Parameter Estimation for Bingham Models

Transcription

1 Dr. Volker Schulz, Dmitriy Logashenko Parameter Estimation for Bingham Models supported by BMBF

2 Parameter Estimation for Bingham Models Industrial application of ceramic pastes Material laws for Bingham fluids Parameter identification technique Discretization of the model Numerical solution of the variational problem Experiment Precision of the technique Shape Optimization of the device Optimal Shape: numerical results

3 Industrial applications: Object: ceramic pastes used, e. g. in production of bricks or bodies of catalytic converters Industrial partner: Braun GmbH (Friedrichshafen) Products: outlets for the extrusion machines Aim of the project: a fast model based measurement technique which allows the simultaneous determination of all model parameters from one experiment.

4 The continuity and impulse equations: divu = 0, ρ u t = divt + f with the stress tensor: T = pi + T E (D), where D = 1 2 ( u + ( u) T ) is the strain tensor. For Bingham fluids T E (D) = 2µ(D)D = 2 where II D is the second invariant of D, ( ) η B + τ F (2II D ) 1 2 D II D = 1 2 ( TrD 2 (TrD) 2) = 1 2 TrD2. This form of T E holds only under a condition II T E > τf 2 else the material is considered as a rigid body: D = 0 for II T E τf 2. Two parameters: η B τ F Bingham viscosity, yield stress.

5 Regularization: µ(d) = η B + τ F (δ + 2II D ) 1 2 δ a regularization parameter. The regularized PDE system for a stationary flow: ( ( divu = 0, ) ) div 2 η B + τ F (δ + 2Tr D 2 ) 1 2 D + p = 0. The stress tensor: T = pi + 2(η B + τ F (δ + 2Tr D 2 ) 1 2) D The boundary conditions of the 3rd kind for describing the wall sliding: n T Tt ku T t = τ G, u T n = 0. Two additional parameters: k τ G wall sliding factor, sliding limit.

6 Parameter identification devices for measuring the Γ 0 normal stress H Γ in Γ 0 L α Γ out h For the used device H = 30 mm, h = 10 mm, L = 244 mm, α The average inflow velocity of the paste is 80 mm s.

7 Let the equation c(u, p, q) = 0 comprise the PDE system with the boundary conditions, where q = (η B, τ F,k, τ G ) T. Note that the pressure p is defined up to a constant. The parameters are determined from the output least squares problem where 7 i=2 1 σ 2 i ((π Pi (u, p, q) π P1 (u, p, q)) (ˆπ i ˆπ 1 )) 2 min s. t. c(u, p, q) = 0, P i are the measurement points ˆπ i are the measured normal stresses π P (u, p, q) = n T P T P(u, p, q)n P are the normal stresses computed on u and p σ i = 0.08(ˆπ i +ˆπ 1 ) are the standard deviations for the difference evaulations, if all measurements are assumed to be independently normally distributed with expectation ˆπ i and standard deviation 0.08ˆπ i.

8 Discretization A collocated finite-volume scheme based on the idea from Schneider G.E., Raw M.J. Control Volume Finite- Element Method for Heat Transfer and Fluid Flow Using Colocated Variables 1. Computational Procedure. Numerical Heat Transfer 11: (1987) The fixed point method for the solution of the discretized system. The discretized system is represented in the form where A is a sparse matrix. A(x, q)x = f, The inner multi-grid method with ILU smoothers for the solution of the linear systems in the fixed point iteration. The model has been implemented on the base of UG toolbox (s. Bastian P., Birken K., Johannsen K., Lang S., Neuß N., Wieners C. UG a flexible software toolbox for solving partial differential equations. Comput Visual Sci 1: (1997)).

9 The discretized problem: f(x, q) min, s. t. c(x, q) = 0, where f : R n 4 R and c : R n 4 R n. The Jacobian J = c x is assumed to be nonsingular. The dimention n of the constraint is very large application of structure exploiting methods to reduce the computation time. Numerical solution by a Reduced SQP method. Idea: Linearization of the constraint by a Taylor expansion: c(x, q) + J(x, q) x + c (x, q) q = 0, q imposing a quadratic subproblem with the approximate projected Hessian of the Lagrangian L(x, q, λ) = f(x, q) λ T c(x, q).

