A spectral stochastic approach to the inverse heat conduction problem

1. A spectral stochastic approach to the inverse heat conduction problem V.A.Badri Narayanan and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email:zabaras@cornell.edu URL:http://www.mae.cornell.edu/zabaras/ C O R N E L L C O R N E L L Materials Process Design and Control Laboratory U N I V E R S I T Y

2. C O R N E L L C O R N E L L U N I V E R S I T Y Outline of the presentation • Deterministic inverse heat conduction – definition, methods for analysis • Stochastic inverse heat conduction problem (SIHCP) - introduction • Uncertainty in engineering systems – nature, representation and methods for analysis • A spectral stochastic framework – mathematical description, stochastic heat conduction example • Stochastic inverse heat conduction problem (SIHCP) – adjoint based functional optimization. • Inverse heat conduction – numerical studies • Conclusions and suggestions for future research Materials Process Design and Control Laboratory

3. C O R N E L L C O R N E L L U N I V E R S I T Y Deterministic inverse heat conduction problem - IHCP W Computation domain Gh Known flux specification G0 GI Unknown flux ? Temperature readings from sensor available for times of interest at all points along the internal boundary GI Materials Process Design and Control Laboratory

4. C O R N E L L C O R N E L L U N I V E R S I T Y Methods addressing deterministic IHCPs • Functional optimization technique –Alifanov - Infinite dimensional optimization - Adjoint problem defined to obtain gradient of objective function - Optimization in function spaces approach • Combined function specification-regularization method –Beck - Finite dimensional optimization - Finite dimensional parametric representation of the design variable • Mollification method • Integral method – Usage of Duhamel’s integral to construct inverse solutions Materials Process Design and Control Laboratory

5. C O R N E L L C O R N E L L U N I V E R S I T Y Stochastic inverse-design heat conduction - definition W Complete PDF of the unknown boundary flux to be reconstructed Gh Uncertainty in sensor readings is directly incorporated Known flux specification G0 GI Unknown flux ? Temperature readings from sensor available with complete statistics along the internal boundary GI Materials Process Design and Control Laboratory

6. C O R N E L L C O R N E L L U N I V E R S I T Y Nature and characterization of uncertainty Uncertainty in engineering systems Intrinsic Extrinsic Experimental setup, measurement, loading conditions, surroundings Parametric Model-form Boundary conditions, material parameters, process parameters Coefficients of phenomenological models, unmodeled dynamics • MONTE-CARLO • Law of large numbers • Perform ensemble of deterministic simulations • Curse of dimensionality • Provides accurate characterization of output PDFs • SENSITIVITY ANALYSIS • Limited experimental data • Linear propagation of uncertainty • Utilized when probabilistic methods are not well defined • Can point to important design variables • FORM/SORM • Limit-state function • Second order/ normal approximation • Tails of PDFs are important • Can incorporate RSMs • Fast computation Materials Process Design and Control Laboratory

7. C O R N E L L C O R N E L L U N I V E R S I T Y Spectral Stochastic Framework Mathematical framework for uncertainty quantification Random variable x MAP S Real line Sample space of elementary events Each outcome is mapped to a corresponding real value Collection of all possible outcomes Materials Process Design and Control Laboratory

8. C O R N E L L C O R N E L L U N I V E R S I T Y Mathematical basis for spectral method Fourier series – spectral representation of square integrable deterministic functions Generalize this idea to develop a spectral representation for random variables = functions in space-time-sample space – set of random variables to be determined – set of deterministic functions in space - time • Spectral stochastic framework involves • Functional interpretation of a random variable • Representation of stochastic processes in terms of spectral series expansions Materials Process Design and Control Laboratory

9. C O R N E L L C O R N E L L U N I V E R S I T Y Spectral series expansions for stochastic processes Karhunen-Loèvè expansion • Based on the spectral decomposition of the covariance kernel of the stochastic process Random process Mean Set of random variables to be found • Need to know covariance • Suitable for inputs • Most optimal spectral decomposition in energy norm • Converges uniformly to any second order process Eigenpairs of covariance kernel Materials Process Design and Control Laboratory

