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Optimal reconstruction of material properties in complex multiphysics phenomena. (English) Zbl 1427.80016

Summary: We develop an optimization-based approach to the problem of reconstructing temperature-dependent material properties in complex thermo-fluid systems described by the equations for the conservation of mass, momentum and energy. Our goal is to estimate the temperature dependence of the viscosity coefficient in the momentum equation based on some noisy temperature measurements, where the temperature is governed by a separate energy equation. We show that an elegant and computationally efficient solution of this inverse problem is obtained by formulating it as a PDE-constrained optimization problem which can be solved with a gradient-based descent method. A key element of the proposed approach, the cost functional gradients are characterized by mathematical structure quite different than in typical problems of PDE-constrained optimization and are expressed in terms of integrals defined over the level sets of the temperature field. Advanced techniques of integration on manifolds are required to evaluate numerically such gradients, and we systematically compare three different methods. As a model system we consider a two-dimensional unsteady flow in a lid-driven cavity with heat transfer, and present a number of computational tests to validate our approach and illustrate its performance.

MSC:

80A23 Inverse problems in thermodynamics and heat transfer
80A20 Heat and mass transfer, heat flow (MSC2010)
80M50 Optimization problems in thermodynamics and heat transfer
65K10 Numerical optimization and variational techniques
76D05 Navier-Stokes equations for incompressible viscous fluids
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
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[1] Volkov, O.; Protas, B.; Liao, W.; Glander, D., Adjoint-based optimization of thermo-fluid phenomena in welding processes, Journal of Engineering Mathematics, 65, 201-220 (2009) · Zbl 1180.76050
[2] Isakov, V., Inverse Problems for Partial Differential Equations (2006), Springer · Zbl 1092.35001
[3] Tarantola, A., Inverse problem theory and methods for model parameter estimation, SIAM (2005) · Zbl 1074.65013
[4] Nashed, M. Z.; Scherzer, O., Inverse Problems, Image Analysis, and Medical Imaging (2002), AMS
[5] Gottlieb, J.; DuChateau, P., Parameter Identification and Inverse Problems in Hydrology, Geology, and Ecology, Kluwer Academic Publishers (1996)
[6] Chavent, G.; Lemonnier, P., Identification de la Identification de la Non-Linearité D’Une Équation Parabolique Quasilineaire, Applied Mathematics and Optimization, 1, 121-162 (1974) · Zbl 0291.35049
[7] Bukshtynov, V.; Volkov, O.; Protas, B., On optimal reconstruction of constitutive relations, Physica D, 240, 1228-1244 (2011) · Zbl 1223.49031
[8] Engquist, B.; Tornberg, A.-K.; Tsai, R., Discretization of Dirac delta functions in level set methods, Journal of Computational Physics, 207, 28-51 (2004) · Zbl 1074.65025
[9] Zahedi, S.; Tornberg, A.-K., Delta functions approximations in level set methods by distance function extension, Journal of Computational Physics, 229, 2199-2219 (2010) · Zbl 1186.65018
[10] Mayo, A., The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM Journal of Numerical Analysis, 21, 285-299 (1984) · Zbl 1131.65303
[11] Smereka, P., The numerical approximation of a delta function with application to level set methods, Journal of Computational Physics, 211, 77-90 (2006) · Zbl 1086.65503
[12] Beale, J. T., A proof that a discrete delta function is second-order accurate, Journal of Computational Physics, 227, 2195-2197 (2008) · Zbl 1136.65017
[13] Towers, J. D., Discretizing delta functions via finite differences and gradient normalization, Journal of Computational Physics, 228, 3816-3836 (2009) · Zbl 1167.65007
[14] Towers, J. D., Two methods for discretizing a delta function supported on a level set, Journal of Computational Physics, 220, 915-931 (2007) · Zbl 1115.65028
[15] Min, C.; Gibou, F., Geometric integration over irregular domains with application to level-set methods, Journal of Computational Physics, 226, 1432-1443 (2007) · Zbl 1125.65021
[16] Min, C.; Gibou, F., Robust second-order accurate discretizations of the multi-dimensional heaviside and Dirac delta functions, Journal of Computational Physics, 227, 9686-9695 (2008) · Zbl 1153.65014
[17] Muschik, W., Aspects of Non-Equilibrium Thermodynamics (1989), World Scientific
[18] Coleman, B. D.; Noll, W., The thermodynamics of elastic materials with heat conduction and viscosity, Archive for Rational Mechanics Analysis, 13, 167-178 (1963) · Zbl 0113.17802
[19] Liu, I.-S., Method of lagrange multipliers for exploitation of the entropy principle, Archive for Rational Mechanics Analysis, 46, 131-148 (1972) · Zbl 0252.76003
[20] Triano, V.; Papenfuss, Ch.; Cimmelli, V. A.; Muschik, W., Exploitation of the second law: Coleman-Noll and Liu procedure in comparison, Journal of Non-Equilibrium Thermodynamics, 33, 47-60 (2008) · Zbl 1186.80004
[21] Kügler, Ph., Identification of a temperature dependent heat conductivity from single boundary measurements, SIAM Journal on Numerical Analysis, 41, 1543-1563 (2003) · Zbl 1061.35169
[22] Ruszczyński, A., Nonlinear Optimization (2006), Princeton University Press · Zbl 1108.90001
[23] Vogel, C. R., Computational Methods for Inverse Problems (2002), SIAM · Zbl 1008.65103
[24] Boyd, S.; Vandenberghe, L., Convex Optimization (2004), Cambridge University Press · Zbl 1058.90049
[25] Luenberger, D., Optimization by Vector Space Methods (1969), John Wiley and Sons · Zbl 0176.12701
[26] Nocedal, J.; Wright, S., Numerical Optimization (2002), Springer
[27] Berger, M. S., Nonlinearity and Functional Analysis (1977), Academic Press
[28] Gunzburger, M. D., Perspectives in flow control and optimization, SIAM (2003) · Zbl 1088.93001
[29] Protas, B.; Bewley, T.; Hagen, G., A comprehensive framework for the regularization of adjoint analysis in multiscale pde systems, Journal of Computational Physics, 195, 49-89 (2004) · Zbl 1049.65059
[30] Engl, H.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer · Zbl 0859.65054
[32] Osher, S.; Sethian, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamiltonian-Jacobi formulations, Journal of Computational Physics, 79, 12-49 (1988) · Zbl 0659.65132
[33] Gockenbach, M. S., Understanding and Implementing the Finite Element Method (2006), SIAM · Zbl 1105.65112
[34] Pérez, C. E.; Thomas, J.-M.; Blancher, S.; Creff, R., The steady Navier-Stokes/energy system with temperature-dependent viscosity – part 1: analysis of the continuous problem, International Journal for Numerical Methods in Fluids, 56, 63-89 (2007) · Zbl 1127.76018
[35] Bruneau, Ch.-H.; Saad, M., The 2D lid-driven cavity problem revisited, Computers & Fluids, 35, 326-348 (2006) · Zbl 1099.76043
[36] Ghia, U.; Ghia, K. N.; Shin, C. T., High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Journal of Computational Physics, 48, 387-411 (1982) · Zbl 0511.76031
[37] Homescu, C.; Navon, I. M.; Li, Z., Suppression of vortex shedding for flow around a circular cylinder using optimal control, International Journal for Numerical Methods in Fluids, 38, 43-69 (2002) · Zbl 1007.76019
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