Analysis of a space-time phase-field fracture complementarity model and its optimal control formulation. (English) Zbl 07920197
Summary: The purpose of this work is the formulation of optimality conditions for phase-field optimal control problems. The forward problem is first stated as an abstract nonlinear optimization problem, and then the necessary optimality conditions are derived. The sufficient optimality conditions are also examined. The choice of suitable function spaces to ensure the regularity of the nonlinear optimization problem is a true challenge here. Afterwards the optimal control problem with a tracking type cost functional is formulated. The constraints are given by the previously derived first order optimality conditions of the forward problem. Herein regularity is proven under certain conditions and first order optimality conditions are formulated.
MSC:
| 49J50 | Fréchet and Gateaux differentiability in optimization |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 74R10 | Brittle fracture |
| 49J40 | Variational inequalities |
| 90Cxx | Mathematical programming |
Keywords:
phase-field fracture propagation; optimal control; necessary optimality conditions; complementarity systemReferences:
| [1] | Ambati, M., Gerasimov, T., and De Lorenzis, L., A review on phase-field models of brittle fracture and a new fast hybrid formulation, Comput. Mech., 55 (2015), pp. 383-405. · Zbl 1398.74270 |
| [2] | Bourdin, B., Numerical implementation of the variational formulation for quasi-static brittle fracture, Interfaces Free Bound., 9 (2007), pp. 411-430. · Zbl 1130.74040 |
| [3] | Bourdin, B., Francfort, G., and Marigo, J.-J., Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids, 48 (2000), pp. 797-826. · Zbl 0995.74057 |
| [4] | Bourdin, B., Francfort, G., and Marigo, J.-J., The variational approach to fracture, J. Elasticity, 91 (2008), pp. 1-148. · Zbl 1176.74018 |
| [5] | Bourdin, B. and Francfort, G. A., Past and present of variational fracture, SIAM News, 52 (2019). |
| [6] | Burke, S., Ortner, C., and Süli, E., An adaptive finite element approximation of a variational model of brittle fracture, SIAM J. Numer. Anal., 48 (2010), pp. 980-1012. · Zbl 1305.74080 |
| [7] | dal Maso, G., Francfort, G. A., and Toader, R., Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), pp. 165-225. · Zbl 1064.74150 |
| [8] | de Borst, R. and Verhoosel, C., Gradient damage vs. phase-field approaches for fracture: Similarities and differences, Comput. Methods Appl. Mech. Engrg., 312 (2016), pp. 78-94. · Zbl 1439.74347 |
| [9] | Desai, J., Allaire, G., and Jouve, F., Topology optimization of structures undergoing brittle fracture, J. Comput. Phys., 458 (2022), 111048. · Zbl 07527708 |
| [10] | Diehl, P., Lipton, R., Wick, T., and Tyagi, M., A comparative review of peridynamics and phase-field models for engineering fracture mechanics, Comput. Mech., 69 (2022), pp. 1259-1293, doi:10.1007/s00466-022-02147-0. · Zbl 1505.74191 |
| [11] | Francfort, G., Variational fracture: Twenty years after, Int. J. Fracture, 237 (2022), pp. 3-13. |
| [12] | Francfort, G. and Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), pp. 1319-1342. · Zbl 0966.74060 |
| [13] | Francfort, G. A. and Larsen, C. J., Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), pp. 1465-1500. · Zbl 1068.74056 |
| [14] | Gerasimov, T. and Lorenzis, L. D., On penalization in variational phase-field models of brittle fracture, Comput. Methods Appl. Mech. Engrg., 354 (2019), pp. 990-1026. · Zbl 1441.74203 |
| [15] | Gerasimov, T., Römer, U., Vondřejc, J., Matthies, H. G., and De Lorenzis, L., Stochastic phase-field modeling of brittle fracture: Computing multiple crack patterns and their probabilities, Comput. Methods Appl. Mech. Engrg., 372 (2020), 113353. · Zbl 1506.74351 |