10 Algorithm 1: The Redused SQP method. (0) Set k := 0; start at some initial guess x 0, q 0. (1) Compute the adjoint variables from the linear system J T (x k, q k ) λ k+1 := x f(x k, q k ); compute the reduced gradient ( ) c T γ k := q f(x k, q k ) q (x k, q k ) λ k+1; determine some approximation B k of the projected Hessian of the Lagrangian. (2) solve B k q k = γ k. (3) compute step on x from the linear system J(x k, q k ) x k := c q (x k, q k ) q k + c(x k, q k ). (4) Set x k+1 := x k + x k, q k+1 := q k + q k. (5) k := k + 1; go to (1) until convergence. The approximate projected Hessian B k by BGFS update formula: B 0 B k+1 = αi, = B k + v kv T k v T k s k with s k := q k q k 1, v k := γ k γ k 1. (Bs k)(bs k ) T s T k Bs k 2-step superlinear local convergence to the minimum Difficulty: necessaty of inverting J and J T.,

11 Algorithm 2: The RSQP method with an approximate Jacobian. (0) Set k := 0; start at some initial guess x 0, q 0. (1) Compute λ k from the linear system A T (x k, q k ) λ k := x f(x k, q k ) J T (x k, q k ) λ k ; compute the reduced gradient ( ) c T γ k := q f(x k, q k ) q (x k, q k ) (λ k + λ k ); determine some approximation B k of the projected Hessian of the Lagrangian. (2) solve B k q k = γ k. (3) compute step on x form the linear system A(x k, q k ) x k := c q (x k, q k ) q k + c(x k, q k ). (4) Set x k+1 := x k + x k, q k+1 := q k + q k and λ k+1 = λ k + λ k. (5) k := k + 1; go to (1) until convergence. Equivalent to: 0 0 A ( T c 0 B k q A c 0 q ) T x q λ = xl q L c.

12 Experiment π i π 1, bar i The differences of the normal stresses. Empty circles the relative measured stresses (ˆπ i ˆπ 1 ), the black circles the computed stresses π h,i (x, q) π h,1 (x, q) Computation time of the parameter identification: 12 min. Computation time of the flow simulation: 5 min. (SGI Indigo 2 with a R4400 Processor, 200 MHz) η B = bar s, k = bar s m, τ F = 3.03 bar, τ G = bar.

13 Precision of the parameters We assume that the measurements at the different measurement points are statistically independent and satisfy the normal distribution. The precision of these measurements is 8%. Let Π(x, q) = (π 2 (x, q) π 1 (x, q),..., π 7 (x, q) π 1 (x, q)) T, ( ) Π Π S(q) = J 1 c q x q I (here x form c(x, q) = 0). D = diag {0.08(ˆπ i 1 ˆπ 1 )}. The covariance matrix of the parameters: [ 1 Cov(q) = (S(q)) T D S(q)] 2. Covariances for the simple device: Value Cov. 95%-conf. int η B ±237.4 τ F ±410.8 k ±428.5 τ G ±17.5

14 Shape optimization of the device h 0 h 1 h 2 h 3 h 4 h 5 h 6 h 7 L (A-optimal design) Numerical solution: Φ(Cov(q)) = 1 Tr Cov(q) min 4 s. t. h 0... h 7, h 0 h min, h 7 h max A gradient-free optimization method Penalty technique for the constraints Assumption that all normal stresses are measured with the equal absolute standard deviations

15 Numerical results for the optimized shape: the pressure field Confidence intervals for the simple and optimized devices: Parameter Simple Optimized η B = bar s ±26.3 ± τ F = 3.03 bar ±38.5 ± k = bar s ±46.2 ± m τ G = bar ±1.48 ± (Standard deviation of the stress measurements: 0.05 bar)

16 Conclusion: Simulation of Flows of the Bingham Fluids Parameter Identification Shape Optimization for the Measurement Device