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Keywords:
curved boundary; error estimate; isoparametric FEM; isogeometric analysis; patch test; local convergence
Summary:
The finite element method (FEM) is popularly used for numerically approximating PDE(s) over complicated domains due to its rich mathematical background, versatility, and ease of implementation. In this article, we investigate one of its important features, i.e., the approximation of PDE(s) over nonpolygonal Lipschitz domains by higher-order simplicial elements in 2D and 3D. This important issue is not well understood and often ignored by engineers due to its mathematical complexity, i.e., the FEM approximation of curved domains results in inexact boundary conditions, which is a variational crime. This article explores the role of approximation at curved boundaries. Further, the effect of incompleteness of the approximation space also contributes to the error induced in the curved elements. A simple benchmark test for errors is proposed. Tests are conducted for subparametric and isoparametric approximations. Comparison with isogeometric analysis (IGA) is also presented to highlight the basic differences and advantages of isoparametric elements.
References:
[1] Agrawal, V., Gautam, S. S.: IGA: A simplified introduction and implementation details for finite element users. J. Inst. Engineers (India), Ser. C 100 (2019), 561-585. DOI 10.1007/s40032-018-0462-6
[2] Argyris, J. H., Scharpf, D. W.: A sequel to Technical Note 13: The curved tetrahedronal and triangular elements TEC and TRIC for the matrix displacement method. Aeronaut. J. 73 (1969), 55-65. DOI 10.1017/s0001924000053574
[3] Arnold, D. N., Boffi, D., Falk, R. S.: Approximation by quadrilateral finite elements. Math. Comput. 71 (2002), 909-922. DOI 10.1090/s0025-5718-02-01439-4 | MR 1898739 | Zbl 0993.65125
[4] Babuška, I.: The stability of the domain of definition with respect to basic problems of the theory of partial differential equations, especially with respect to the theory of elasticity. II. Czech. Math. J. 11 (1961), 165-203 Russian. DOI 10.21136/cmj.1961.100453 | MR 0125326 | Zbl 0126.11401
[5] Babuška, I., Strouboulis, T.: The Finite Element Method and Its Reliability. Oxford University Press, Oxford (2001). DOI 10.1093/oso/9780198502760.001.0001 | MR 1857191 | Zbl 0995.65501
[6] Bartels, S., Carstensen, C., Hecht, A.: P2Q2Iso2D=2D isoparametric FEM in Matlab. J. Comput. Appl. Math. 192 (2006), 219-250. DOI 10.1016/j.cam.2005.04.032 | MR 2228811 | Zbl 1091.65112
[7] Berger, A. E.: Error Estimates for the Finite Element Method: Ph.D. Thesis. Massachusetts Institute of Technology, Cambridge (1972). MR 2940236
[8] Berger, A. E.: Two types of piecewise quadratic spaces and their order of accuracy for Poisson's equation. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations Academic Press, New York (1972), 757-761. DOI 10.1016/b978-0-12-068650-6.50033-2 | MR 0416064 | Zbl 0282.65078
[9] Berger, A. E.: $L^2$ error estimates for finite elements with interpolated boundary conditions. Numer. Math. 21 (1973), 345-349. DOI 10.1007/bf01436388 | MR 0343655 | Zbl 0287.65060
[10] Berger, A. E., Scott, R., Strang, G.: Approximate boundary conditions in the finite element method. Symposia Mathematica. Vol. X Academic Press, London (1972), 295-313. MR 0403258 | Zbl 0266.73050
[11] Bernardi, C.: Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989), 1212-1240. DOI 10.1137/0726068 | MR 1014883 | Zbl 0678.65003
[12] Blair, J. J.: Approximate Solution of Elliptic and Parabolic Boundary Value Problems: Ph.D. Thesis. University of California, Berkeley (1970). MR 2619649
[13] Blair, J. J.: Bounds for the change in the solutions of second order elliptic PDE's when the boundary is perturbed. SIAM J. Appl. Math. 24 (1973), 277-285. DOI 10.1137/0124029 | MR 0317557 | Zbl 0252.35021
[14] Blair, J. J.: Higher order approximations to the boundary conditions for the finite element method. Math. Comput. 30 (1976), 250-262. DOI 10.2307/2005966 | MR 0398123 | Zbl 0342.65068
[15] Botti, L.: Influence of reference-to-physical frame mappings on approximation properties of discontinuous piecewise polynomial spaces. J. Sci. Comput. 52 (2012), 675-703. DOI 10.1007/s10915-011-9566-3 | MR 2948713 | Zbl 1255.65222
[16] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15. Springer, New York (2008). DOI 10.1007/978-0-387-75934-0 | MR 2373954 | Zbl 1135.65042
[17] Chessa, J.: Programing the finite element method with Matlab. Available at https://www.math.purdue.edu/ {caiz/math615/matlab_fem.pdf} (2002).
