Open Access
Issue
ESAIM: COCV
Volume 30, 2024
Article Number 68
Number of page(s) 37
DOI https://doi.org/10.1051/cocv/2024052
Published online 12 September 2024
  1. A. Hehl and I. Neitzel, Second-order optimality conditions of an optimal control problem governed by a regularized phase-field fracture propagation model. Optimization 72 (2023) 1665–1689. [CrossRef] [MathSciNet] [Google Scholar]
  2. I. Neitzel, T. Wick and W. Wollner, An optimal control problem governed by a regularized phase-field fracture propagation model. SIAM J. Control Optim. 55 (2017) 2271–2288. [CrossRef] [MathSciNet] [Google Scholar]
  3. I. Neitzel, T. Wick and W. Wollner, An optimal control problem governed by a regularized phase-field fracture propagation model. Part II: The regularization limit. SIAM J. Control Optim. 57 (2019) 1672–1690. [CrossRef] [MathSciNet] [Google Scholar]
  4. B. Bourdin, G.A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797–826. [Google Scholar]
  5. B. Bourdin, G.A. Francfort and J.-J. Marigo, The variational approach to fracture. J. Elast. 91 (2008) 1–148. [CrossRef] [Google Scholar]
  6. G.A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. [Google Scholar]
  7. L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Unione Math. Ital. 1 (1992) 105–123. [Google Scholar]
  8. D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23 (2013) 565–616. [Google Scholar]
  9. C. Meyer, A. Rademacher and W. Wollner, Adaptive optimal control of the obstacle problem. SIAM J. Sci. Comput. 37 (2015) A918–A945. [CrossRef] [Google Scholar]
  10. A. Hehl, M. Mohammadi, I. Neitzel and W. Wollner, Optimizing fracture propagation using a phase-field approach, in Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization, edited by M. Hintermüller et al. Springer International Publishing, Cham (2022) 329–351. [CrossRef] [Google Scholar]
  11. M. Mohammadi and W. Wollner, A priori error estimates for a linearized fracture control problem. Optim. Eng. 22 (2021) 2127–2149. [CrossRef] [MathSciNet] [Google Scholar]
  12. D. Khimin, M.C. Steinbach and T. Wick, Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems. J. Comput. Phys. 470 (2022) 111554. [CrossRef] [Google Scholar]
  13. D. Khimin, M.C. Steinbach and T. Wick, Optimal control for phase-field fracture: algorithmic concepts and Computations, in Current Trends and Open Problems in Computational Mechanics, edited by F. Aldakheel, B. Hudobivnik, M. Soleimani, H. Wessels, C. Weißenfels and M. Marino. Springer International Publishing, Cham (2022) 247–255. [CrossRef] [Google Scholar]
  14. H. Goldberg and F. Tröltzsch, On a Lagrange–Newton method for a nonlinear parabolic boundary control problem. Optim. Methods Softw. 8 (1998) 225–247. [CrossRef] [MathSciNet] [Google Scholar]
  15. F. Tröltzsch, An SQP method for optimal control of a nonlinear heat equation. Control Cybern. 23 (1994) 267–288. [Google Scholar]
  16. F. Tröltzsch, On the Lagrange–Newton-SQP method for the optimal control of semilinear parabolic equations. SIAM J. Control Optim. 38 (1999) 294–312. [CrossRef] [MathSciNet] [Google Scholar]
  17. F. Tröltzsch and S. Volkwein, The SQP method for control constrained optimal control of the burgers equation. ESAIM: Cont. Optim. Calc. Var. 6 (2001) 649–674. [CrossRef] [EDP Sciences] [Google Scholar]
  18. A. Unger, Hinreichende Optimalitätsbedingungen 2. Ordnung und Konvergenz des SQP-Verfahrens für semilineare elliptische Randsteuerprobleme. Ph.D. thesis, Tech. U. Chemnitz (1997). [Google Scholar]
  19. R. Griesse, N. Metla and A. Rösch, Convergence analysis of the SQP method for nonlinear mixed-constrained elliptic optimal control problems. ZAMM Z. Angew. Math. Mech. 88 (2008) 776–792. [CrossRef] [MathSciNet] [Google Scholar]
  20. R. Griesse, N. Metla and A. Rösch, Local quadratic convergence of SQP for elliptic optimal control problems with mixed control-state constraints. Control Cybern. 39 (2010) 717–738. [Google Scholar]
