Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have proposed to use various finite elements discretizations. We provide in this paper a mathematical analysis which ensures the convergence of the finite element method, even with finite elements with discontinuous derivatives such as the quadratic 6 node Lagrange triangles and the cubic Hermite triangles. More precisely, we prove the -convergence of the discretized problems towards the continuous limit analysis problem. Numerical results illustrate the relevance of this analysis for the yield design of both homogeneous and non-homogeneous materials.
DOI: 10.1051/m2an/2015040
Keywords: Bounded Hessian functions, Finite element method, Γ-convergence
@article{M2AN_2016__50_1_215_0, author = {Bleyer, J\'er\'emy and Carlier, Guillaume and Duval, Vincent and Mirebeau, Jean-Marie and Peyr\'e, Gabriel}, title = {A $\Gamma{}${-Convergence} {Result} for the {Upper} {Bound} {Limit} {Analysis} of {Plates}}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {215--235}, publisher = {EDP-Sciences}, volume = {50}, number = {1}, year = {2016}, doi = {10.1051/m2an/2015040}, zbl = {1353.74068}, mrnumber = {3460107}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015040/} }
TY - JOUR AU - Bleyer, Jérémy AU - Carlier, Guillaume AU - Duval, Vincent AU - Mirebeau, Jean-Marie AU - Peyré, Gabriel TI - A $\Gamma{}$-Convergence Result for the Upper Bound Limit Analysis of Plates JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 215 EP - 235 VL - 50 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015040/ DO - 10.1051/m2an/2015040 LA - en ID - M2AN_2016__50_1_215_0 ER -
%0 Journal Article %A Bleyer, Jérémy %A Carlier, Guillaume %A Duval, Vincent %A Mirebeau, Jean-Marie %A Peyré, Gabriel %T A $\Gamma{}$-Convergence Result for the Upper Bound Limit Analysis of Plates %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 215-235 %V 50 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015040/ %R 10.1051/m2an/2015040 %G en %F M2AN_2016__50_1_215_0
Bleyer, Jérémy; Carlier, Guillaume; Duval, Vincent; Mirebeau, Jean-Marie; Peyré, Gabriel. A $\Gamma{}$-Convergence Result for the Upper Bound Limit Analysis of Plates. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 1, pp. 215-235. doi : 10.1051/m2an/2015040. http://www.numdam.org/articles/10.1051/m2an/2015040/
R.A. Adams, Sobolev Spaces. Academic Press New York (1975). | MR | Zbl
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr. Oxford University Press (2000). | MR | Zbl
Existence result and discontinuous finite element discretization for a plane stresses hencky problem. Math. Meth. Appl. Sci. 11 (1989) 169–184. | DOI | MR | Zbl
and ,On the performance of non-conforming finite elements for the upper bound limit analysis of plates. Int. J. Numer. Methods Eng. 94 (2013) 308–330. | DOI | MR | Zbl
and ,M.W. Bræstrup, Yield-line Theory and Limit Analysis of Plates and Slabs. Danmarks Tekniske Højskole, Afdelingen for Bærende Konstruktioner (1971).
A. Braides, Gamma-convergence for Beginners. Oxford University Press, USA (2002). | MR | Zbl
S.C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. Texts Appl. Math. Springer (2008). | MR | Zbl
Limit analysis of plates- a finite element formulation. Struct. Eng. Mech. 8 (1999) 325–341. | DOI
and ,Justification of the 2-dimensional linear plate model. J. Mécanique 18 (1979) 315–344. | MR | Zbl
and ,G. Dal Maso, An Introduction to -Convergence. Vol. 8 of Progr. Nonlin. Differ. Eq. Appl. Birkhäuser, Boston, MA (1993). | MR | Zbl
Problèmes variationnels en plasticité parfaite des plaques. Numer. Funct. Anal. Optim. 6 (1983) 73–119. | DOI | MR | Zbl
,Fonctions à hessien borné. Ann. Inst. Fourier 34 (1984) 155–190. | DOI | Numdam | MR | Zbl
,Compactness theorems for spaces of functions with bounded derivatives and applications to limit analysis problems in plasticity. Archive for Rational Mechanics and Anal. 105 (1989) 123–161. | DOI | MR | Zbl
,Convex function of a measure and its applications. Indiana Univ. Math. J. 33 (1984) 673–709. | DOI | MR | Zbl
and ,Limit analysis for plates: the exact solution for a clamped square plate of isotropic homogeneous material obeying the square yield criterion and loaded by uniform pressure. Philos. Trans. Roy. Soc. London. Ser. A, Math. Phys. Sci. 277 (1974) 121–155. | Zbl
,Fonction convexe d’une mesure. C.R. Acad. Sci. Paris 301 (1985) 687–690. | MR | Zbl
,Étude dans d’un modèle de plaques élastoplastiques comportant une non-linéarité géométrique. ESAIM: M2AN 19 (1985) 235–283. | DOI | Numdam | MR | Zbl
,T. Hadhri, Présentation et analyse mathématique de quelques modèles pour des structures élastoplastiques homogènes ou hétérogènes. Thèse d’état, Université Pierre et Marie Curie, Paris (1986).
Prise en compte d’une force linéique de frontière dans un modèle de plaques de Hencky comportant une non-linéarité géométrique. ESAIM: M2AN 22 (1988) 457–468. | DOI | Numdam | MR | Zbl
,Numerical methods for the limit analysis of plates. J. Appl. Mech. 35 (1968) 796. | DOI | Zbl
and ,On the identification of yield-line collapse mechanisms. Eng. Struct. 18 (1996) 332–337. | DOI
,Mechanism determination by automated yield-line analysis. Struct. Eng. 72 (1994) 323–323.
.Limit analysis of plates using the EFG method and second-order cone programming. Int. J. Numer. Methods Eng. 78 (2009) 1532–1552. | DOI | MR | Zbl
, and ,Upper and lower bound limit analysis of plates using FEM and second-order cone programming. Comput. Struct. 88 (2010) 65–73. | DOI
, and ,Mosek, The Mosek optimization toolbox for MATLAB manual (2008).
Yield line method by finite elements and linear programming. Struct. Eng. 56 (1978) 37–44.
and ,On the convergence of adaptive nonconforming finite element methods for a class of convex variational problems. SIAM J. Numer. Anal. 49 (2011) 346–367. | DOI | MR | Zbl
and ,W. Prager, An Introduction to Plasticity. Addison-Wesley series in the engineering sciences: Mechanics and thermodynamics. Addison-Wesley Pub. Co. (1959). | MR | Zbl
T. Rockafellar, Convex Analysis. Vol. 224 of Grundlehren der Mathematischen Wissenschaften. Princeton University Press, 2nd edition (1983). | MR
M.A. Save, C.E. Massonnet and G. de Saxce, Plastic Limit Analysis of Plates, Shells, and Disks, vol. 43. North Holland, 1997. | MR | Zbl
Epi-limit on HB and homogenization of heterogeneous plastic plates. Nonlinear Anal. 25 (1995) 499–529. | DOI | MR | Zbl
,R. Temam, Mathematical Problems in Plasticity. Gauthier-Villars Paris (1985). | MR
Upper bound limit analysis of plates utilizing the C1 natural element method. Comput. Mech. 50 (2012) 543–561. | DOI | MR | Zbl
, and ,Cited by Sources: