April  2014, 10(2): 503-519. doi: 10.3934/jimo.2014.10.503

On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints

1. 

Niederrhein University of Applied Sciences, Chair for Applied Mathematics and Numerical Simulation, Reinarzstr. 49, 47805 Krefeld, Germany

2. 

Fraunhofer Institute for Laser Technology, Steinbachstr. 15, 52074 Aachen, Germany

Received  December 2012 Revised  August 2013 Published  October 2013

A mathematical model for laser cutting with time-dependent cutting velocity is presented. The model involves two coupled nonlinear partial differential equations for the interacting dynamical behaviors of the free melt boundaries during the process. We define a measurement for the roughness of a cutting surface and introduce an optimal control problem for minimizing the roughness with respect to the cutting velocity and the laser beam intensity along the free melt surface. The optimal control problem involves an additional finite-dimensional averaging constraint. Necessary optimality conditions will be deduced and illustrated by means of numerical examples with data from industrial applications.
Citation: Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial & Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503
References:
[1]

M. K. Bernauer and R. Herzog, Optimal control of the classical two-phase Stefan problem in level set formulation,, SIAM J. Sci. Comput., 33 (2011), 342.  doi: 10.1137/100783327.  Google Scholar

[2]

R. Fourer, D. M. Gay and B. Kernighan, A modeling language for mathematical programming,, Management Science, 36 (1990), 519.   Google Scholar

[3]

R. Friedrich, G. Radons, T. Ditzinger and A. Henning, Ripple formation through an interface instability from moving growth and erosion sources,, Phys. Rev. Lett., 85 (2000), 4884.   Google Scholar

[4]

M. Hinze and S. Ziegenbalg, Optimal control of the free boundary in a two-phase Stefan problem,, J. Comput. Phys., 223 (2007), 657.  doi: 10.1016/j.jcp.2006.09.030.  Google Scholar

[5]

K. Hirano and R. Fabbro, Experimental observation of hydrodynamics of melt layer and striation generation during laser cutting of steel,, Physics Procedia, 12 (2011), 555.   Google Scholar

[6]

K. Hirano and R. Fabbro, Possible explanations for different surface quality in laser cutting with 1 and 10 $\mu$m beams,, Journal of Laser Application, 24 (2012).   Google Scholar

[7]

J. D. Jackson, Classical Electrodynamics,, 3rd Edition, (1999).   Google Scholar

[8]

G. Lamé and B. D. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe liquide,, Ann Chimie Physique, 47 (1831), 250.   Google Scholar

[9]

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Vol. 6. Fluid Mechanics,, 2nd edition, (1987).   Google Scholar

[10]

M. Nießen, Numerische Modellierung freier Randwertaufgaben und Anwendung auf das Laserschneiden,, (in German) Ph.D thesis, (2005).   Google Scholar

[11]

R. Poprawe, W. Schulz and R. Schmitt, Hydrodynamics of material removal by melt expulsion: Perspectives of laser cutting and drilling,, Phys. Procedia, 5 (2010), 1.   Google Scholar

[12]

S. Repke, N. Marheineke and R. Pinnau, On Adjoint-Based Optimization of a Free Surface Stokes Flow,, Fraunhofer ITWM Bericht, (2010).   Google Scholar

[13]

W. Schulz, M. Nießen, U. Eppelt and K. Kowalick, Simulation of laser cutting,, in The Theory of Laser Materials Processing: Heat and Mass Transfer in Modern Technology (ed. J. M. Dowden), (2009).   Google Scholar

[14]

W. Schulz, V. Kostrykin, M. Nießen, J. Michel, D. Petring, E. W. Kreutz and R. Poprawe, Dynamics of ripple formation and melt flow in laser beam cutting,, J. Phys D: Appl. Phys., 32 (1999), 1219.   Google Scholar

[15]

K. Theißen, Optimale Steuerprozesse unter partiellen Differentialgleichungs-Restriktionen mit linear eingehender Steuerfunktion,, (in German) Ph.D thesis, (2006).   Google Scholar

[16]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications,, Graduate Studies in Mathematics, (2010).   Google Scholar

[17]

O. Volkov, B. Protas, W. Liao and D. W. Glander, Adjoint-based optimization of thermo-fluid phenomena in welding processes,, J. Eng. Math., 65 (2009), 201.  doi: 10.1007/s10665-009-9292-0.  Google Scholar

