×

zbMATH — the first resource for mathematics

Computational analysis of flow-driven string dynamics in turbomachinery. (English) Zbl 1390.76011
Summary: We focus on computational analysis of flow-driven string dynamics. The objective is to understand how the strings carried by a fluid interact with the solid surfaces present and get stuck on or around those surfaces. Our target application is turbomachinery, such as understanding how strings get stuck on or around the blades of a fan. The components of the method we developed for this purpose are the space-time variational multiscale (ST-VMS) and ST slip interface (ST-SI) methods for the fluid dynamics, and a one-way-dependence model and the isogeometric analysis (IGA) for the string dynamics. The ST-VMS method is the core computational technology and it also has the features of a turbulence model. The ST-SI method allows in a consistent fashion slip at the interface between the mesh covering a spinning solid surface and the mesh covering the rest of the domain, and with this, we maintain high-resolution representation of the boundary layers near spinning solid surfaces such as fan blades. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow, which we expect to be small. The IGA for the string dynamics gives us not only a higher-order method and smoothness in the structure shape, but also smoothness in the fluid dynamics forces calculated on the string. To demonstrate how the method can be used in computational analysis of flow-driven string dynamics, we present the pilot computations we carried out, for a duct with cylindrical obstacles and for a ventilating fan.

MSC:
76-04 Software, source code, etc. for problems pertaining to fluid mechanics
76T20 Suspensions
Software:
SUPG
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Tezduyar, T. E.; Takizawa, K.; Moorman, C.; Wright, S.; Christopher, J., Space-time finite element computation of complex fluid-structure interactions, Int J Numer Methods Fluids, 64, 1201-1218, (2010) · Zbl 1427.76148
[2] Takizawa, K.; Tezduyar, T. E., Computational methods for parachute fluid-structure interactions, Arch Comput Methods Eng, 19, 125-169, (2012) · Zbl 1354.76113
[3] Bazilevs, Y.; Takizawa, K.; Tezduyar, T. E., Computational fluid-structure interaction: methods and applications, (2013), Wiley, 978-0470978771
[4] Corsini, A.; Rispoli, F.; Sheard, A. G.; Takizawa, K.; Tezduyar, T. E.; Venturini, P., A variational multiscale method for particle-cloud tracking in turbomachinery flows, Comput Mech, 54, 1191-1202, (2014) · Zbl 1311.76030
[5] Rispoli, F.; Delibra, G.; Venturini, P.; Corsini, A.; Saavedra, R.; Tezduyar, T. E., Particle tracking and particle-shock interaction in compressible-flow computations with the V-SGS stabilization and YZβ shock-capturing, Comput Mech, 55, 1201-1209, (2015) · Zbl 1325.76121
[6] Takizawa, K.; Tezduyar, T. E., Multiscale space-time fluid-structure interaction techniques, Comput Mech, 48, 247-267, (2011) · Zbl 1398.76128
[7] Takizawa, K.; Tezduyar, T. E., Space-time fluid-structure interaction methods, Math Models Methods Appl Sci, 22, supp02, 1230001, (2012) · Zbl 1248.76118
[8] Tezduyar, T. E., Stabilized finite element formulations for incompressible flow computations, Adv Appl Mech, 28, 1-44, (1992) · Zbl 0747.76069
