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  • 1.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Karlsson, Matts
    Linköpings universitet, Institutionen för ekonomisk och industriell utveckling, Mekanisk värmeteori och strömningslära. Linköpings universitet, Tekniska fakulteten. Linköpings universitet, Centrum för medicinsk bildvetenskap och visualisering, CMIV.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Nazarov, Sergei A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia.
    A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients2018Rapport (Övrigt vetenskapligt)
    Abstract [en]

    One dimensional models for fluid flow in tubes are frequently used tomodel complex systems, such as the arterial tree where a large numberof vessels are linked together at bifurcations. At the junctions transmission conditions are needed. One popular option is the classic Kirchhoffconditions which means conservation of mass at the bifurcation andprescribes a continuous pressure at the joint.

    In reality the boundary layer phenomena predicts fast local changesto both velocity and pressure inside the bifurcation. Thus it is not appropriate for a one dimensional model to assume a continuous pressure. In this work we present a modification to the classic Kirchhoff condi-tions, with a symmetric pressure drop matrix, that is more suitable forone dimensional flow models. An asymptotic analysis, that has beencarried out previously shows that the new transmission conditions hasen exponentially small error.

    The modified transmission conditions take the geometry of the bifurcation into account and can treat two outlets differently. The conditions can also be written in a form that is suitable for implementationin a finite difference solver. Also, by appropriate choice of the pressuredrop matrix we show that the new transmission conditions can producehead loss coefficients similar to experimentally obtained ones.

  • 2.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Ghosh, Arpan
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Nazarov, S. A.
    St Petersburg State Univ, Russia; Inst Problems Mech Engn RAS, Russia.
    A one dimensional model of blood flow through a curvilinear artery2018Ingår i: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 63, s. 633-643Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier-Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow. (C) 2018 Elsevier Inc. All rights reserved.

    Publikationen är tillgänglig i fulltext från 2020-07-17 13:46
  • 3.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Mpinganzima, Lydie
    Univ Rwanda, Rwanda.
    Turesson, Bengt-Ove
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Robin-Dirichlet algorithms for the Cauchy problem for the Helmholtz equation2018Ingår i: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 26, nr 7, s. 1062-1078Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Mazya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

  • 4.
    Jahedi, Mohammad
    et al.
    University of Gävle, Sweden.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Wren, Joakim
    Linköpings universitet, Institutionen för ekonomisk och industriell utveckling, Mekanisk värmeteori och strömningslära. Linköpings universitet, Tekniska fakulteten.
    Moshfegh, Bahram
    Linköpings universitet, Institutionen för ekonomisk och industriell utveckling, Energisystem. Linköpings universitet, Tekniska fakulteten. University of Gavle, Sweden.
    Transient inverse heat conduction problem of quenching a hollow cylinder by one row of water jets2018Ingår i: International Journal of Heat and Mass Transfer, ISSN 0017-9310, E-ISSN 1879-2189, Vol. 117, s. 748-756Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this study, a two-dimensional linear transition inverse heat conduction problem (IHCP) was solved using the Generalized Minimal Residual Method (GMRES) in quenching process by water jets. The inverse solution method was validated by set of artificial data and solution sensitivity analysis was done on data noise level, regularization parameter, cell size, etc. An experimental study has been carried out on quenching a rotary hollow cylinder by one row of subcooled water jets. The inverse solution approach enabled prediction of surface temperature and heat flux distribution of test specimen in the quenching experiments by using measured internal specimen temperature. Three different boiling curves were defined in the quenching process of a rotary cylinder. Result obtained by the inverse solution showed clear footprint of rotation in surface temperature and heat flux on each revolution of cylinder and temperature variation damping from quenching surface toward interior of specimen. (C) 2017 Elsevier Ltd. All rights reserved.

  • 5.
    Evarest Sinkwembe, Emanuel
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Singull, Martin
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Yang, Xiangfeng
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Weather Derivatives Pricing Using Regime Switching Model2018Ingår i: Monte Carlo Methods and Applications, ISSN 0929-9629, Vol. 24, nr 1, s. 13-27Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with nonzero drift. We develop mathematical formulas for pricing futures and option contracts on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We use the Monte Carlo simulation method for pricing weather derivatives call option contracts.

