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Mathematical Modelling of Light Scattering in Paper and Print
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics. (FSCN - Fibre Science and Communication Network)ORCID iD: 0000-0002-0529-1009
Responsible organisation
2004 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

A problem formulation and a solution method are outlined for the radiative transfer problem in vertically inhomogeneous scattering and absorbing media, using discrete ordinate model geometry. The treatment spans from the physical problem via a continuous formulation, a discretization and a numerical analysis, to an implementation with performance evaluation and application to real-world problems. The thesis clearly illustrates how considerations in one step affect other steps, and thus provides an example of an overall treatment of mathematical modeling of a large applied problem. A selection of different steps is brought together. First all the steps necessary to get a numerically stable solution procedure are treated, and then methods are introduced to increase the speed by a factor of several thousand. The solution procedure is implemented in MATLAB under the name of DORT2002, and is adapted primarily to light scattering simulations in paper and print. A confined presentation is given of the effect of the steps that are needed, or possible, to make any discrete ordinate radiative transfer solution method numerically efficient. This is done through studies of the numerical performance of DORT2002. Performance tests show that the steps that are included to improve stability and speed of DORT2002 are very successful. Together they give an unconditionally stable solution method to a problem previously considered numerically intractable, and decrease computation time compared to a naive implementation with a factor of 1 000 � 10 000 in typical cases and with a factor up to and beyond 10 000 000 in extreme cases. It is also shown that the speed increasing steps are not introduced at the cost of reduced accuracy, and that DORT2002 converges to the true value as the discretization is made finer. It is shown by the use of DORT2002 that when a medium has a finite thickness, the light distribution deviates from the perfectly diffuse even under the theoretically ideal conditions for which the Kubelka-Munk model was created. This effect, which is in opposition to what one would intuitively expect, is caused by light escaping through the lower boundary of the medium, and causes errors in Kubelka-Munk reflectance calculations that can be up to 20% and more, even for a grammage of 80 g/m2. The magnitude of the error shows a strong dependence on the degree of absorption, with higher absorption giving greater error. This confirms previously reported problems with Kubelka-Munk for strongly absorbing media, and DORT2002 offers a partial explanation of these problems, as it can describe this effect and quantify the Kubelka-Munk errors. It is argued that DORT2002 could well be considered for increased understanding in cases where the level of accuracy of Kubelka-Munk reflectance calculations is not acceptable. A comprehensive list of advantages for the applied user of a model with higher dimensionality is supplied.

Place, publisher, year, edition, pages
Sundsvall: Mittuniversitetet , 2004.
Series
Mid Sweden University licentiate thesis, ISSN 1652-8948 ; 6
Keyword [en]
mathematical modeling, radiative transfer, solution method, numerical stability, speed, light scattering, light absorption, Kubelka-Munk, errors, reflectance calculations
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-5670Local ID: 1961ISBN: 91-87908-87-5 (print)OAI: oai:DiVA.org:miun-5670DiVA: diva2:30703
Presentation
(English)
Supervisors
Available from: 2008-09-30 Created: 2009-07-10 Last updated: 2009-07-10Bibliographically approved
List of papers
1. A Fast and Stable Solution Method for the Radiative Transfer Problem
Open this publication in new window or tab >>A Fast and Stable Solution Method for the Radiative Transfer Problem
2005 (English)In: SIAM Review, ISSN 0036-1445, Vol. 47, no 3, 447-468 p.Article in journal (Refereed) Published
Abstract [en]

Radiative transfer theory considers radiation in turbid media, and is used in a wide range of applications. This paper outlines a problem formulation and a solution method for the radiative transfer problem in multilayer scattering and absorbing media, using discrete ordinate model geometry. A selection of different steps is brought together. The main contribution here is the synthesis of these steps, all of which have been used in different areas, but never all together in one method. First all necessary steps to get a numerically stable solution procedure are treated, and then methods are introduced to increase the speed by a factor of several thousand. This includes methods for handling strongly forward-scattering media. The method is shown to be unconditionally stable, whilst the problem was previously considered numerically intractable.

