The Nitsche method applied to a class of mixed-dimensional coupling problems. (English) Zbl 1296.74012
Summary: A computational approach for the mixed-dimensional modeling of time-harmonic waves in elastic structures is proposed. A two-dimensional (2D) structure is considered, that includes a part which is assumed to behave in a one-dimensional (1D) way. The 2D and 1D structural regions are discretized using 2D and 1D finite element formulations. The coupling of the 2D and 1D regions is performed weakly, by using the Nitsche method. The advantage of using the Nitsche method to impose boundary and interface conditions has been demonstrated by various authors; here this advantage is shown in the context of mixed-dimensional coupling. The computational aspects of the method are discussed, and it is compared to the slightly simpler penalty method, both theoretically and numerically. Numerical examples are presented in various configurations: where the 1D model is either confined laterally or laterally free, and where the 2D part is either simply connected or doubly connected. The performance is investigated for various wave numbers and various extents of the 1D region. Varying material properties and distributed loads in the 1D and 2D parts are also considered. It is concluded that the Nitsche method is a viable technique for mixed-dimensional coupling of elliptic problems of this type.
MSC:
| 74B05 | Classical linear elasticity |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74J05 | Linear waves in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Software:
F.E.MReferences:
| [1] | Donaldson, B. K., Analysis of Aircraft Structures (1993), McGraw-Hill: McGraw-Hill New York |
| [2] | Szabó, B.; Babuška, I., Introduction to Finite Element Analysis: Formulation, Verification and Validation (2011), Wiley: Wiley New York · Zbl 1410.65003 |
| [3] | Chong, C. S.; Senthil Kumar, A.; Lee, K. H., Automatic solid decomposition and reduction for non-manifold geometric model generation, Comput. Aided Des., 36, 1357-1369 (2004) |
| [4] | Cuillière, J-C.; Bournival, S.; François, V., A mesh-geometry-based solution to mixed-dimensional coupling, Comput. Aided Des., 42, 509-522 (2010) |
| [5] | Bournival, S.; Cuillière, J-C.; François, V., A mesh-geometry-based method for coupling 1D and 3D elements, Adv. Eng. Software, 41, 838-858 (2010) · Zbl 1346.74165 |
| [6] | Osawa, N.; Hashimoto, K.; Sawamura, J.; Nakai, T.; Suzuki, S., Study on shell-solid coupling FE analysis for fatigue assessment of ship structure, Mar. Struct., 20, 143-163 (2007) |
| [7] | Surana, K. S., Transition finite elements for three-dimensional stress analysis, Int. J. Numer. Methods Eng., 15, 991-1020 (1980) · Zbl 0443.73059 |
| [8] | Gmür, T. C.; Kauten, R. H., Three-dimensional solid-to-beam transition elements for stuctural dynamics analysis, Int. J. Numer. Methods Eng., 36, 1429-1444 (1993) · Zbl 0775.73254 |
| [9] | Kim, J.; Varadan, V. V.; Varadan, V. K., Finite element modelling of structures including piezoelectric active devices, Int. J. Numer. Methods Eng., 40, 817-832 (1997) · Zbl 0913.73064 |
| [10] | Garusi, E.; Tralli, A., A hybrid stress-assumed transition element for solid-to-beam and plate-to-beam connections, Comput. Struct., 80, 105-115 (2002) |
| [11] | McCune, R. W.; Armstrong, C. G.; Robinson, D. J., Mixed-dimensional coupling in finite element models, Int. J. Numer. Methods Eng., 49, 725-750 (2000) · Zbl 0967.74067 |
| [12] | Shim, W.; Monaghan, D. J.; Armstrong, C. G., Mixed dimensional coupling in finite element stress analysis, Eng. Comput., 18, 241-252 (2002) |
| [13] | Haas, M.; Kuhn, G., Mixed-dimensional, symmetric coupling of FEM and BEM, Eng. Anal. Boundary Elem., 27, 575-582 (2003) · Zbl 1054.74729 |
| [14] | Haas, M.; Helldörfer, B.; Kuhn, G., Improved coupling of finite shell elements and 3D boundary elements, Arch. Appl. Mech., 75, 649-663 (2006) · Zbl 1168.74452 |
| [15] | Curiskis, J. I.; Valiappan, S., A solution algorithm for linear constraint equations in finite element analysis, Comput. Struct., 8, 117-124 (1978) · Zbl 0365.65063 |
| [16] | Ho, R. J.; Meguid, S. A.; Zhu, Z. H.; Sauvè, R. G., Consistent element coupling in nonlinear static and dynamic analyses using explicit solvers, Int. J. Mech. Mater. Des., 6, 319-330 (2010) |
| [17] | Mendes, L. A.M.; Castro, L. M.S. S., Using hybrid discretizations in paralellized structural dynamic analyses, Adv. Eng. Software, 56, 72-83 (2013) |
| [18] | Panasenko, G. P., Method of asymptotic partial decomposition of domain, Math. Models Methods Appl. Sci., 8 (1998) · Zbl 0940.35026 |
| [19] | Panasenko, G. P., Method of asymptotic partial decomposition of rod structures, Int. J. Comput. Civil Struct. Eng., 1, 57-70 (2000) |
| [20] | Fontvieille, F.; Panasenko, G. P.; Pousin, J., FEM implementation for the asymptotic partial decomposition, Appl. Anal., 86, 519-536 (2007) · Zbl 1115.65114 |
