×

zbMATH — the first resource for mathematics

Non-body-fitted fluid-structure interaction: divergence-conforming B-splines, fully-implicit dynamics, and variational formulation. (English) Zbl 1416.74026
Summary: Immersed boundary (IB) methods deal with incompressible visco-elastic solids interacting with incompressible viscous fluids. A long-standing issue of IB methods is the challenge of accurately imposing the incompressibility constraint at the discrete level. We present the divergence-conforming immersed boundary (DCIB) method to tackle this issue. The DCIB method leads to completely negligible incompressibility errors at the Eulerian level and various orders of magnitude of increased accuracy at the Lagrangian level compared to other IB methods. Furthermore, second-order convergence of the incompressibility error at the Lagrangian level is obtained as the discretization is refined. In the DCIB method, the Eulerian velocity-pressure pair is discretized using divergence-conforming B-splines, leading to inf-sup stable and pointwise divergence-free Eulerian solutions. The Lagrangian displacement is discretized using non-uniform rational B-splines, which enables to robustly handle large mesh distortions. The data transfer needed between the Eulerian and Lagrangian descriptions is performed at the quadrature level using the same spline basis functions that define the computational meshes. This conduces to a fully variational formulation, sharp treatment of the fluid-solid interface, and a 0.5 increase in the convergence rate of the Eulerian velocity and the Lagrangian displacement measured in \(L^2\) norm in comparison with using discretized Dirac delta functions for the data transfer. By combining the generalized-\(\alpha\) method and a block-iterative solution strategy, the DCIB method results in a fully-implicit discretization, which enables to take larger time steps. Various two- and three-dimensional problems are solved to show all the aforementioned properties of the DCIB method along with mesh-independence studies, verification of the numerical method by comparison with the literature, and measurement of convergence rates.

MSC:
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S05 Finite element methods applied to problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
92C10 Biomechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
BoomerAMG; ML; PetIGA
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Peskin, C., Flow patterns around heart valves: a numerical method, J. Comput. Phys., 10, 252-271, (1972) · Zbl 0244.92002
[2] Peskin, C., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 3, 220-252, (1977) · Zbl 0403.76100
[3] Peskin, C., The immersed boundary method, Acta Numer., 11, 479-517, (2002) · Zbl 1123.74309
[4] Boffi, D.; Gastaldi, L.; Heltai, L.; Peskin, C. S., On the hyper-elastic formulation of the immersed boundary method, Comput. Methods Appl. Mech. Eng., 197, 25, 2210-2231, (2008) · Zbl 1158.74523
[5] Boffi, D.; Gastaldi, L., A finite element approach for the immersed boundary method, Comput. Struct., 81, 8, 491-501, (2003)
[6] Zhang, L.; Gerstenberger, A.; Wang, X.; Liu, W. K., Immersed finite element method, Comput. Methods Appl. Mech. Eng., 193, 2051-2067, (2004) · Zbl 1067.76576
[7] Liu, W. K.; Liu, Y.; Farrell, D.; Zhang, L.; Wang, X.; Fukui, Y.; Patankar, N.; Zhang, Y.; Bajaj, C.; Lee, J.; Hong, J.; Chen, X.; Hsu, H., Immersed finite element method and its applications to biological systems, Comput. Methods Appl. Mech. Eng., 195, 1722-1749, (2006) · Zbl 1178.76232
[8] Wang, C.; Eldredge, J. D., Strongly coupled dynamics of fluids and rigid-body systems with the immersed boundary projection method, J. Comput. Phys., 295, 87-113, (2015) · Zbl 1349.76280
[9] Kallemov, B.; Bhalla, A.; Griffith, B.; Donev, A., An immersed boundary method for rigid bodies, Commun. Appl. Math. Comput. Sci., 11, 1, 79-141, (2016) · Zbl 1382.76191
[10] Boffi, D.; Cavallini, N.; Gastaldi, L., Finite element approach to immersed boundary method with different fluid and solid densities, Math. Models Methods Appl. Sci., 21, 12, 2523-2550, (2011) · Zbl 1242.76190
[11] Fai, T. G.; Griffith, B. E.; Mori, Y.; Peskin, C. S., Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers I: numerical method and results, SIAM J. Sci. Comput., 35, 5, B1132-B1161, (2013) · Zbl 1282.76088
[12] Du, J.; Guy, R. D.; Fogelson, A. L., An immersed boundary method for two-fluid mixtures, J. Comput. Phys., 262, 231-243, (2014) · Zbl 1349.76873
[13] Guo, Y.; Wu, C.-Y.; Thornton, C., Modeling gas-particle two-phase flows with complex and moving boundaries using DEM-CFD with an immersed boundary method, AIChE J., 59, 4, 1075-1087, (2013)
[14] Griffith, B. E.; Luo, X., Hybrid finite difference/finite element immersed boundary method, Int. J. Numer. Methods Biomed. Eng., 33, 12, 1-31, (2017)
[15] Casquero, H.; Bona-Casas, C.; Gomez, H., A NURBS-based immersed methodology for fluid-structure interaction, Comput. Methods Appl. Mech. Eng., 284, 943-970, (2015)
[16] Casquero, H.; Liu, L.; Bona-Casas, C.; Zhang, Y.; Gomez, H., A hybrid variational-collocation immersed method for fluid-structure interaction using unstructured T-splines, Int. J. Numer. Methods Eng., 105, 11, 855-880, (2016)
[17] Sigüenza, J.; Mendez, S.; Ambard, D.; Dubois, F.; Jourdan, F.; Mozul, R.; Nicoud, F., Validation of an immersed thick boundary method for simulating fluid-structure interactions of deformable membranes, J. Comput. Phys., 322, 723-746, (2016) · Zbl 1351.76080
[18] Tian, F.-B.; Luo, H.; Zhu, L.; Liao, J. C.; Lu, X.-Y., An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments, J. Comput. Phys., 230, 19, 7266-7283, (2011) · Zbl 1327.76106
[19] Li, Z.; Favier, J.; D’Ortona, U.; Poncet, S., An immersed boundary-lattice Boltzmann method for single- and multi-component fluid flows, J. Comput. Phys., 304, 424-440, (2016) · Zbl 1349.76708
[20] Griffith, B. E., Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions, Int. J. Numer. Methods Biomed. Eng., 28, 3, 317-345, (2012) · Zbl 1243.92017
[21] Borazjani, I., Fluid-structure interaction, immersed boundary-finite element method simulations of bio-prosthetic heart valves, Comput. Methods Appl. Mech. Eng., 257, 103-116, (2013) · Zbl 1286.74030
[22] 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
[23] Gao, H.; Feng, L.; Qi, N.; Berry, C.; Griffith, B.; Luo, X., A coupled mitral valve-left ventricle model with fluid-structure interaction, arXiv preprint
[24] Cordasco, D.; Bagchi, P., Dynamics of red blood cells in oscillating shear flow, J. Fluid Mech., 800, 484-516, (2016)
[25] Casquero, H.; Bona-Casas, C.; Gomez, H., NURBS-based numerical proxies for red blood cells and circulating tumor cells in microscale blood flow, Comput. Methods Appl. Mech. Eng., 316, 646-667, (2017)
[26] Balogh, P.; Bagchi, P., A computational approach to modeling cellular-scale blood flow in complex geometry, J. Comput. Phys., 334, 280-307, (2017)
[27] Lushi, E.; Peskin, C. S., Modeling and simulation of active suspensions containing large numbers of interacting micro-swimmers, Comput. Struct., 122, 239-248, (2013)
[28] Hoover, A. P.; Griffith, B. E.; Miller, L. A., Quantifying performance in the medusan mechanospace with an actively swimming three-dimensional jellyfish model, J. Fluid Mech., 813, 1112-1155, (2017) · Zbl 1383.76578
[29] Ge, M.; Chua, K.; Shu, C.; Yang, W., Analytical and numerical study of tissue cryofreezing via the immersed boundary method, Int. J. Heat Mass Transf., 83, 1-10, (2015)
[30] Huang, W.-X.; Chang, C. B.; Sung, H. J., Three-dimensional simulation of elastic capsules in shear flow by the penalty immersed boundary method, J. Comput. Phys., 231, 8, 3340-3364, (2012) · Zbl 1404.74159
[31] Hu, W.-F.; Kim, Y.; Lai, M.-C., An immersed boundary method for simulating the dynamics of three-dimensional axisymmetric vesicles in Navier-Stokes flows, J. Comput. Phys., 257, 670-686, (2014) · Zbl 1349.76612
[32] Kempe, T.; Fröhlich, J., An improved immersed boundary method with direct forcing for the simulation of particle laden flows, J. Comput. Phys., 231, 9, 3663-3684, (2012) · Zbl 1402.76143
[33] 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
[34] Galvin, K. J.; Linke, A.; Rebholz, L. G.; Wilson, N. E., Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Eng., 237, 166-176, (2012) · Zbl 1253.76057
[35] Glowinski, R.; Pan, T.; Hesla, T.; Joseph, D.; Periaux, J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. Comput. Phys., 169, 2, 363-426, (2001) · Zbl 1047.76097
[36] Baaijens, F. P., A fictitious domain/mortar element method for fluid-structure interaction, Int. J. Numer. Methods Fluids, 35, 7, 743-761, (2001) · Zbl 0979.76044
[37] Van Loon, R.; Anderson, P. D.; De Hart, J.; Baaijens, F. P., A combined fictitious domain/adaptive meshing method for fluid-structure interaction in heart valves, Int. J. Numer. Methods Fluids, 46, 5, 533-544, (2004) · Zbl 1060.76582
[38] Kamensky, D.; Hsu, M.-C.; Schillinger, D.; Evans, J. A.; Aggarwal, A.; Bazilevs, Y.; Sacks, M. S.; Hughes, T. J.R., An immersogeometric variational framework for fluid-structure interaction: application to bioprosthetic heart valves, Comput. Methods Appl. Mech. Eng., 284, 1005-1053, (2015)
[39] Kamensky, D.; Evans, J. A.; Hsu, M.-C.; Bazilevs, Y., Projection-based stabilization of interface Lagrange multipliers in immersogeometric fluid-thin structure interaction analysis, with application to heart valve modeling, Comput. Math. Appl., 74, 9, 2068-2088, (2017) · Zbl 1397.65274
[40] Peskin, C. S.; Printz, B. F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys., 105, 1, 33-46, (1993) · Zbl 0762.92011
[41] Griffith, B. E., On the volume conservation of the immersed boundary method, Commun. Comput. Phys., 12, 2, 401-432, (2012) · Zbl 1373.74098
[42] Boilevin-Kayl, L.; Fernández, M. A.; Gerbeau, J.-F., Numerical methods for immersed FSI with thin-walled structures, Comput. Fluids, (2018)
[43] Wang, X.; Zhang, L. T., Interpolation functions in the immersed boundary and finite element methods, Comput. Mech., 45, 4, 321-334, (2010) · Zbl 1362.74035
[44] Alauzet, F.; Fabrèges, B.; Fernández, M. A.; Landajuela, M., Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures, Comput. Methods Appl. Mech. Eng., 301, 300-335, (2016)
[45] Bao, Y.; Donev, A.; Griffith, B. E.; McQueen, D. M.; Peskin, C. S., An immersed boundary method with divergence-free velocity interpolation and force spreading, J. Comput. Phys., 347, 183-206, (2017) · Zbl 1380.76078
[46] Kamensky, D.; Hsu, M.-C.; Yu, Y.; Evans, J. A.; Sacks, M. S.; Hughes, T. J., Immersogeometric cardiovascular fluid-structure interaction analysis with divergence-conforming B-splines, Comput. Methods Appl. Mech. Eng., 314, 408-472, (2017)
[47] Buffa, A.; Rivas, J.; Sangalli, G.; Vázquez, R., Isogeometric discrete differential forms in three dimensions, SIAM J. Numer. Anal., 49, 2, 818-844, (2011) · Zbl 1225.65100
[48] Evans, J. A.; Hughes, T. J., Isogeometric divergence-conforming B-splines for the Darcy-Stokes-Brinkman equations, Math. Models Methods Appl. Sci., 23, 04, 671-741, (2013) · Zbl 1355.76064
[49] Evans, J. A.; Hughes, T. J., Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations, Math. Models Methods Appl. Sci., 23, 08, 1421-1478, (2013) · Zbl 1383.76337
[50] Evans, J. A.; Hughes, T. J., Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations, J. Comput. Phys., 241, 141-167, (2013) · Zbl 1349.76054
[51] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194, 4135-4195, (2005) · Zbl 1151.74419
[52] Piegl, L.; Tiller, W., The NURBS book, (2012), Springer Science & Business Media
[53] Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15, 1-155, (2006) · Zbl 1185.65204
[54] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric analysis toward integration of CAD and FEA, (2009), Wiley · Zbl 1378.65009
[55] Hsu, M.-C.; Kamensky, D.; Xu, F.; Kiendl, J.; Wang, C.; Wu, M.; Mineroff, J.; Reali, A.; Bazilevs, Y.; Sacks, M., Dynamic and fluid-structure interaction simulations of bioprosthetic heart valves using parametric design with T-splines and Fung-type material models, Comput. Mech., 55, 6, 1211-1225, (2015) · Zbl 1325.74048
[56] Heltai, L.; Kiendl, J.; DeSimone, A.; Reali, A., A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling, Comput. Methods Appl. Mech. Eng., 316, 522-546, (2017)
[57] Maestre, J.; Pallares, J.; Cuesta, I.; Scott, M. A., A 3D isogeometric BE-FE analysis with dynamic remeshing for the simulation of a deformable particle in shear flows, Comput. Methods Appl. Mech. Eng., 326, 70-101, (2017)
[58] Bazilevs, Y.; Moutsanidis, G.; Bueno, J.; Kamran, K.; Kamensky, D.; Hillman, M.; Gomez, H.; Chen, J., A new formulation for air-blast fluid-structure interaction using an immersed approach: part II - coupling of IGA and meshfree discretizations, Comput. Mech., 1-16, (2017) · Zbl 1386.74047
[59] Kadapa, C.; Dettmer, W.; Perić, D., A fictitious domain/distributed Lagrange multiplier based fluid-structure interaction scheme with hierarchical B-spline grids, Comput. Methods Appl. Mech. Eng., 301, 1-27, (2016)
[60] Kadapa, C.; Dettmer, W.; Perić, D., A stabilised immersed framework on hierarchical b-spline grids for fluid-flexible structure interaction with solid-solid contact, Comput. Methods Appl. Mech. Eng., 335, 472-489, (2018)
[61] Akkerman, I.; Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Hulshoff, S., The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech., 41, 371-378, (2008) · Zbl 1162.76355
[62] Lipton, S.; Evans, J.; Bazilevs, Y.; Elguedj, T.; Hughes, T. J.R., Robustness of isogeometric structural discretizations under severe mesh distortion, Comput. Methods Appl. Mech. Eng., 199, 357-373, (2010) · Zbl 1227.74112
[63] Boffi, D.; Gastaldi, L.; Heltai, L., Numerical stability of the finite element immersed boundary method, Math. Models Methods Appl. Sci., 17, 10, 1479-1505, (2007) · Zbl 1186.76661
[64] Boffi, D.; Gastaldi, L.; Heltai, L., On the CFL condition for the finite element immersed boundary method, Comput. Struct., 85, 11, 775-783, (2007)
[65] Sugiyama, K.; Ii, S.; Takeuchi, S.; Takagi, S.; Matsumoto, Y., A full Eulerian finite difference approach for solving fluid-structure coupling problems, J. Comput. Phys., 230, 3, 596-627, (2011) · Zbl 1283.74010
[66] Boffi, D.; Brezzi, F.; Fortin, M., Mixed finite element methods and applications, vol. 44, (2013), Springer · Zbl 1277.65092
[67] Auricchio, F.; Boffi, D.; Gastaldi, L.; Lefieux, A.; Reali, A., A study on unfitted 1D finite element methods, Comput. Math. Appl., 68, 12, 2080-2102, (2014) · Zbl 1369.65095
[68] Chung, J.; Hulbert, G., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, J. Appl. Mech., 60, 2, 371-375, (1993) · Zbl 0775.73337
[69] Jansen, K.; Whiting, C.; Hulbert, G., Generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Comput. Methods Appl. Mech. Eng., 190, 305-319, (2000) · Zbl 0973.76048
[70] Bazilevs, Y.; Takizawa, K.; Tezduyar, T. E., Computational fluid-structure interaction: methods and applications, (2012), John Wiley & Sons
[71] Dalcin, L.; Collier, N.; Vignal, P.; Côrtes, A.; Calo, V., Petiga: a framework for high-performance isogeometric analysis, Comput. Methods Appl. Mech. Eng., 308, 151-181, (2016)
[72] Sarmiento, A. F.; Côrtes, A. M.; Garcia, D.; Dalcin, L.; Collier, N.; Calo, V. M., Petiga-MF: a multi-field high-performance toolbox for structure-preserving B-splines spaces, J. Comput. Sci., 18, 117-131, (2017)
[73] Côrtes, A.; Dalcin, L.; Sarmiento, A.; Collier, N.; Calo, V., A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem, Comput. Methods Appl. Mech. Eng., 316, 839-858, (2017)
[74] Espath, L.; Sarmiento, A.; Vignal, P.; Varga, B.; Cortes, A.; Dalcin, L.; Calo, V., Energy exchange analysis in droplet dynamics via the Navier-Stokes-Cahn-Hilliard model, J. Fluid Mech., 797, 389-430, (2016) · Zbl 1422.76037
[75] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Eijkhout, V.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; May, D. A.; McInnes, L. C.; Rupp, K.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H., Petsc web page, (2017)
[76] Yang, U. M., Boomeramg: a parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41, 1, 155-177, (2002) · Zbl 0995.65128
[77] Gee, M. W.; Siefert, C. M.; Hu, J. J.; Tuminaro, R. S.; Sala, M. G., ML 5.0 smoothed aggregation User’s guide, (2006), Sandia National Laboratories, Tech. rep., SAND2006-2649
[78] Brune, P. R.; Knepley, M. G.; Smith, B. F.; Tu, X., Composing scalable nonlinear algebraic solvers, SIAM Rev., 57, 4, 535-565, (2015) · Zbl 1336.65030
[79] Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869, (1986) · Zbl 0599.65018
[80] Towns, J.; Cockerill, T.; Dahan, M.; Foster, I.; Gaither, K.; Grimshaw, A.; Hazlewood, V.; Lathrop, S.; Lifka, D.; Peterson, G. D., XSEDE: accelerating scientific discovery, Comput. Sci. Eng., 16, 5, 62-74, (2014)
[81] Zhao, H.; Freund, J. B.; Moser, R. D., A fixed-mesh method for incompressible flow-structure systems with finite solid deformations, J. Comput. Phys., 227, 6, 3114-3140, (2008) · Zbl 1329.74313
[82] Roy, S.; Heltai, L.; Costanzo, F., Benchmarking the immersed finite element method for fluid-structure interaction problems, Comput. Math. Appl., 69, 10, 1167-1188, (2015)
[83] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Hughes, T. J.R.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Eng., 197, 173-201, (2007) · Zbl 1169.76352
[84] Auricchio, F.; Beirao Da Veiga, L.; Hughes, T. J.R.; Reali, A.; Sangalli, G., Isogeometric collocation methods, Math. Models Methods Appl. Sci., 20, 2075-2107, (2010) · Zbl 1226.65091
[85] Casquero, H.; Liu, L.; Zhang, Y.; Reali, A.; Gomez, H., Isogeometric collocation using analysis-suitable T-splines of arbitrary degree, Comput. Methods Appl. Mech. Eng., 301, 164-186, (2016)
[86] Wang, Y.; Jimack, P. K.; Walkley, M. A., A one-field monolithic fictitious domain method for fluid-structure interactions, Comput. Methods Appl. Mech. Eng., 317, 1146-1168, (2017)
[87] Griffith, B. E.; Peskin, C. S., On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys., 208, 1, 75-105, (2005) · Zbl 1115.76386
[88] Yu, Y.; Kamensky, D.; Hsu, M.-C.; Lu, X. Y.; Bazilevs, Y.; Hughes, T. J., Error estimates for dynamic augmented Lagrangian boundary condition enforcement, with application to immersogeometric fluid-structure interaction, (2017), The University of Texas at Austin, ICES REPORT 17-21
[89] Liao, C.-C.; Hsiao, W.-W.; Lin, T.-Y.; Lin, C.-A., Simulations of two sedimenting-interacting spheres with different sizes and initial configurations using immersed boundary method, Comput. Mech., 55, 6, 1191-1200, (2015) · Zbl 1325.70011
[90] Pivkin, I. V.; Peng, Z.; Karniadakis, G. E.; Buffet, P. A.; Dao, M.; Suresh, S., Biomechanics of red blood cells in human spleen and consequences for physiology and disease, Proc. Natl. Acad. Sci. USA, (2016)
[91] Gounley, J.; Draeger, E. W.; Randles, A., Numerical simulation of a compound capsule in a constricted microchannel, Proc. Comput. Sci., 108, 175-184, (2017)
[92] Serrano-Alcalde, F.; García-Aznar, J. M.; Gómez-Benito, M. J., The role of nuclear mechanics in cell deformation under creeping flows, J. Theor. Biol., 432, 25-32, (2017)
[93] Mills, J.; Qie, L.; Dao, M.; Lim, C.; Suresh, S., Nonlinear elastic and viscoelastic deformation of the human red blood cell with optical tweezers, MCB, 1, 169-180, (2004)
[94] Skalak, R.; Tozeren, A.; Zarda, R.; Chien, S., Strain energy function of red blood cell membranes, Biophys. J., 13, 3, 245-264, (1973)
[95] Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Zhang, Y., Isogeometric fluid-structure interaction: theory, algorithms, and computations, Comput. Mech., 43, 3-37, (2008) · Zbl 1169.74015
[96] Bueno, J.; Casquero, H.; Bazilevs, Y.; Gomez, H., Three-dimensional dynamic simulation of elastocapillarity, Meccanica, 53, 6, 1221-1237, (2018) · Zbl 1391.76138
[97] Takizawa, K.; Tezduyar, T. E.; Terahara, T., Ram-air parachute structural and fluid mechanics computations with the space-time isogeometric analysis (ST-IGA), Comput. Fluids, 141, 191-200, (2016) · Zbl 1390.76359
[98] Pozrikidis, C., Modeling and simulation of capsules and biological cells, (2003), CRC Press · Zbl 1026.92002
[99] Zhao, H.; Spann, A. P.; Shaqfeh, E. S., The dynamics of a vesicle in a wall-bound shear flow, Phys. Fluids, 23, 12, (2011)
[100] Sauer, R. A.; Duong, T. X.; Mandadapu, K. K.; Steigmann, D. J., A stabilized finite element formulation for liquid shells and its application to lipid bilayers, J. Comput. Phys., 330, 436-466, (2017) · Zbl 1378.74066
[101] Casquero, H.; Liu, L.; Zhang, Y.; Reali, A.; Kiendl, J.; Gomez, H., Arbitrary-degree T-splines for isogeometric analysis of fully nonlinear Kirchhoff-love shells, Comput. Aided Des., 82, 140-153, (2017)
[102] van Opstal, T. M.; Yan, J.; Coley, C.; Evans, J. A.; Kvamsdal, T.; Bazilevs, Y., Isogeometric divergence-conforming variational multiscale formulation of incompressible turbulent flows, Comput. Methods Appl. Mech. Eng., 316, 859-879, (2017)
[103] Beirao da Veiga, L.; Buffa, A.; Sangalli, G.; Vazquez, R., Analysis suitable T-splines of arbitrary degree: definition, linear independence, and approximation properties, Math. Models Methods Appl. Sci., 23, 11, 1979-2003, (2013) · Zbl 1270.65009
[104] Lorenzo, G.; Scott, M.; Tew, K.; Hughes, T.; Gomez, H., Hierarchically refined and coarsened splines for moving interface problems, with particular application to phase-field models of prostate tumor growth, Comput. Methods Appl. Mech. Eng., 319, 515-548, (2017)
[105] Buffa, A.; Sangalli, G.; Vázquez, R., Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations, J. Comput. Phys., 257, 1291-1320, (2014) · Zbl 1351.78036
[106] Evans, J. A.; Scott, M. A.; Shepherd, K.; Thomas, D.; Vazquez, R., Hierarchical B-spline complexes of discrete differential forms, arXiv preprint
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.