Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

Boos, Everton; Luchesi, Vanda Maria; Bazán, Fermín S. V.

doi:10.1080/17415977.2020.1801675

Cited by 2 publications

(18 citation statements)

References 42 publications

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“…The conclusion to be drawn in the latter case is that any attempt to reconstruct values of k 11 and k 22 at the corners from temperature data that do not depend on the conductivities at these locations will be unsuccessful or the results will be inaccurate. We believe this explains why the reconstructions of conductivities at the corners reported in Boos et al (2020) were somewhat imprecise. The minimization problem can be accomplished in different ways by methods that use derivatives and methods that do not.…”

Section: Inverse Problemmentioning

confidence: 97%

“…A significant change from standard LMM implementations is that the version presented here uses a scaling matrix that is singular and of the form D=scriptDTD. Examples that show the gain of using this type of scaling matrix in some reconstruction problems can be seen in Boos et al (2020), Bazán et al (2017) and Ismailov et al (2018).…”

Section: Inverse Problemmentioning

confidence: 99%

“…In particular, the estimation of thermal properties of composite materials is an ill-posed inverse heat conduction problem (IHCP) and presents difficulties because it is intrinsically unstable and thus very sensitive to the inaccuracy of input data (Baralić et al , 2019; Cao et al , 2019; Özişik, 1993; Mahmood and Lesnic, 2019). Contrary to IHCP, transient heat conduction problems used for the determination of temperature distribution are well-posed and treated numerically by a number of integration methods such as finite difference method (FDM), the finite element method (FEM), the boundary element method (BEM) and most recently, collocation pseudospectral methods (CPM) (Bazán et al , 2017; Ismailov et al , 2018; Cao et al , 2019; Mahmood and Lesnic, 2019; Mohebbi and Sellier, 2016; Pasdunkorale and Turner, 2003; Boos et al , 2020; Telejko and Malinowski, 2004).…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, a method for estimating 2D orthotropic material bodies is proposed. As in Boos et al (2020), the method relies on the minimization of a square sum of differences between the measured temperatures and the calculated temperatures by a pseudospectral method. The resulting minimization problem is solved by using the LMM in which the calculation of the Jacobian matrix is based on the differentiation of the semi-discrete system generated by the discretization of spatial derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…The resulting minimization problem is solved by using the LMM in which the calculation of the Jacobian matrix is based on the differentiation of the semi-discrete system generated by the discretization of spatial derivatives. The main goal of the work is thus to overcome the difficulties of the method by Boos et al (2020), as well as to develop a strategy that makes the proposed method functional when dealing with real-world problems where available data is collected only in some locations inside or in the boundary spatial domain. This work was motivated by a machining process reported by Luchesi and Coelho (2012), where the objective is to estimate a moving heat source from experimental collected temperature data at some internal workpiece locations, using a transient 2D heat conduction model governed by a partial differential equation (PDE) with appropriate initial and boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Thermal conductivity reconstruction method with application in a face milling operation

Boos

Bazán

Luchesi

2023

HFF

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Purpose This paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source. Design/methodology/approach The discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method. Findings The paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted. Originality/value The paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

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Section: Inverse Problemmentioning

confidence: 97%

Section: Inverse Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Thermal conductivity reconstruction method with application in a face milling operation

Boos

Bazán

Luchesi

2023

HFF

Self Cite

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Reconstrução de condutividade térmica ortotrópica com método pseudoespectral de Chebyshev e aplicações

Boos,

Bazán,

Luchesi

2023

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Em trabalhos recentes, estudamos um problema de condução de calor em um domínio retangular modelado por uma EDP com condições de fronteira mistas e condição inicial, e envolvendo parâmetros tais como capacidade térmica, condutividade, entre outros. Conhecer a condutividade térmica de um material é assunto de importância em processos industriais e tem se tornado um tópico ativo de pesquisa das últimas décadas. Fornecemos uma forma de discretizar o modelo original através do método pseudo-espectral de Chebyshev nas variáveis espaciais, pelas suas boas propriedades de aproximação com baixo custo numérico, e a regra do trapézio na variável temporal, pela segunda ordem de convergência e estabilidade absoluta. O problema inverso de aproximar a condutividade térmica a partir de dados capturados da temperatura é então elaborado como um problema de mínimos quadrados não lineares, que faz uso recorrente da discretização pregressa. A minimização é feita através de uma versão do método de Levenberg-Marquardt (LMM) com matrizes de scaling singular escolhidas para representarem operadores de derivação discretos de primeira e segunda ordens, com a intenção de introduzir suavidade nos iterados construídos. A motivação para tal vem de bons resultados de técnicas similares em problemas lineares através, por exemplo, da regularização de Tikhonov. Para amenizar o efeito de imprecisões nos dados de temperatura fornecidos, o princípio da discrepância (DP) é utilizado como critério de parada. Resultados numéricos sintéticos e para dados experimentais ilustram o potencial da técnica proposta, com reconstruções de qualidade a um baixo custo operacional, mesmo em situações com medições restritas.Palavras-chave. Problemas inversos. Métodos iterativos. Condutividade térmica. Método pseudo-espectral de Chebyshev. Método de Levenberg-Marquardt com scaling singular. Princípio da discrepância.

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Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

Cited by 2 publications

References 42 publications

Thermal conductivity reconstruction method with application in a face milling operation

Thermal conductivity reconstruction method with application in a face milling operation

Reconstrução de condutividade térmica ortotrópica com método pseudoespectral de Chebyshev e aplicações

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