  • 6.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Karlsson, Matts
    Linköpings universitet, Institutionen för ekonomisk och industriell utveckling, Mekanisk värmeteori och strömningslära. Linköpings universitet, Tekniska fakulteten.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Nazarov, Sergey A.
    St Petersburg State University, St Petersburg State Polytechnical University, and Institute of Problems of Mechanical Engineering RAS, Russia..
    A one-dimensional model of a false aneurysm2017Ingår i: International Journal of Research in Engineering and Science (IJRES), ISSN 2320-9356, Vol. 5, nr 6, s. 61-73Artikel i tidskrift (Refereegranskat)
    Abstract [en]

     A false aneurysm is a hematoma, i.e. collection ofblood outside of a blood vessel, that forms due to a hole  in the wall of an artery . This represents a serious medical condition that needs to be monitored and, under certain conditions, treatedurgently. In this work a one-dimensional model of a false aneurysm isproposed. The new model is based on a one-dimensional model of anartery previously presented by the authors and it takes into accountthe interaction between the hematoma  and the surrounding musclematerial. The model equations are derived  using rigorous asymptoticanalysis for the case of a simplified geometry.   Even though the model is simple it still supports a realisticbehavior for the system consisting of the vessel and the  hematoma. Using numerical simulations we illustrate the behavior ofthe model. We also investigate the effect  of changing the size of the hematoma. The simulations show that ourmodel can reproduce realistic solutions. For instance we show thetypical strong pulsation of an aneurysm by blood entering the hematoma during the work phase of the cardiac cycle, and the blood returning tothe vessel during the resting phase. Also we show that the aneurysmgrows  if the pulse rate is increased due to, e.g., a higher work load. 

  • 7.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Chen, Lin
    State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.
    Xu, Tao
    State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China; CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, China.
    Wokiyi, Dennis
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, Makerere University, Kampala, Uganda.
    An efficient regularization method for a large scale ill-posed geothermal problem2017Ingår i: Computers & Geosciences, ISSN 0098-3004, E-ISSN 1873-7803, Vol. 105, s. 1-9Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The inverse geothermal problem consists of estimating the temperature distribution below the earth's surface using measurements on the surface. The problem is important since temperature governs a variety of geologic processes, including the generation of magmas and the deformation style of rocks. Since the thermal properties of rocks depend strongly on temperature the problem is non-linear.

    The problem is formulated as an ill-posed operator equation, where the righthand side is the heat-flux at the surface level. Since the problem is ill-posed regularization is needed. In this study we demonstrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a well-posed problem related to the above mentioned operator. The algorithm is designed in such a way that it can deal with both 2D and 3D calculations.

    Numerical results, for 2D domains, show that the algorithm works well and the inverse problem can be solved accurately with a realistic noise level in the surface data.

  • 8.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Orlof, Anna
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Thim, Johan
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots2017Ingår i: Numerical Functional Analysis and Optimization, ISSN 0163-0563, E-ISSN 1532-2467, Vol. 38, nr 3, s. 293-305Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron–electron interactions in the potential.

  • 9.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Mpinganzima, L.
    University of Rwanda, Rwanda.
    Turesson, Bengt-Ove
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation2017Ingår i: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 73, nr 1, s. 163-172Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regularization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H-1/2)* spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well. (C) 2016 Elsevier Ltd. All rights reserved.

  • 10.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Ohlson, Martin
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    More on Estimation of Banded and Banded Toeplitz Covariance Matrices2017Rapport (Övrigt vetenskapligt)
    Abstract [en]

    In this paper we consider two different linear covariance structures, e.g., banded and bended Toeplitz, and how to estimate them using different methods, e.g., by minimizing different norms.

    One way to estimate the parameters in a linear covariance structure is to use tapering, which has been shown to be the solution to a universal least squares problem. We know that tapering not always guarantee the positive definite constraints on the estimated covariance matrix and may not be a suitable method. We propose some new methods which preserves the positive definiteness and still give the correct structure.

    More specific we consider the problem of estimating parameters of a multivariate normal p–dimensional random vector for (i) a banded covariance structure reflecting m–dependence, and (ii) a banded Toeplitz covariance structure.

  • 11.
    Evarest, Emanuel
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Singull, Martin
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Charles, Wilson M.
    Department od Mathematics, University of Dar el Salaam, Tanzania.
    Regime Switching models on Temperature Dynamics2017Ingår i: International Journal of Applied Mathematics and Statistics, ISSN 0973-1377, E-ISSN 0973-7545, Vol. 56, nr 2Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Two regime switching models for predicting temperature dynamics are presented in this study for the purpose to be used for weather derivatives pricing. One is an existing model in the literature (Elias model) and the other is presented in this paper. The new model we propose in this study has a mean reverting heteroskedastic process in the base regime and a Brownian motion in the shifted regime. The parameter estimation of the two models is done by the use expectation-maximization (EM) method using historical temperature data. The performance of the two models on prediction of temperature dynamics is compared using historical daily average temperature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The expected HDDs, CDDs and CAT of the models are compared to the true indices from the real data. Results from the expected HDDs, CDDs and CAT together with their corresponding daily average plots demonstrate that, our model captures temperature dynamics relatively better than Elias model.

  • 12.
    Wikström, Patrik
    et al.
    SSAB Europe, Sweden.
    Nilsson, Christer
    SSAB Europe, Sweden.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Thermomechanical modeling of cooling slabs under a hood after continuous casting2017Ingår i: MATERIALS RESEARCH EXPRESS, ISSN 2053-1591, Vol. 4, nr 11, artikel-id 116517Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Subsequent cooling of slabs after continuous casting is an important step to control before further processing such as hot-rolling. In this work the stresses that occur in slabs during the cooling process were investigated by using a finite element model. In the specific situation that we model a stack of slabs are left to cool under a hood, insulated by a ceramic material, which makes the cooling slower and the temperature distribution more even during the cooling process. The accuracy of the numerical model was validated by a comparison with temperature measurements, and also by a comparison with hot tensile tests for the specific cooling situation under investigation. The results show that the highest stress levels achieved during cooling did not exceed the mechanical strength of the material at the temperatures investigated.

  • 13.
    Evarest Sinkwembe, Emanuel
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Singull, Martin
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Yang, Xiangfeng
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Weather derivatives pricing using regim switching models2017Rapport (Övrigt vetenskapligt)
    Abstract [en]

    In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with mean different from zero. We develop the mathematical formulas for pricing futures contract on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. We also present the mathematical expressions for pricing the corresponding options on futures contracts for the same temperature indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We provide the description of Montecarlo simulation method for pricing weather derivatives under this model and use it to price a few weather derivatives call option contracts.

  • 14.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Karlsson, Matts
    Linköpings universitet, Institutionen för ekonomisk och industriell utveckling, Mekanisk värmeteori och strömningslära. Linköpings universitet, Tekniska fakulteten. Linköpings universitet, Centrum för medicinsk bildvetenskap och visualisering, CMIV.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Nazarov, Sergey A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia; RAS, Russia.
    A one-dimensional model of viscous blood flow in an elastic vessel2016Ingår i: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 274, s. 125-132Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessels wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel. (C) 2015 Elsevier Inc. All rights reserved.

  • 15.
    Evarest, Emanuel
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Singull, Martin
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Charles, Wilson
    Department of Mathematics, University of Dar es Salaam, Tanzania.
    Regime Switching models on Temperature Dynamics2016Rapport (Övrigt vetenskapligt)
    Abstract [en]

    Two regime switching models for predicting temperature dynamics are presented in this study for the purpose to be used for weather derivatives pricing. One is an existing model in the literature (Elias model) and the other is presented in this paper. The new model we propose in this study has a mean reverting heteroskedastic process in the base regime and a Brownian motion in the shifted regime. The parameter estimation of the two models is done by the use expectation-maximization (EM) method using historical temperature data. The performance of the two models on prediction of temperature dynamics is compared using historical daily average temperature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The expected HDDs, CDDs and CAT of the models are compared to the true indices from the real data. Results from the expected HDDs, CDDs and CAT together with their corresponding daily average plots demonstrate that, our model captures temperature dynamics relatively better than Elias model.

  • 16.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Mpinganzima, Lydie
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Turesson, Bengt-Ove
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation2014Ingår i: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 68, nr 1-2, s. 44-60Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by the authors for solving this problem but the convergence has been rather slow. We demonstrate how to instead use conjugate gradient methods for accelerating the convergence. The main idea is to introduce an artificial boundary in the interior of the domain. This addition of the interior boundary allows us to derive an inner product that is natural for the application and that gives us a proper framework for implementing the steps of the conjugate gradient methods. The numerical results performed using the finite difference method show that the conjugate gradient based methods converge considerably faster than the modified alternating iterative procedure studied previously.

  • 17.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Mpinganzima, Lydie
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Turesson, Bengt-Ove
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    An alternating iterative procedure for the Cauchy problem for the Helmholtz equation2014Ingår i: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 22, nr 1, s. 45-62Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method.

  • 18.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Baravdish, George
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska högskolan.
    Coefficient identification in PDEs applied to image inpainting2014Ingår i: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 242, s. 227-235Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we introduce the concept of parameter identification problems, which are inverse problems, as a methodology to inpainting. More specifically, as a first study in this new direction, we generalize the method of harmonic inpainting by studying an elliptic equation in divergence form where we assume that the diffusion coefficient is unknown. As a first step, this unknown coefficient is determined from the information obtained by the known data in the image. Next, we fill in the region with missing data by solving an elliptic equation in divergence form using this obtained diffusion coefficient. An error analysis shows that this approach is promising and our numerical experiments produces better results than the harmonic inpainting.

  • 19.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    Kozlov, Vladimir A.
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Mpinganzima, Lydie
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan. University of Rwanda.
    Turesson, Bengt-Ove
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Numerical Solution of the Cauchy Problem for the Helmholtz Equation2014Rapport (Övrigt vetenskapligt)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.

  • 20.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Kozlov, Vladimir
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Mpinganzima, Lydie
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Turesson, Bengt-Ove
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation2014Manuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers in the Helmholtz equation. We prove here that if we alternate Robin and Dirichlet boundary conditions instead of Neumann and Dirichlet boundary conditions, then the algorithm will converge. We present also another algorithm based on the same idea, which converges for large wavenumbers. Numerical implementations obtained using the finite difference method are presented. Numerical results illustrate that the algorithms suggested in this paper, produce a convergent iterative sequences.

  • 21.
    Chen, Lin
    et al.
    Chinese Academy of Sciences, Beijing, China.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Zhang, Zhongjie
    Chinese Academy of Sciences, Beijing, China.
    Wang, Peng
    Chinese Academy of Sciences, Guangzhou, China; University of Chinese Academy of Sciences, Beijing, China.
    Wu, Jing
    Chinese Academy of Sciences, Beijing, China.
    Xu, Tao
    Chinese Academy of Sciences, Beijing, China.
    Seismically constrained thermo-rheological structure of the eastern Tibetan margin: Implication for lithospheric delamination2014Ingår i: Tectonophysics, ISSN 0040-1951, E-ISSN 1879-3266, Vol. 627, s. 122-134Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The eastern Tibetan margin bordered by the Longmen Shan range exhibits significant lateral differences in the lithospheric structure and thermal state. To investigate the roles of these differences in mountain building, we construct a thermo-rheological model along a wide-angle seismic profile across the eastern Tibetan margin based on recent seismic and thermal observations. The thermal modeling is constrained by the surface heat flow data and crustal P wave velocity model. The construction of the theological envelopes is based on rock mechanics results, and involves two types of rheology: a weak case where the lower crust is felsic granulite and the lithospheric mantle is wet peridotite, and a strong case where the lower crust is mafic granulite and the lithospheric mantle is dry peridotite. The results demonstrate: (1) one high-temperature anomaly exists within the uppermost mantle beneath eastern Tibet, indicating that the crust in eastern Tibet is remarkably warmer than that in the Sichuan basin, and (2) the rheological strength of the lithospheric mantle beneath eastern Tibet is considerably weaker than that beneath the Sichuan basin. The rheological profiles are in accord with seismicity distribution. By combining these results with the observed crustal/lithospheric architecture, Pn velocity distribution and magmatism in the eastern Tibetan margin, we suggest that the delamination of a thickened lithospheric mantle root beneath eastern Tibet is responsible for the growth of the eastern Tibetan margin.

  • 22.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska högskolan.
    An Inverse Heat Conduction Problem and Improving Shielded Thermocouple Accuracy2012Ingår i: Numerical Heat Transfer, Part A Applications, ISSN 1040-7782, E-ISSN 1521-0634, Vol. 61, nr 10, s. 754-763Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A shielded thermocouple is a measurement device used for monitoring the temperature in chemically, or mechanically, hostile environments. The sensitive parts of the thermocouple are protected by a shielding layer. In order to improve the accuracy of the measurement device, we study an inverse heat conduction problem where the temperature on the surface of the shielding layer is sought, given measured temperatures in the interior of the thermocouple. The procedure is well suited for real-time applications where newly collected data is continuously used to compute current estimates of the surface temperature. Mathematically we can formulate the problem as a Cauchy problem for the heat equation, in cylindrical coordinates, where data is given along the line r = r 1 and the solution is sought at r 1 < r ≤ r 2. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. Thus, regularization techniques are needed. The ill–posedness of the problem is analyzed and a numerical method is proposed. Numerical experiments demonstrate that the proposed method works well.

