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Uncertainty quantification of critical speed for railway vehicle dynamics PhD student: D. Bigoni 1 Supervisors: A. P. Engsig-Karup 1 , H. True 1 , J.S. Hesthaven 2 1 The Technical University of Denmark, 2 Brown University Railway vehicle dynamics -

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Uncertainty quantification of critical speed for railway vehicle dynamics

PhD student: D. Bigoni1 Supervisors: A. P. Engsig-Karup1, H. True1, J.S. Hesthaven2

1The Technical University of Denmark, 2Brown University

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Railway vehicle dynamics - Euler’s formulation1

m¨

  • x1 + 2D2 ˙
  • x1 + 2k4

x1 + 2FX(ξx1, ξy1) + 2FX(ξx2, ξy2) = 0 I¨

  • x2 + k6

x2 + 2ha [FX(ξx1, ξy1) − FX(ξx2, ξy2)]+ +a [FY (ξx1, ξy1) + FY (ξx2, ξy2)] = 0 where FX and FY are the creep forces, and determine a non-linear coupling of x1 and

  • x2. Among other

components, these forces involve also the running velocity v of the vehicle, the conicity of the wheels and the wheel-rail friction.

1H.True and C.Kaas-Petersen 1983

2 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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Railway vehicle dynamics - Hunting

2 4 6 8 10 Time 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x1(t) 1e 4 2 4 6 8 10 Time 1.5 1.0 0.5 0.0 0.5 1.0 x2(t) 1e 5 Speed 50.0m/s 5 10 15 20 25 30 Time 4 3 2 1 1 2 3 4 x1(t) 1e 2 5 10 15 20 25 30 Time 1.0 0.5 0.0 0.5 1.0 x2(t) 1e 2 Speed 105.0m/s

80 60 40 20 Re(λ) 20 15 10 5 5 10 15 20 Im(λ)

Speed = 110m/s

3 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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Railway vehicle dynamics - Stochastic Model

Let’s now assume that the suspension components k6, k4 and D2 are known within a certain level of accuracy and model this by: k6 ∼ N(3.44 · 106, 2.96 · 1010), (std. of approx. 5%) k4 ∼ N(9.12 · 104, 4.15 · 107), (std. of approx. 7%) D2 ∼ N(1.46 · 104, 1.07 · 106), (std. of approx. 7%)

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 K6 1e6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PDF 1e 6 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 K4 1e5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1e 5 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 D2 1e4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1e 4

What are the dynamics of the system under these conditions?

4 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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Uncertainty Quantification - Traditional Approaches

Analytical Methods

  • Moment Equation
  • Perturbation Method

Pros.: recover the exact solution Cons.: problem-dependent, cumbersome Sampling Methods

  • (MC) Monte Carlo – O
  • N −1/2
  • (QMC) Quasi Monte Carlo – O
  • (log N)d /

√ N

  • (MCMC) Markov Chain Monte Carlo

Pros.: general applicability, MC convergence indepen- dent from dimensionality d Cons.: very slow convergence

1.5 1.6 1.7 1.8 1.9 2.0 2.1 K4 1e5 2.2 2.4 2.6 2.8 3.0 K6 1e6 2.4 2.6 2.8 3.0 3.2 3.4 D2 1e4 2.2 2.4 2.6 2.8 3.0 K6 1e6 2.4 2.6 2.8 3.0 3.2 3.4 D2 1e4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 K4 1e5 0.85 0.90 0.95 1.00 1.05 1.10 Linear Critical Speed 1e2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 PDF 1e 1

Figure: Linear critical speed distribution using 104 realizations for MC method.

5 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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UQ - Generalized Polynomial Chaos (gPC)2

Let Y be a r.v. with CDF FY (y). Use the Nth-degree gPC expansion of the random parameters and the solution YN =

N

  • k=0

ˆ akΦk(Z), ˆ ak = 1 γk

  • IZ

F −1

Y

(FZ(z))Φk(z)dFZ(z) uN(t, Z) =

N

  • k=0

ˆ uk(t)Φk(Z)

  • E [∂tuN(t, Z)Φk(Z)] = E [f(uN)Φk(Z)] ,

D × (0, T] ˆ uk(0) =

1 γk E [u(0, Z)Φk(Z)] ,

D × {t = 0} µu(t) ≈ E [uN(t, Z)] = ˆ u0(t) Var [u(t, Z)] ≈ Var [uN(t, Z)] = N

k=1 γk ˆ

u2

k(x, t)

where E [f(Z)] =

  • IZ f(z)dFZ(z) and {φi(Z)}N

i=0 are proper orthonormal basis. 2D.Xiu and G.Karniadakis 2004

6 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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UQ - gPC on Railway Vehicle Dynamics

The N-th order gPC expansion of the problem is given by                    E

  • ∂tu1,Nφk
  • =

E

  • u2,Nφk
  • E
  • ∂tu2,Nφk
  • =

−2E

  • D2,Nu2,Nφk
  • − 2E
  • k4,Nu1,Nφk
  • −2E [(FX(ξx1, ξy1) + FX(ξx2, ξy2)) φk]

E

  • ∂tu3,Nφk
  • =

E

  • u4,Nφk
  • E
  • ∂tu4,Nφk
  • =

−E

  • k6,Nu3,Nφk
  • − 2haE [(FX(ξx1, ξy1) − FX(ξx2, ξy2)) φk]