[9] Tezduyar, T. E., Computation of moving boundaries and interfaces and stabilization parameters, Int J Numer Methods Fluids, 43, 555-575, (2003) · Zbl 1032.76605
[10] Tezduyar, T. E.; Sathe, S., Modeling of fluid-structure interactions with the space-time finite elements: solution techniques, Int J Numer Methods Fluids, 54, 855-900, (2007) · Zbl 1144.74044
[11] Hughes, T. J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles, and the origins of stabilized methods, Comput Methods Appl Mech Eng, 127, 387-401, (1995) · Zbl 0866.76044
[12] Hughes, T. J.R.; Oberai, A. A.; Mazzei, L., Large eddy simulation of turbulent channel flows by the variational multiscale method, Phys Fluids, 13, 1784-1799, (2001) · Zbl 1184.76237
[13] 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
[14] Bazilevs, Y.; Akkerman, I., Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method, J Comput Phys, 229, 3402-3414, (2010) · Zbl 1290.76037
[15] Takizawa, K.; Bazilevs, Y.; Tezduyar, T. E., Space-time and ALE-VMS techniques for patient-specific cardiovascular fluid-structure interaction modeling, Arch Comput Methods Eng, 19, 171-225, (2012) · Zbl 1354.92023
[16] Bazilevs, Y.; Hsu, M.-C.; Takizawa, K.; Tezduyar, T. E., ALE-VMS and ST-VMS methods for computer modeling of wind-turbine rotor aerodynamics and fluid-structure interaction, Math Models Methods Appl Sci, 22, supp02, 1230002, (2012) · Zbl 1404.76187
[17] Hughes, T. J.R.; Liu, W. K.; Zimmermann, T. K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput Methods Appl Mech Eng, 29, 329-349, (1981) · Zbl 0482.76039
[18] 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
[19] 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
[20] Bazilevs, Y.; Calo, V. M.; Zhang, Y.; Hughes, T. J.R., Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Comput Mech, 38, 310-322, (2006) · Zbl 1161.74020
[21] Bazilevs, Y.; Hughes, T. J.R., NURBS-based isogeometric analysis for the computation of flows about rotating components, Comput Mech, 43, 143-150, (2008) · Zbl 1171.76043
[22] Bazilevs, Y.; Gohean, J. R.; Hughes, T. J.R.; Moser, R. D.; Zhang, Y., Patient-specific isogeometric fluid-structure interaction analysis of thoracic aortic blood flow due to implantation of the jarvik 2000 left ventricular assist device, Comput Methods Appl Mech Eng, 198, 3534-3550, (2009) · Zbl 1229.74096
[23] Bazilevs, Y.; Hsu, M.-C.; Benson, D.; Sankaran, S.; Marsden, A., Computational fluid-structure interaction: methods and application to a total cavopulmonary connection, Comput Mech, 45, 77-89, (2009) · Zbl 1398.92056
[24] Bazilevs, Y.; Hsu, M.-C.; Akkerman, I.; Wright, S.; Takizawa, K.; Henicke, B., 3D simulation of wind turbine rotors at full scale. part I: geometry modeling and aerodynamics, Int J Numer Methods Fluids, 65, 207-235, (2011) · Zbl 1428.76086
[25] Bazilevs, Y.; Hsu, M.-C.; Kiendl, J.; Wüchner, R.; Bletzinger, K.-U., 3D simulation of wind turbine rotors at full scale. part II: fluid-structure interaction modeling with composite blades, Int J Numer Methods Fluids, 65, 236-253, (2011) · Zbl 1428.76087
[26] Hsu, M.-C.; Akkerman, I.; Bazilevs, Y., Wind turbine aerodynamics using ALE-VMS: validation and role of weakly enforced boundary conditions, Comput Mech, 50, 499-511, (2012) · Zbl 06128533
