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Konstruktion av en optimal balk med Tikhonovregularisering
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
2015 (Swedish)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [sv]

I detta kandidatexamensarbete studeras ett inverst problem: att för en jämnt fördelad last numeriskt minimera energin som tas upp i en fritt upplagd balk. I detta problem antas balken vara indelad i ett givet antal lika långa stycken med konstant böjstyvhet och minimeringen sker med avseende på dessa styckens böjstyvheter. För en fin indelning av balken riskerar den numeriska metoden att bli instabil. Målet med denna studie är undersöka om Tikhonovregularisering kan stabilisera den numeriska metoden i fallet med en fin indelning. I lösningen av balkproblemet används Bernoulli-Eulers ekvation och Lagranges multiplikatormetod för att härleda den målfunktion som skall minimeras. Minimering utförs iterativt genom gradientmetoden. Problemet löses för en indelning av balken i tre och 24 stycken. För fallet med 24 stycken konvergerar inte iterationerna i gradientmetoden och negativa värden på böjstyvheternaerhålls. Tikhonovregularisering visar sig stabilisera iterationernaså att konvergens uppnås och böjstyvheterna hålls positiva.

Abstract [en]

In this bachelor thesis the problem of numerically minimizing the elastic energy of a uniformly loaded, simply supported beam is studied. In this problem, the beam is assumed to be divided into a given number of equally long pieces with constant flexural rigidity and the minimization is done with respect to the flexural rigidities of the pieces. For a fine division of the beam there is a risk that the numerical method becomes unstable. The aim of this study is to investigate whether Tikhonov regularization can stabilize the numerical method in the case of a fine division. In the solution of the beam problem the Bernoulli-Euler equation and Lagrange’s multiplier method areused to derive the function to be minimized. Minimization is done iteratively by the gradient method. The problem is solved for a division of the beam into three and 24 pieces. In the case of 24 pieces, the iterations in the gradient method do not converge and negative values of the flexural rigidities are obtained. Tikhonov regularization is shown to stabilize the iterations so that convergence is attained and the flexural rigidities remain positive.

Place, publisher, year, edition, pages
2015. , 25 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-167634OAI: oai:DiVA.org:kth-167634DiVA: diva2:813372
Supervisors
Available from: 2015-05-22 Created: 2015-05-22 Last updated: 2015-05-22Bibliographically approved

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