Model order reduction of mechanical systems subjected to moving loads by the approximation of the input

Transcription

1 Model order reduction of mechanical systems subjected to moving loads by the approximation of the input Alexander Vasilyev, Tatjana Stykel Institut für Mathematik, Universität Augsburg ModRed 2013 MPI Magdeburg, December 11-13, 2013

2 I. Inrtoduction Elastic multibody systems (EMBS) EMBS with moving loads Vehicle-bridge interaction Working gears 2 Cableways etc.

3 I. Inrtoduction FEM PDEs 3 ODEs Example 4 2 a0 4 w(x,t )+ a1 2 w (x,t)+ a2 w(x,t )= ρ (x, t)u (t) t x t FEM [] b1 (t) b(t )= b2 (t), b N (t) l bi (t)= ρ (x, t) φ i (x)dx, i=1,..., N 0 M q (t)+ D q (t )+ K q(t) = b(t)u(t) y (t) = C (t)q(t) M, D, K ℝ N N, q ℝ N, b(t) ℝ N, C (t ) ℝ p N, y (t) ℝ p many forces input matrix B (t)ℝ N m instead of b(t)

4 I. Inrtoduction High computational cost 4 Model order reduction (MOR) MOR by projection: q (t) V q (t ), q ℝr, r N W T M V q (t )+W T D V q (t)+w T K V q (t) = W T B(t)u(t) r r M ℝ r r D ℝ r r K ℝ (t) ℝ r m B y (t) = C (t)v q (t) y (t) ℝ p (t) ℝ p r C Systems with time-varying input and/or output matrices: V,W???

5 II. Approximation of the input matrix ρ (x, t)=g(x ζ (t)), ζ (t) - a position of a «centre of force» at t ζ (t) Ω [0, l] [] b1 (t) b(t )= b2 (t), b N (t) l bi (t)= g (x ζ (t)) φ i (x )dx, i=1,..., N 0 Consider a SISO system M q (t)+ D q (t )+ K q(t) = b(t)u(t) T y (t) = b (t)q(t) Naive approach: M q (t)+ D q (t )+ K q(t) = I (b (t)u (t)) T y (t) = b (t)q(t) Difficulty: many inputs with t [0,T ] 5

6 II. Approximation of the input matrix Goal: approximate b(t) in a lower dimension subspace n χ (ζ(t))= b(t ) B bi χ i (ζ (t)), n N i=1 M q (t)+ D q (t )+ K q (t) = B u (t) with T y (t) = B q (t ) u (t)= χ (ζ (t))u(t) T Note: y (t) χ ( ζ (t)) y (t) Error bound: [ ] [ ] q (t) q(t ) q (t ) q (t) χ η b B Two approximation approaches:, find the vector χ (ζ ) B 2. given the vector χ (ζ ), find the matrix B 1. given the matrix such that b B χ min 6

7 II. Approximation of the input matrix [ ] [ ][ b11 ϕ 1 (ζ (t)) b(t )= ϕ 2 ( ζ (t)) b21 b N 1 ϕ N (ζ (t)) b 1 n b2 n bn n ] χ 1 (ζ (t)) = B χ (ζ (t)) χ n (ζ (t)) n approximation by polynomial expansion ϕ i(x ) bij P j 1( x), i=1,..., N, j=1 where P0 (x),..., Pn 1 (x) are orthogonal polynomials; n B-spline interpolation ϕ i( x ) bij β j 2 ( x), i=1,..., N j=1 where β 1 (x),..., β n 2 (x) are B-splines; n (n) linear least square method (LLSM) ϕ i(x )=ϕ (x) b ij φ j (x ), i=1,..., N (N ) i j=1 (n) (n ) where φ 1 (x),..., φ n (x) are FEM basis functions on a coarse grid; empirical interpolation method (EIM) [Barrault, Maday, Nguyen, Patera, 2004] 7

8 III. Model order reduction of mechanical systems 8 Balanced truncation solving Lyapunov equations is required use SO-LR-ADI method specially adapted for second-order systems [Benner, Kürschner, Saak, 2012] But, for mechanical systems with a weak damping, this method converges very slowly Krylov subspace methods SOAR [Bai, Su, 2005; Salimbahrami, 2005] SOR-IRKA, SO-IRKA [Wyatt, 2012] AORA [Lee, Chu, Feng, 2004; Bodendiek, Bollhöfer, 2013] MIRKA [Soppa, 2011] Choice of interpolation points and directions for second-order systems is still unclear Our approach: subspace acceleration poles finding combined with an extention of a frequency range

9 IV. Numerical experiments 9 Test model with a moving load: 1D Euler-Bernoulli beam equation 2 4 ρ A 2 w (x, t)+2 ρ A ω d w(x, t)+ EI 4 w (x,t)=δ (x vt)u t t x ( x, t) ( 0,l) (0,T ) (has an analytical solution) w (x, t) is a vertical deflection of the beam δ (x ζ (t )) is a point force density v is a velocity of the moving load ζ (t)=vt is an instantaneous position of a force u is a magnitude of the moving load ρ A ωd E I is a mass density is a cross section area is a circular frequency of damping is an Young modulus is an area moment of inertia with simply supported ends of the beam w (0, t)=0, w (l, t)=0, 2 w (0, t)=0, x2 2 w (l, t)=0 x2 and initial conditions w (x,0)=0, [Fryba, 1999] w (x,0)=0 t

10 IV. Numerical experiments 10 input distribution vector [] b1 (t) b(t )= b2 (t), b N (t) l bi (t )= δ(x ζ (t )) φ i (x)dx=φ i (ζ (t )), i=1,.., N, 0 where φ i (x ) is a finite element method basis function corresponded to some node and ζ (t) Ω =[0, l] M q (t)+ D q (t )+ K q(t) = b(t)u y (t) = b T (t)q(t) w b(t ) - moving input distribution b(t )u - moving load x c (t)=bt (t) - moving output distribution y (t) - moving observation

11 IV. Numerical experiments Approximation of FEM basis functions n φ i (ζ (t)) b ij χ j (t), j=1,..., N, n N j=1 x i 1 xi x i+1 x i 1 FEM basis functions φ i (x) xi x i+1 11

12 IV. Numerical experiments Approximated output by approximations of the input n=50 deflection / m absolute error / m N=5000, time / sec. time / sec. 50 φ i (ζ (t)) b ij χ j (t), i=1,...,5000 j=1 12

13 IV. Numerical experiments Approximated output by approximations of the input n=50 deflection / m absolute error / m N=50, time / sec. time / sec. 13

14 IV. Numerical experiments 14 Approximated output of the reduced system with moving load n=50, r =20 deflection / m absolute error / m N=5000, time / sec. time / sec. Reduction is carried out by the software MatMorembs Eberhard, Lehner, Fehr, Nowakowski, Fischer, Kürschner et al.

15 V. Conclusion It was considered: second-order systems with moving load approximation of time-varying input matrix model reduction methods for mechanical systems Further work: search of optimal methods to reduction of mechanical systems search of new approaches to model reduction of systems with moving load 15

16 V. Conclusion Further work: consideration of more realistic models for example, a coupled bridge-vehicle system beam subjected to a moving two-axle system Thank you for your attention! 16