# An hp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 6, page 1081-1113
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topNupur, Gupta, and Neela, Nataraj. "An hp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel." ESAIM: Mathematical Modelling and Numerical Analysis 45.6 (2011): 1081-1113. <http://eudml.org/doc/276342>.

@article{Nupur2011,

abstract = {
In this paper, we discuss an hp-discontinuous Galerkin finite
element method (hp-DGFEM) for the laser surface hardening of
steel, which is a constrained optimal control problem governed by a
system of differential equations, consisting of an ordinary
differential equation for austenite formation and a semi-linear
parabolic differential equation for temperature evolution. The space
discretization of the state variable is done using an hp-DGFEM,
time and control discretizations are based on a discontinuous
Galerkin method. A priori error estimates are developed at
different discretization levels. Numerical experiments
presented justify the theoretical order of convergence obtained.
},

author = {Nupur, Gupta, Neela, Nataraj},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Laser surface hardening of steel; semi-linear parabolic
equation; optimal control; error estimates; discontinuous Galerkin
finite element method; laser surface hardening of steel; semi-linear parabolic equation; discontinuous Galerkin finite element method; numerical experiments},

language = {eng},

month = {6},

number = {6},

pages = {1081-1113},

publisher = {EDP Sciences},

title = {An hp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel},

url = {http://eudml.org/doc/276342},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Nupur, Gupta

AU - Neela, Nataraj

TI - An hp-Discontinuous Galerkin Method for the Optimal Control Problem of Laser Surface Hardening of Steel

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/6//

PB - EDP Sciences

VL - 45

IS - 6

SP - 1081

EP - 1113

AB -
In this paper, we discuss an hp-discontinuous Galerkin finite
element method (hp-DGFEM) for the laser surface hardening of
steel, which is a constrained optimal control problem governed by a
system of differential equations, consisting of an ordinary
differential equation for austenite formation and a semi-linear
parabolic differential equation for temperature evolution. The space
discretization of the state variable is done using an hp-DGFEM,
time and control discretizations are based on a discontinuous
Galerkin method. A priori error estimates are developed at
different discretization levels. Numerical experiments
presented justify the theoretical order of convergence obtained.

LA - eng

KW - Laser surface hardening of steel; semi-linear parabolic
equation; optimal control; error estimates; discontinuous Galerkin
finite element method; laser surface hardening of steel; semi-linear parabolic equation; discontinuous Galerkin finite element method; numerical experiments

