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Element length calculation in B-spline meshes for complex geometries. (English) Zbl 07185496
Summary: Variational multiscale methods, and their precursors, stabilized methods, have been playing a core-method role in semi-discrete and space-time (ST) flow computations for decades. These methods are sometimes supplemented with discontinuity-capturing (DC) methods. The stabilization and DC parameters embedded in most of these methods play a significant role. Various well-performing stabilization and DC parameters have been introduced in both the semi-discrete and ST contexts. The parameters almost always involve some element length expressions, most of the time in specific directions, such as the direction of the flow or solution gradient. Until recently, stabilization and DC parameters originally intended for finite element discretization were being used also for isogeometric discretization. Recently, element lengths and stabilization and DC parameters targeting isogeometric discretization were introduced for ST and semi-discrete computations, and these expressions are also applicable to finite element discretization. The key stages of deriving the direction-dependent element length expression were mapping the direction vector from the physical (ST or space-only) element to the parent element in the parametric space, accounting for the discretization spacing along each of the parametric coordinates, and mapping what has been obtained back to the physical element. Targeting B-spline meshes for complex geometries, we introduce here new element length expressions, which are outcome of a clear and convincing derivation and more suitable for element-level evaluation. The new expressions are based on a preferred parametric space and a transformation tensor that represents the relationship between the integration and preferred parametric spaces. The test computations we present for advection-dominated cases, including 2D computations with complex meshes, show that the proposed element length expressions result in good solution profiles.

MSC:
74 Mechanics of deformable solids
Software:
SUPG
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[109] Suito, H.; Takizawa, K.; Huynh, VQH; 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
[110] Suito, H.; Takizawa, K.; Huynh, VQH; Sze, D.; Ueda, T.; Tezduyar, TE; Bazilevs, Y.; Takizawa, K., A geometrical-characteristics study in patient-specific FSI analysis of blood flow in the thoracic aorta, Advances in computational fluid-structure interaction and flow simulation: new methods and challenging computations, modeling and simulation in science, engineering and technology, 379-386 (2016), New York: Springer, New York · Zbl 1356.76471
[111] Takizawa, K.; Tezduyar, TE; Uchikawa, H.; Terahara, T.; Sasaki, T.; Shiozaki, K.; Yoshida, A.; Komiya, K.; Inoue, G.; Tezduyar, TE, Aorta flow analysis and heart valve flow and structure analysis, Frontiers in computational fluid-*structure interaction and flow simulation: research from lead investigators under forty—2018, modeling and simulation in science, engineering and technology, 29-89 (2018), New York: Springer, New York
[112] Takizawa, K.; Tezduyar, TE; Uchikawa, H.; Terahara, T.; Sasaki, T.; Yoshida, A., Mesh refinement influence and cardiac-cycle flow periodicity in aorta flow analysis with isogeometric discretization, Comput Fluids, 179, 790-798 (2019) · Zbl 1411.76184
[113] Takizawa, K.; Bazilevs, Y.; Tezduyar, TE; Hsu, M-C, Computational cardiovascular flow analysis with the variational multiscale methods, J Adv Eng Comput, 3, 366-405 (2019)
[114] Takizawa, K.; Tezduyar, TE; Buscher, A.; Asada, S., Space-time fluid mechanics computation of heart valve models, Comput Mech, 54, 973-986 (2014) · Zbl 1311.74083
[115] Takizawa, K.; Tezduyar, TE; Bazilevs, Y.; Takizawa, K., New directions in space-time computational methods, Advances in computational fluid-structure interaction and flow simulation: new methods and challenging computations, modeling and simulation in science, engineering and technology, 159-178 (2016), New York: Springer, New York · Zbl 1356.76291
[116] Takizawa K, Tezduyar TE, Terahara T, Sasaki T (2018) Heart valve flow computation with the space-time slip interface topology change (ST-SI-TC) method and isogeometric analysis (IGA). In: Wriggers P, Lenarz T (eds) Biomedical technology: modeling, experiments and simulation. Lecture notes in applied and computational mechanics. Springer, New York, pp 77-99. 10.1007/978-3-319-59548-1_6 ISBN: 978-3-319-59547-4
[117] Takizawa, K.; Tezduyar, TE; Terahara, T.; Sasaki, T., Heart valve flow computation with the integrated space-time VMS, slip interface, topology change and isogeometric discretization methods, Comput Fluids, 158, 176-188 (2017) · Zbl 1390.76944
[118] Yu Y, Zhang YJ, Takizawa K, Tezduyar TE, Sasaki T (October 2019) Anatomically realistic lumen motion representation in patient-specific space-time isogeometric flow analysis of coronary arteries with time-dependent medical-image data. Comput Mech. published online. 10.1007/s00466-019-01774-4
[119] Takizawa, K.; Montes, D.; McIntyre, S.; Tezduyar, TE, 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
[120] Takizawa, K.; Tezduyar, TE; Kuraishi, T.; Tabata, S.; Takagi, H., Computational thermo-fluid analysis of a disk brake, Comput Mech, 57, 965-977 (2016) · Zbl 1382.74044
[121] Takizawa, K.; Tezduyar, TE; Hattori, H., Computational analysis of flow-driven string dynamics in turbomachinery, Comput Fluids, 142, 109-117 (2017) · Zbl 1390.76011
[122] Komiya K, Kanai T, Otoguro Y, Kaneko M, Hirota K, Zhang Y, Takizawa K, Tezduyar TE, Nohmi M, Tsuneda T, Kawai M, Isono M (2019) Computational analysis of flow-driven string dynamics in a pump and residence time calculation. In: IOP conference series earth and environmental science, vol 240. 10.1088/1755-1315/240/6/062014 · Zbl 1425.76139
[123] Kanai, T.; Takizawa, K.; Tezduyar, TE; Komiya, K.; Kaneko, M.; Hirota, K.; Nohmi, M.; Tsuneda, T.; Kawai, M.; Isono, M., Methods for computation of flow-driven string dynamics in a pump and residence time, Math Models Methods Appl Sci, 29, 839-870 (2019) · Zbl 1425.76139
[124] Takizawa, K.; Tezduyar, TE; Otoguro, Y.; Terahara, T.; Kuraishi, T.; Hattori, H., Turbocharger flow computations with the space-time isogeometric analysis (ST-IGA), Comput Fluids, 142, 15-20 (2017) · Zbl 1390.76689
[125] Otoguro, Y.; Takizawa, K.; Tezduyar, TE, Space-time VMS computational flow analysis with isogeometric discretization and a general-purpose NURBS mesh generation method, Comput Fluids, 158, 189-200 (2017) · Zbl 1390.76345
[126] Otoguro Y, Takizawa K, Tezduyar TE (2018) A general-purpose NURBS mesh generation method for complex geometries. In: Tezduyar TE (ed) Frontiers in computational fluid-structure interaction and flow simulation: research from lead investigators under forty—2018, Modeling and simulation in science, engineering and technology. Springer, New York, pp 399-434. 10.1007/978-3-319-96469-0_10. ISBN: 978- 3-319-96468-3
[127] Otoguro, Y.; Takizawa, K.; Tezduyar, TE; Nagaoka, K.; Mei, S., Turbocharger turbine and exhaust manifold flow computation with the space-time variational multiscale method and isogeometric analysis, Comput Fluids, 179, 764-776 (2019) · Zbl 1411.76070
[128] Otoguro, Y.; Takizawa, K.; Tezduyar, TE; Nagaoka, K.; Avsar, R.; Zhang, Y., Space-time VMS flow analysis of a turbocharger turbine with isogeometric discretization: computations with time-dependent and steady-inflow representations of the intake/exhaust cycle, Comput Mech, 64, 1403-1419 (2019) · Zbl 07147411
[129] Takizawa, K.; Tezduyar, TE; Asada, S.; Kuraishi, T., Space-time method for flow computations with slip interfaces and topology changes (ST-SI-TC), Comput Fluids, 141, 124-134 (2016) · Zbl 1390.76358
[130] Kuraishi, T.; Takizawa, K.; Tezduyar, TE; Tezduyar, TE, Space-time computational analysis of tire aerodynamics with actual geometry, road contact and tire deformation, Frontiers in computational fluid-structure interaction and flow simulation: research from lead investigators under forty—2018, modeling and simulation in science, engineering and technology, 337-376 (2018), New York: Springer, New York
[131] Kuraishi, T.; Takizawa, K.; Tezduyar, TE, Tire aerodynamics with actual tire geometry, road contact and tire deformation, Comput Mech, 63, 1165-1185 (2019) · Zbl 07053716
[132] Kuraishi, T.; Takizawa, K.; Tezduyar, TE, Space-time computational analysis of tire aerodynamics with actual geometry, road contact, tire deformation, road roughness and fluid film, Comput Mech, 64, 1699-1718 (2019) · Zbl 07147427
[133] Kuraishi, T.; Takizawa, K.; Tezduyar, TE, Space-time isogeometric flow analysis with built-in Reynolds-equation limit, Math Models Methods Appl Sci, 29, 871-904 (2019) · Zbl 1425.76142
[134] Takizawa, K.; Tezduyar, TE; 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
[135] Takizawa, K.; Tezduyar, TE; Kanai, T., Porosity models and computational methods for compressible-flow aerodynamics of parachutes with geometric porosity, Math Models Methods Appl Sci, 27, 771-806 (2017) · Zbl 1361.76017
[136] Kanai, T.; Takizawa, K.; Tezduyar, TE; Tanaka, T.; Hartmann, A., Compressible-flow geometric-porosity modeling and spacecraft parachute computation with isogeometric discretization, Comput Mech, 63, 301-321 (2019) · Zbl 07037442
[137] Tezduyar, TE; Aliabadi, SK; Behr, M.; Mittal, S., Massively parallel finite element simulation of compressible and incompressible flows, Comput Methods Appl Mech Eng, 119, 157-177 (1994) · Zbl 0848.76040
[138] Hughes, TJR; Cottrell, JA; 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
[139] Takizawa, K.; Tezduyar, TE, Space-time computation techniques with continuous representation in time (ST-C), Comput Mech, 53, 91-99 (2014)
[140] Takizawa, K.; Takagi, H.; Tezduyar, TE; Torii, R., Estimation of element-based zero-stress state for arterial FSI computations, Comput Mech, 54, 895-910 (2014) · Zbl 1398.74096
[141] Takizawa, K.; Torii, R.; Takagi, H.; Tezduyar, TE; Xu, XY, Coronary arterial dynamics computation with medical-image-based time-dependent anatomical models and element-based zero-stress state estimates, Comput Mech, 54, 1047-1053 (2014) · Zbl 1311.76158
[142] Takizawa K, Tezduyar TE, Sasaki T (2018) Estimation of element-based zero-stress state in arterial FSI computations with isogeometric wall discretization. In: Wriggers P, Lenarz T (eds) Biomedical technology: modeling, experiments and simulation. Lecture notes in applied and computational mechanics. Springer, New York, pp 101-122. 10.1007/978-3-319-59548-1_7. ISBN: 978-3-319-59547-4
[143] Takizawa, K.; Tezduyar, TE; Sasaki, T., Aorta modeling with the element-based zero-stress state and isogeometric discretization, Comput Mech, 59, 265-280 (2017)
[144] Sasaki, T.; Takizawa, K.; Tezduyar, TE, Aorta zero-stress state modeling with T-spline discretization, Comput Mech, 63, 1315-1331 (2019) · Zbl 07053724
[145] Sasaki, T.; Takizawa, K.; Tezduyar, TE, Medical-image-based aorta modeling with zero-stress-state estimation, Comput Mech, 64, 249-271 (2019) · Zbl 07073977
[146] Takizawa, K.; Tezduyar, TE; Sasaki, T., Isogeometric hyperelastic shell analysis with out-of-plane deformation mapping, Comput Mech, 63, 681-700 (2019) · Zbl 07053688
[147] Akin, JE; Tezduyar, T.; Ungor, M.; Mittal, S., Stabilization parameters and Smagorinsky turbulence model, J Appl Mech, 70, 2-9 (2003) · Zbl 1110.74311
[148] Akin, JE; Tezduyar, TE, Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements, Comput Methods Appl Mech Eng, 193, 1909-1922 (2004) · Zbl 1067.76557
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[150] Tezduyar, TE; Osawa, Y., Finite element stabilization parameters computed from element matrices and vectors, Comput Methods Appl Mech Eng, 190, 411-430 (2000) · Zbl 0973.76057
[151] Hsu, M-C; Bazilevs, Y.; Calo, VM; Tezduyar, TE; Hughes, TJR, 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
[152] Tezduyar TE (2001) Adaptive determination of the finite element stabilization parameters. In: Proceedings of the ECCOMAS computational fluid dynamics conference 2001. CD-ROM), Swansea, Wales
[153] Tezduyar TE (2004) Finite element methods for fluid dynamics with moving boundaries and interfaces, In: Stein E, Borst RD, Hughes TJR (eds) Encyclopedia of computational mechanics, volume 3: fluids, Chapter 17, Wiley, New York. 10.1002/0470091355.ecm069. ISBN: 978-0-470-84699-5 · Zbl 1130.76369
[154] Tezduyar TE (2004) Determination of the stabilization and shock-capturing parameters in SUPG formulation of compressible flows, In: Proceedings of the European congress on computational methods in applied sciences and engineering, ECCOMAS 2004. (CD-ROM), Jyvaskyla
[155] Tezduyar, TE, Finite elements in fluids: stabilized formulations and moving boundaries and interfaces, Comput Fluids, 36, 191-206 (2007) · Zbl 1177.76202
[156] Rispoli, F.; Corsini, A.; Tezduyar, TE, Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD), Comput Fluids, 36, 121-126 (2007) · Zbl 1181.76098
[157] Tezduyar, TE; 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
[158] Tezduyar, TE; Senga, M., SUPG finite element computation of inviscid supersonic flows with YZ \(\beta\) shock-capturing, Comput Fluids, 36, 147-159 (2007) · Zbl 1127.76029
[159] Tezduyar, TE; Senga, M.; Vicker, D., Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZ \(\beta\) shock-capturing, Comput Mech, 38, 469-481 (2006) · Zbl 1176.76077
[160] Corsini, A.; Menichini, C.; Rispoli, F.; Santoriello, A.; Tezduyar, TE, A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reactionlike terms, J Appl Mech, 76, 021211 (2009)
[161] Rispoli, F.; Saavedra, R.; Menichini, F.; Tezduyar, TE, Computation of inviscid supersonic flows around cylinders and spheres with the V-SGS stabilization and YZ \(\beta\) shock-capturing, J Appl Mech, 76, 021209 (2009)
[162] Corsini, A.; Iossa, C.; Rispoli, F.; Tezduyar, TE, A DRD finite element formulation for computing turbulent reacting flows in gas turbine combustors, Comput Mech, 46, 159-167 (2010) · Zbl 1301.76045
[163] Corsini, A.; Rispoli, F.; Tezduyar, TE, Stabilized finite element computation of NOx emission in aero-engine combustors, Int J Numer Methods Fluids, 65, 254-270 (2011) · Zbl 1426.76240
[164] Corsini, A.; Rispoli, F.; Tezduyar, TE, Computer modeling of wave-energy air turbines with the SUPG/PSPG formulation and discontinuity-capturing technique, J Appl Mech, 79, 010910 (2012)
[165] Corsini, A.; Rispoli, F.; Sheard, AG; Tezduyar, TE, Computational analysis of noise reduction devices in axial fans with stabilized finite element formulations, Comput Mech, 50, 695-705 (2012) · Zbl 1311.76121
[166] Kler, PA; Dalcin, LD; Paz, RR; Tezduyar, TE, SUPG and discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems, Comput Mech, 51, 171-185 (2013) · Zbl 1312.76062
[167] Corsini, A.; Rispoli, F.; Sheard, AG; Takizawa, K.; Tezduyar, TE; Venturini, P., A variational multiscale method for particle-cloud tracking in turbomachinery flows, Comput Mech, 54, 1191-1202 (2014) · Zbl 1311.76030
[168] Rispoli, F.; Delibra, G.; Venturini, P.; Corsini, A.; Saavedra, R.; Tezduyar, TE, Particle tracking and particle-shock interaction in compressible-flow computations with the V-SGS stabilization and YZ \(\beta\) shock-capturing, Comput Mech, 55, 1201-1209 (2015) · Zbl 1325.76121
[169] Cardillo, L.; Corsini, A.; Delibra, G.; Rispoli, F.; Tezduyar, TE, Flow analysis of a wave-energy air turbine with the SUPG/PSPG stabilization and discontinuity-capturing directional dissipation, Comput Fluids, 141, 184-190 (2016) · Zbl 1390.76295
[170] Castorrini, A.; Corsini, A.; Rispoli, F.; Venturini, P.; Takizawa, K.; Tezduyar, TE, Computational analysis of wind-turbine blade rain erosion, Comput Fluids, 141, 175-183 (2016) · Zbl 1390.76298
[171] Castorrini, A.; Corsini, A.; Rispoli, F.; Venturini, P.; Takizawa, K.; Tezduyar, TE, Computational analysis of performance deterioration of a wind turbine blade strip subjected to environmental erosion, Comput Mech, 64, 1133-1153 (2019) · Zbl 07119155
[172] Takizawa, K.; Tezduyar, TE; Otoguro, Y., Stabilization and discontinuity-capturing parameters for space-time flow computations with finite element and isogeometric discretizations, Comput Mech, 62, 1169-1186 (2018) · Zbl 06981055
[173] Takizawa K, Ueda Y, Tezduyar TE (2019) A node-numbering-invariant directional length scale for simplex elements. Math Models Methods Appl Sci. November 2019, published online. 10.1142/S0218202519500581
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