  • 23.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Mpinganzima, Lydie
    National University of Rwanda, Box 117, Butare, Rwanda.
    A Data Assimilation Approach to Coefficient Identification2011Rapport (Övrigt vetenskapligt)
    Abstract [en]

    The thermal conductivity properties of a material can be determined experimentally by using temperature measurements taken at specified locations inside the material. We examine a situation where the properties of a (previously known) material changed locally. Mathematically we aim to find the coefficient k(x) in the stationary heat equation (kTx)x = 0;under the assumption that the function k(x) can be parametrized using only a few degrees of freedom.

    The coefficient identification problem is solved using a least squares approach; where the (non-linear) control functional is weighted according to the distribution of the measurement locations. Though we only discuss the 1D case the ideas extend naturally to 2D or 3D. Experimentsdemonstrate that the proposed method works well.

     

     

     

     

  • 24.
    Wikstrom, P.
    et al.
    Wikström, P., Department of Materials Science and Engineering, Division of Energy and Furnace Technology, Royal Institute of Technology (KTH), Brinellvägen 23, S-100 44 Stockholm, Sweden.
    Blasiak, W.
    Department of Materials Science and Engineering, Division of Energy and Furnace Technology, Royal Institute of Technology (KTH), Brinellvägen 23, S-100 44 Stockholm, Sweden.
    Berntsson, Fredrik
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method2007Ingår i: Applied Thermal Engineering, ISSN 1359-4311, E-ISSN 1873-5606, Vol. 27, nr 14-15, s. 2463-2472Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the steel industry it is of great importance to be able to control the surface temperature and heating- or cooling rates during heat treatment processes. An experiment was performed in which a steel slab was heated up to 1250 °C in a fuel fired test furnace. The transient surface temperature and heat flux of a steel slab is calculated using a model for inverse heat conduction. That is, the time dependent local surface temperature and heat flux of a slab is calculated on the basis of temperature measurements in selected points of its interior by using a model of inverse heat conduction. Time- and temperature histories were measured at three points inside a steel slab. Measured temperature histories at the two lower locations of the slab were used as input to calculate the temperature at the position of the third location. A comparison of the experimentally measured and the calculated temperature histories was made to verify the model. The results showed very good agreement and suggest that this model can be applied to similar applications in the Steel industry or in other areas where the target of investigation for some reason is inaccessible to direct measurements. © 2007 Elsevier Ltd. All rights reserved.

  • 25. Wikstrom, Patrik
    et al.
    Blasiak, Wlodzimierz
    Berntsson, Fredrik
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Estimation of the transient surface temperature, heat flux and effective heat transfer coefficient of a slab in an industrial reheating furnace by using an inverse method2007Ingår i: STEEL RESEARCH INTERNATIONAL, ISSN 1611-3683, Vol. 78, nr 1, s. 63-70Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the steel industry it is of great importance to be able to control the surface temperature and heating or cooling rates during heat treatment processes. In this paper, a steel slab is heated up to 1300 degrees C in an industrial reheating furnace and the temperature data are recorded during the reheating process. The transient local surface temperature, heat flux and effective heat transfer coefficient of the steel slab ares calculated using a model for inverse heat conduction. The calculated surface temperatures are compared with the temperatures achieved by using a model of the heating process with the help of the software STEELTEMP (R) 2D. The results obtained show very good agreement and suggest that the inverse method can be applied to similar high temperature applications with very good accuracy.

  • 26.
    Elden, Lars
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Berntsson, Fredrik
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    A stability estimate for a Cauchy problem for an elliptic partial differential equation2005Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 21, nr 5, s. 1643-1653Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y ≤ 0, and boundary data are for x ≤ 0 and x ≤ 1. The elliptic operator is self-adjoint with non-constant, smooth coefficients. The solution for y ≤ 1 is sought. This Cauchy problem is ill-posed in an L2-setting. A stability functional is defined, for which a differential inequality is derived. Using this inequality a stability result of Hölder type is proved. It is demonstrated explicitly how the stability depends on the smoothness of the coefficients. The results can also be used for rectangle-like regions that can be mapped conformally onto a rectangle. © 2005 IOP Publishing Ltd.

  • 27.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Eldén, Lars
    Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
    Numerical Solution of Cauchy Problems for Elliptic Equations in ``Rectangle-like'' Geometries2005Ingår i: Proceedings for the FEMLAB Conference 2005, 2005Konferensbidrag (Övrigt vetenskapligt)
    Abstract [en]

    We consider two dimensional inverse steady state heat conductionproblems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on onepart of the boundary and we want to find the solution in the wholedomain. The problem is ill--posed in the sense that the solution doesnot depend continuously on the data.

