SFB 837 Interaction Modeling Mechanized Tunneling S Freitag, B T Cao, J Ninić & G Meschke Institute for Structural Mechanics Ruhr University Bochum 1
Content 1 Motivation 2 Process-oriented FE model for mechanized tunneling 3 Numerical simulation with polymorphic uncertain data 4 Surrogate modeling for uncertain processes 5 Examples 6 Conclusion / Outlook 2
Motivation Computational models advanced finite element simulation a priori tunnel design during construction steering Uncertainties limited information geometry and heterogeneity of soil layers construction process uncertain parameters 3
Process-oriented simulation model subsystems Subsystem I: geology, geotechnical model Subsystem II: face support Subsystem III: soil excavation Subsystem IV: tunnel boring machine, steering Subsystem VI: existing buildings Subsystem V: lining 4
Process-oriented simulation model steering 5
Process-oriented simulation model interaction with buildings surrogate models for buildings element for pile foundation Embedded beam element independent of the soil mesh arbitrary number and orientation per element target elem ent reference pile deform ed pile soil σ N g u rel u T skin friction, pile tip resistance J Ninic, J Stascheit & G Meschke, Int J Num Anal Meth Geomechanics 2013 6
Simulation with uncertain data Numerical structural analysis with uncertain data modeling loading geometry material behavior numerical analysis uncertain structural responses structural reliability 7
Simulation with uncertain data uncertainty aleatory randomness / variability not reducible objective assessment epistemic lack of knowledge reducible subjective / objective assessment f(x) stochastic numbers pdf intervals, fuzzy numbers (x) ~ x 1 s x 0 s,l x s,u x x 8
Simulation with uncertain data Generalized uncertainty models probability boxes F(x) 1 cdf l cdf u fuzzy stochastic numbers f(x) pdf =0 =1 F(x) 1 x F(x) 1 x cdf l cdf u cdf =0 =1 0 x 0 x 9
Simulation with uncertain data Stochastic analysis inputs outputs stochastic numbers X Z stochastic structural responses stochastic analysis deterministic analysis 10
Simulation with uncertain data Interval analysis inputs intervals _ interval arithmetic optimization x interval analysis deterministic analysis _ z outputs interval structural responses 11
Simulation with uncertain data Fuzzy analysis inputs fuzzy numbers ~ fuzzy arithmetic -level optimization x fuzzy analysis deterministic analysis ~ z outputs fuzzy structural responses 12
Simulation with uncertain data Fuzzy stochastic analysis inputs stochastic fuzzy stochastic interval stochastic fuzzy numbers intervals X ~ Z ~ outputs fuzzy stochastic structural responses fuzzy analysis stochastic analysis deterministic simulation surrogate models for deterministic simulation 13
Simulation with uncertain data Time-dependent behavior process simulation with time constant uncertain inputs inputs X ~ outputs ~ Z(t) process simulation with time varying uncertain inputs inputs ~ X(t) outputs ~ Z(t) 14
Neural network based surrogate models Feed forward neural network deterministic simulation x 1 input layer hidden layers output layer (1) (2) (m) (M) z 1 x J z K hidden neuron 15
Neural network based surrogate models Feed forward neural network deterministic simulation hidden neuron m 1 x m w m kj k b k m k m x k neuron k x m 1 J m x w b m m m1 m k k j1 j kj k 16
Neural network based surrogate models Feed forward neural network deterministic simulation x 1 input layer hidden layers output layer (1) (2) (m) (M) z 1 x J z K t hidden neuron 17
Neural network based surrogate models Recurrent neural network deterministic simulation [n] x 1 input layer hidden layers output layer (1) (2) (m) (M) [n] z 1 [n] x J [n] z K hidden neuron context neuron 18