UR - http://eudml.org/doc/276342

ER -

## References

top- V. Arnăutu, D. Hömberg and J. Sokołowski, Convergence results for a nonlinear parabolic control problem. Numer. Funct. Anal. Optim.20 (1999) 805–824. Zbl0951.49005
- D.N. Arnold, An interior penalty method for discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760. Zbl0482.65060
- D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779. Zbl1008.65080
- I. Babuška, The finite element method with penalty. Math. Comput.27 (1973) 221–228. Zbl0299.65057
- C. Bernardi, M. Dauge and Y. Maday, Polynomials in the Sobolev world. Preprint of the Laboratoire Jacques-Louis LionsR03038 (2003).
- P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). Zbl0383.65058
- M. Crouzeix, V. Thomee, L.B. Wahlbin, Error estimates for spatial discrete approximation of semilinear parabolic equation with initial data of low regularity. Math. Comput.53 187 (1989) 25–41. Zbl0674.65071
- J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing Methods in Applied Sciences, Lecture Notes in Phys.58. Springer-Verlag, Berlin (1976).
- K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal.28 (1991) 43–77. Zbl0732.65093
- K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems II: optimal error estimates in ${L}_{\infty}{L}_{2}$ and ${L}_{\infty}{L}_{\infty}$. SIAM J. Numer. Anal.32 (1995) 706–740. Zbl0830.65094
- L.C. Evans, Partial Differential Equations. American Mathematics Society, Providence, Rhode Island (1998). Zbl0902.35002
- T. Gudi, N. Nataraj and A.K. Pani, Discontinuous Galerkin methods for quasi-linear elliptic problems of nonmonotone type. SIAM J. Numer. Anal.45 (2007) 163–192. Zbl1140.65082
- T. Gudi, N. Nataraj and A.K. Pani, hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer. Math.109 (2008) 233–268. Zbl1146.65076
- T. Gudi, N. Nataraj and A.K. Pani, An hp-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type. Math. Comput.77 (2008) 731–756. Zbl1147.65093
- N. Gupta, N. Nataraj and A.K. Pani, An optimal control problem of laser surface hardening of steel. Int. J. Numer. Anal. Model.7 (2010). Zbl05773761
- N. Gupta, N. Nataraj and A.K. Pani, A priori error estimates for the optimal control of laser surface hardening of steel. Paper communicated. Zbl05773761
- D. Hömberg, A mathematical model for the phase transitions in eutectoid carbon steel. IMA J. Appl. Math.54 (1995) 31–57. Zbl0830.65126
- D. Hömberg, Irreversible phase transitions in steel. Math. Methods Appl. Sci.20 (1997) 59–77. Zbl0876.35118
- D. Hömberg and J. Fuhrmann, Numerical simulation of surface hardening of steel. Int. J. Numer. Meth. Heat Fluid Flow9 (1999) 705–724. Zbl1073.76597
- D. Hömberg and J. Sokolowski, Optimal control of laser hardening. Adv. Math. Sci.8 (1998) 911–928. Zbl0937.35186
- D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition. Math. Comput. Model.37 (2003) 1003–1028. Zbl1044.49028
- D. Hömberg and W. Weiss, PID-control of laser surface hardening of steel. IEEE Trans. Control Syst. Technol.14 (2006) 896–904.
- P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal.39 (2002) 2133–2163. Zbl1015.65067
- A. Lasis and E. Süli, hp-version discontinuous Galerkin finite element methods for semilinear parabolic problems. Report No. 03/11, Oxford university computing laboratory (2003). Zbl1155.65073
- A. Lasis and E. Süli, Poincaré-type inequalities for broken Sobolev spaces. Isaac Newton Institute for Mathematical Sciences, Preprint No. NI03067-CPD (2003).
- J.B. Leblond and J. Devaux, A new kinetic model for anisothermal metallurgical transformations in steels including effect of austenite grain size. Acta Metall.32 (1984) 137–146.
- V.I. Mazhukin and A.A. Samarskii, Mathematical modelling in the technology of laser treatments of materials. Surveys Math. Indust.4 (1994) 85-149. Zbl0805.35165
- D. Meidner, and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints. SIAM J. Control Optim.47 (2007) 1150–1177. Zbl1161.49026
- J.A. Nitsche, Über ein Variationprinzip zur Lösung Dirichlet-Problemen bei Verwen-dung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. Zbl0229.65079
- J.T. Oden, I. Babuška and C.E. Baumann, A discontinuous hp finite element method for diffusion problems. J. Comput. Phys.146 (1998) 491–519. Zbl0926.65109
- S. Prudhomme, F. Pascal and J.T. Oden, Review of error estimation for discontinuous Galerkin method. TICAM-report 00-27 (2000).
- B. Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Frontiers in Mathematics35. SIAM 2008. ISBN: 978-0-898716-56-6. Zbl1153.65112
- B. Rivière and M.F. Wheeler, A discontinuous Galerkin method applied to nonlinear parabolic equations. The Center for Substance Modeling, TICAM, The University of Texas, Austin TX 78712, USA. Zbl0946.65078
- B. Rivière, M.F. Wheeler and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal.39 (2001) 902–931. Zbl1010.65045
- V. Thomée, Galerkin finite element methods for parabolic problems. Springer (1997). Zbl0884.65097
- S. Volkwein, Non-linear conjugate gradient method for the optimal control of laser surface hardening, Optim. Methods Softw.19 (2004) 179–199. Zbl1062.49034
- M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal.15 (1978) 152–161. Zbl0384.65058

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.