    Using an orthogonal coordinate transformation the domain is mappedonto a rectangle. The Cauchy problem can then be solved by replacing one derivative by a bounded approximation. The resulting well--posed problem can then be solved by a method of lines. A bounded approximation of the derivative can be obtained by differentiating a cubic spline, that approximate the function in theleast squares sense. This particular approximation of the derivativeis computationally efficient and flexible in the sense that its easy to handle different kinds of boundary conditions.This inverse problem arises in iron production, where the walls of amelting furnace are subject to physical and chemical wear. Temperature and heat--flux data are collected by several thermocouples locatedinside the walls. The shape of the interface between the molten ironand the walls can then be determined by solving an inverse heatconduction problem.  In our work we make extensive use of Femlab for creating testproblems. By using FEMLAB we solve relatively complex model problems for the purpose of creating numerical test data used for validating our methods. For the types of problems we are intressted in numerical artefacts appear, near corners in the domain, in the gradients that Femlab calculates. We demonstrate why this happen and also how we deal with the problem.

  • 28.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Elden, Lars
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Numerical Solution of Cauchy Problems for Elliptic Equations in "Rectangle-like" Geometries2005Ingår i: FEMLAB Conference,2005, Stockholm: Comsol AB , 2005Konferensbidrag (Övrigt vetenskapligt)
    Abstract [en]

    We consider two dimensional inverse steady state heat conduction problems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on one part of the boundary and we want to find the solution in the whole domain. The problem is ill--posed in the sense that the solution does not depend continuously on the data. Using an orthogonal coordinate transformation the domain is mapped onto a rectangle. The Cauchy problem can then be solved by replacing one derivative by a bounded approximation. The resulting well--posed problem can then be solved by a method of lines. A bounded approximation of the derivative can be obtained by differentiating a cubic spline, that approximate the function in the least squares sense. This particular approximation of the derivative is computationally efficient and flexible in the sense that its easy to handle different kinds of boundary conditions. This inverse problem arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. Temperature and heat--flux data are collected by several thermocouples located inside the walls. The shape of the interface between the molten iron and the walls can then be determined by solving an inverse heat conduction problem. In our work we make extensive use of Femlab for creating test problems. By using Femlab we solve relatively complex model problems for the purpose of creating numerical test data used for validating our methods. For the types of problems we are intressted in numerical artefacts appear, near corners in the domain, in the gradients that Femlab calculates. We demonstrate why this happen and also how we deal with the problem.

  • 29.
    Berntsson, Fredrik
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Sequential solution of the sideways heat equation by windowing of the data2003Ingår i: Inverse Problems in Engineering, ISSN 1068-2767, E-ISSN 1029-0281, Vol. 11, nr 2, s. 91-103Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The sideways heat equation is a one-dimensional model of a problem, where one wants to determine the temperature on the surface of a body using interior measurements. More precisely, we consider a heat conduction problem, where temperature and heat-flux data are available along the line x = 1 and the solution is sought in the interval 0 = x < 1. The problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation with a bounded spectral approximation. The cut off level in the spectral approximation acts as a regularization parameter, that controls the degree of smoothness in the solution. In certain applications one wants to solve the sideways heat equation in real time, i.e. to constantly update the solution as new measurements are recorded. For this case sequential solution methods are required.

  • 30.
    Berntsson, Fredrik
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.
    Boundary identification for an elliptic equation2002Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 18, nr 6, s. 1579-1592Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider an inverse problem for the two-dimensional steady-state heat equation. More precisely, the heat equation is valid in a domain O, that is a subset of the unit square. Temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and 1 are assumed to be insulated. From these we wish to determine the temperature in the domain O. Furthermore, a part of the boundary ?O is considered to be unknown, and must also be determined. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least-squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equations, which can be solved using standard numerical methods. The inverse problem that we consider in this paper arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an inverse heat conduction problem.

  • 31.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    A survey of methods for determinig surface temperatures using interior measurements2001Ingår i: Trends in Heat, Mass & Momentum Transfer, ISSN 0973-2446, Vol. 7, nr pp, s. 105-128Artikel i tidskrift (Refereegranskat)
  • 32.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Boundary identification for an elliptic equation2001Rapport (Övrigt vetenskapligt)
    Abstract [en]

    We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.

    The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.

    The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.

  • 33.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Numerical methods for inverse heat conduction problems2001Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    In many industrial applications one wishes to determine the temperature history on the surface of a body, where the surface itself is inaccessible for measurements. The sideways heat equation is a model of this situation. In a one-dimensional setting this is formulated mathematically as a Cauchy problem for the heat equation, where temperature and heat--flux data are available along the line x=1, and a solution is sought for 0 ≤ x< 1. This problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation by a bounded approximation. We consider both spectral and wavelet approximations of the derivative. The resulting problem is a system of ordinary differential equations in the space variable, that can be solved using standard methods, e.g. a Runge-Kutta method. The methods are analyzed theoretically, and error estimates are derived, that can be used for selecting the appropriate level of regularization. The numerical implementation of the proposed methods is discussed. Numerical experiments demonstrate that the proposed methods work well, and can be implemented efficiently. Furthermore, the numerical methods can easily be adapted to solve problems with variable coefficients, and also non-linear equations. As test problems we take model equations, with constant and variable coefficients. Also, we solve problems from applications, with actual measured data.

    Inverse problems for the stationary heat equation are also discussed. Suppose that the Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are given on the outer boundary, and we wish to compute the steady state temperature on the inner boundary. A standard approach is to discretize the equation by finite differences, and use Tikhonov's method for stabilizing the discrete problem, which leads to a large sparse least squares problem. Alternatively, we propose to use a conformal mapping to transform the domain into an annulus, where the equivalent problem can be solved using separation of variables. The ill-posedness is dealt with by filtering away high frequencies from the solution. Numerical results using both methods are presented. A closely related problem is that of determining the stationary temperature inside a body, from temperature and heat-flux measurements on a part of the boundary. In practical applications it is sometimes the case that the domain, where the differential equation is valid, is partly unknown. In such cases we want to determine not only the temperature, but also the shape of the boundary of the domain. This problem arises, for instance, in iron production, where the walls of a melting furnace is subject to both physical and chemical wear. In order to avoid a situation where molten metal breaks out through the walls the thickness of the walls should be constantly monitored. This is done by solving an inverse problem for the stationary heat equation, where temperature and heat-flux data are available at certain locations inside the walls of the furnace. Numerical results are presented also for this problem.

    Delarbeten
    1. Wavelet and Fourier methods for solving the sideways heat equation
    Öppna denna publikation i ny flik eller fönster >>Wavelet and Fourier methods for solving the sideways heat equation
    2000 (Engelska)Ingår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 21, nr 6, s. 2187-2205Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.

    We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

    Nationell ämneskategori
    Teknik och teknologier
    Identifikatorer
    urn:nbn:se:liu:diva-47660 (URN)10.1137/S1064827597331394 (DOI)
    Tillgänglig från: 2009-10-11 Skapad: 2009-10-11 Senast uppdaterad: 2017-12-13
    2. A spectral method for solving the sideways heat equation
    Öppna denna publikation i ny flik eller fönster >>A spectral method for solving the sideways heat equation
    1999 (Engelska)Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 15, nr 4, s. 891-906Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.

    Nationell ämneskategori
    Matematik
    Identifikatorer
    urn:nbn:se:liu:diva-68233 (URN)10.1088/0266-5611/15/4/305 (DOI)
    Tillgänglig från: 2011-05-13 Skapad: 2011-05-13 Senast uppdaterad: 2017-12-11
    3. An inverse heat conduction problem and an application to heat treatment of aluminium
    Öppna denna publikation i ny flik eller fönster >>An inverse heat conduction problem and an application to heat treatment of aluminium
    2000 (Engelska)Ingår i: Inverse Problems in Engineering Mechanics II / [ed] Masataka Tanaka, G.S. Dulikravich, 2000, s. 99-106Konferensbidrag, Publicerat paper (Refereegranskat)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors.

    Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation.

    Nationell ämneskategori
    Matematik
    Identifikatorer
    urn:nbn:se:liu:diva-68235 (URN)978-0-08-043693-7 (ISBN)008053516X (ISBN)
    Konferens
    International Symposium on Inverse Problems in Engineering Mechanics, Nagano, Japan, March 2000
    Tillgänglig från: 2011-05-13 Skapad: 2011-05-13 Senast uppdaterad: 2014-12-15
    4. Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation
    Öppna denna publikation i ny flik eller fönster >>Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation
    2001 (Engelska)Rapport (Övrigt vetenskapligt)
    Abstract [en]

    Numerical procedures for solving a non-Characteristic Cauchy problem for the heat equation are discussed. More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted 0 ≤ x <1. The problem is often referred to as the sideways heat equation.

    The problem is analyzed, using both Fourier analysis and the singular value decomposition, and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the IHCP. In this paper we attempt to give an overview that covers the most popular methods that have been considered.

    Numerical examples that illustrate the numerical algorithms are given.

