Trefftz method for a polynomial-based boundary identification in two-dimensional Laplacian problems

Hożejowski, Leszek

doi:10.15632/jtam-pl.54.3.935

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…We find such an approach insufficient. It seems a better way to repeat such an experiment with noisy data a number of times (Hożejowski, 2016a), even a large number of times and then use statistical tools to calculate the proper confidence interval for the boundary location.…”

Section: Sensitivity Analysismentioning

confidence: 99%

“…A very convenient tool for use in boundary reconstruction is the Trefftz method and the method of fundamental solutions (MFS), which are mathematically equivalent for the Laplacian problems. There have been proposed some solutions based on the Trefftz method and referring to Laplace's equation (Fan and Chan, 2011;Fan et al, 2012;Karageorghis et al, 2014;Ho_ zejowski, 2016aHo_ zejowski, , 2016b, including those employing MSF (Marin et al, 2011;Lesnic and Bin-Mohsin, 2012;Sun and Ma, 2015). The approach was successfully applied to the problems governed by the biharmonic equation (Chan and Fan, 2013) or the modified Helmholtz equation (Fan et al, 2014).…”

Section: Introductionmentioning

confidence: 99%

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A boundary determination problem in steady-state heat conduction – a solution and sensitivity analysis

Hożejowski

2019

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Purpose The purpose of this paper is to propose a numerical procedure for discrete identification of the missing part of the domain boundary in a heat conduction problem. A new approach to sensitivity analysis is intended to give a better understanding of the influence of measurement error on boundary reconstruction. Design/methodology/approach The solution of Laplace’s equation is obtained using the Trefftz method, and then each of the sought boundary points can be derived numerically from a nonlinear equation. The sensitivity analysis comes down to the analytical evaluation of a sensitivity factor. Findings The proposed method very accurately recovers the unknown boundary, including irregular shapes. Even a very large number of the boundary points can be determined without causing computational problems. The sensitivity factor provides quantitative assessment of the relationship between the temperature measurement errors and boundary identification errors. The numerical examples show that some boundary reconstruction problems are error-sensitive by nature but such problems can be recognized with the use of a sensitive factor. Originality/value The present approach based on the Trefftz method separates, in terms of computation, specification of the coefficients appearing in the Trefftz method and missing coordinates of the sought boundary points. Due to introducing a sensitivity factor, a more profound sensitivity analysis was successfully conducted.

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Section: Sensitivity Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A boundary determination problem in steady-state heat conduction – a solution and sensitivity analysis

Hożejowski

2019

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“…In the paper of Liu and Wang (2018), the Cauchy problem for the Laplace’s equation was solved with the use of the method of fundamental solutions and the energy regularization technique to choose the source points. The Laplace’s equation was also solved with the use of iterative algorithms (Frąckowiak et al , 2015a, 2015b), of the Trefftz method (Ciałkowski and Frąckowiak, 2002; Ciałkowski and Grysa, 2010; Grysa et al , 2012; Hożejowski, 2016; Lin et al , 2018), of the method of fundamental solution (Kołodziej and Mierzwiczak, 2008; Mierzwiczak et al , 2015; Mierzwiczak and Kołodziej, 2011) and of the collocation method (Joachimiak et al , 2016). In many cases, the regularization of the inverse problem concerns the problem of choosing the regularization parameter.…”

Section: Introductionmentioning

confidence: 99%

Choice of the regularization parameter for the Cauchy problem for the Laplace equation

Joachimiak

2020

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Purpose In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy. Design/methodology/approach The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied. Findings Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm Nmax. Practical implications Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary. Originality/value The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

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“…In such cases, it is possible to determine the distribution of temperature on the boundary through solving the inverse boundary problem or the inverse Cauchy-type problem. There are many methods for solving inverse problems, such as method of fundamental solutions (MFS) (Karageorghis and Lesnic, 2018; Kołodziej and Mierzwiczak, 2008; Liu and Wang, 2018; Marin, 2005; Mierzwiczak and Kołodziej, 2011; Wei and Chen, 2012), singular boundary method (Mierzwiczak et al , 2015), boundary element method (BEM) (Lesnic et al , 1997; Sun, 2017), singular value decomposition (SVD) (Hasanov and Mukanova, 2015; Louis, 1989), iterative algorithms (Frąckowiak et al , 2015a; Frąckowiak et al , 2015b), method of Trefftz functions (Ciałkowski and Frąckowiak, 2002; Grysa et al , 2012; Hożejowski, 2016), collocation method (Kołodziej and Mierzwiczak, 2008), modified collocation Trefftz method (Liu, 2008), Fourier method (Mukanova, 2013) and other. In the reference paper (Frąckowiak et al , 2008), the application of the modified finite element method to solving the Cauchy problem for the Laplace’s equation in an annulus is presented.…”

Section: Introductionmentioning

confidence: 99%

Stable method for solving the Cauchy problem with the use of Chebyshev polynomials

Joachimiak

Ciałkowski

Frąckowiak

2019

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Purpose The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ1) and find proper method for the problem regularization. Design/methodology/approach The solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle. Findings Calculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ1 was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization. Practical implications Discussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion. Originality/value In this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.

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Trefftz method for a polynomial-based boundary identification in two-dimensional Laplacian problems

Cited by 5 publications

References 21 publications

A boundary determination problem in steady-state heat conduction – a solution and sensitivity analysis

A boundary determination problem in steady-state heat conduction – a solution and sensitivity analysis

Choice of the regularization parameter for the Cauchy problem for the Laplace equation

Stable method for solving the Cauchy problem with the use of Chebyshev polynomials

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