Parameter Estimation for Bingham Models

Dr. Volker Schulz, Dmitriy Logashenko

Parameter Estimation for Bingham Models

supported by BMBF

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

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 tech-

The continuity and impulse equations:

div u = 0, ρ ∂ u

∂t = div T + 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 

η _B + τ _F (2II _D ) ⁻

 D where II _D is the second invariant of D ,

II _D = ¹ ₂ Tr D ² − (Tr D ) ²

= ¹ ₂ Tr D ² . This form of T ^E holds only under a condition

|II T

| > τ _F ²

else the material is considered as a rigid body:

D = 0 for |II T

| ≤ τ _F ² . Two parameters:

η _B Bingham viscosity,

τ _F yield stress .

Regularization:

µ( D ) = η _B + τ _F (δ + 2II _D ) ⁻

δ — a regularization parameter.

The regularized PDE system for a stationary flow:

div u = 0,

−div  2 

η _B + τ _F (δ + 2Tr D ² ) ⁻



· D 

+ ∇p = 0.

The stress tensor:

T = −pI + 2(η _B + τ _F (δ + 2Tr D ² ) ⁻

) · D

The boundary conditions of the 3rd kind for describing the wall sliding:

n ^T Tt _{− k} u ^T t = τ _G , u ^T n = 0.

Two additional parameters:

k wall sliding factor ,

τ _G sliding limit .

Parameter identification

HH   H H 

 6

?

H ⁶

? h

-

 L

Γ _in α

Γ _out Γ 0 _{HH j}

Γ 0 ^{ *}

devices for measuring the normal stress

 

 

 )

For the used device H = 30 mm, h = 10 mm, L = 244 mm,

α ≈ 2.35 ⁰ . The average inflow velocity of the paste is 80 ^mm _s .

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

7

X

i=2

1

σ _i ² ((π _P

( u , p, q) − π _P

( u , p, q)) − (ˆ π _i − ˆ π 1 )) ² → min s . t . c ( u , p, q) = 0,

where

• P i are the measurement points

• ˆ π _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 com- puted on u and p

• σ i = 0.08(ˆ π _i + ˆ π 1 ) are the standard deviations for the dif- ference evaulations, if all measurements are assumed to be independently normally distributed with expectation ˆ

π _i and standard deviation 0.08ˆ π _i .

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 Us- ing Colocated Variables — 1. Computational Procedure.

Numerical Heat Transfer 11: 363–390 (1987)

• The fixed point method for the solution of the discretized system. The discretized system is represented in the form

A ( x , q) x = f , where A is a sparse matrix.

• 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: 27–40

(1997)).

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

∂q ( x , q) ∆q = 0,

imposing a quadratic subproblem with the approximate pro- jected Hessian of the Lagrangian

L( x , q, λ) = f ( x , q) − λ ^T c ( x , q).

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 γ _k := ∇ q f ( x _k , q _k ) −

 ∂ c

∂q ( x _k , q _k )

 T

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 up- date formula:

B 0 = αI,

B _k+1 = B _k + v _k v _k ^T

v _k ^T s _k − (Bs _k )(Bs _k ) ^T s ^T _k Bs _k , with s _k := q _k − q _k−1 , v _k := γ _k − γ _k−1 .

2-step superlinear local convergence to the minimum

Difficulty: necessaty of inverting J and J ^T .

Algorithm 2: The RSQP method with an approximate Ja- cobian.

(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

γ _k := ∇ q f ( x _k , q _k ) −

 ∂ c

∂q ( x _k , q _k )

 T

(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

0 B _k

 ∂ c

∂q

 T

A ^∂ c

∂q 0

















∆ x

∆q

∆λ



 =





∇ x L

∇ q L

− c



 .

Experiment

- 6

0 2 4 6 8 10 π _i − π 1 , bar

1 ⁱ 2 3 4 5 6 7 i

i i

i i

i i

v v

v v

v v

v

The differences of the normal stresses.

Empty circles — the relative measured stresses (ˆ π _i − ˆ π 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 = 0.302 bar · s, τ _F = 3.03 bar,

k = 0.497 ^bar·s _m , τ _G = 0.180 bar.

We assume that the measurements at the different measure- ment points are statistically independent and satisfy the nor- mal 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 ,

S(q) =

 ∂Π

∂ x

∂Π

∂q







−J ⁻¹ ∂c

∂q I





(here x form c ( x , q) = 0).

D = diag {0.08(ˆ π _i−1 − ˆ π 1 )}.

The covariance matrix of the parameters:

Cov (q) = h

(S(q)) ^T D ² S(q) i −1

. Covariances for the simple device:

Value Cov. 95%-conf. int

η _B 0.302 5568.20 ±237.4

τ _F 3.03 16669.2 ±410.8

k 0.497 18052.5 ±428.5

τ 0.180 30.1338 ±17.5

t

t 6

h ₀

t

t 6

h 1

t

t 6

h 2

t

t 6

h 3

t

t 6

h 4 ^t

t 6 h ₅

t

t

6 h ₆ ^t

t 6 h 7

-

 L

Φ( Cov (q)) = ¹ ₄ Tr Cov (q) → min s . t . h 0 ≥ . . . ≥ h 7 ,

h 0 ≥ h min , h 7 ≤ h max

(A-optimal design) Numerical solution:

• A “gradient-free” optimization method

• Penalty technique for the constraints

• Assumption that all normal stresses are measured with

the equal absolute standard deviations

Numerical results for the optimized shape: the pressure field

Confidence intervals for the simple and optimized de- vices:

Parameter Simple Optimized η _B = 0.302 bar · s ±26.3 ±0.00421 τ _F = 3.03 bar ±38.5 ±0.0520 k = 0.497 ^bar·s _m ±46.2 ±0.0640 τ _G = 0.180 bar ±1.48 ±0.0819

(Standard deviation of the stress measurements: 0.05 bar)

Conclusion:

• Simulation of Flows of the Bingham Fluids

• Parameter Identification

• Shape Optimization for the Measurement Device