[27] Hsu, M.-C.; Bazilevs, Y., Fluid-structure interaction modeling of wind turbines: simulating the full machine, Comput Mech, 50, 821-833, (2012) · Zbl 1311.74038
[28] Bazilevs, Y.; Takizawa, K.; Tezduyar, T. E., Challenges and directions in computational fluid-structure interaction, Math Models Methods Appl Sci, 23, 215-221, (2013) · Zbl 1261.76025
[29] Korobenko, A.; Hsu, M.-C.; Akkerman, I.; Tippmann, J.; Bazilevs, Y., Structural mechanics modeling and FSI simulation of wind turbines, Math Models Methods Appl Sci, 23, 249-272, (2013) · Zbl 1261.74011
[30] Korobenko, A.; Hsu, M.-C.; Akkerman, I.; Bazilevs, Y., Aerodynamic simulation of vertical-axis wind turbines, J Appl Mech, 81, 021011, (2013)
[31] Bazilevs, Y.; Takizawa, K.; Tezduyar, T. E.; Hsu, M.-C.; Kostov, N.; McIntyre, S., Aerodynamic and FSI analysis of wind turbines with the ALE-VMS and ST-VMS methods, Arch Comput Methods Eng, 21, 359-398, (2014) · Zbl 1348.74095
[32] Bazilevs, Y.; Korobenko, A.; Deng, X.; Yan, J.; Kinzel, M.; Dabiri, J. O., FSI modeling of vertical-axis wind turbines, J Appl Mech, 81, 081006, (2014)
[33] Hsu, M.-C.; Akkerman, I.; Bazilevs, Y., Finite element simulation of wind turbine aerodynamics: validation study using NREL phase VI experiment, Wind Energy, 17, 461-481, (2014)
[34] Long, C. C.; Esmaily-Moghadam, M.; Marsden, A. L.; Bazilevs, Y., Computation of residence time in the simulation of pulsatile ventricular assist devices, Comput Mech, 54, 911-919, (2014) · Zbl 1311.74041
[35] Long, C. C.; Marsden, A. L.; Bazilevs, Y., Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk, Comput Mech, 54, 921-932, (2014) · Zbl 1314.74056
[36] Hsu, M.-C.; Kamensky, D.; Bazilevs, Y.; Sacks, M. S.; Hughes, T. J.R., Fluid-structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation, Comput Mech, 54, 1055-1071, (2014) · Zbl 1311.74039
[37] Augier, B.; Yan, J.; Korobenko, A.; Czarnowski, J.; Ketterman, G.; Bazilevs, Y., Experimental and numerical FSI study of compliant hydrofoils, Comput Mech, 55, 1079-1090, (2015)
[38] Bazilevs, Y.; Korobenko, A.; Yan, J.; Pal, A.; Gohari, S. M.I.; Sarkar, S., ALE-VMS formulation for stratified turbulent incompressible flows with applications, Math Models Methods Appl Sci, 25, 2349-2375, (2015) · Zbl 1329.76050
[39] Bazilevs, Y.; Takizawa, K.; Tezduyar, T. E., New directions and challenging computations in fluid dynamics modeling with stabilized and multiscale methods, Math Models Methods Appl Sci, 25, 2217-2226, (2015) · Zbl 1329.76007
[40] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 32, 199-259, (1982) · Zbl 0497.76041
[41] Takizawa, K.; Henicke, B.; Tezduyar, T. E.; Hsu, M.-C.; Bazilevs, Y., Stabilized space-time computation of wind-turbine rotor aerodynamics, Comput Mech, 48, 333-344, (2011) · Zbl 1398.76127
[42] Takizawa, K.; Henicke, B.; Montes, D.; Tezduyar, T. E.; Hsu, M.-C.; Bazilevs, Y., Numerical-performance studies for the stabilized space-time computation of wind-turbine rotor aerodynamics, Comput Mech, 48, 647-657, (2011) · Zbl 1334.74032
[43] Takizawa, K.; Tezduyar, T. E.; McIntyre, S.; Kostov, N.; Kolesar, R.; Habluetzel, C., Space-time VMS computation of wind-turbine rotor and tower aerodynamics, Comput Mech, 53, 1-15, (2014) · Zbl 1398.76129
[44] Takizawa, K.; Bazilevs, Y.; Tezduyar, T. E.; Hsu, M.-C.; Øiseth, O.; Mathisen, K. M., Engineering analysis and design with ALE-VMS and space-time methods, Arch Comput Methods Eng, 21, 481-508, (2014) · Zbl 1348.74104
[45] Takizawa, K., Computational engineering analysis with the new-generation space-time methods, Comput Mech, 54, 193-211, (2014) · Zbl 06327161
[46] Takizawa, K.; Tezduyar, T. E.; Mochizuki, H.; Hattori, H.; Mei, S.; Pan, L., Space-time VMS method for flow computations with slip interfaces (ST-SI), Math Models Methods Appl Sci, 25, 2377-2406, (2015) · Zbl 1329.76345
[47] Takizawa, K.; Henicke, B.; Puntel, A.; Spielman, T.; Tezduyar, T. E., Space-time computational techniques for the aerodynamics of flapping wings, J Appl Mech, 79, 010903, (2012)
[48] Takizawa, K.; Henicke, B.; Puntel, A.; Kostov, N.; Tezduyar, T. E., Space-time techniques for computational aerodynamics modeling of flapping wings of an actual locust, Comput Mech, 50, 743-760, (2012) · Zbl 1286.76179
[49] Takizawa, K.; Kostov, N.; Puntel, A.; Henicke, B.; Tezduyar, T. E., Space-time computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle, Comput Mech, 50, 761-778, (2012) · Zbl 1286.76180
[50] Takizawa, K.; Henicke, B.; Puntel, A.; Kostov, N.; Tezduyar, T. E., Computer modeling techniques for flapping-wing aerodynamics of a locust, Comput Fluids, 85, 125-134, (2013) · Zbl 1290.76170
[51] Takizawa, K.; Tezduyar, T. E.; Buscher, A.; Asada, S., Space-time interface-tracking with topology change (ST-TC), Comput Mech, 54, 955-971, (2014) · Zbl 1311.74045
[52] Takizawa, K.; Tezduyar, T. E.; Kostov, N., Sequentially-coupled space-time FSI analysis of bio-inspired flapping-wing aerodynamics of an MAV, Comput Mech, 54, 213-233, (2014) · Zbl 06327162
[53] Takizawa, K.; Tezduyar, T. E.; Buscher, A., Space-time computational analysis of MAV flapping-wing aerodynamics with wing clapping, Comput Mech, 55, 1131-1141, (2015)
[54] Takizawa, K.; Schjodt, K.; Puntel, A.; Kostov, N.; Tezduyar, T. E., Patient-specific computer modeling of blood flow in cerebral arteries with aneurysm and stent, Comput Mech, 50, 675-686, (2012) · Zbl 1311.76157
[55] Takizawa, K.; Schjodt, K.; Puntel, A.; Kostov, N.; Tezduyar, T. E., Patient-specific computational analysis of the influence of a stent on the unsteady flow in cerebral aneurysms, Comput Mech, 51, 1061-1073, (2013) · Zbl 1366.76106
[56] Takizawa, K.; Bazilevs, Y.; Tezduyar, T. E.; Long, C. C.; Marsden, A. L.; Schjodt, K., ST and ALE-VMS methods for patient-specific cardiovascular fluid mechanics modeling, Math Models Methods Appl Sci, 24, 2437-2486, (2014) · Zbl 1296.76113
[57] Suito, H.; Takizawa, K.; Huynh, V. Q.H.; Sze, D.; Ueda, T., FSI analysis of the blood flow and geometrical characteristics in the thoracic aorta, Comput Mech, 54, 1035-1045, (2014) · Zbl 1311.74044
[58] Takizawa, K.; Tezduyar, T. E.; Buscher, A.; Asada, S., Space-time fluid mechanics computation of heart valve models, Comput Mech, 54, 973-986, (2014) · Zbl 1311.74083
[59] Takizawa, K.; Montes, D.; Fritze, M.; McIntyre, S.; Boben, J.; Tezduyar, T. E., Methods for FSI modeling of spacecraft parachute dynamics and cover separation, Math Models Methods Appl Sci, 23, 307-338, (2013) · Zbl 1261.76013
[60] Takizawa, K.; Montes, D.; McIntyre, S.; Tezduyar, T. E., Space-time VMS methods for modeling of incompressible flows at high Reynolds numbers, Math Models Methods Appl Sci, 23, 223-248, (2013) · Zbl 1261.76037
[61] Takizawa, K.; Tezduyar, T. E.; Kuraishi, T., Multiscale ST methods for thermo-fluid analysis of a ground vehicle and its tires, Math Models Methods Appl Sci, 25, 2227-2255, (2015) · Zbl 1325.76139
[62] Otoguro, Y.; Terahara, T.; Takizawa, K.; Tezduyar, T. E.; Kuraishi, T.; Hattori, H., A higher-order ST-VMS method for turbocharger analysis, Proceedings of the 13th Asian international conference on fluid machinery, paper no. AICFM13-153; Tokyo, Japan, (2015)
[63] Takizawa, K.; Tezduyar, T. E., Space-time computation techniques with continuous representation in time (ST-C), Comput Mech, 53, 91-99, (2014) · Zbl 06327072
[64] Kalro, V.; Tezduyar, T. E., Parallel finite element computation of 3D incompressible flows on mpps, (Habashi, W. G., Solution techniques for large-scale CFD problems, (1995), John Wiley & Sons)
[65] Tezduyar, T.; Aliabadi, S.; Behr, M.; Johnson, A.; Kalro, V.; Litke, M., Flow simulation and high performance computing, Comput Mech, 18, 397-412, (1996) · Zbl 0893.76046
[66] Behr, M.; Tezduyar, T., The shear-slip mesh update method, Comput Methods Appl Mech Eng, 174, 261-274, (1999) · Zbl 0959.76037
[67] Bazilevs, Y.; Korobenko, A.; Deng, X.; Yan, J., Novel structural modeling and mesh moving techniques for advanced FSI simulation of wind turbines, Int J Numer Methods Fluids, 102, 766-783, (2015) · Zbl 1352.76033
[68] Raknes, S. B.; Deng, X.; Bazilevs, Y.; Benson, D. J.; Mathisen, K. M.; Kvamsdal, T., Isogeometric rotation-free bending-stabilized cables: statics, dynamics, bending strips and coupling with shells, Comput Methods Appl Mech Eng, 263, 127-143, (2013) · Zbl 1286.74119
[69] Tezduyar, T. E.; Ganjoo, D. K., Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: applications to transient convection-diffusion problems, Comput Methods Appl Mech Eng, 59, 49-71, (1986) · Zbl 0604.76077
[70] Le Beau, G. J.; Ray, S. E.; Aliabadi, S. K.; Tezduyar, T. E., SUPG finite element computation of compressible flows with the entropy and conservation variables formulations, Comput Methods Appl Mech Eng, 104, 397-422, (1993) · Zbl 0772.76037
[71] Tezduyar, T. E., Finite elements in fluids: stabilized formulations and moving boundaries and interfaces, Comput Fluids, 36, 191-206, (2007) · Zbl 1177.76202
[72] Tezduyar, T. E.; Senga, M., Stabilization and shock-capturing parameters in SUPG formulation of compressible flows, Comput Methods Appl Mech Eng, 195, 1621-1632, (2006) · Zbl 1122.76061
[73] Tezduyar, T. E.; Senga, M., SUPG finite element computation of inviscid supersonic flows with YZβ shock-capturing, Comput Fluids, 36, 147-159, (2007) · Zbl 1127.76029
[74] Tezduyar, T. E.; Senga, M.; Vicker, D., Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZβ shock-capturing, Comput Mech, 38, 469-481, (2006) · Zbl 1176.76077
[75] Tezduyar, T. E.; Sathe, S., Enhanced-discretization selective stabilization procedure (EDSSP), Comput Mech, 38, 456-468, (2006) · Zbl 1187.76712
[76] Corsini, A.; Rispoli, F.; Santoriello, A.; Tezduyar, T. E., Improved discontinuity-capturing finite element techniques for reaction effects in turbulence computation, Comput Mech, 38, 356-364, (2006) · Zbl 1177.76192