  21. M. Heinkenschloss, Formulation and analysis of a sequential quadratic programming method. Optim. Control Appl. Optim. 15 (1998) 178–203. [CrossRef] [Google Scholar]
  22. M. Hintermüller and M. Hinze, A SQP-semismooth Newton-type algorithm applied to control of the instationary Navier–Stokes system subject to control constraints. SIAM J. Optim. 16 (2006) 1177–1200. [CrossRef] [MathSciNet] [Google Scholar]
  23. M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow. SIAM J. Optim. 40 (2001) 925–946. [CrossRef] [MathSciNet] [Google Scholar]
  24. D. Wachsmuth, Analysis of the SQP-method for optimal control problems governed by the non-stationary Navier–Stokes equations. ESAIM Cont. Optim. Calc. Var. 12 (2007) 93–119. [Google Scholar]
  25. M. Heinkenschloss, The numerical solution of a control problem governed by a phase field model. Optim. Methods Softw. 7 (1997) 178–203. [Google Scholar]
  26. M. Heinkenschloss and F. Tröltzsch, Analysis of the Lagrange-SQP-Newton method for the control of a phase-field equation. Control Cybern. 28 (1999) 177–211. [Google Scholar]
  27. F. Hoppe and I. Neitzel, Convergence of the SQP method for quasilinear parabolic optimal control problems. Optim. Eng. 22 (2021). [Google Scholar]
  28. W. Alt, The Lagrange–Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim. 11 (1990) 201–224. [CrossRef] [MathSciNet] [Google Scholar]
  29. W. Alt, Local convergence of the Lagrange–Newton method with applications to optimal control. Control Cybern. 1–2 (1994) 87–105. [Google Scholar]
  30. N.H. Josephy, Newton’s method for generalized equations. Tech. Summary Report, Mathematics Research Center, University of Wisconsin, Madison, WI (1979). [Google Scholar]
  31. S. Robinson, Strongly regular generalized equations. Math. Oper. Res. 5 (1980) 43–62. [CrossRef] [MathSciNet] [Google Scholar]
  32. A. Dontchev, Implicit function theorems for generalized equations. Math. Program. 70 (1995) 91–106. [Google Scholar]
  33. E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: Improving results expected from abstract theory. SIAM J. Optim. 22 (2012) 261–279. [CrossRef] [MathSciNet] [Google Scholar]
  34. E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math.-Ver. 117 (2015) 3–44. [Google Scholar]
  35. M.H. Farshbaf-Shaker and C. Heinemann, A phase-field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015) 2749–2793. [CrossRef] [MathSciNet] [Google Scholar]
  36. M.H. Farshbaf-Shaker and C. Heinemann, Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D. ESAIM Control Optim. Calc. Var. 24 (2018) 479–603. [Google Scholar]
  37. G. Allaire, F. Jouve and N. van Goethem, Damage and fracture evolution in brittle materials by shape optimization methods. J. Comput. Phys. 230 (2011) 5010–5044. [CrossRef] [MathSciNet] [Google Scholar]
  38. A. Münch and P. Pedregal, Relaxation of an optimal design problem in fracture mechanic: The anti-plane case. ESAIM: Control Optim. Calc. Var. 16 (2010) 719–743. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  39. P. Destuynder, Remarks on a crack propagation control for stationary loaded structures. CR Acad. Sci. Paris, Ser. IIb 308 (1989) 697–701. [Google Scholar]
  40. P. Hild, A. Münch and Y. Ousset, On the active control of crack growth in elastic media. Comput. Methods Appl. Mech. Eng. 198 (2008) 407–419. [CrossRef] [Google Scholar]
  41. L.M. Betz, Second-order sufficient optimality conditions for optimal control of nonsmooth, semilinear parabolic equations. SIAM J. Control Optim. 57 (2019) 4033–4062. [CrossRef] [MathSciNet] [Google Scholar]
  42. L.M. Susu, Optimal control of a viscous two-field gradient damage model. GAMM-Rep. 40 (2018) 287–311. [CrossRef] [Google Scholar]
