Optimal Experiment Design for the estimation of moisture material properties

Size: px

Start display at page:

Download "Optimal Experiment Design for the estimation of moisture material properties"

Bartholomew Barrett
5 years ago
Views:

1 Optimal Experiment Design for the estimation of moisture material properties Julien Berger in collaboration with: T. Busser, D. Dutykh, N. Mendes, H. Le Meur Journée SFT, Paris, Mai 2 nd, 218

2 Context Comparison numerical prediction VS. experimental observations phase d'adsorption phase de desorption Some discrepancies observed [1] : for different materials, for different facilities, at different scales : material & wall. expérimental simulation Possible explanation : estimation of material properties according to standards { } gravimetric method ISO standards = = steady state measurements cup method ISO / 24

3 Issues Objective : Estimating the material properties using transient measurements. Methodology, 1. Define a configuration 2. Define a physical model : c u = ( k u ), t x x ( u x = Bi u u ( t ) ), x = material 3. Obtain the experimental observations u obs ( t ) with local sensor(s) 4. Solve the inverse problem : k = arg min u ( X, t, k ) u obs (X, t ) k 2 3 / 24

4 Problem statement Estimation of material properties However, Where to place the sensors? What variations for u ( t )? what amplitude? what frequency? Using the Optimal Experimental Design (OED) Some references : Fedorov 1972 [2], Beck and Arnold 1977 [3], Walter et al. 199 [4, 5], Ucinski 24 [6]. With applications in : Artyukhin et al [7], Nenarokomov et al. 25 [8], Karalashvili et al. 215 [9]. 4 / 24

5 Searching the OED Parameter estimation problem for P = [ p m ], m { 1,..., M } 1. Compute the sensitivity functions : Θ m (x, t) = σ p σ u 2. Compute the Fisher information matrix : u, m { 1,..., M } p m F = [ Φ i j ], (i, j) { 1,..., M } 2, Φ i j = N n = 1 τ Θ i (X n, t) Θ j (X n, t) dt, 3. Maximize the criteria Ψ : Ψ = det [ F ( π ) ], as a function of the design π = { X, u } 5 / 24

6 Synthesis of the methodology Boundary conditions Sensor locations Searching the OED différencier partie exp, numérique, pd inv. Unknown parameter Numerical model Sensitivity functions matrix Optimal Exp. Design Experimental campaign Parameter estimation 6 / 24

7 The facility Independent chamber Chamber 1 Chamber 2 samples salt solutions samples salt solutions Sensors Salt solution Sample Airlock Samples 7 / 24

8 The physical model Vapor transfer in porous material [1] ( ) c( u ) u = Fo d( u ) u t x x Pe u, and the boundary conditions : d( u ) u x Pe u = Bi ( u u ), x =. material? Coefficient c( u ) is unknown and parameterized as : c( u ) = 1 + c 1 u + c 2 u 2 Parameters to be estimated : Fo, c 1 and c 2. 8 / 24

9 The possible designs : single step material Configuration : single step of relative humidity, one sensor to place inside the material, estimation of one parameter among Fo, c 1 and c 2. Design Initial cond. Boundary cond. φ i φ Design 2 Design Design 3 Design 4 9 / 24

10 The sensitivity functions Design φ i φ Θ = ( u Fo, u c 1, ) u c 2 1 / 24

11 Searching the OED Design φ i φ OED : Estimation of Fo and c 1 Design 2 Estimation of c 2 Design 4 Location of the sensor X [.9, 1 ]. 11 / 24

12 Single step experiments However, only for the estimation of one parameter, strong correlation of sensitivity functions : ( ) [ ] Cor Fo, c 1.94,.99, ( ) [ ] Cor c 1, c 2.92,.99, ( ) [ ] Cor Fo, c 2.71, for the estimation of the three parameters Fo, c 1 and c 2 need other experimental data 12 / 24

13 The possible designs : multiple steps Configuration : multiple steps of relative humidity, variable duration of the step τ, one sensor to place inside the material, ( ) estimation of the two parameters Fo, c 2. material Des. Initial cond. Boundary cond. φ i φ, 1 φ, 2 φ, [ ] τ 1, 8 days Designs 5-12 Designs / 24

14 The sensitivity functions Design φ i φ, 1 φ, 2 φ, 3 τ (days) / 24

15 Searching the OED OED : Estimation of ( Fo, c 2 ) Design 2 Location of the sensor X [.9, 1 ]. Design φ i φ, 1 φ, 2 φ, 3 τ (days) / 24

16 Solving the parameter estimation problem Performing the experiments : one sensor located at X = 1. OED for single step of relative humidity estimation of c 1 Single step Design φ i φ OED.1.75 OED for multiple steps of relative humidity, ( ) Multiple steps estimation of Fo, c 2 Design φ i φ, 1 φ, 2 φ, 3 τ (days) OED Solving the parameter estimation problem : demonstration of the formal identifiability (Strucutral Global Identifiability), interior point algorithm with fmincon Matlab TM function. 16 / 24

17 Solving the parameter estimation problem / 24

18 Solving the parameter estimation problem The importance of sensitivity functions : the probability of the unknown parameter p m can be approximated : { } p m p m F ( p m ) = P = F ( u + Θ m ( ) ) p m p m / 24

19 Conclusion Optimal Experimental Design approach : within parameter estimation problem context, numerical method for the definition of experimental design, the importance of the sensitivity functions. Limitations : depends on the physical model, depends on the a priori parameters values. Outlooks : taking into account hysteresis effects in the physical model. 19 / 24

20 Merci pour votre attention 2 / 24

21 Validation OED Problem : c u t = ( ( k + k 1 u ) ) u, x x k ( u ) u x = A sin( 2 π ω t ), x =. Validation : for a fixed ω, for a given p m { c, k, k 1 }, observation generated numerically with a noise u obs, N e = 1 inverse problem solved to estimate p m, computation of the error : ε 2 = 1 N e ( p m ) p m / 24

22 T. Busser, J.Berger, A. Piot, M. Pailha, and M. Woloszyn. Experimental validation of hygrothermal models for building materials and walls : an analysis of recent trends. V. Fedorov. Theory of Optimal Experiments Designs. Academic Press, J. V. Beck and K. J. Arnold. Parameter Estimation in Engineering and Science. John Wiley and Sons, New York, E. Walter and Y. Lecourtier. Global approaches to identifiability testing for linear and nonlinear state space models. Mathematics and Computers in Simulation, 24(6) : , E. Walter and L. Pronzato. Qualitative and quantitative experiment design for phenomenological models ; a survey. Automatica, 26(2) : , / 24

23 D. Ucinski. Optimal Measurement Methods for Distributed Parameter System Identification. CRC Press, New York, 24. E. A. Artyukhin and S. A. Budnik. Optimal planning of measurements in numerical experiment determination of the characteristics of a heat flux. Journal of Engineering Physics, 49(6) : , A. V. Nenarokomov and D. V. Titov. Optimal experiment design to estimate the radiative properties of materials. Journal of Quantitative Spectroscopy and Radiative Transfer, 93(1 3) : , 25. M. Karalashvili, W. Marquardt, and A. Mhamdi. Optimal experimental design for identification of transport coefficient models in convection diffusion equations. Computers and Chemical Engineering, 8 :11 113, / 24

24 A.V. Luikov. Heat and Mass Transfer in Capillary-porous Bodies. Pergamon Press Ltd, London, / 24

An efficient method to estimate sorption isotherm curve coefficients

An efficient method to estimate sorption isotherm curve coefficients arxiv:174.6565v3 [physics.app-ph] 27 Jun 218 Julien Berger Polytech Annecy Chambéry, LOCIE, France Thomas Busser Polytech Annecy Chambéry, LOCIE, France Denys Dutykh CNRS LAMA, Université Savoie Mont Blanc,