Adjoint computational methods for 2D inverse design of linear transport equations on unstructured grids
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(2019) 38:168
Adjoint computational methods for 2D inverse design of linear transport equations on unstructured grids M. Morales-Hernández1,2
· E. Zuazua3,4,5
Received: 26 March 2018 / Revised: 7 June 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract We address the problem of inverse design of linear hyperbolic transport equations in 2D heterogeneous media. We develop numerical algorithms based on gradient-adjoint methodologies on unstructured grids. While the flow equation is compulsorily solved by means of a second order upwind scheme so to guarantee sufficient accuracy, the necessity of using the same order of approximation when solving the sensitivity or adjoint equation is examined. Two test cases, including Doswell frontogenesis, are analysed. We show the convenience of using a low order method for the adjoint resolution, both in terms of accuracy and efficiency. An analytical explanation for this fact is also provided in the sense that, when employing higher order schemes for the adjoint problem, spurious high frequency numerical components slow down the convergence process. Keywords Linear transport · Inverse design · Sensitivity · First and second order schemes · Gradient descent method · Adjoint Mathematics Subject Classification 35L04 · 49M04 · 93B00
1 Introduction Adjoint methods have been systematically associated to the optimal control design (Herty et al. 2015) and their applications to aerodynamics (Carpentieri et al. 2007; Castro et al.
Communicated by Cristina Turner.
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M. Morales-Hernández [email protected]
1
Fluid Mechanics, LIFTEC-EINA,CSIC-Universidad Zaragoza, Maria de Luna 3, 50018 Zaragoza, Spain
2
Department of Soil and Water, EEAD-CSIC, Avda. Montañana 1005, 50059 Zaragoza, Spain
3
DeustoTech, University of Deusto, 48007 Bilbao, Basque Country, Spain
4
Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, Madrid 28049, Spain
5
Facultad Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country, Spain
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2007; Giles and Pierce 2000; Ulbrich 2001). During the last decades, several works were oriented to develop a robust control theory based on the concepts of observability, optimality and controllability for linear and non-linear equations and systems of equations (Castro et al. 2008; Zuazua 2002, 2005, 2007). At the same time, and focusing on applications related to computational fluid dynamics, many contributions can be found in the literature concerning inverse design, parameter identification or optimization in general for engineering problems (Power et al. 2006) and in aeronautical applications in particular (Castro et al. 2007; Jameson 1988). The use of adjoint equations and gradient methods for this purposes is widely justified via the minimization of a functional or cost function (Huang and Ascher 2014; Nocedal and Wright 1999), resulting a suitable way of analysing the sensitivity of a complex
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