Neural network based surrogate models Recurrent neural network deterministic simulation hidden neuron context neuron n m 1 x m k w kj m b k m k m k m n xk m k m n yk n1 m y i m c ki neuron k context neuron k m n xk n m m x k k j m1 J 1 n m1 m m x w b m I i1 j n1 m m y i kj c ki k n m m n m m n1 m m y k k x k k m k y k k 19
Example RNN surrogate model (fuzzy stochastic analysis) Simulation model Soil behaviour: Drucker-Prager yield surface and isotropic hardening Poisson ratio ν 03 cohesion c (kpa) 100 hardening modulus H (MPa) 8 Investigated parameters Min Max Young modulus E (MPa) 10 100 internal friction angle ϕ ( ) 30 40 20
Example RNN surrogate model (fuzzy stochastic analysis) Neural network based surrogate model recurrent neural network network architecture 2 8 1 E v(t) ϕ 9 context neurons for history dependent computation network training and validation 50 training, 50 validation patterns with 22 time steps modified backpropagation algorithm 10 7 runs 21
Example RNN surrogate model (fuzzy stochastic analysis) Surrogate model training results 22
Example RNN surrogate model (fuzzy stochastic analysis) Surrogate model validation results 23
Example RNN surrogate model (fuzzy stochastic analysis) Reliability analysis uncertain soil material parameters fuzzy stochastic number for modulus of elasticity E fuzzy number for internal friction angle ϕ fuzzy stochastic analysis for each time step settlement limit state of 2 cm α-level optimization for lower and upper bound P f Monte Carlo simulation with 10 6 samples 24
Example RNN surrogate model (fuzzy stochastic analysis) fuzzy analysis (E) 1 0 stochastic analysis F(E) 1 fuzzy mean value of E ~ fuzzy number for ϕ ~ -level optimization 597 60 603 E [MN/m²] fuzzy stochastic number for E ~ (f) 1 Monte Carlo Simulation with 10 6 realisations 0 30 33 36 f [ ] (P f ) 1 failure probability 0 E [MN/m²] for limit state tunneling simulation with RNN surrogate model v = 2 cm 0 fuzzy failure probability P f B Möller und M Beer Fuzzy Randomness, 2004 25
Example RNN surrogate model (fuzzy stochastic analysis) Time varying fuzzy failure probability trajectories result of stochastic analysis 26
Example RNN surrogate model (fuzzy stochastic analysis) Time varying fuzzy failure probability memberships 27
Proper Orthogonal Decomposition One snapshot U i Snapshot response matrix U POD matrix Φ u 1 u i u n 1 u m 1 n u i u n 1 k<<m n Parameter matrix Z Radial Basis Functions (interpolation functions) + 1 m p POD RBF network 28
Proper Orthogonal Decomposition Mathematical point of view: Given set of functions u k L 2 Ω k = 1,, N -> Exists eigenfunctions φ i i=1 u k = i=1 A i k φ i Eigenvalue λ i is the average energy in the projection of the ensemble onto the basis function φ i Total energy is given by i=1 λ i Select the first n POD modes to get the approximation u k u k = n j=1 A i k φ i POD: Method for computing the optimal linear basis (modes) for representing a sample set of data (snapshots) Goal: Construct the set of orthonormal vectors Φ (POD modes, POD basis vectors) 29
Proper Orthogonal Decomposition POD can be performed by Singular Value Decomposition (SVD) or solving eigenvalue problem Given matrix A: Compute SVD A = U Σ V T Basis vector: φ i 2 = U,i and λ i = ii Compute eigenvalue of AA T V = Λ V Basis vector: φ i = V,i and λ i = Λ ii Compute eigenvalue of A T A V = Λ V Basis vector: φ i = A V,i / λ i and λ i = Λ ii Algorithm: POD procedure to find the basis vectors capture desired accuracy Input: Snapshots matrix U, desired accuracy E Output: Truncated POD basis vectors Φ Compute covariance matrix C = U T U Compute eigenvalue decomposition Ψ, Λ = eig C Set Φ i = U Ψ,i / λ i Set λ i = Λ ii for i = 1,, M Define K based on desired accuracy E as: i=1 K λ i / M i=1 λ i E Return: truncated POD basis Φ by taking the first K columns of Φ 30