  43. T. Wick, Multiphysics Phase-Field Fracture: Modeling, Adaptive Discretizations, and Solvers. De Gruyter, Berlin (2020). [CrossRef] [Google Scholar]
  44. L. Blank, M.H. Farshbaf-Shaker, H. Garcke, C. Rupprecht and V. Styles, Multi-material phase field approach to structural topology Optimization, in Trends in PDE Constrained Optim., edited by G. Leugering, P. Benner, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher and S. Ulbrich. Springer International Publishing, Cham (2014) 231–246. [CrossRef] [Google Scholar]
  45. L. Blank, M.H. Farshbaf-Shaker, C. Hecht, J. Michl and C. Rupprecht, Optimal control of Allen-Cahn systems, in Trends in PDE Constrained Optim., edited by G. Leugering, P. Benner, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher and S. Ulbrich. Springer International Publishing, Cham (2014) 11–26. [CrossRef] [Google Scholar]
  46. L. Blank, H. Garcke, C. Hecht and C. Rupprecht, Sharp interface limit for a phase field model in structural optimization. SIAM J. Control Optim. 54 (2016) 1559–1584. [Google Scholar]
  47. H. Garcke, K.F. Lam and A. Signori, On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects. Nonlinear Anal.: Real World Appl. 57 (2021) 103192. [CrossRef] [Google Scholar]
  48. H. Garcke and D. Trautwein, Numerical analysis for a Cahn–-Hilliard system modelling tumour growth with chemotaxis and active transport. J. Num. Math. 30 (2022) 295–324. [CrossRef] [Google Scholar]
  49. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42 (1989) 577–685. [CrossRef] [Google Scholar]
  50. B. Bourdin, Image segmentation with a finite element method. ESAIM: M2AN 33 (1999) 229–244. [CrossRef] [EDP Sciences] [Google Scholar]
  51. B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609–646. [CrossRef] [MathSciNet] [Google Scholar]
  52. O. Scherzer, editor, Handbook of Mathematical Methods in Imaging. Springer Berlin Heidelberg, Berlin, Heidelberg (2020). [CrossRef] [Google Scholar]
  53. L. De Lorenzis and T. Gerasimov, Numerical implementation of phase-field models of brittle fracture, in Modeling in Engineering Using Innovative Numerical Methods for Solids and Fluids, edited by L. De Lorenzis and A. Duöster. Springer International Publishing, Cham (2020) 75–101. [CrossRef] [Google Scholar]
  54. K. Gröger, A W1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679–687. [Google Scholar]
  55. R. Rannacher, Probleme der Kontinuumsmechanik und ihre numerische Behandlung. Heidelberg University Publishing, Heidelberg, Germany (2017). [Google Scholar]
  56. R. Haller-Dintelmann, H. Meinlschmidt and W. Wollner, Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions. Ann. Math. Pura Appl. 198 (2018) 1227–1241. [Google Scholar]
  57. F. Tröltzsch, Lipschitz stability of solutions to linear-quadratic parabolic control problems with respect to perturbations. Dynam. Cont. Disc. Impul. Syst. 7 (2000). [Google Scholar]
  58. F. Tröltzsch, Optimal control of partial differential equations: theory, methods and applications. Vol. 112 of Grad. Stud. Math. American Math. Soc, Providence, RI (2010). [CrossRef] [Google Scholar]
  59. W. Alt, Discretization and mesh independence of Newton’s method for generalized equations. Math. Program. Data Perturbations (1990) 1–30. [Google Scholar]
  60. L. Bonifacius and I. Neitzel, Second order optimality conditions for optimal control of quasilinear parabolic equations. Math. Control Relat. Fields 8 (2018) 1–34. [CrossRef] [MathSciNet] [Google Scholar]
  61. F. Tröltzsch and D. Wachsmuth, Second-order sufficient optimality conditions for the optimal control of Navier–Stokes equations. ESAIM: Control Optim. Calc. Var. 12 (2006) 93–119. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  62. E. Casas and F. Tröltzsch, A general theorem on error estimates with application to a quasilinear elliptic control problem. Comput. Optim. Appl. 53 (2012) 173–206. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.