Boundary Identification in the Domain of a Parabolic Partial Differential Equation

The background and inspiration for this study is previous work with applications of boundary identification in the metals processing industry. With an efficient boundary identification method, safety margins can be narrowed and maintenance intervals extended for high-temperature industrial equipment, without increasing the risk of failure and/or casualties. Ideally, such a method would be based on monitoring some indirectly influenced variable which is routinely measured or can be measured at a low cost. Furnace wall temperature at different positions is such a variable, suitable as input for a boundary identification method assessing the instantaneous lining wall thickness of the furnace. We give a background and motivation for the choice of a spatially one-dimensional dynamical model for the boundary, i.e., a thickness description, by briefly considering the limitations of spatially two-dimensional models. Dynamics being more important for our applications than adding a spatial dimension and the advantages of the simplest formulation in possible industrial use made us choose the model formulation described in the latter parts of this work. The sideways heat equation is a related topic with many characteristics carrying over to boundary identification, which is why we considered solution properties, noise contamination, regularization and ill-posedness more generally for the sideways heat equation. For boundary identification we study a set of three different methods of estimating the boundary position, of which the two first ones have been developed from a purely mathematical standpoint and the third one from a more applied background. These methods have different characteristics with benefits and disadvantages. For the purely mathematically based methods we observed good accuracy and low computation cost, however, with low flexibility for dealing with nonlinear coefficients in the governing partial differential equation (PDE). For the third, more applied method, accuracy suffered from the ill-posedness of the more flexible formulation allowing (at least in principle) nonlinear and discontinuous coefficients in the governing PDE. Finally, we attempted to derive an error estimate, which was observed to conform with our experience from computational trials with the method. We consider the present study as a good starting point and a mathematical basis for developing industrial applications of boundary identification, particularly toward dealing with real, nonlinear and discontinuous, material properties and sudden changes due to material “fall-off”. With the methods outlined here, it seems feasible to obtain a robust, rapid and sufficiently accurate method using a reasonable amount of mathematical machinery. SAMMANDRAG ABSTRACT IN SWEDISH iii Sammandrag Abstract in Swedish Bakgrunden och inspirationen till foreliggande studie ar tidigare forskning i tillampningar pa randidentifiering i metallindustrin. Effektiv randidentifiering mojliggor mindre sakerhetsmarginaler och langre serviceintervall for apparaturen i industriella hogtemperaturprocesser, utan okad risk for materielhaverier. I idealfallet vore en metod for randidentifiering baserad pa uppfoljning av nagon indirekt variabel som kan matas rutinmassigt eller till en ringa kostnad. En dylik variabel for smaltugnar ar temperaturen i olika positioner i vaggen. Denna kan utnyttjas som insignal till en randidentifieringsmetod for att overvaka ugnens vaggtjocklek. Vi ger en bakgrund och motivering till valet av den geometriskt endimensionella dynamiska modellen for randidentifiering, som diskuteras i arbetets senare del, framom en flerdimensionell geometrisk beskrivning. I de aktuella industriella tillampningarna ar dynamiken samt fordelarna med en enkel modellstruktur viktigare an exakt geometrisk beskrivning. Losningsmetoder for den s.k. sidledes varmeledningsekvationen har manga saker gemensamt med randidentifiering. Darfor studerar vi egenskaper hos losningarna till denna ekvation, inverkan av matfel och nagot som brukar kallas fororening av matbrus, regularisering och allmannare foljder av icke-valstalldheten hos sidledes varmeledningsekvationen. Vi studerar en uppsattning av tre olika metoder for randidentifiering, av vilka de tva forsta ar utvecklade fran en strikt matematisk och den tredje fran en mera tillampad utgangspunkt. Metoderna har olika egenskaper med specifika fordelar och nackdelar. De rent matematiskt baserade metoderna karakteriseras av god noggrannhet och lag numerisk kostnad, dock till priset av lag flexibilitet i formuleringen av den modellbeskrivande partiella differentialekvationen. Den tredje, mera tillampade, metoden kannetecknas av en samre noggrannhet fororsakad av en hogre grad av icke-valstalldhet hos den mera flexibla modellen. For denna gjordes aven en ansats till feluppskattning, som senare kunde observeras overensstamma med praktiska berakningar med metoden. Studien kan anses vara en god startpunkt och matematisk bas for utveckling av industriella tillampningar av randidentifiering, speciellt mot hantering av olinjara och diskontinuerliga materialegenskaper och plotsliga forandringar orsakade av “nedfallande” vaggmaterial. Med de behandlade metoderna forefaller det mojligt att uppna en robust, snabb och tillrackligt noggrann metod av begransad komplexitet for randidentifiering.