Summary
This work introduces a mathematical model for laser cutting which involves two coupled nonlinear partial differential equations. The model will be investigated by linear stability analysis to study the occurence of ripple formations at a cutting surface. We define a measurement for the roughness of the cutting surface and give a method for minimizing the roughness with respect to process parameters. A numerical solution of this nonlinear optimization problem will be presented and compared with the results of the linear stability analysis.
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References
Colombo RM, Mercier M, Rosini MD (2009) Stability estimates on general scalar balance laws. C R Acad Sci Paris, Ser I 347
Fourer R, Gay DM, Kernighan (1990) A Modeling Language for Mathematical Programming. Management Science 36:519554
Friedrich R, Radons G, Ditzinger T, Henning A (2000) Ripple Formation through an Interface Instability from Moving Growth and Erosion Sources. Phys Rev Lett 85:4884-4887
Lamé G, Clapeyron BD (1831) Mémoire sur la solidification par refroidissement d’un globe liquide. Ann Chimie Physique, 47:250-256
Lax PD, Wendroff B (1960) Systems of conservation laws. Commun Pure Appl Math 13:217-237
Kevorkian J (2000) Partial Differential Equations: Analytical Solution Techniques. 2nd Edition. Springer, New York
Nießen M (2005) Numerische Modellierung freier Randwertaufgaben und Anwendung auf das Laserschneiden. PhD thesis, RWTH Aachen University
Schulz W (2003) Die Dynamik des thermischen Abtrags mit Grenzschichtcharakter. Aachen, Shaker-Verlag, Habilitation thesis, RWTH Aachen University
Schulz W, Nießen M, Eppelt U, Kowalick K (2009) Simulation of Laser Cutting. In: Dowden JM (ed) The Theory of Laser Materials Processing: Heat and Mass Transfer in Modern Technology. Springer Series in Materials Science 119
Schulz W, Kostrykin V, Nießen M, Michel J, Petring D, Kreutz EW, Poprawe R (1999) Dynamics of Ripple Formation and Melt Flow in Laser Beam Cutting. J Phys D: Appl. Phys. 32:1219-1228
Theißen K (2006) Optimale Steuerprozesse unter partiellen Differentialgleichungs- Restriktionen mit linear eingehender Steuerfunktion. PhD thesis, University of Münster
Vossen G, Schulz W (2009) Multiple Scale Methods for Nonlinear Stability Analysis in Laser Cutting Processes. Technical Report, RWTH Aachen. Online: http://www.nld.rwth-aachen.de/
Wächter A, Biegler LT (2006) On the Implementation of an Interior-Point Filter Line-Search Algorithm for Large-Scale Nonlinear Programming. Mathematical Programming 106(1):25-57
Acknowledgement
The research related to this paper is supported by the German Research Foundation DFG as part of the Cluster of Excellence “Integrative Production Technology for High-Wage Countries” at RWTH Aachen University.
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Vossen, G., Schüttler, J., Nießen, M. (2010). Optimization of Partial Differential Equations for Minimizing the Roughness of Laser Cutting Surfaces. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds) Recent Advances in Optimization and its Applications in Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12598-0_46
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DOI: https://doi.org/10.1007/978-3-642-12598-0_46
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