−aE [(FY (ξx1, ξy1) + FY (ξx2, ξy2)) φk] where k is a multi index such that ui,N(t, Z) =

  • |k|≤N

ˆ uk(t)Φk(Z), i = 1, . . . , 4 We obtain a system of K = N

i=0

i + (d − 1) (d − 1)

  • coupled equations that can be treated

using standard ODE solvers. The following table shows how this number scales:

N 1 2 3 4 5 6 7 d = 1 2 3 4 5 6 7 8 d = 2 3 6 10 15 21 28 36 d = 3 4 10 20 35 56 84 120 d = 4 5 15 35 70 126 210 330 d = 5 6 21 56 126 252 462 792 7 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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UQ - gPC on Railway Vehicle Dynamics

1 2 3 4 5 Time 1.0 0.5 0.0 0.5 1.0 1.5 Lateral Displacement 1e 2

MC vs gPC(N=5) - v=70m/s Monte Carlo gPC

(a) Mean and variance

1 2 3 4 5 Time 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Lateral Displacement - Variance 1e 7

MC vs gPC(N=5) - v=70m/s Monte Carlo gPC

(b) Variance

50 100 150 200 250 N 10

  • 5

10

  • 4

kE[u]−µ max

u

k2 50 100 150 200 250 N 10

  • 9

10

  • 8

10

  • 7

kVar[u]−σ max

u

k2

Pros: Elegant fomulation, one single solution of the system,

  • ptimal accuracy

Cons: Intrusive and cumbersome to implement, non-linearities must be treated carefully, weak on time-dependent problems (but there exist improvements).

8 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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UQ - Probabilistic Collocation Methods (PCM)

Solve the deterministic ODE on a ”proper” set ΘM =

  • Z(j)M

j=1 of nodes in the

random space:

  • ∂tu(t, Z(j)) = f(u),

D × (0, T] u(0) = u0, D × {t = 0} This will give u(j) = u(t, Z(j)) solutions on which we can apply interpolation rules or projection rules. Let’s consider the discrete projection: uN(Z) =

  • |k|≤N

ˆ uk(t)Φk(Z) ˆ uk(t) = 1 γk E [u(t, Z)φk(Z)] = 1 γk

  • u(z)φk(z)dFZ(z)

where the integral can be computed by cubature rules using the ”properly” selected set

  • f nodes ΘM. Then statistics can be easily obtained:

µu(t) ≈ E [uN(t, Z)] = ˆ u0(t) Var [u(t, Z)] ≈ Var [uN(t, Z)] =

  • |k|≤N

γk ˆ u2

k(x, t)

Target: obtain the ”best” statistics out of the smallest number of simulation!

9 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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UQ - PCM on Railway Vehicle Dynamics

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 K4 1e6 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 K6 1e5 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 D2 1e6 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 K6 1e4 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 D2 1e5 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 K4 1e4

(c) Collocation points, N = 3

1 2 3 4 5 Time 1.0 0.5 0.0 0.5 1.0 1.5 Lateral Displacement 1e 2 MC vs PCM(4th-ord) - v=70m/s

Monte Carlo Collocation

(d) Mean and variance

1 2 3 4 5 Time 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Lateral Displacement - Variance 1e 7 MC vs PCM(4th-ord) - v=70m/s

Monte Carlo Collocation

(e) Variance Figure: PCM vs. Monte Carlo

Hermite polynomials are chosen as basis for the projection/cubature. Projection with these polynomials can be highly accurate, using proper Gauss quadrature nodes and weights, for which analytical formulas exist.

50 100 150 200 250 300 350 N 10

  • 7

10

  • 6

10

  • 5

10

  • 4

kE[u]−µ max

u

k2 50 100 150 200 250 300 350 N 10

  • 10

10

  • 9

10

  • 8

10

  • 7

10

  • 6

kVar[u]−σ max

u

k2

Figure: PCM convergence to highest accuracy (mean and variance).

10 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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UQ - PCM for Critical Speed statistics

Let’s extend the dynamical system in order to obtain a “controlled” ramping method:

                             ˙

  • u1 =
  • u2

m ˙

  • u2 =

−2D2 u2 − 2k4 u1 − 2FX (ξx1, ξy1) − 2FX (ξx2, ξy2) ˙

  • u3 =
  • u4

I ˙

  • u4 =

−k6 u3 − 2ha

  • FX (ξx1, ξy1) − FX (ξx2, ξy2)
  • −

−a

  • FY (ξx1, ξy1) + FY (ξx2, ξy2)
  • ˙
  • v =

     if t < tst ∨ u2 < εmin − u2 if u2 < εmax −εmax

  • therwise

50 100 150 200 250 300 350 Time (s) 75 80 85 90 95 100 105 110 Speed (m/s)

PCM Mean

83.45 83.50 83.55 83.60 83.65 83.70 Mean (m/s)

Monte Carlo

50 100 150 200 250

PCM

5000 10000 15000 20000 N of simulations 4.55 4.60 4.65 4.70 4.75 4.80 4.85 Variance (m/s)^2

Monte Carlo PCM

11 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

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Outlook - Future work

UQ on Railway vehicle dynamics

  • Uncertainty quantification with Sparse Grids
  • Uncertainty quantification on a realistic model
  • Parameter space compression and compressed sensing

UQ on Free Water Wave Dynamics

  • Parametrization of random fields

Other applications of Uncertainty Quantification

12 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics