zbMATH — the first resource for mathematics

An adjoint based pressure boundary optimal control approach for fluid-structure interaction problems. (English) Zbl 1411.76058
Summary: In this work, we investigate a new pressure boundary optimal control approach to the fluid-structure interaction problem based on Lagrangian multipliers and adjoint variables. We consider the steady FSI problem written in variational monolithic form in order to balance automatically solid and liquid forces at the interface and propose a pressure boundary optimal control method with the purpose to control the solid deformation in a well defined region by changing the fluid pressure on domain boundaries. The optimality system is obtained by imposing the first order necessary condition to the Lagrangian functional. In order to couple also the adjoint variables, we must introduce a fictitious velocity field in the solid region that balances automatically interface adjoint forces as well. The system is solved in a segregated approach with different optimization schemes, such as the steepest descent and the quasi-Newton methods. We implement the algorithms in a finite element code with mesh-moving capabilities for the study of large solid displacements. In order to support the proposed approach, we perform numerical tests in two and three-dimensional spaces.
76M10 Finite element methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76L05 Shock waves and blast waves in fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Full Text: DOI
[1] Turek, S.; Hron, J., Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow, Fluid-structure interaction, 371-385, (2006), Springer · Zbl 1323.76049
[2] Formaggia, L.; Quarteroni, A.; Veneziani, A., Cardiovascular mathematics, vol. 1, (2009), Springer, Heidelberg
[3] Le Tallec, P.; Mouro, J., Fluid structure interaction with large structural displacements, Comput Meth Appl Mech Eng, 190, 24, 3039-3067, (2001) · Zbl 1001.74040
[4] Bukač, M.; Čanić, S.; Muha, B., A partitioned scheme for fluid-composite structure interaction problems, J Comput Phys, 281, 493-517, (2015) · Zbl 1351.76321
[5] Gilmanov, A.; Le, T. B.; Sotiropoulos, F., A numerical approach for simulating fluid structure interaction of flexible thin shells undergoing arbitrarily large deformations in complex domains, J Comput Phys, 300, 814-843, (2015) · Zbl 1349.74323
[6] Calderer, A.; Kang, S.; Sotiropoulos, F., Level set immersed boundary method for coupled simulation of air/water interaction with complex floating structures, J Comput Phys, 277, 201-227, (2014) · Zbl 1349.76438
[7] Aulisa, E.; Cervone, A.; Manservisi, S.; Seshaiyer, P., A multilevel domain decomposition approach for studying coupled flow applications, Comm Comput Phys, 6, 2, 319-341, (2009) · Zbl 1364.76079
[8] Wu, Y.; Cai, X.-C., A fully implicit domain decomposition based ale framework for three-dimensional fluid-structure interaction with application in blood flow computation, J Comput Phys, 258, 524-537, (2014) · Zbl 1349.76552
[9] Cerroni, D.; Da Vià, R.; Manservisi, S., A projection method for coupling two-phase vof and fluid structure interaction simulations, J Comput Phys, 354, 646-671, (2018) · Zbl 1380.74029
[10] Cerroni, D.; Giommi, D.; Manservisi, S.; Menghini, F., Preliminary monolithic fluid structure interaction model for ventricle contraction, Lect Notes Appl Comput Mech, 84, 217-231, (2018)
[11] Chinchuluun, A.; Pardalos, P. M.; Enkhbat, R.; Tseveendorj, I., Optimization and optimal control, (2010), Springer
[12] Noack, B. R.; Morzynski, M.; Tadmor, G., Reduced-order modelling for flow control, 528, (2011), Springer Science & Business Media
[13] Leugering, G.; Engell, S.; Griewank, A.; Hinze, M.; Rannacher, R.; Schulz, V., Constrained optimization and optimal control for partial differential equations, 160, (2012), Springer Science & Business Media
[14] Gunzburger, M.; Manservisi, S., Analysis and approximation for linear feedback control for tracking the velocity in navier-stokes flows, Comput Methods Appl Mech Eng, 189, 3, 803-823, (2000) · Zbl 0969.76025
[15] Nocedal, J.; Wright, S., Numerical optimization, (2006), Springer Science & Business Media · Zbl 1104.65059
[16] Gunzburger, M. D., Perspectives in flow control and optimization, 5, (2003), Siam · Zbl 1088.93001
[17] Rowley, C. W.; Williams, D. R., Dynamics and control of high-Reynolds-number flow over open cavities, Annu Rev Fluid Mech, 38, 251-276, (2006) · Zbl 1101.76019
[18] Kim, J.; Bewley, T. R., A linear systems approach to flow control, Annu Rev Fluid Mech, 39, 383-417, (2007) · Zbl 1296.76074
[19] Yan, Y.; Keyes, D. E., Smooth and robust solutions for dirichlet boundary control of fluid-solid conjugate heat transfer problems, J Comput Phys, 281, 759-786, (2015) · Zbl 1351.49033
[20] Gunzburger, M.; Manservisi, S., The velocity tracking problem for navier-stokes flows with bounded distributed controls, SIAM J Control Optim, 37, 6, 1913-1945, (1999) · Zbl 0938.35118
[21] Failer, L.; Meidner, D.; Vexler, B., Optimal control of a linear unsteady fluid-structure interaction problem, J Optim Theory Appl, 170, 1, 1-27, (2016) · Zbl 1346.35033
[22] Bazilevs, Y.; Hsu, M.-C.; Bement, M., Adjoint-based control of fluid-structure interaction for computational steering applications, Procedia Comput Sci, 18, 1989-1998, (2013)
[23] Richter, T.; Wick, T., Optimal control and parameter estimation for stationary fluid-structure interaction problems, SIAM J Scient Comput, 35, 5, B1085-B1104, (2013) · Zbl 1282.35287
[24] Adams, R. A., Sobolev spaces, (1975), Academic Press, New York · Zbl 0314.46030
[25] Brenner, S.; Scott, R., The mathematical theory of finite element methods, 15, (2007), Springer Science
[26] Hron, J.; Turek, S., A monolithic fem/multigrid solver for an ale formulation of fluid-structure interaction with applications in biomechanics, Fluid-structure interaction, 146-170, (2006), Springer · Zbl 1323.74086
[27] Chirco, L.; Da Vià, R.; Manservisi, S., An optimal control method for fluid structure interaction systems via adjoint boundary pressure, J Phys Conf Ser, 923, 012026, (2017)
[28] Sokolowski, J.; Zolesio, J.-P., Introduction to shape optimization, Introduction to shape optimization, 5-12, (1992), Springer
[29] Armijo, L., Minimization of functions having lipschitz continuous first partial derivatives, Pacific J Math, 16, 1, 1-3, (1966) · Zbl 0202.46105
[30] Aulisa, E.; Manservisi, S.; Seshaiyer, P., A computational multilevel approach for solving 2d navier-stokes equations over non-matching grids, Comput Meth Appl Mech Eng, 195, 33, 4604-4616, (2006) · Zbl 1124.76024
[31] Bramble, J. H., Multigrid methods, 294, (1993), CRC Press · Zbl 0786.65094
[32] Cerroni, D.; Manservisi, S.; Menghini, F., An improved monolithic multigrid fluid- structure interaction solver with a new moving mesh technique, Internat J Math Mod Meth Appl Sci, 9, 227-234, (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.