| [16] | Gräser, C., Kienle, D., and Sander, O., Truncated nonsmooth Newton multigrid for phase-field brittle-fracture problems, with analysis, Comput. Mech., 72 (2023), pp. 1059-1089, doi:10.1007/s00466-023-02330-x. · Zbl 1528.74101 |
| [17] | Gröger, K., A \({W}^{1,p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), pp. 679-687, doi:10.1007/BF01442860. · Zbl 0646.35024 |
| [18] | Hehl, A., Mohammadi, M., Neitzel, I., and Wollner, W., Optimizing Fracture Propagation Using a Phase-Field Approach, Springer, Cham, Switzerland, 2022, pp. 329-351. · Zbl 1502.49026 |
| [19] | Hehl, A. and Neitzel, I., Second order optimality conditions for an optimal control problem governed by a regularized phase-field fracture propagation model, Optimization, 72 (2023), pp. 1665-1689, doi:10.1080/02331934.2022.2034814. · Zbl 1517.49013 |
| [20] | Heister, T., Wheeler, M. F., and Wick, T., A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach, Comput. Methods Appl. Mech. Engrg., 290 (2015), pp. 466-495. · Zbl 1423.76239 |
| [21] | Khimin, D., Steinbach, M. C., and Wick, T., Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems, J. Comput. Phys., 470 (2022), 111554, doi:10.1016/j.jcp.2022.111554. · Zbl 07599599 |
| [22] | Khimin, D., Steinbach, M. C., and Wick, T., Space-time mixed system formulation of phase-field fracture optimal control problems, J. Optim. Theory Appl., 199 (2023), pp. 1222-1248, doi:10.1007/s10957-023-02272-7. · Zbl 1538.74125 |
| [23] | Knees, D., Rossi, R., and Zanini, C., A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), pp. 565-616. · Zbl 1262.74030 |
| [24] | Kopanicakova, A. and Krause, R., A recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture, Comput. Methods Appl. Mech. Engrg., 360 (2020), 112720, doi:10.1016/j.cma.2019.112720. · Zbl 1441.74208 |
| [25] | Kuhn, C. and Müller, R., A continuum phase field model for fracture, Eng. Fract. Mech., 77 (2010), pp. 3625-3634, doi:10.1016/j.engfracmech.2010.08.009. |
| [26] | Kumar, A., Bourdin, B., Francfort, G. A., and Lopez-Pamies, O., Revisiting nucleation in the phase-field approach to brittle fracture, J. Mech. Phys. Solids, 142 (2020), 104027, doi:10.1016/j.jmps.2020.104027. |
| [27] | Kurcyusz, S., On the existence and nonexistence of Lagrange multipliers in Banach spaces, J. Optim. Theory Appl., 20 (1976), pp. 81-110. · Zbl 0309.49010 |
| [28] | Lazzaroni, G., Rossi, R., Thomas, M., and Toader, R., Rate-independent damage in thermo-viscoelastic materials with inertia, J. Dynam. Differential Equations, 30 (2018), pp. 1311-1364, doi:10.1007/s10884-018-9666-y. · Zbl 1412.35325 |
| [29] | Loeb, P. A. and Talvila, E., Lusin’s theorem and Bochner integration, Sci. Math. Jpn., 60 (2004), pp. 113-120. · Zbl 1069.28007 |
| [30] | Mang, K., Wick, T., and Wollner, W., A phase-field model for fractures in nearly incompressible solids, Comput. Mech., 65 (2020), pp. 61-78. · Zbl 1477.65233 |
| [31] | Maurer, H., First and second order sufficient optimality conditions in mathematical programming and optimal control, Math. Program. Stud., 14 (1981), pp. 163-177. · Zbl 0448.90069 |
| [32] | Maurer, H. and Zowe, J., First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Program., 16 (1979), pp. 98-110. · Zbl 0398.90109 |
| [33] | Miehe, C., Hofacker, M., and Welschinger, F., A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Comput. Meth. Appl. Mech. Engrg., 199 (2010), pp. 2765-2778. · Zbl 1231.74022 |
| [34] | Miehe, C., Welschinger, F., and Hofacker, M., Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83 (2010), pp. 1273-1311. · Zbl 1202.74014 |