| [22] | Halliday, P. J.; Grosh, K., Dynamic response of complex structural intersections using hybrid methods, ASME J. Appl. Mech., 66, 653-659 (1999) |
| [23] | Hughes, T. J.R., The Finite Element Method (1987), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0634.73056 |
| [24] | Avdeev, I. V.; Borovkov, A. I.; Kiylo, O. L.; Lovell, M. R.; Onipede, D., Mixed 2D and beam formulation for modeling sandwich structures, Eng. Comput., 19, 451-466 (2002) · Zbl 1169.74586 |
| [25] | Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräuman, die keinen Randbedingungen unterworfen sind, Abh. Math. Univ. Hamburg, 36, 9-15 (1971) · Zbl 0229.65079 |
| [26] | Stenberg, R., On some techniques for approximating boundary conditions in the finite element method, J. Comput. Appl. Math., 63, 139-148 (1995) · Zbl 0856.65130 |
| [27] | Embar, A.; Dolbow, J.; Harari, I., Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements, Int. J. Numer. Methods Eng., 83, 877-898 (2010) · Zbl 1197.74178 |
| [28] | Hansbo, A.; Hansbo, P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., 191, 5537-5552 (2002) · Zbl 1035.65125 |
| [29] | Becker, R.; Hansbo, P.; Stenberg, R., A finite element for domain decomposition with non-matching grids, ESAIM Math. Model. Numer. Anal., 37, 209-225 (2003) · Zbl 1047.65099 |
| [30] | Dolbow, J.; Harari, I., An efficient finite element method for embedded interface problems, Int. J. Numer. Methods Eng., 78, 229-252 (2009) · Zbl 1183.76803 |
| [31] | Annavarapu, C.; Hautefeuille, M.; Dolbow, J. E., A robust Nitsche’s formulation for interface problems, Comput. Methods Appl. Mech. Eng., 225, 44-54 (2012) · Zbl 1253.74096 |
| [32] | Sanders, J. D.; Dolbow, J. E.; Laursen, T. A., On methods for stabilizing constraints over enriched interfaces in elasticity, Int. J. Numer. Methods Eng., 78, 1009-1036 (2008) · Zbl 1183.74313 |
| [33] | Coon, E. T.; Shaw, B. E.; Spiegelman, M., A Nitsche-extended finite element method for earthquake rupture on complex fault systems, Comput. Methods Appl. Mech. Eng., 200, 2859-2870 (2011) · Zbl 1230.74176 |
| [34] | Hansbo, P.; Hermansson, J., Nitsche’s method for coupling non-matching meshes in fluid-structure vibration problems, Comput. Mech., 32, 134-139 (2003) · Zbl 1035.74055 |
| [35] | Hansbo, P.; Hermansson, J.; Svedberg, T., Nitsche’s method combined with space-time finite elements for ALE fluid-structure interaction problems, Comput. Methods Appl. Mech. Eng., 193, 4195-4206 (2004) · Zbl 1175.74082 |
| [36] | Hansbo, A.; Hansbo, P.; Larson, M. G., A finite element method on composite grids based on Nitsche’s method, ESAIM Math. Model. Numer. Anal., 37, 495-514 (2003) · Zbl 1031.65128 |
| [37] | Sanders, J. D.; Laursen, T. A.; Puso, M. A., A Nitsche embedded mesh method, Comput. Mech., 49, 243-257 (2012) · Zbl 1366.74075 |
| [38] | Massing, A.; Larson, M. G.; Logg, A., Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions, SIAM J. Sci. Comput., 35, C23-C47 (2013) · Zbl 1264.65194 |
| [39] | Becker, R.; Burman, E.; Hansbo, P., A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Eng., 198, 3352-3360 (2009) · Zbl 1230.74169 |
| [40] | Burman, E.; Hansbo, P., Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method, Appl. Numer. Math., 62, 328-341 (2012) · Zbl 1316.65099 |
| [41] | Hansbo, P.; Larson, M. G., Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comput. Methods Appl. Mech. Eng., 191, 1895-1908 (2002) · Zbl 1098.74693 |
| [42] | Hansbo, P., Nitsche’s method for interface problems in computational mechanics, GAMM Mitt., 28, 183-206 (2005) · Zbl 1179.65147 |
| [43] | Wriggers, P.; Zavarise, G., A fromulation for frictionless contact problems using a weak form introduced by Nitsche, Comput. Mech., 41, 407-420 (2008) · Zbl 1162.74419 |
| [44] | Chouly, F.; Hild, P., A Nitsche-based method for unilateral contact problems: numerical analysis, SIAM J. Numer. Anal., 51, 295-1307 (2013) · Zbl 1268.74033 |
| [46] | Chouly, F., An adaptation of Nitsche’s method to the Tresca friction problem, J. Math. Anal. Appl., 411, 329-339 (2014) · Zbl 1311.74112 |
| [47] | Andersen, L., Linear Elastodynamic Analysis (2006), Aalborg University |
| [48] | Krylov, S.; Harari, I.; Gadasi, D., Consistent loading in structural reduction procedures for beam models, Int. J. Multiscale Comput. Eng., 4, 559-583 (2006) |
| [49] | Annavarapu, C.; Hautefeuille, M.; Dolbow, J. E., A robust Nitsche’s formulation for interface problems, Comput. Methods. Appl. Mech. Eng., 225, 44-54 (2012) · Zbl 1253.74096 |
| [50] | Annavarapu, C.; Hautefeuille, M.; Dolbow, J. E., Stable imposition of stiff constraints in explicit dynamics for embedded finite element methods, Int. J. Numer. Methods Eng., 92, 206-228 (2012) · Zbl 1352.74314 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
