20th June 2016 Janov nad Nisou 1/24

Návrh robustního experimentu založený na globální citlivosti

Anna KučerováJan Sýkora

Eliška JanouchováJan Chleboun

Daniela Jarušková

Czech Technical University in PragueFaculty of Civil EngineeringDepartment of Mechanics

Experiment design problem

Material parameters to be identified:

● - values of thermal conductivity, heat capacity or water vapour resistance

Design variables to be optimised:

● - loading types (e.g. prescribed temperature, moisture flux or heat transfer at a boundary)

and loading magnitude as well as positions of sensors (thermometers, hygrometers etc.)

Noise variables to be considered:

● - measurement errors,

● - inaccurate positioning of sensors or imperfections in prescribed loading conditions

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Inverse analysis

After realisation of an experiment according to selected values of

Nonlinear regression with random parameters

- random = noise parameters follow prescribed probability distribution corresponding

to distribution of their values in a set of repeated experiments

- assumption of more repeated experiments

Bayesian inference

- distribution of noise parameters is updated along with the distribution of material properties

- assumption of only one realisation of the experiment, no repetitions

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Goals of experiment design

● Maximise information content on the investigated material parameters

● Minimise influence of experimental errors

(sensitivity of observations to material properties)

(sensitivity of observations to noise variables)

(expert on material model specifies feasible intervals)

(expert on experimentation specifies feasible intervals for design variables and mean and variance of noise variables)

Bayesian approach

- maximising Kullback-Leibner divergence

- quantifying information content of experiment by comparing prior and posterior distribution

- computationally very exhaustive

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Nonlinear regression with random parameters

Local sensitivity matrix – 1st order derivatives in Taylor development of the model:

Explicit solution of a least square problem:

With neglecting and considering measurement errors

to be i.i.d. variables with variance :

Optimality criterion:

[Ruffio,Saury,Petit, 2012, IJHMT]

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F-optimality:

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Robustness of solution

Worst case approach

● Noise parameters fixed at mean (zero) values

● Material parameters evaluated in central star design points

Requires starting guess about which is apriori unknown!

[Ruffio,Saury,Petit, 2012, IJHMT]

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Robust optimisation of by particle swarm optimisation algorithm.

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Formulation of using global sensitivity matrix

Polynomial chaos-based approximation of the model

All problem variables need to be transformed into the standardised variables of a chosen family via linear or nonlinear transformation

According to Doob-Dynkin lemma (Bobrovski, 2005), model response is also random vector expressed as a function of the same random variables.

According to (Xiu and Karniadakis, 2002, SIAM JSC), orthogonal polynomials w.r.t. given probability measures are the most suitable choice for the approximation.

● Hermite polynomials for Gaussian variables

● Legendre polynomials for uniform variables etc.

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Formulation of using global sensitivity matrix

Construction of PCE-based approximation – computation of PC coefficients● Linear regression● Stochastic collocation method● Stochastic Galerkin method

We use linear regression based

on Latin Hypercube sampling derived

from Halton sequence - very easy to implement

Global sensitivity matrix – variance-based Sobol' indices

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Formulation of using global sensitivity matrix

Global sensitivity matrix for given values of design variables:

Complete global sensitivity matrix:

Global sensitivity to material properties:

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Optimality measures:

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Linear non-stationary heat problem

Illustrative example – physical problem

Discretized system of equations

Energy balance equationInitial scheme

Solution at time step i+1

+ Boundary conditions+ Initial conditions

Material parameters:

Initial temperature:

Other properties:

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Illustrative example – physical problem

Linear non-stationary heat problem, numerical example

Evolution of temperature

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Polynomial chaos – error analysis

Polynomial degree p=5

x=y=0t=60

x=y=0t=60

x=y=0t=60

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Experiment design problem definition

Initial schemeMaterial parameters to be identified:

● - values of thermal conductivities and heat

capacity (3 variables) :

Design variables to be optimised:

● - positions of 3 sensors (6 variables) :

Noise variables to be considered:

● all zero-mean normal variables

● - measurement errors – at 3 sensors at 60

time steps: 180 i.i.d. variables,

● - inaccurate positions of sensors, 6 i.i.d. var.,

,

● - imperfection in loading

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FEM & Variable sensor positions

● design variables: sensor positions

● noise variables: error in sensor positions

FE discretisation of model response

● localisation of k-th element

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Experiment design problem definition

Material parameters to be identified:

● - values of thermal conductivities and heat

capacity (3 variables) :

Design variables to be optimised:

● - positions of 3 sensors (6 variables) :

Noise variables to be considered:

● - measurement errors,

● - inaccurate positions of sensors,

● - imperfection in loading

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Linearised inverse analysis, global sensitivityCoarse mesh

Fine mesh

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Considering only measurement errors:

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Linearised inverse analysis, global sensitivityCoarse mesh

Fine mesh

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Considering only measurement errors:

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Linearised inverse analysis

[Ruffio, Saury, Petit, 2012, IJHMT]

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Global sensitivity

x2

Local sensitivity

[Sawaf et al., 1995, IJHMT]

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Linearised inverse analysis

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Testing of robustness:

- statistics over discretised space of material parameters

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MCMC-based Bayesian inference

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E-opt. design U2 design

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Std = 1.097%

Std = 0.119%

Std = 1.364%

Std = 0.376%

Std = 2.069%

Std = 0.286%

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Wider bounds

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Linearised inverse analysis

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Testing of robustness:

- statistics over discretised space of material parameters

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MCMC-based Bayesian inference

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E-opt. design S2 design

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Std = 1.097%

Std = 0.119%

Std = 0.541%

Std = 0.090%

Std = 0.849% Std = 0.103%

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Conclusions

Proposed method:

● Global sensitivity matrix – no need for good guess about parameters to

be identified, only wide intervals are sufficient

● Better suited for cases, where response is nonlinear w.r.t. material

properties

● Involve two approximations: FEM & PCE & total Sobol

● Cause inaccuracy

● Decrease computational demands – allow to solve more complex tasks

Future work:

● Investigation of optimality criteria – multi-objective optimisation,

minimisation of correlation among the identified parameters etc.

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Thank you for your attention

The financial support of this research by the Czech Science

Foundation, project No. P15-07299S, is gratefully acknowledged.

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