| [35] | Mielke, A., Chapter 6 - Evolution of rate-independent systems, in Handbook of Differential Equations Evolutionary Equations, Vol. 2, Dafermos, C. M. and Feireisl, E. eds., Elsevier/North-Holland, 2004, pp. 461-559. · Zbl 1120.47062 |
| [36] | Mielke, A. and Roubíček, T., Rate-Independent Systems, Springer, New York, 2015, doi:10.1007/978-1-4939-2706-7. · Zbl 1339.35006 |
| [37] | Mielke, A., Roubíček, T., and Zeman, J., Complete damage in elastic and viscoelastic media and its energetics, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1242-1253, doi:10.1016/j.cma.2009.09.020. · Zbl 1227.74058 |
| [38] | Mikelić, A., Wheeler, M. F., and Wick, T., A quasi-static phase-field approach to pressurized fractures, Nonlinearity, 28 (2015), pp. 1371-1399. · Zbl 1316.35287 |
| [39] | Mohammadi, M. and Wollner, W., A priori error estimates for a linearized fracture control problem, Optim. Eng., 22 (2021), pp. 2127-2149. · Zbl 1481.49029 |
| [40] | Neitzel, I., Wick, T., and Wollner, W., An optimal control problem governed by a regularized phase-field fracture propagation model, SIAM J. Control Optim., 55 (2017), pp. 2271-2288, doi:10.1137/16M1062375. · Zbl 1370.49003 |
| [41] | Neitzel, I., Wick, T., and Wollner, W., An optimal control problem governed by a regularized phase-field fracture propagation model. Part II: The regularization limit, SIAM J. Control Optim., 57 (2019), pp. 1672-1690. · Zbl 1420.49006 |
| [42] | Partmann, K., Thomas, M., Tornquist, S., Weinberg, K., and Wieners, C., Dynamic phase-field fracture in viscoelastic materials using a first-order formulation, PAMM Proc. Appl. Math. Mech., 22 (2023), doi:10.1002/pamm.202200249. |
| [43] | Pham, K., Amor, H., Marigo, J.-J., and Maurini, C., Gradient damage models and their use to approximate brittle fracture, Int. J. Damage Mech., 20 (2011), pp. 618-652, doi:10.1177/1056789510386852. |
| [44] | Quarteroni, A. and Valli, A., Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1994, doi:10.1007/978-3-540-85268-1. · Zbl 0803.65088 |
| [45] | Robinson, S. M., First order conditions for general nonlinear optimization, SIAM J. Appl. Math., 30 (1976), pp. 597-607. · Zbl 0364.90093 |
| [46] | Thomas, M., Bilgen, C., and Weinberg, K., Analysis and simulations for a phase-field fracture model at finite strains based on modified invariants, ZAMM Z. Angew. Math. Mech., 100 (2020), e201900288, doi:10.1002/zamm.201900288. · Zbl 07812950 |
| [47] | Wambacq, J., Ulloa, J., Lombaert, G., and François, S., Interior-point methods for the phase-field approach to brittle and ductile fracture, Comput. Methods Appl. Mech. Engrg., 375 (2021), 113612. · Zbl 1506.74026 |
| [48] | Wheeler, M., Wick, T., and Wollner, W., An augmented-Lagangrian method for the phase-field approach for pressurized fractures, Comput. Methods Appl. Mech. Engrg., 271 (2014), pp. 69-85. · Zbl 1296.65170 |
| [49] | Wick, T., An error-oriented Newton/inexact augmented Lagrangian approach for fully monolithic phase-field fracture propagation, SIAM J. Sci. Comput., 39 (2017), pp. B589-B617, doi:10.1137/16M1063873. · Zbl 1403.74131 |
| [50] | Wick, T., Multiphysics Phase-Field Fracture: Modeling, Adaptive Discretizations, and Solvers, De Gruyter, Berlin, 2020, doi:10.1515/9783110497397. · Zbl 1448.74003 |
| [51] | Wu, J.-Y., Nguyen, V. P., Thanh Nguyen, C., Sutula, D., Bordas, S., and Sinaie, S., Phase field modelling of fracture, Adv. Appl. Mech., 53 (2020), pp. 1-183, doi:10.1016/bs.aams.2019.08.001. |
| [52] | Zeidler, E., Nonlinear Functional Analysis and its Applications. I, Springer, New York, 1986, https://link.springer.com/book/9780387909141. · Zbl 0583.47050 |
| [53] | Zowe, J. and Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), pp. 49-62. · Zbl 0401.90104 |
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