[77] Rispoli, F.; Corsini, A.; Tezduyar, T. E., Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD), Comput Fluids, 36, 121-126, (2007) · Zbl 1181.76098
[78] Tezduyar, T. E.; Ramakrishnan, S.; Sathe, S., Stabilized formulations for incompressible flows with thermal coupling, Int J Numer Methods Fluids, 57, 1189-1209, (2008) · Zbl 1140.76024
[79] Rispoli, F.; Saavedra, R.; Corsini, A.; Tezduyar, T. E., Computation of inviscid compressible flows with the V-SGS stabilization and YZβ shock-capturing, Int J Numer Methods Fluids, 54, 695-706, (2007) · Zbl 1207.76104
[80] Bazilevs, Y.; Calo, V. M.; Tezduyar, T. E.; Hughes, T. J.R., YZβ discontinuity-capturing for advection-dominated processes with application to arterial drug delivery, Int J Numer Methods Fluids, 54, 593-608, (2007) · Zbl 1207.76049
[81] Corsini, A.; Menichini, C.; Rispoli, F.; Santoriello, A.; Tezduyar, T. E., A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reactionlike terms, J Appl Mech, 76, 021211, (2009)
[82] Rispoli, F.; Saavedra, R.; Menichini, F.; Tezduyar, T. E., Computation of inviscid supersonic flows around cylinders and spheres with the V-SGS stabilization and YZβ shock-capturing, J Appl Mech, 76, 021209, (2009)
[83] Corsini, A.; Iossa, C.; Rispoli, F.; Tezduyar, T. E., A DRD finite element formulation for computing turbulent reacting flows in gas turbine combustors, Comput Mech, 46, 159-167, (2010) · Zbl 1301.76045
[84] Hsu, M.-C.; Bazilevs, Y.; Calo, V. M.; Tezduyar, T. E.; Hughes, T. J.R., Improving stability of stabilized and multiscale formulations in flow simulations at small time steps, Comput Methods Appl Mech Eng, 199, 828-840, (2010) · Zbl 1406.76028
[85] Corsini, A.; Rispoli, F.; Tezduyar, T. E., Stabilized finite element computation of nox emission in aero-engine combustors, Int J Numer Methods Fluids, 65, 254-270, (2011) · Zbl 1426.76240
[86] Corsini, A.; Rispoli, F.; Tezduyar, T. E., Computer modeling of wave-energy air turbines with the SUPG/PSPG formulation and discontinuity-capturing technique, J Appl Mech, 79, 010910, (2012)
[87] Corsini, A.; Rispoli, F.; Sheard, A. G.; Tezduyar, T. E., Computational analysis of noise reduction devices in axial fans with stabilized finite element formulations, Comput Mech, 50, 695-705, (2012) · Zbl 1311.76121
[88] Kler, P. A.; Dalcin, L. D.; Paz, R. R.; Tezduyar, T. E., SUPG and discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems, Comput Mech, 51, 171-185, (2013) · Zbl 1312.76062
[89] Takizawa, K.; Spielman, T.; Tezduyar, T. E., Space-time FSI modeling and dynamical analysis of spacecraft parachutes and parachute clusters, Comput Mech, 48, 345-364, (2011) · Zbl 1398.74095
[90] Sathe, S.; Tezduyar, T. E., Modeling of fluid-structure interactions with the space-time finite elements: contact problems, Comput Mech, 43, 51-60, (2008) · Zbl 1297.74129
[91] Kudo, H.; Kawahara, T.; Kanai, H.; Miyagawa, K.; Saito, S.; Isono, M., Study on clogging mechanism of fibrous materials in a pump by experimental and computational approaches, Design and optimization of hydraulic machines Montreal, Canada, vol. 22, 012011, (2014)
[92] Saad, Y.; Schultz, M., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J Sci Stat Comput, 7, 856-869, (1986) · Zbl 0599.65018
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.