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On the differentiability of fluid-structure interaction problems with respect to the problem data. (English) Zbl 1418.35316
Summary: A coupled system of stationary fluid-structure equations in an arbitrary Lagrangian-Eulerian framework is considered in this work. Existence results presented in the literature are extended to show differentiability of the solutions to a stationary fluid-structure interaction problem with respect to the given data, volume forces and boundary values, provided a small data assumption holds. Numerical experiments are used to substantiate the theoretical findings.

MSC:
35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Software:
deal.ii; DOpElib
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[1] Allen, M.; Maute, K., Reliability-based shape optimization of structures undergoing fluid-structure interaction phenomena, Comput. Methods Appl. Mech. Eng., 194, 3472-3495, (2005) · Zbl 1100.74046
[2] Avalos, G.; Lasiecka, I.; Triggiani, R., Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, Georgian Math. J., 15, 403-437, (2010) · Zbl 1157.35085
[3] Bangerth, W.; Hartmann, R.; Kanschat, G., deal.II—a general purpose object oriented finite element library, ACM Trans. Math. Softw., 33, 24/1-24/27, (2007) · Zbl 1365.65248
[4] Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations, 1st edn. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2003) · Zbl 1020.65058
[5] Barbu, V.; Grujic, Z.; Lasiecka, I.; Tuffaha, A., Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57, 1173-1207, (2008) · Zbl 1147.74016
[6] Bazilevs, Y., Takizawa, K., Tezduyar, T.: Computational Fluid-Structure Interaction: Methods and Applications. Wiley, London (2013) · Zbl 1286.74001
[7] Becker, R.; Rannacher, R., A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math., 4, 237-264, (1996) · Zbl 0868.65076
[8] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation, Acta Numer., 2001, 1-102, (2001) · Zbl 1105.65349
[9] Bodnar, T., Galdi, G., Necasova, S.: Fluid-Structure Interaction and Biomedical Applications. Birkhaeuser, Basel (2014) · Zbl 1300.76003
[10] Boulakia, M., Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid, C. R. Math. Acad. Sci. Paris, 336, 985-990, (2003) · Zbl 1129.74306
[11] Bucci, F.; Lasiecka, I., Optimal boundary control with critical penalization for a pde model of fluid-solid interactions, Calc. Var. Partial Differ. Equ., 37, 217-235, (2010) · Zbl 1198.35278
[12] Bungartz, H.-J., Schäfer, M. (eds.): Fluid-Structure Interaction: Modelling, Simulation, Optimisation, vol. 53 of Lecture Notes in Computational Science and Engineering. Springer, Berlin (2006)
[13] Ciarlet, P.G.: Mathematical Elasticity. Volume 1: Three Dimensional Elasticity. North-Holland, Amsterdam (1984)
[14] Coutand, D.; Shkoller, S., Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176, 25-102, (2005) · Zbl 1064.74057
[15] Coutand, D.; Shkoller, S., The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., 179, 303-352, (2006) · Zbl 1138.74325
[16] Deneuvy, AC, Theoretical study and optimization of a fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 22, 75-92, (1988) · Zbl 0663.76096
[17] Desjardins, B.; Esteban, MJ, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146, 59-71, (1999) · Zbl 0943.35063
[18] Desjardins, B.; Esteban, MJ; Grandmont, C.; Tallec, P., Weak solutions for a fluid-structure interaction problem, Rev. Mat. Complut., 14, 523-538, (2001) · Zbl 1007.35055
[19] Santos, ND; Gerbeau, J-F; Bourgat, JF, A partitioned fluid-structure algorithm for elastic thin valves with contact, Comput. Methods Appl. Mech. Eng., 197, 1750-1761, (2008) · Zbl 1194.74383
[20] Dunne, Th.; Rannacher, R.; Richter, Th., NUMERICAL SIMULATION OF FLUID-STRUCTURE INTERACTION BASED ON MONOLITHIC VARIATIONAL FORMULATIONS, 1-75, (2010)
[21] Failer, L.; Meidner, D.; Vexler, B., Optimal control of a linear unsteady fluid-structure interaction problem, J. Optim. Theory Appl., 170, 1-27, (2017) · Zbl 1346.35033
[22] Failer, L.; Wick, T., Adaptive time-step control for nonlinear fluid-structure interaction, J. Comput. Phys., 366, 448-477, (2018) · Zbl 1406.76048
[23] Formaggia, L.; Nobile, F., A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements, East-West J. Numer. Math., 7, 105-132, (1999) · Zbl 0942.65113
[24] Formaggia, L., Quarteroni, A., Veneziani, A. (eds.): Cardiovascular Mathematics, vol. 1 of MS&A. Modeling, Simulation and Applications. Modeling and Simulation of the Circulatory System. Springer, Milan (2009) · Zbl 1300.92005
[25] Frei, S., Holm, B., Richter, T., Wick, T., Yang, H.: Fluid-Structure Interaction: Modeling, Adaptive Discretisations and Solvers. Walter de Gruyter, Berlin (2017) · Zbl 1390.74004
[26] Galdi, G., Rannacher, R.: Fundamental Trends in Fluid-Structure Interaction. World Scientific, Singapore (2010) · Zbl 1410.76010
[27] Goll, C.; Wick, T.; Wollner, W., DOpElib: differential equations and optimization environment; a goal oriented software library for solving PDEs and optimization problems with PDEs, Arch. Numer. Softw., 5, 1-14, (2017)
[28] Grandmont, C., Existence et unicité de solutions d’un problème de couplage fluide-structure bidimensionnel stationnaire, C. R. Acad. Sci. Paris, 326, 651-656, (1998) · Zbl 0919.73139
[29] Grandmont, C., Existence for a three-dimensional steady state fluid-structure interaction problem, J. Math. Fluid Mech., 4, 76-94, (2002) · Zbl 1009.76016
[30] Grandmont, C., Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40, 716-737, (2008) · Zbl 1158.74016
[31] Grätsch, T.; Bathe, K-J, Goal-oriented error estimation in the analysis of fluid flows with structural interactions, Comput. Methods Appl. Mech. Eng., 195, 5673-5684, (2006) · Zbl 1122.76055
[32] Helenbrook, B., Mesh deformation using the biharmonic operator, Int. J. Numer. Methods Eng., 56, 1-30, (2001)
[33] Heywood, JG; Rannacher, R.; Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 22, 325-352, (1996) · Zbl 0863.76016
[34] Hirth, C.; Amsden, A.; Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 14, 227-253, (1974) · Zbl 0292.76018
[35] Hron, J., Turek, S.: Proposal for Numerical Benchmarking of Fluid-Structure Interaction Between an Elastic Object and Laminar Incompressible Flow, vol. 53, pp. 146-170. Springer, Berlin (2006) · Zbl 1323.76049
[36] Hughes, T.; Liu, W.; Zimmermann, T., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng., 29, 329-349, (1981) · Zbl 0482.76039
[37] Langer, U.; Yang, H., Partitioned solution algorithms for fluid-structure interaction problems with hyperelastic models, J. Comput. Appl. Math., 276, 47-61, (2015) · Zbl 1297.74039
[38] Lund, E.; Moller, H.; Jakobsen, LA, Shape design optimization of stationary fluid-structure interaction problems with large displacements and turbulence, Struct. Multidiscip. Optim., 25, 383-392, (2003)
[39] Murea, CM; Vázquez, C., Sensitivity and approximation of coupled fluid-structure equations by virtual control method, Appl. Math. Optim., 52, 183-218, (2005) · Zbl 1136.74319
[40] Nobile, F.: Numerical Approximation of Fluid-Structure Interaction Problems with Applications to Haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2001)
[41] Nobile, F.; Vergara, C., An effective fluid-structure interaction formulation for vascular dynamics by generalized robin conditions, SIAM J. Sci. Comput., 30, 731-763, (2008) · Zbl 1168.74038
[42] Noh, W.: A Time-Dependent Two-Space-Dimensional Coupled Eulerian-Lagrangian Code, Vol. 3 of Methods Comput. Phys, pp. 117-179. Academic Press, New York (1964)
[43] Perego, M.; Veneziani, A.; Vergara, C., A variational approach for estimating the compliance of the cardiovascular tissue: an inverse fluid-structure interaction problem, SIAM J. Sci. Comput., 33, 1181-1211, (2011) · Zbl 1227.92010
[44] Piperno, S.; Farhat, C., Paritioned procedures for the transient solution of coupled aeroelastic problems—part ii: energy transfer analysis and three-dimensional applications, Comput. Methods Appl. Mech. Eng., 190, 3147-3170, (2001) · Zbl 1015.74009
[45] Quaini, A.; Canic, S.; Glowinski, R.; Igo, S.; Hartley, CJ; Zoghbi, W.; Little, S., Validation of a 3d comoputational fluid-structure interaction model simulating flow through elastic aperature, J. Biomech., 45, 310-318, (2012)
[46] Raymond, J-P, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48, 5398-5443, (2010) · Zbl 1213.93177
[47] Richter, T., Goal-oriented error estimation for fluid-structure interaction problems, Comput. Methods Appl. Mech. Eng., 223-224, 38-42, (2012) · Zbl 1253.74037
[48] Richter, T.: Fluid-Structure Interactions: Models, Analysis, and Finite Elements. Springer, Berlin (2017) · Zbl 1374.76001
[49] Richter, T.; Wick, T., Optimal control and parameter estimation for stationary fluid-structure interaction problems, SIAM J. Sci. Comput., 35, b1085-b1104, (2013) · Zbl 1282.35287
[50] Schäfer, M., Sternel, D., Becker, G., Pironkov, P.: Efficient Numerical Simulation and Optimization of Fluid-Structure Interaction, vol. 53, pp. 133-160. Springer, Berlin (2010) · Zbl 1215.74019
[51] Stein, K.; Tezduyar, T.; Benney, R., Mesh moving techniques for fluid-structure interactions with large displacements, J. Appl. Mech., 70, 58-63, (2003) · Zbl 1110.74689
[52] Takizawa, K.; Tezduyar, T., Computational methods for parachute fluid-structure interactions, Arch. Comput. Methods Eng., 19, 125-169, (2012) · Zbl 1354.76113
[53] Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland Publishing Co., Amsterdam (1977) · Zbl 0383.35057
[54] Turek, S., Hron, J., Razzaq, M., Wobker, H., Schäfer, M.: Numerical Benchmarking of Fluid-Structure Interaction: A Comparison of Different Discretization and Solution Approaches, Fluid Structure Interaction II: Modelling, Simulation, Optimization, pp. 413-424. Springer, Heidelberg (2010) · Zbl 1215.76061
[55] Zee, K.; Brummelen, E.; Borst, R., Goal-oriented error estimation for Stokes flow interacting with a flexible channel, Int. J. Numer. Methods Fluids, 56, 1551-1557, (2008) · Zbl 1136.76032
[56] Wick, T., Fluid-structure interactions using different mesh motion techniques, Comput. Struct., 89, 1456-1467, (2011)
[57] Wick, T., Goal-oriented mesh adaptivity for fluid-structure interaction with application to heart-valve settings, Arch. Mech. Eng., 59, 73-99, (2012)
[58] Winslow, A., Numerical solution of the quasi-linear poisson equation in a nonuniform triangle mesh, J. Comput. Phys., 1, 149-172, (1967) · Zbl 0254.65069
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