10. C O R N E L L C O R N E L L U N I V E R S I T Y Spectral series expansions for stochastic processes Polynomial chaos expansion • Based on Hilbert space of random variables and projection theorem Random process Projections Basis for the Hilbert space of random variables • Involves more computation • Used to represent non-Gaussian processes or when covariance is not available apriori • Can use different random bases to obtain desired convergence Materials Process Design and Control Laboratory

11. C O R N E L L C O R N E L L U N I V E R S I T Y Polynomial chaos functions Hermite polynomials in underlying Gaussian variables Generating function Functional form of individual polynomial chaos functions Constant Gaussian Chi-square • For a 1D-underlying Gaussian distribution, the first two polynomial chaos functions capture all the Gaussian components of the distribution. • Higher polynomial chaos functions capture nonlinearities in uncertainty propagation Materials Process Design and Control Laboratory

12. C O R N E L L C O R N E L L U N I V E R S I T Y Transient stochastic heat conduction Motivation behind the example Mathematical model • Most basic form of heat transfer in solids • Driven by thermal conductivity. Investigation of uncertainty in thermal conductivity is important • Do steady state patterns in heat conduction change as a result of uncertainty in physical parameters like heat capacity and thermal conductivity Gh Temperature specified Neumann conditions specified here W Gg Governing equation Symbols used with meanings Random temperature field Dirichlet boundary condition Random heat capacity Neumann boundary condition Random thermal conductivity Computational time domain Initial condition Member of sample space of elementary events Materials Process Design and Control Laboratory

13. C O R N E L L C O R N E L L U N I V E R S I T Y Direct stochastic heat conduction – weak form Materials Process Design and Control Laboratory

14. C O R N E L L C O R N E L L U N I V E R S I T Y Spectral stochastic FEM formulation Pseudo-spectral finite element basis functions The final discretized weak form can be written as follows Materials Process Design and Control Laboratory

15. C O R N E L L C O R N E L L U N I V E R S I T Y Implementation issues Karhunen-Loeve expansion for heat capacity and thermal conductivity Polynomial chaos expansion for temperature Mass matrix can be rewritten as Materials Process Design and Control Laboratory

16. C O R N E L L C O R N E L L U N I V E R S I T Y 2D heat conduction and non-Gaussian processes (1,1) T0 L (0,0) T0 • Non-Gaussian temperature field • lognormal with parameters 1, 0.1 • Uncertainty considered in k Analytical solution to the deterministic problem is given by Materials Process Design and Control Laboratory

17. C O R N E L L C O R N E L L U N I V E R S I T Y 2D example with non-Gaussian applied temperature • Physical interpretations • Various coefficients in the polynomial chaos expansion of temperature are plotted • The values of the coefficients decrease as the order of approximation increases • In all the off-mean terms, the effect of uncertainty in k is predominant during the transient phase • From the decreasing trend observed in the first off-mean term, it is clear that the final distribution is not Gaussian. Further, increasing tend observed in the second coefficient indicates that the final distribution is similar to a chi-squared distribution Mean First off-mean coefficient Second off-mean coefficient Materials Process Design and Control Laboratory

18. C O R N E L L C O R N E L L U N I V E R S I T Y Stochastic inverse heat conduction - recap W Gh Known flux specification G0 GI Unknown flux ? Temperature readings from sensor available with complete statistics along the internal boundary GI Materials Process Design and Control Laboratory

19. C O R N E L L C O R N E L L U N I V E R S I T Y A functional optimization approach to SIHCP Definition of the objective function as a norm in an appropriate function space Define gradient of objective from directional derivative Sensitivity of temperature with respect to perturbation in flux Materials Process Design and Control Laboratory

