Estimation of the number of forward time steps for the sequential Beck approach used for solving inverse heat-conduction problems

Abstract

Direct heat-conduction problems are those whose boundary conditions, initial state and material properties are known and the entire temperature field in a model can be computed. In contrast, an inverse problem is defined as the determination of the unknown causes based on the observation of their effects. The inverse heat-conduction method is often used for problems where the boundary conditions cannot be measured directly but are computed from the recorded temperature history inside the model. A very effective method for solving this difficult problem is the sequential Beck approach. To stabilize this inverse problem, a proper regularization parameter must be used. For this method, the regularization parameter is the number of the forward time steps that stabilize the inverse computation. This paper describes two methods for computing the number of the recommended forward time steps for nonlinear heat-conduction models with temperature-dependent material properties. The first method is based on tracking the sensitivity (at the interior point of a measurement) to the Dirac heat-flux pulse on the surface. The second method determines the number of the forward time steps from the residual function computed from the heat fluxes obtained from the inverse computation. The stability and noise (in the results) of several variants of these methods are compared. The results showed that the first method is much less computationally intensive and gives a slightly higher value of the number of forward time steps than the second method. Keywords: inverse heat-conduction problem, Beck approach, number of forward time steps Neposredni problemi prevajanja toplote so tisti pri katerih so poznani robni pogoji, za~etno stanje in lastnosti materiala ter mo`nost izra~una temperaturnega polja znotraj modela. Nasprotno pa je inverzni problem definiran kot dolo~anje nepoznanih vzrokov na osnovi opazovanja njihovih vplivov. Metoda inverznega prevajanja toplote se pogosto uporabi pri problemih, kjer se robni pogoji ne morejo neposredno izmeriti, temve~se jih izra~una iz zabele`enega poteka temperature znotraj modela. Zelo u~inkovita metoda za re{evanje tovrstnega problema je sekven~ni Beckov pribli`ek. Za stabilizacijo tak{nega inverznega problema se mora uporabiti ustrezen regulirni parameter. Pri tej metodi je regulirni parameter {tevilo priporo~enih~asovnih korakov, ki stabilizirajo inverzni izra~un.^lanek opisuje dve metodi za izra~un {tevila priporo~enih~asovnih korakov za nelinearni model prenosa toplote, s temperaturno odvisnimi lastnostmi materiala. Prva metoda temelji na iskanju ob~utljivosti, na notranji to~ki merjenja, do Dirac utripa toplotnega toka na povr{ini. Druga metoda dolo~a {tevilo vnaprej{njih~asovnih korakov iz preostale funkcije izra~unane iz toplotnih tokov, ki so dobljeni z inverznim izra~unom. V rezultatih je primerjana stabilnost {uma pri ve~variantah teh metod. Rezultati so pokazali, da je prva metoda mnogo manj ra~unsko intenzivna in daje rahlo ve~jo vrednost {tevila predhodnih~asovnih korakov kot druga metod...

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