Keyword
radiative transfer, discrete ordinates, solution method, numerical stability, speed
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-2798 (URN)10.1137/S0036144503438718 (DOI)000231301400004 ()2321 (Local ID)2321 (Archive number)2321 (OAI)
Projects
DPC - Digital printing center
Available from: 2008-09-30 Created: 2009-07-10 Last updated: 2009-07-30Bibliographically approved
2. Numercal performance of the DORT2002 radiative transfer Solution method
Open this publication in new window or tab >>Numercal performance of the DORT2002 radiative transfer Solution method
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-9346 (URN)
Note
Submitted to applied Numerical mathematics (2004)Available from: 2009-07-10 Created: 2009-07-10 Last updated: 2010-01-14Bibliographically approved
3. Comparison of the DORT2002 Radiative Transfer Solution Method and the Kubelka-Munk Model
Open this publication in new window or tab >>Comparison of the DORT2002 Radiative Transfer Solution Method and the Kubelka-Munk Model
2004 (English)In: Nordic Pulp & Paper Research Journal, ISSN 0283-2631, E-ISSN 2000-0669, Vol. 19, no 3, 397-403 p.Article in journal (Refereed) Published
Abstract [en]

The need for optical modeling of paper is obvious to provide connections between its optical response and the actual properties of the paper. It is argued that modern solution methods from radiative transfer theory could be considered instead of the Kubelka-Munk model, and a specific example, DORT2002, is tested. It is shown that Kubelka-Munk is a simple special case of DORT2002, and the two models and their coefficients are compared. A comprehensive list of advantages for the applied user of a model with higher dimensionality is supplied. It is shown, by the use of DORT2002, that when the medium has finite thickness, the light distribution deviates from the perfectly diffuse even under the theoretically ideal conditions for which Kubelka-Munk was created. This effect causes errors in Kubelka-Munk reflectance calculations that may be up to 20% and more, even for a grammage of 80 g/m2. The magnitude of the error shows a strong dependence on the degree of light absorption, with higher absorption giving greater error. DORT2002 can well be considered for increased understanding in cases where the level of accuracy of Kubelka-Munk reflectance calculations is not sufficient.

Keyword
light scattering, mathematical models, radiative transfer, Kubelka-Munk, light absorption, errors, reflectance calculations
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-2686 (URN)10.3183/NPPRJ-2004-19-03-p397-403 (DOI)000224636500019 ()2-s2.0-6344280895 (Scopus ID)2118 (Local ID)2118 (Archive number)2118 (OAI)
Projects
DPC - Digital printing center
Available from: 2008-09-30 Created: 2009-07-10 Last updated: 2016-09-26Bibliographically approved
4. Quantification of the intrinsic error of the kubelka–munk model caused by strong light absorption
Open this publication in new window or tab >>Quantification of the intrinsic error of the kubelka–munk model caused by strong light absorption
2003 (English)In: Journal of Pulp and Paper Science (JPPS), ISSN 0826-6220, Vol. 29, no 11, 386-390 p.Article in journal (Refereed) Published
Abstract [en]

The Kubelka-Munk (KM) model is widely used within the paper industry to interpret diffuse reflectance factor measurements of paper and its components. It has been found in the literature that the addition of a dye colorant to a paper sheet not only increases its KM light absorption coefficient, but for strong absorption also decreases its KM light scattering coefficient. This effect has previously been attributed to the intrinsic error of the KM model induced by light absorption that tends to orient of the light fluxes perpendicular to the sheet. In the present work we have mapped the intrinsic error of the KM model by comparing light scattering calculations from the KM model with the more accurate Discrete Ordinate Radiative Transfer model DORT2002. We found that the models agree within 2.3% in reflectance, and that the intrinsic error in the KM model explains about 1/5 of the previously observed interdependence of the KM coefficients for heavily dyed sheets.

Keyword
light scattering, light absorption, reflectance, Kubelka-Munk, DORT2002, optical properties, errors
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-1799 (URN)000187624200006 ()507 (Local ID)507 (Archive number)507 (OAI)
Available from: 2008-09-30 Created: 2008-09-30Bibliographically approved

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