[18] Ciarlet, P. G., Raviart, P.-A.: Interpolation theory over curved elements, with applications to finite element methods. Computer Methods Appl. Mech. Engin. 1 (1972), 217-249. DOI 10.1016/0045-7825(72)90006-0 | MR 0375801 | Zbl 0261.65079
[19] Ciarlet, P. G., Raviart, P.-A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations Academic Press, New York (1972), 409-474. DOI 10.1016/b978-0-12-068650-6.50020-4 | MR 0421108 | Zbl 0262.65070
[20] Cottrell, J. A., Hughes, T. J. R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons, Hoboken (2009). DOI 10.1002/9780470749081 | MR 3618875 | Zbl 1378.65009
[21] Dey, S.: Curvilinear Mesh generation in 3D. Proceedings of the 8th International Meshing Roundtable South Lake Tahoe, California, USA (1999), 407-417.
[22] Ergatoudis, I., Irons, B. M., Zienkiewicz, O. C.: Curved, isoparametric, `quadrilateral' elements for finite element analysis. Int. J. Solids Struct. 4 (1968), 31-42. DOI 10.1016/0020-7683(68)90031-0 | Zbl 0152.42802
[23] Fortunato, M., Persson, P.-O.: High-order unstructured curved mesh generation using the Winslow equations. J. Comput. Phys. 307 (2016), 1-14. DOI 10.1016/j.jcp.2015.11.020 | MR 3448195 | Zbl 1352.65607
[24] Geuzaine, C., Johnen, A., Lambrechts, J., Remacle, J.-F., Toulorge, T.: The generation of valid curvilinear meshes. IDIHOM: Industrialization of High-Order Methods -- A Top-Down Approach Notes on Numerical Fluid Mechanics and Multidisciplinary Design 128. Springer, Cham (2015), 15-39. DOI 10.1007/978-3-319-12886-3_2
[25] Geuzaine, C., Remacle, J.-F.: Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79 (2009), 1309-1331. DOI 10.1002/nme.2579 | MR 2566786 | Zbl 1176.74181
[26] 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 (2005), 4135-4195. DOI 10.1016/j.cma.2004.10.008 | MR 2152382 | Zbl 1151.74419
[27] Hussain, F., Karim, M. S., Ahamad, R.: Appropriate Gaussian quadrature formulae for triangles. Int. J. Appl. Math. Comput. 4 (2012), 24-38.
[28] Johnen, A., Remacle, J.-F., Geuzaine, C.: Geometrical validity of curvilinear finite elements. J. Comput. Phys. 233 (2013), 359-372. DOI 10.1016/j.jcp.2012.08.051 | MR 3000936
[29] Johnen, A., Remacle, J.-F., Geuzaine, C.: Geometrical validity of high-order triangular finite elements. Eng. Comput. (Lond.) 30 (2014), 375-382. DOI 10.1007/s00366-012-0305-7 | MR 3000936
[30] Jordan, W. B.: Plane Isoparametric Structural Element. Knolls Atomic Power Laboratory, New York (1970). DOI 10.2172/4157041
[31] Lenoir, M.: Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986), 562-580. DOI 10.1137/0723036 | MR 0842644 | Zbl 0605.65071
[32] Luo, X., Shephard, M. S., Remacle, J.-F.: The influence of geometric approximation on the accuracy of high order methods. Available at https://www.researchgate.net/publication/228599667 (2002), 11 pages.
[33] McLeod, R., Mitchell, A. R.: The construction of basis functions for curved elements in the finite element method. J. Inst. Math. Appl. 10 (1972), 382-393. DOI 10.1093/imamat/10.3.382 | MR 0440959 | Zbl 0254.65071
[34] McLeod, R. J. Y., Mitchell, A. R.: The use of parabolic arcs in matching curved boundaries in the finite element method. J. Inst. Math. Appl. 16 (1975), 239-246. DOI 10.1093/imamat/16.2.239 | MR 0400747 | Zbl 0308.65065
[35] Minakowski, P., Richter, T.: Finite element error estimates on geometrically perturbed domains. J. Sci. Comput. 84 (2020), Article ID 30, 19 pages. DOI 10.1007/s10915-020-01285-y | MR 4127419 | Zbl 1458.65152
[36] Mitchell, A. R., Phillips, G., Wachspress, E.: Forbidden shapes in the finite element method. J. Inst. Math. Appl. 8 (1971), 260-269. DOI 10.1093/imamat/8.2.260 | MR 0292262 | Zbl 0229.65082
[37] Moxey, D., Sastry, S. P., Kirby, R. M.: Interpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinement. J. Sci. Comput. 78 (2019), 1045-1062. DOI 10.1007/s10915-018-0795-6 | MR 3918679 | Zbl 1417.65207
[38] Nguyen, V. P., Anitescu, C., Bordas, S. P. A., Rabczuk, T.: Isogeometric analysis: An overview and computer implementation aspects. Math. Comput. Simul. 117 (2015), 89-116. DOI 10.1016/j.matcom.2015.05.008 | MR 3372009 | Zbl 1540.65492
[39] Piegl, L., Tiller, W.: The $\mathcal{N}\mathcal{U}\mathcal{R}\mathcal{B}\mathcal{S}$ Book. Monographs in Visual Communication. Springer, Berlin (1995). DOI 10.1007/978-3-642-97385-7 | Zbl 0828.68118
[40] Poya, R., Sevilla, R., Gil, A. J.: A unified approach for a posteriori high-order curved mesh generation using solid mechanics. Comput. Mech. 58 (2016), 457-490. DOI 10.1007/s00466-016-1302-2 | MR 3533501 | Zbl 1398.74472
[41] Rektorys, K.: Variational Methods in Mathematics, Science and Engineering. D. Reidel, Dordrecht (1977). DOI 10.1007/978-94-011-6450-4 | MR 0487653 | Zbl 0371.35002
[42] Rogers, D. F.: An Introduction to NURBS: With Historical Perspective. Elsevier, Amsterdam (2000). DOI 10.1016/b978-1-55860-669-2.x5000-3
[43] Ruiz-Gironés, E., Sarrate, J., Roca, X.: Generation of curved high-order meshes with optimal quality and geometric accuracy. Procedia Eng. 163 (2016), 315-327. DOI 10.1016/j.proeng.2016.11.108
[44] Ruiz-Gironés, E., Sarrate, J., Roca, X.: Measuring and improving the geometric accuracy of piece-wise polynomial boundary meshes. J. Comput. Phys. 443 (2021), Article ID 110500, 22 pages. DOI 10.1016/j.jcp.2021.110500 | MR 4273818 | Zbl 07515413
[45] Sastry, S. P., Kirby, R. M.: On interpolation errors over quadratic nodal triangular finite elements. Proceedings of the 22nd International Meshing Roundtable Springer, Cham (2014), 349-366. DOI 10.1007/978-3-319-02335-9_20
[46] Scott, L. R.: Finite-Element Techniques for Curved Boundaries: Ph.D. Thesis. Massachusetts Institute of Technology, Cambridge (1973). MR 2940387
[47] Scott, L. R.: Interpolated boundary conditions in the finite element method. SIAM J. Numer. Anal. 12 (1975), 404-427. DOI 10.1137/0712032 | MR 0386304 | Zbl 0357.65082
[48] Sevilla, R., Fernández-Méndez, S.: Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM. Finite Elem. Anal. Des. 47 (2011), 1209-1220. DOI 10.1016/j.finel.2011.05.011 | MR 2817724
[49] Sevilla, R., Fernández-Méndez, S., Huerta, A.: NURBS-enhanced finite element method. European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006 Delft University of Technology, Delft (2006), 1-13. MR 2455923