20. C O R N E L L C O R N E L L U N I V E R S I T Y Adjoint formulation to evaluate gradient of objective Given the temperature residual how to evaluate gradient of objective function? Adjoint to sensitivity Temperature sensitivity variable Simplification of the above using integration by parts yield the SPDEs governing adjoint problem • Solution of adjoint problem at the boundary with unknown flux specification is defined as the gradient of the objective function • Adjoint problem is driven by the temperature residual Materials Process Design and Control Laboratory

21. C O R N E L L C O R N E L L U N I V E R S I T Y Continuum stochastic sensitivity Definition of stochastic parameter sensitivity –Gateaux differential of the parametric stochastic field w.r.t perturbations in parameters is the stochastic parameter sensitivity field is the infinitesimal perturbation in the vector of stochastic design parameters • Observations • Mathematically rigorous definition of a continuum stochastic sensitivity field • Deterministic sensitivity information is contained in the direct simulation. Stochastic sensitivity is concerned with perturbations in the PDF Materials Process Design and Control Laboratory

22. C O R N E L L C O R N E L L U N I V E R S I T Y Stochastic sensitivity- physical interpretation PDF obtained after perturbing the design parameter vector Original PDF • The key question is how much a perturbation in PDF of the design variable affect the PDF of the solution • Need to characterize the perturbation in PDF of the solution process Realization of T • Physical observations and implementation issues • An accurate differentiate-then-discretize approach is used here • The temperature field is design differentiated to obtain the sensitivity field. Since we assume C1 stochastic processes, this approach is valid • Sensitivities are essentially linear, since we are interested only in infinitesimal perturbations in PDF of design variables Materials Process Design and Control Laboratory

23. C O R N E L L C O R N E L L U N I V E R S I T Y Parametric problems DIRECT Gh W Known flux applied here Guess flux q0 applied here G0 Gh SENSITIVITY W Flux perturbation applied here G0 Insulated boundary Insulated boundary W Gh Flux jump condition ADJOINT GI Insulated boundary G0 Materials Process Design and Control Laboratory

24. C O R N E L L C O R N E L L U N I V E R S I T Y 1D inverse problem – example 1 Required design temperature readings with specified confidence intervals Schematic of computational domain Temperature sensor readings Unknown flux Insulated X = 0 X = L Individual examples vary based on the kind of Design temperature sensor readings specified Reconstructed heat flux with comparisons to analytical mean • Observations • Given the temperature confidence intervals, the confidence intervals of the unknown flux can be reconstructed. This shows the methods capability to handle reliability based optimization problems • More importantly complete flux PDF can be generated Materials Process Design and Control Laboratory

25. C O R N E L L C O R N E L L U N I V E R S I T Y 1D inverse problem – example 2 Design specification for temperature statistics in terms of coefficients of PCE PCE coefficients of the reconstructed flux • Observations • End condition problem is carried over to the stochastic case also • Excellent reproduction of temperature statistics is seen • Four orders of magnitude faster than Monte Carlo based optimization • Good convergence statistics Materials Process Design and Control Laboratory

26. C O R N E L L C O R N E L L U N I V E R S I T Y Triangular flux problem – example 2 Reference – Beck JV and Blackwell B, St. Clair C. Inverse heat conduction: Ill-posed problems. Wiley, New York, 1985. Triangular heat flux profile is used as a basic template on which unknown flux probability distributions are built • Non-Gaussian thermal conductivity • A chi-square distribution is chosen for thermal conductivity, this ensures that k > 0 • Based on temperature readings from sensor at x = 0.3, triangular flux is reconstructed • Estimation of temperature based on reconstructed flux is done at x = 0.5 Materials Process Design and Control Laboratory

