookup table Read matrices from text files = time: s s Solve reduced system POD and Lookup table α? α 1 α 2 α 3 α m K ~ ( α R ~ 1) ( α1) K ~ ( α R ~ 2 ) ( α2 ) K ~ ( α R ~ 3 ) ( α3 ) K ~ ( α ) R ~ m ( αm ) 2.3 s Compare: Solve full system in Abaqus s 18

19 The BMBF Research Project SNiMoRed Multidisziplinäre Simulation, nichtlineare Modellreduktion und proaktive Regelung in der Fahrzeugdynamik Fraunhofer ITWM Technische Universität Kaiserslautern Martin-Luther-Universität Halle-Wittenberg Universität der Bundeswehr München John Deere Werke Mannheim AUDI AG Dr. K. Dreßler, Dr. S. Herkt, U. Becker Prof. Dr. B. Simeon, Prof. Dr. R. Pinnau Prof. Dr. M. Arnold Prof. Dr. M. Gerdts Dr. C. von Holst Dr. O. Schlicht Funded by BMBF 19

20 SNiMoRed: Aims Development of mathematical methods to include nonlinear components into multibody simulations of full vehicles for analyses of comfort and durability Application: tyres, rubber bushings, hydro bushings Create detailed component models Enhance nonlinear model reduction methods Develop adapted time integration methods 20

21 Creation of Detailed Component Models Tyres, rubber bushings: Nonlinear viscoelastic material behaviour Hydro bushing: Multidisciplinary modelling Fluid structure interaction Electrorheological fluids: multi field problems Co-simulation Control of active components 21

22 Nonlinear Viscoelastic Material Behaviour Relaxation of stresses under given strain Description of material behaviour yields memory integral t d F ( t) = R( x( t), t s) ds dt 0 R = Relaxation function with amplitude dependent coefficients Approach following Pipkin & Rodgers: t t τ ( ( ) ) ( ) ( ) 1 i τ 3 i, = i i R x t t C C e x t C C e x t i i F(t), x(t) x(t) F(t) 2 x a =0.2 x a = 2 x(t) 1.5 F(t) F(t), x(t) F(t) normiert F(x a = 0.1) F(x a = 1) F(x a = 2) Zeit t [s] Zeit t [s] x(t) normiert

23 Nonlinear Model Reduction and Inclusion into MBS Automatisation of the method: Optimal training excitation Enhancement of existing nonlinear model reduction methods: Contact Viscoelasticity, fluid-structure, Adaptive meshing Error definition and estimate concerning durability Adaptation of time integration methods 23