    Förlag
    s. 33
    Serie
    LiTH-MAT-R, ISSN 0348-2960 ; 17
    Nationell ämneskategori
    Teknik och teknologier
    Identifikatorer
    urn:nbn:se:liu:diva-88733 (URN)
    Tillgänglig från: 2013-02-15 Skapad: 2013-02-15 Senast uppdaterad: 2013-02-15
    5. Numerical solution of a Cauchy problem for the Laplace equation
    Öppna denna publikation i ny flik eller fönster >>Numerical solution of a Cauchy problem for the Laplace equation
    2001 (Engelska)Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, nr 4, s. 839-853Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

    Nationell ämneskategori
    Teknik och teknologier
    Identifikatorer
    urn:nbn:se:liu:diva-47296 (URN)10.1088/0266-5611/17/4/316 (DOI)
    Tillgänglig från: 2009-10-11 Skapad: 2009-10-11 Senast uppdaterad: 2017-12-13
    6. Boundary identification for an elliptic equation
    Öppna denna publikation i ny flik eller fönster >>Boundary identification for an elliptic equation
    2001 (Engelska)Rapport (Övrigt vetenskapligt)
    Abstract [en]

    We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.

    The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.

    The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.

    Förlag
    s. 16
    Serie
    LiTH-MAT-R, ISSN 0348-2960 ; 23
    Nationell ämneskategori
    Teknik och teknologier
    Identifikatorer
    urn:nbn:se:liu:diva-88734 (URN)
    Tillgänglig från: 2013-02-15 Skapad: 2013-02-15 Senast uppdaterad: 2013-02-15
  • 34.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation2001Rapport (Övrigt vetenskapligt)
    Abstract [en]

    Numerical procedures for solving a non-Characteristic Cauchy problem for the heat equation are discussed. More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted 0 ≤ x <1. The problem is often referred to as the sideways heat equation.

    The problem is analyzed, using both Fourier analysis and the singular value decomposition, and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the IHCP. In this paper we attempt to give an overview that covers the most popular methods that have been considered.

    Numerical examples that illustrate the numerical algorithms are given.

  • 35.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Eldén, Lars
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Numerical solution of a Cauchy problem for the Laplace equation2001Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, nr 4, s. 839-853Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

  • 36.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Eldén, Lars
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    An inverse heat conduction problem and an application to heat treatment of aluminium2000Ingår i: Inverse Problems in Engineering Mechanics II / [ed] Masataka Tanaka, G.S. Dulikravich, 2000, s. 99-106Konferensbidrag (Refereegranskat)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors.

    Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation.

  • 37.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Eldén, Lars
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Numerical Solution of an Inverse Steady State Heat Conduction Problem2000Konferensbidrag (Övrigt vetenskapligt)
  • 38.
    Elden, Lars
    et al.
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    Reginska, Teresa
    Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland.
    Wavelet and Fourier methods for solving the sideways heat equation2000Ingår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 21, nr 6, s. 2187-2205Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.

    We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

  • 39.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen. Linköpings universitet, Tekniska högskolan.
    A spectral method for solving the sideways heat equation1999Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 15, nr 4, s. 891-906Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a solution is sought in the interval 0 ≤ x < 1.

    The problem is ill-posed, in the sense that the solution does not depend continuously on the data. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.

  • 40.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Numerical Methods for an Inverse Heat Conduction Problem1998Konferensbidrag (Övrigt vetenskapligt)
  • 41.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Eldén, Lars
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Spectral and Wavelet Methods for Solving an Inverse Heat Conduction Problem1998Ingår i: Inverse Problems in Engineering Mechanics: International Symposium on Inverse Problems in Engineering Mechanics, 1998 (ISIP 98) / [ed] M. Tanaka and G.S. Dulikravich, Oxford: Elsevier Science , 1998, s. 3-10Konferensbidrag (Övrigt vetenskapligt)
  • 42.
    Berntsson, Fredrik
    et al.
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Eldén, Lars
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Loyd, Dan
    Linköpings universitet, Institutionen för konstruktions- och produktionsteknik, Mekanisk värmeteori och strömningslära. Linköpings universitet, Tekniska högskolan.
    Garcia-Padrón, Ricardo
    Linköpings universitet, Institutionen för konstruktions- och produktionsteknik. Linköpings universitet, Tekniska högskolan.
    A Comparison of Three Numerical Methods for an Inverse Heat Conduction Problem and an Industrial Application1997Konferensbidrag (Övrigt vetenskapligt)
  • 43.
    Berntsson, Fredrik
    Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.
    Simulation Tools for Injection Moulding1997Konferensbidrag (Övrigt vetenskapligt)
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