[50] Sevilla, R., Fernández-Méndez, S., Huerta, A.: NURBS-enhanced finite element method for Euler equations. Int. J. Numer. Methods Fluids 57 (2008), 1051-1069. DOI 10.1002/fld.1711 | MR 2435082 | Zbl 1140.76023
[51] Sevilla, R., Fernández-Méndez, S., Huerta, A.: NURBS-enhanced finite element method (NEFEM). Int. J. Numer. Methods Eng. 76 (2008), 56-83. DOI 10.1002/nme.2311 | MR 2455923 | Zbl 1162.65389
[52] Sevilla, R., Fernández-Méndez, S., Huerta, A.: Comparison of high-order curved finite elements. Int. J. Numer. Methods Eng. 87 (2011), 719-734. DOI 10.1002/nme.3129 | MR 2858254 | Zbl 1242.65244
[53] Sevilla, R., Fernández-Méndez, S., Huerta, A.: 3D NURBS-enhanced finite element method (NEFEM). Int. J. Numer. Methods Eng. 88 (2011), 103-125. DOI 10.1002/nme.3164 | MR 2835747 | Zbl 1242.78032
[54] Sevilla, R., Fernández-Méndez, S., Huerta, A.: NURBS-enhanced finite element method (NEFEM): A seamless bridge between CAD and FEM. Arch. Comput. Methods Eng. 18 (2011), 441-484. DOI 10.1007/s11831-011-9066-5 | MR 2851386
[55] Strang, G.: Variational crimes in the finite element method. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations Academic Press, New York (1972), 689-710. DOI 10.1016/b978-0-12-068650-6.50030-7 | MR 0413554 | Zbl 0264.65068
[56] Strang, G., Berger, A. E.: The change in solution due to change in domain. Partial Differential Equations Proceedings of Symposia in Pure Mathematics 23. AMS, Providence (1973), 199-205. DOI 10.1090/pspum/023/0337023 | MR 0337023 | Zbl 0259.35020
[57] Szabó, B., I.\.Babuška: Introduction to Finite Element Analysis: Formulation, Verification and Validation. John Wiley & Sons, Hoboken (2011). DOI 10.1002/9781119993834 | MR 1164869 | Zbl 1410.65003
[58] Xie, Z., Sevilla, R., Hassan, O., Morgan, K.: The generation of arbitrary order curved meshes for 3D finite element analysis. Comput. Mech. 51 (2013), 361-374. DOI 10.1007/s00466-012-0736-4 | MR 3029066
[59] Xue, D.: Control of Geometry Error in $hp$ Finite Element (FE) Simulations of Electromagnetic (EM) Waves: Ph.D. Thesis. The University of Texas at Austin, Austin (2005). MR 2707661
[60] Xue, D., Demkowicz, L.: Control of geometry induced error in $hp$ finite element (FE) simulations. I. Evaluation of FE error for curvilinear geometries. Int. J. Numer. Anal. Model. 2 (2005), 283-300. MR 2112649 | Zbl 1073.65122
[61] Zienkiewicz, O. C., Taylor, R. L.: The finite element patch test revisited: A computer test for convergence, validation and error estimates. Comput. Methods Appl. Mech. Eng. 149 (1997), 223-254. DOI 10.1016/s0045-7825(97)00085-6 | MR 1486242 | Zbl 0918.73134
[62] Zlámal, M.: A finite element procedure of the second order of accuracy. Numer. Math. 14 (1970), 394-402. DOI 10.1007/bf02165594 | MR 0256577 | Zbl 0209.18002
[63] Zlámal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10 (1973), 229-240. DOI 10.1137/0710022 | MR 0395263 | Zbl 0285.65067
[64] Zlámal, M.: The finite element method in domains with curved boundaries. Int. J. Numer. Methods Eng. 5 (1973), 367-373. DOI 10.1002/nme.1620050307 | MR 0395262 | Zbl 0254.65073
[65] Zlámal, M.: Curved elements in the finite element method. II. SIAM J. Numer. Anal. 11 (1974), 347-362. DOI 10.1137/0711031 | MR 0343660 | Zbl 0277.65064
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