[18]

G. Vossen and J. Schüttler, Mathematical modelling and stability analysis for laser cutting,, Math. Comput. Model. Dyn. Syst., 18 (2012), 439.  doi: 10.1080/13873954.2011.642387.  Google Scholar

[19]

G. Vossen, J. Schüttler and T. Hermanns, Analysis and optimal control for free melt flow boundaries in laser cutting with distributed radiation,, submitted, (2011).   Google Scholar

[20]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

show all references

References:
[1]

M. K. Bernauer and R. Herzog, Optimal control of the classical two-phase Stefan problem in level set formulation,, SIAM J. Sci. Comput., 33 (2011), 342.  doi: 10.1137/100783327.  Google Scholar

[2]

R. Fourer, D. M. Gay and B. Kernighan, A modeling language for mathematical programming,, Management Science, 36 (1990), 519.   Google Scholar

[3]

R. Friedrich, G. Radons, T. Ditzinger and A. Henning, Ripple formation through an interface instability from moving growth and erosion sources,, Phys. Rev. Lett., 85 (2000), 4884.   Google Scholar

[4]

M. Hinze and S. Ziegenbalg, Optimal control of the free boundary in a two-phase Stefan problem,, J. Comput. Phys., 223 (2007), 657.  doi: 10.1016/j.jcp.2006.09.030.  Google Scholar

[5]

K. Hirano and R. Fabbro, Experimental observation of hydrodynamics of melt layer and striation generation during laser cutting of steel,, Physics Procedia, 12 (2011), 555.   Google Scholar

[6]

K. Hirano and R. Fabbro, Possible explanations for different surface quality in laser cutting with 1 and 10 $\mu$m beams,, Journal of Laser Application, 24 (2012).   Google Scholar

[7]

J. D. Jackson, Classical Electrodynamics,, 3rd Edition, (1999).   Google Scholar

[8]

G. Lamé and B. D. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe liquide,, Ann Chimie Physique, 47 (1831), 250.   Google Scholar

[9]

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Vol. 6. Fluid Mechanics,, 2nd edition, (1987).   Google Scholar

[10]

M. Nießen, Numerische Modellierung freier Randwertaufgaben und Anwendung auf das Laserschneiden,, (in German) Ph.D thesis, (2005).   Google Scholar

[11]

R. Poprawe, W. Schulz and R. Schmitt, Hydrodynamics of material removal by melt expulsion: Perspectives of laser cutting and drilling,, Phys. Procedia, 5 (2010), 1.   Google Scholar

[12]

S. Repke, N. Marheineke and R. Pinnau, On Adjoint-Based Optimization of a Free Surface Stokes Flow,, Fraunhofer ITWM Bericht, (2010).   Google Scholar

[13]

W. Schulz, M. Nießen, U. Eppelt and K. Kowalick, Simulation of laser cutting,, in The Theory of Laser Materials Processing: Heat and Mass Transfer in Modern Technology (ed. J. M. Dowden), (2009).   Google Scholar

[14]

W. Schulz, V. Kostrykin, M. Nießen, J. Michel, D. Petring, E. W. Kreutz and R. Poprawe, Dynamics of ripple formation and melt flow in laser beam cutting,, J. Phys D: Appl. Phys., 32 (1999), 1219.   Google Scholar

[15]

K. Theißen, Optimale Steuerprozesse unter partiellen Differentialgleichungs-Restriktionen mit linear eingehender Steuerfunktion,, (in German) Ph.D thesis, (2006).   Google Scholar

[16]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications,, Graduate Studies in Mathematics, (2010).   Google Scholar

[17]

O. Volkov, B. Protas, W. Liao and D. W. Glander, Adjoint-based optimization of thermo-fluid phenomena in welding processes,, J. Eng. Math., 65 (2009), 201.  doi: 10.1007/s10665-009-9292-0.  Google Scholar

[18]

G. Vossen and J. Schüttler, Mathematical modelling and stability analysis for laser cutting,, Math. Comput. Model. Dyn. Syst., 18 (2012), 439.  doi: 10.1080/13873954.2011.642387.  Google Scholar

[19]

G. Vossen, J. Schüttler and T. Hermanns, Analysis and optimal control for free melt flow boundaries in laser cutting with distributed radiation,, submitted, (2011).   Google Scholar

[20]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

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