Proper Orthogonal Decomposition Radial Basis Functions Consider the truncated POD basis vectors Φ of snapshots matrix U U and a single snapshot of U can be approximated as: U Φ A U i Φ A i A and A i are the truncated amplitude matrix and vector, respectively (constant) Continuous response: Introduce A and A i as nonlinear function of the vector of input parameters Z A i = B F i F i = f 1 Z j f j Z j f M (Z j ) T f j Z : radial basis functions as interpolation functions Finally, an approximation of the output with an arbitrary set of input parameters: U i Φ B F i 31
Gappy Proper Orthogonal Decomposition One snapshot U i Snapshot response matrix U POD matrix Φ u 1 u i u n 1 u m 1 n u i u n 1 k<<m n A snapshot with missing elements U* u 1?? u 1 Gappy POD Behaviour can be characterised with U u i??? u i u n Repaired 32
Gappy Proper Orthogonal Decomposition The repaired vector U can be approximated as: U = Φ A To compute coefficients A the error E between original and repair vectors must be minimized E = U U 2 Only available data are compared Solution is given by a linear system of equations in the form M = Φ T, Φ R = Φ T, U M A = R Finally, replacing missing elements of U by corresponding calculated elements of U 33
Verification Example GPOD a 1 6 11 P x EI = 100000 l = 10 11 nodes x = 0,1,2,,10 knm m m y l P = 10,40,70,100 kn a = 2,4,6,8 m Analytical solutions : y x = P l a x 6lEI l 2 x 2 l a 2, for 0 < x < a y x = P l a 6lEI l l a x a 3 + l 2 l a 2 x x 3, for a < x <l 34
Verification Example GPOD 16 snapshots (4 P x 4 a) Truncated POD basis vectors Φ U = 0 0 0 0 05 19 22 32 09 34 43 11 03 13 33 47 0 0 0 0 11 nodes Φ = 0 0 0 01 03 03 03 04 04 01 03 03 0 0 0 11 rows 16 snapshots U*: solution of P = 125 kn and a = 25 m U =? 668? 179? 483? Node 2 Node 6 Node 10 GPOD U = 0 6 68 1803 17 9 1617 4 83 0 3 columns Error comparing with analytical solution is 034% 35
Hybrid RNN-GPOD surrogate model available data Prediction [1] t [n] t [n+1] t t [1] t recurrent neural network [i] t [n+1] t Gappy Proper Orthogonal Decomposition [n] t 36
Example hybrid RNN-GPOD surrogate model FE model 85m 6m 2525m uncertain geotechnical parameter Drucker-Prager Model Parameter E-Modul E 2 (MN/m²) 3275m Min 40 Max 130 48m hybrid process-oriented surrogate model recurrent neural network in combination with Proper Orthogonal Decomposition Method (POD) E 2 GP(t) v(t) time-varying settlement field E 2 : stochastic number 37
Example hybrid RNN-GPOD surrogate model Time varying grouting pressure Grouting pressure [kpa] 240 220 200 180 160 Grouting pressure scenarios scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 training validation prediction 140 120 1 20 22 24 Excavation steps 38
Example hybrid RNN-GPOD surrogate model Displacement processes of 7 monitoring points Prediction with RNN 0,002 Point 1-FE 0-0,002-0,004-0,006 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Point 2-FE Point 3-FE Point 4-FE Point 5-FE Point 6-FE Point 7-FE Point 1-RNN Point 2-RNN Point 3-RNN Point 4-RNN Point 5-RNN Point 6-RNN -0,008 training Verification Prediction Point 7-RNN 39
Example hybrid RNN-GPOD surrogate model (stochastic analysis) Displacement field predictions by GPOD-RNN (Case 36 : E 2 = 90 MPa, scenario 6) 40
Example hybrid RNN-GPOD surrogate model (stochastic analysis) cdf: F E 2 = 1 1 + e E 2 a b a = 80 MPa (mean value) b = 3 Mpa [n] P : scenario 6 limit state: 10 mm stochastic reliability analysis 41
Example hybrid RNN-GPOD surrogate model (interval analysis) Displacement field predictions by GPOD-RNN (Case 54 : E 2 = 70 MPa, scenario 5) 42
Example hybrid RNN-GPOD surrogate model (interval analysis) Interval E 2 = 74, 86 MPa Particle Swarm Optimization Number of particles: 20 c 1 = c 2 = 1494 c 3 = 0729 43
Summary / Outlook Process-oriented FE model for mechanized tunneling simulations Numerical simulation with uncertain geotechnical parameters Hybrid RNN-GPOD based surrogate model Further objectives: damage risk assessment of existing buildings stress reduction in tunnel lining investigation of tunnel face stability Real-time predictions to support steering P-box approach for polymorphic uncertain data 44