zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. (English) Zbl 1151.74419
Summary: The concept of isogeometric analysis is proposed. Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element $h$- and $p$-refinement schemes are presented and a new, more efficient, higher-order concept, $k$-refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. A $k$-refinement strategy is shown to converge toward monotone solutions for advection--diffusion processes with sharp internal and boundary layers, a very surprising result. It is argued that isogeometric analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.

MSC:
74S05Finite element methods in solid mechanics
74S30Other numerical methods in solid mechanics
76M10Finite element methods (fluid mechanics)
76M25Other numerical methods (fluid mechanics)
65N50Mesh generation and refinement (BVP of PDE)
Software:
CUBIT; D-NURBS
WorldCat.org
Full Text: DOI
References:
[1] Rogers, D. F.: An introduction to NURBS with historical perspective. (2001)
[2] T.J. Barth, Simplified numerical methods for gas dynamics systems on triangulated domains, Ph.D. Thesis, Department of Aeronautics and Astronautics, Stanford University (1998).
[3] M. McMullen, A. Jameson, J.J. Alonso, Application of a non-linear frequency domain solver to the Euler and Navier-Stokes equations, 40th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2002-0120, Reno, NV, 2002.
[4] M. Gee, W.A. Wall, E. Ramm, Parallel multilevel solutions of nonlinear shell structures, Comput. Methods Appl. Mech. Engrg. (accepted). · Zbl 1097.74057
[5] G.M. Stanley, Continuum-based shell elements, Ph.D. Thesis, Division of Applied Mechanics, Stanford University, 1985.
[6] Barth, T. J.: Numerical methods for gasdynamic systems on unstructured meshes. Lecture notes in computational science and engineering 5, 195-285 (1998)
[7] S. Yunus, private communication.
[8] Szabó, B.; Düster, A.; Rank, E.: The p-version of the finite element method. Encyclopedia of computational mechanics 1 (2004)
[9] Szabó, B.; Babuška, I.: Finite element analysis. (1991) · Zbl 0792.73003
[10] E. Rank, A. Düster, V. Nübel, K. Preusch, O.T. Bruhns, High order finite elements for shells, Comput. Methods Appl. Mech. Engrg. (accepted). · Zbl 1082.74056
[11] Abhyankar, S.; Bajaj, C.: Automatic parameterization of rational curves and surfaces I: Conics and conicoids. Comput.-aided des. 19, 11-14 (1987) · Zbl 0655.65017
[12] Abhyankar, S.; Bajaj, C.: Automatic parameterization of rational curves and surfaces II: Cubics and cubicoids. Comput.-aided des. 19, 499-502 (1987) · Zbl 0655.65018
[13] Abhyankar, S.; Bajaj, C.: Automatic parameterization of rational curves and surfaces III: Algebraic plane curves. Comput.-aided geomet. Des. 5, 309-321 (1988) · Zbl 0655.65019
[14] Abhyankar, S.; Bajaj, C.: Automatic parameterization of rational curves and surfaces IV: Algebraic space curves. ACM transactions on graphics 4, 325-334 (1989) · Zbl 0746.68104
[15] Bajaj, C.; Chen, J.; Holt, R.; Netravali, A.: Energy formulations of A-splines. Comput.-aided geomet. Des. 16, 39-59 (1999) · Zbl 0908.68176
[16] Bajaj, C.; Chen, J.; Xu, G.: Modeling with cubic A-patches. ACM transactions on graphics 14, 103-133 (1995)
[17] Cirak, F.; Ortiz, M.: Fully C1-conforming subdivision elements for finite deformation thin shell analysis. Int. J. Numer. methods engrg. 51, 813-833 (2001) · Zbl 1039.74045
[18] Cirak, F.; Ortiz, M.; Schröder, P.: Subdivision surfaces: a new paradigm for thin shell analysis. Int. J. Numer. methods engrg. 47, 2039-2072 (2000) · Zbl 0983.74063
[19] Cirak, F.; Scott, M. J.; Antonsson, E. K.; Ortiz, M.; Schröder, P.: Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision. Comput.-aided des. 34, 137-148 (2002)
[20] Hughes, T. J. R.: The finite element method: linear static and dynamic finite element analysis. (2000) · Zbl 1191.74002
[21] Piegl, L.; Tiller, W.: The NURBS book (Monographs in visual communication). (1997) · Zbl 0868.68106
[22] Kagan, P.; Fischer, A.; Bar-Yoseph, P. Z.: Mechanically based models: adaptive refinement for B-spline finite element. Int. J. Numer. methods engrg. 57, 1145-1175 (2003) · Zbl 1062.74615
[23] Bonet, J.; Kulasegaram, S.; Rodriguez-Paz, M. X.; Profit, M.: Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems. Comput. methods appl. Mech. engrg. 193, 1245-1256 (2004) · Zbl 1060.76637
[24] Breitkopf, P.; Rassineux, A.; Savignat, J.; Villon, P.: Integration constraint in diffuse element method. Comput. methods appl. Mech. engrg. 193, 1203-1220 (2004) · Zbl 1060.74661
[25] Chen, J. S.; Dongdong, W.; Dong, S. B.: An extended meshfree method for boundary value problems. Comput. methods appl. Mech. engrg. 193, 1085-1103 (2004) · Zbl 1103.74367
[26] Chen, J. S.; Kotta, V.; Lu, H.; Wang, D.; Moldovan, D.; Wolf, D.: A variational formulation and a double-grid method for meso-scale modeling of stressed grain growth in polycrystalline materials. Comput. methods appl. Mech. engrg. 193, 1277-1303 (2004) · Zbl 1060.74662
[27] Dongdong, W.; Chen, J. S.: Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput. methods appl. Mech. engrg. 193, 1065-1083 (2004) · Zbl 1060.74675
[28] Fasshauer, G. E.: Toward approximate moving least squares approximation with irregularly spaced centers. Comput. methods appl. Mech. engrg. 193, 1231-1243 (2004) · Zbl 1060.65041
[29] Fernandez-Mendez, S.; Huerta, A.: Imposing essential boundary conditions in mesh-free methods. Comput. methods appl. Mech. engrg. 193, 1257-1275 (2004) · Zbl 1060.74665
[30] Huerta, A.; Fernandez-Mendez, S.; Liu, W. K.: A comparison of two formulations to blend finite elements and mesh-free methods. Comput. methods appl. Mech. engrg. 193, 1105-1117 (2004) · Zbl 1059.65104
[31] Huerta, A.; Vidal, Y.; Villon, P.: Pseudo-divergence-free element free Galerkin method for incompressible fluid flow. Comput. methods appl. Mech. engrg. 193, 1119-1136 (2004) · Zbl 1060.76626
[32] Li, G.; Aluru, N. R.: Positivity conditions in meshless collocation methods. Comput. methods appl. Mech. engrg. 193, 1171-1202 (2004) · Zbl 1060.74667
[33] Li, S.; Lu, H.; Han, W.; Liu, W. K.; Jr., D. Simkins: Reproducing kernel element method. Part II: Globally conforming im/cn hierarchies. Comput. methods appl. Mech. engrg. 193, 953-987 (2004) · Zbl 1093.74062
[34] Liu, W. K.; Han, W.; Lu, H.; Li, S.; Cao, J.: Reproducing kernel element method. Part I: Theoretical formulation. Comput. methods appl. Mech. engrg. 193, 933-951 (2004) · Zbl 1060.74670
[35] Lu, H.; Li, S.; Jr., D. Simkins; Liu, W. K.; Cao, J.: Reproducing kernel element method. Part III: Generalized enrichment and applications. Comput. methods appl. Mech. engrg. 193, 989-1011 (2004) · Zbl 1060.74671
[36] Rabczuk, T.; Belytschko, T.; Xiao, S. P.: Stable particle methods based on Lagrangian kernels. Comput. methods appl. Mech. engrg. 193, 1035-1063 (2004) · Zbl 1060.74672
[37] Jr., D. Simkins; Li, S.; Lu, H.; Liu, W. K.: Reproducing kernel element method. Part IV: Globally compatible cn (n\geqslant1) triangular hierarchy. Comput. methods appl. Mech. engrg. 193, 1013-1034 (2004) · Zbl 1093.74064
[38] Sulsky, D.; Kaul, A.: Implicit dynamics in the material-point method. Comput. methods appl. Mech. engrg. 193, 1137-1170 (2004) · Zbl 1060.74674
[39] Wang, X.; Liu, W. K.: Extended immersed boundary method using FEM and RKPM. Comput. methods appl. Mech. engrg. 193, 1305-1321 (2004) · Zbl 1060.74676
[40] Wu, Z.: Dynamically knots setting in meshless method for solving time dependent propagations equation. Comput. methods appl. Mech. engrg. 193, 1221-1229 (2004) · Zbl 1060.76633
[41] Gould, P. L.: Introduction to linear elasticity. (1999)
[42] Timoshenko, S.; Woinowsky-Krieger, S.: Theory of plates and shells (Engineering societies monograph). (1959) · Zbl 0114.40801
[43] Bischoff, M.; A., Wall W.; Bletzinger, K. U.; Ramm, E.: Models and finite elements for thin-walled structures. Encyclopedia of computational mechanics 2 (2004)
[44] C. Felippa, Course notes for advanced finite element methods, http://caswww.colorado.edu/Felippa.d/FelippaHome.d/Home.html.
[45] Belytschko, T.; Stolarski, H.; Liu, W. K.; Carpenter, N.; Ong, J. S. -J.: Stress projection for membrane and shear locking in shell finite elements. Comput. methods appl. Mech. engrg. 51, 221-258 (1985) · Zbl 0581.73091
[46] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods in fluid dynamics. (1988) · Zbl 0658.76001
[47] Moin, P.: Fundamentals of engineering numerical analysis. (2001) · Zbl 0993.65003
[48] Hughes, T. J. R.; Franca, L. P.: A mixed finite element formulation for Reissner-Mindlin plate theory: uniform convergence of all higher order spaces. Comput. methods appl. Mech. engrg. 67, 223-240 (1988) · Zbl 0611.73077
[49] Engel, G.; Garikipati, K.; Hughes, T. J. R.; Larson, M. G.; Mazzei, L.: Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. methods appl. Mech. engrg. 191, 3669-3750 (2002) · Zbl 1086.74038
[50] Oñate, E.; Cervera, M.: Derivation of thin plate bending elements with one degree of freedom per node: a simple three node triangle. Engrg. comput. 10, 543-561 (1993)
[51] Oñate, E.; Zarate, F.: Rotation-free triangular plate and shell elements. Int. J. Numer. methods engrg. 47, 557-603 (2000) · Zbl 0968.74070
[52] Phaal, R.; Calladine, C. R.: A simple class of finite-elements for plate and shell problems: 1. Elements for beams and thin flat plates. Int. J. Numer. methods engrg. 35, 955-977 (1992) · Zbl 0775.73285
[53] Phaal, R.; Calladine, C. R.: A simple class of finite-elements for plate and shell problems: 2. An element for thin shells with only translational degrees of freedom. Int. J. Numer. methods engrg. 35, 979-996 (1992) · Zbl 0775.73286
[54] 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. engrg. 32, 199-259 (1982) · Zbl 0497.76041
[55] Akin, J. E.; Tezduyar, T. E.: Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements. Comput. methods appl. Mech. engrg. 193, 1909-1922 (2004) · Zbl 1067.76557
[56] Bischoff, M.; Bletzinger, K.: Improving stability and accuracy of Reissner-Mindlin plate finite elements via algebraic subgrid scale stabilization. Comput. methods appl. Mech. engrg. 193, 1491-1516 (2004) · Zbl 1079.74633
[57] Bochev, P. B.; Gunzburger, M. D.; Shadid, J. N.: On inf-sup stabilized finite element methods for transient problems. Comput. methods appl. Mech. engrg. 193, 1471-1489 (2004) · Zbl 1079.76577
[58] Burman, E.; Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. methods appl. Mech. engrg. 193, 1437-1453 (2004) · Zbl 1085.76033
[59] Codina, R.; Soto, O.: Approximation of the incompressible Navier-Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes. Comput. methods appl. Mech. engrg. 193, 1403-1419 (2004) · Zbl 1079.76579
[60] Coutinho, A. L. G.A.; Diaz, C. M.; Alvez, J. L. D.; Landau, L.; Loula, A. F. D.; Malta, S. M. C.; Castro, R. G. S.; Garcia, E. L. M.: Stabilized methods and post-processing techniques for miscible displacements. Comput. methods appl. Mech. engrg. 193, 1421-1436 (2004) · Zbl 1079.76580
[61] Gravemeier, V.; Wall, W. A.; Ramm, E.: A three-level finite element method for the instationary incompressible Navier-Stokes equations. Comput. methods appl. Mech. engrg. 193, 1323-1366 (2004) · Zbl 1085.76038
[62] Harari, I.: Stability of semidiscrete formulations for parabolic problems at small time steps. Comput. methods appl. Mech. engrg. 193, 1491-1516 (2004) · Zbl 1079.76597
[63] Hauke, G.; Valiño, L.: Computing reactive flows with a field Monte Carlo formulation and multi-scale methods. Comput. methods appl. Mech. engrg. 193, 1455-1470 (2004) · Zbl 1079.76616
[64] Koobus, B.; Farhat, C.: A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes - application to vortex shedding. Comput. methods appl. Mech. engrg. 193, 1367-1383 (2004) · Zbl 1079.76567
[65] Masud, A.; Khurram, R. A.: A multiscale/stabilized finite element method for the advection-diffusion equation. Comput. methods appl. Mech. engrg. 193, 1997-2018 (2004) · Zbl 1067.76570
[66] Tezduyar, T. E.; Sathe, S.: Enhanced-discretization space-time technique (EDSTT). Comput. methods appl. Mech. engrg. 193, 1385-1401 (2004) · Zbl 1079.76585
[67] Chalot, F. L.: Industrial aerodynamics. Encyclopedia of computational mechanics 3 (2004)
[68] Johnson, C.; Nävert, U.; Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. methods appl. Mech. engrg. 45, 285-312 (1984) · Zbl 0526.76087
[69] Hughes, T. J. R.; Scovazzi, G.; Franca, L. P.: Multiscale and stabilized methods. Encyclopedia of computational mechanics 3 (2004)
[70] 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. engrg. 127, 387-401 (1995) · Zbl 0866.76044
[71] Hughes, T. J. R.; Feijóo., G.; Mazzei, L.; Quincy, J. B.: The variational multiscale method--a paradigm for computational mechanics. Comput. methods appl. Mech. engrg. 166, 3-24 (1998) · Zbl 1017.65525
[72] Wahlbin, L. B.: Local behavior in finite element methods. Handbook of numerical analysis 2, 353-522 (1991) · Zbl 0875.65089
[73] T.D. Blacker, CUBIT mesh generation environment users manual vol. 1, Technical Report, Sandia National Laboratories, Albuquerque, NM, 1994.
[74] Qin, H.; Terzopoulos, D.: Triangular NURBS and their dynamic generalizations. Comput.-aided geomet. Des. 14, 325-347 (1996) · Zbl 0906.68166
[75] Höllig, K.: Finite element methods with B-splines. (2003) · Zbl 1020.65085
[76] Kravchenko, A. G.; Moin, P.; Moser, R.: Zonal embedded grids for numerical simulation of wall-bounded turbulent flows. J. computat. Phys. 127, 412-423 (1996) · Zbl 0862.76062
[77] Kravchenko, A. G.; Moin, P.; Shariff, K.: B-spline method and zonal grids for simulation of complex turbulent flows. J. computat. Phys. 151, 757-789 (1999) · Zbl 0942.76058
[78] Kwok, W. Y.; Moser, R. D.; Jiménez, J.: A critical evaluation of the resolution properties of B-spline and compact finite difference methods. J. computat. Phys. 174, 510-551 (2001) · Zbl 0995.65089
[79] Shariff, K.; Moser, R. D.: Two-dimensional mesh embedding for B-spline methods. J. computat. Phys. 145, 471-488 (1998) · Zbl 0910.65083
[80] Laursen, T. A.: Computational contact and impact mechanics. (2002) · Zbl 0996.74003
[81] Wriggers, P.: Computational contact mechanics. (2002) · Zbl 1104.74002
[82] Qin, H.; Terzopoulos, D.: D-NURBS: a physics-based framework for geometric design. IEEE trans. Visual. comput. Graph. 2, 85-96 (1996) · Zbl 0873.68198