A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems. (English) Zbl 1270.74201
Summary: The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke (FdV) functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems: the numerical evaluation of the integrals over the domain \(A\) and the enforcement of boundary conditions of the type \(u_i=\tilde{u}_i\) on \(S_u\). In the absence of body forces (\(F_i=0)\), it will be shown that the calculation of integrals of the type \(\int\nolimits_A \cdot\mathrm{d}A\) can be avoided and that boundary conditions of the type \(u_i=\overset\sim u_i\) on \(S_u\) can be imposed in the average sense in general and exactly if \(\overset\sim u_i\) is linear between two contour nodes, which is obviously the case for \(\overset\sim u_i=0\).
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
Keywords:
natural neighbour method; meshless method; Fraeijs de Veubeke variational principle; discretization; Voronoi cellReferences:
| [1] | Cuteo E., Sukumar N., Calvo B., Cegonino J., Doblare M.: Overview and recent advances in natural neighbour Galerkin methods. Arch. Comput. Methods Eng. 10(4), 307–384 (2003) · Zbl 1050.74001 · doi:10.1007/BF02736253 |
| [2] | Yoo J., Moran B., Chen J.S.: Stabilized conforming nodal integration in the natural element method. Int. J. Numer. Methods Eng. 60, 861–890 (2004) · Zbl 1060.74677 · doi:10.1002/nme.972 |
| [3] | Sukumar N., Moran B., Semenov Y., Belikov B.: Natural neighbor Galerkin methods. Int. J. Numer. Methods Eng. 50, 1–27 (2001) · Zbl 1082.74554 · doi:10.1002/1097-0207(20010110)50:1<1::AID-NME14>3.0.CO;2-P |
| [4] | Sukumar N., Moran B., Belytschko T.B.: The natural element method in solid mechanics. Int. J. Numer. Methods Eng. 43, 839–887 (1998) · Zbl 0940.74078 · doi:10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R |
| [5] | Lancaster P., Salkauskas K.: Curve and surface fitting: an introduction. Academic Press, London (1990) · Zbl 0649.65012 |
| [6] | Atluri S.N., Kim H.G., Cho J.Y.: A critical assessment of the truly meshless local Petrov-Galerkin and local boundary integral equation methods. Comput. Mech. 24, 348–372 (1999) · Zbl 0977.74593 · doi:10.1007/s004660050457 |
| [7] | Atluri S.N., Zhu T.: New concepts in meshless methods. Int. J. Numer. Methods Eng. 47, 537–556 (2000) · Zbl 0988.74075 · doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<537::AID-NME783>3.0.CO;2-E |
| [8] | Chen J.S., Wu C.T., Yoon S., You Y.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001) · Zbl 1011.74081 · doi:10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A |
| [9] | Yvonnet J., Ryckelynck D., Lorong Ph., Chinesta F.: A new extension of the natural element method for non-convex and discontinuous domains: the constrained natural element method (c-nem). Int. J. Numer. Methods Eng. 60, 1451–1474 (2004) · Zbl 1060.74678 · doi:10.1002/nme.1016 |
| [10] | Yvonnet, J.: Nouvelles approches sans maillage basées sur la méthode des éléments naturels pour la simulation numérique des procédés de mise en forme. [Ph.D. Thesis], Ecole Nationale Supérieure d’Arts et Métiers (2004) |
| [11] | Gonzalez D., Cueto E., Martinez M.A., Doblare M.: Numerical integration in natural neighbour Galerkin methods. Int. J. Numer. Methods Eng. 60, 2077–2104 (2004) · Zbl 1070.74048 · doi:10.1002/nme.1038 |
| [12] | Yagawa, G., Matsubara, H.: Enriched element method and its application to solid mechanics. In: Computational methods in Engineering and Science 2006, pp. 15–18. Proc. EPMESC X, Aug. 21–23, Tsinghua University Press, Springer |
| [13] | Nayroles B., Touzot G., Villon P.: Generalizing the finite element method: diffuse approximation and diffuse elements. J. Comput. Mech. 10, 307–318 (1992) · Zbl 0764.65068 · doi:10.1007/BF00364252 |
| [14] | Kronghauz Y., Belytchko T.: Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput. Methods Appl. Mech. Eng. 131, 133–145 (1996) · Zbl 0881.65098 · doi:10.1016/0045-7825(95)00954-X |
| [15] | Belytchko T., Gu L., Lu Y.Y.: Fracture and crack growth by element free Galerkin methods. Model. Simul. Mater. Sci. Eng. 2, 519–534 (1994) · doi:10.1088/0965-0393/2/3A/007 |
| [16] | Gavette L., Benito J.J., Falcon S., Ruiz A.: Penalty functions in constrained variational principles for element free Galerkin method. Eur. J. Mech. A/Solids 19, 699–720 (2000) · Zbl 0992.74078 · doi:10.1016/S0997-7538(00)00168-6 |
| [17] | Belytchko T., Lu Y.Y., Gu L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205 |
| [18] | Lu Y.Y., Belytchko T., Gu L.: A new implementation of the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 113, 397–414 (1994) · Zbl 0847.73064 · doi:10.1016/0045-7825(94)90056-6 |
| [19] | Kaljevic I., Saigal S.: An improved element-free Galerkin formulation. Int. J. Numer. Methods Eng. 40, 2953–2974 (1997) · Zbl 0895.73079 · doi:10.1002/(SICI)1097-0207(19970830)40:16<2953::AID-NME201>3.0.CO;2-S |
| [20] | Ghosh S., Moorthy S.: Particle fracture simulation in non-uniform microstructures of metal-matrix composites. Acta Mater. 46, 965–982 (1998) · doi:10.1016/S1359-6454(97)00289-9 |
| [21] | Moorthy S., Ghosh S.: A Voronoi cell finite element model for particle cracking in elastic-plastic composite materials. Comput. Methods Appl. Mech. Eng. 151, 377–400 (1998) · Zbl 0907.73066 · doi:10.1016/S0045-7825(97)00160-6 |
| [22] | Hu C., Bai J., Ghosh S.: Micromechanical and macroscopic models of ductile fracture in particle reinforced metallic materials. Model. Simul. Mater. Sci. Eng. 15(4), 5377–5392 (2007) |
| [23] | Zhang H.W., Wang H., Chen B.S., Xie Z.Q.: Analysis of Cosserat materials with Voronoi cell finite element method and parametric variational principle. Comput. Methods Appl. Mech. Eng. 197, 741–755 (2008) · Zbl 1169.74330 · doi:10.1016/j.cma.2007.09.003 |
| [24] | Vu-Quoc L., Tan X.G.: Optimal solid shells for non-linear analyses of multilayer composites. I. Statics. Comput. Methods Appl. Mech. Eng. 192, 975–1016 (2003) · Zbl 1091.74524 · doi:10.1016/S0045-7825(02)00435-8 |
| [25] | Herrmann, L.R., Roms, R.M.: A reformulation of the elastic field equations in terms of displacements valid for all admissible values of Poisson’s ratio. Trans. ASME, Int. J. Appl. Mech. 140–141 (1964) |
| [26] | Fraeijs de Veubeke, B. M.: Diffusion des inconnues hyperstatiques dans les voilures à longerons couplés. Bull. Serv. Technique de l’Aéronautique N{\(\deg\)} 24, Imprimerie Marcel Hayez, Bruxelles, 56 (1951) |
| [27] | Belikov V.V., Ivanov V.D., Kontorovich V.K., Korytnik S.A., Semonov A.Y.: The non Sibsonian interpolation: a new method of interpolation of the values of a function on an arbitrary set of points. Comput. Math. Math. Physics 37(1), 9–15 (1997) · Zbl 0948.65005 |
| [28] | Hiyoshi H., Sugihara K.: Two generations of an interpolant based on Voronoi diagrams. Int. J. Shape Modelling 5(2), 219–231 (1999) · Zbl 0957.52502 · doi:10.1142/S0218654399000186 |
| [29] | Sibson R.: A vector identity for the Dirichlet tessellation. Math. Proc. Camb. Philos. Soc. 87, 151–155 (1980) · Zbl 0466.52010 · doi:10.1017/S0305004100056589 |
| [30] | Rossi, B., Pascon, F., Habraken, A.M.: Theoretical and numerical evaluation of residual stresses in thin-walled sections, Report N{\(\deg\)} AG2007-02/01, ArGEnCo Department, Division of Mechanics of Solids, Fluids and Structures (2007) · Zbl 1173.74389 |
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.