27. C O R N E L L C O R N E L L U N I V E R S I T Y Triangular flux problem – cases Based on the generation of sensor readings Data generated using a SSFEM direct analysis Data generated to simulate experimental conditions • Solve a deterministic direct problem with the triangular flux applied at the left end of the bar. • Extract the temperature readings at the sensor point x = 0.3 • Add a Gaussian noise to the temperature readings. This simulates approximately the experimental noise level. • This case provides insight into how uncertainty in material data and boundary conditions affect quantities in the physical domain – ( estimated heat flux, information loss). • Solve a SSFEM direct problem with a Gaussian heat flux applied at the left end of the bar • The stochastic temperature solution at the sensor point x = 0.3 is extracted as the sensor readings • This case provides exact comparisons for the heat flux since the solution to the inverse problem should yield a Gaussian flux Materials Process Design and Control Laboratory

28. C O R N E L L C O R N E L L U N I V E R S I T Y Case – Data generated using SSFEM • First term in PCE • Least iterations to converge • Second term in PCE • Maximum iterations to converge (iter = 80) • Mean optimal flux • Least iterations to converge The objective function is L2, quadratic. Thus convergence of statistics of higher order than variance takes large number of iterations Improving the optimization techniques in stochastic space and introduction of new objective functions may lead to optimal convergence Materials Process Design and Control Laboratory

29. C O R N E L L C O R N E L L U N I V E R S I T Y Case – Data generated using SSFEM First term in PCE of temperature readings Second term in PCE of temperature readings Mean temperature readings Materials Process Design and Control Laboratory

30. C O R N E L L C O R N E L L U N I V E R S I T Y Case – Data to simulate real experiments Mean temperature readings + large measurement noise Mean temperature readings + small measurement noise Mean temperature readings Observations • Estimation is closely related to the accuracy of temperature readings • Unlike deterministic inverse problems, large error in measured data does not affect estimation of mean flux Materials Process Design and Control Laboratory

31. C O R N E L L C O R N E L L U N I V E R S I T Y 2D inverse problem - definition (1,1) • Distance d is used to vary sensor distances • Uncertainty considered in sensor readings and k • 80-by-80 grid used for computation G0 d GI (0,0) G0 Materials Process Design and Control Laboratory

32. C O R N E L L C O R N E L L U N I V E R S I T Y 2D inverse problem – example 1 Reconstructed optimal non-dimensional heat flux Flux mean Flux first term • Both mean and the off-mean contributions are captured accurately • Better grid resolution was needed to capture off-mean contributions Materials Process Design and Control Laboratory

33. C O R N E L L C O R N E L L U N I V E R S I T Y 2D inverse problem – example 1 Non-dimensional temperature at sensor location computed using optimal heat flux Mean First term • Near the edges of the grid, the temperature solution is not very accurate. Regularization needs to be incorporated in the algorithm • The functional forms of regularization needs to be investigated ( off-mean terms also tend to regularize the mean). Materials Process Design and Control Laboratory

34. C O R N E L L C O R N E L L U N I V E R S I T Y Conclusions • Observations on the stochastic inverse heat conduction • Direct incorporation of uncertainty in inverse problem formulation • Uncertainty characterized in a spectral framework • Parametric study conducted to identify key input variables • Inverse problem cast in an infinite dimensional optimization framework • Adjoint formulation extended to an appropriate probability space to evaluate gradient of the objective function • Complete statistical reconstruction of the stochastic flux • Superior to other robust inverse techniques in both level of description and computational performance Materials Process Design and Control Laboratory

35. C O R N E L L C O R N E L L U N I V E R S I T Y Related future investigations • Suggestions for future investigations • Parametric characterization of non-linear uncertainty propagations, e.g. thermal conductivity as a function of temperature • Investigating the role of regularization in these problems. Contributions due to off-mean terms cannot be neglected • Finite dimensional optimization theory for stochastic inverse problems • Devising Bayesian stochastic parameter estimation framework for obtaining material properties from experimental data • Extension of the spectral stochastic finite element techniques to investigate convection-diffusion problems • Extension of these techniques to more complex fluid-flow, convection-diffusion inverse problems Materials Process Design and Control Laboratory