11th. World Congress on Computational Mechanics Convergence analysis of configurational forces for brittle cracks modeled through C k -Generalized FEM Diego Amadeu F. Torres Clovis S. de Barcellos Department of Mechanical Engineering Paulo de Tarso R. Mendonça Federal University of Santa Catarina Brazil
Presentation topics C k -GFEM - Construction of continuous Partition of Unity at C - Construction of a smooth approximation subspace - Elshebian mechanics as tool to post-processing of J-integral - Quality assessment through global and local measures - Concluding remarks
GFEM/XFEM versus C k -GFEM Continuous partition of unity with C k -GFEM Defining an approximation subspace - Regularity; Quality assessment through global - Polynomial reproducibility; measures Configurational forces - Efficient enrichment patterns; method Quality assessment - Flat-top property; through local measures - Integration cost; Smoothness, enrichments - Integration of C 0 and C k -GFEM. 3
C k partition of unity – convex clouds C Arbitrary patch shape; Arbitrary element shape; free of coordinate mapping; Continuous partition of unity with C k -GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method k = ∞ Quality assessment through local measures Smoothness, k = p -1 enrichments Concluding remarks Edwards, C ∞ finite element basis functions , Report 45, Institute for Computational Engineering and Sciences – The University of Texas at Austin, 1996 Duarte, Kim and Quaresma, Arbitrarily smooth generalized finite element approximations . Computer Methods in Applied Mechanics and Engineering, 196 (2006) Barcellos, Mendonça and Duarte, A Ck continuous generalized finite element formulation applied to 4 laminated Kirchhoff plate model . Computational Mechanics, 44 (2009)
C ∞ partition of unity – convex clouds C Continuous partition of unity with C k -GFEM Defining an approximation α subspace Quality assessment through global M α = 6 measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Some improvements beyond... 5
Galerkin aproximation Continuous partition of unity with C k -GFEM Defining an if p=3 approximation subspace Quality assessment for reducing mesh through global e.g. measures dependences Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks Belytschko and Black, Elastic crack growth in finite elements with minimal remeshing . International 6 Journal for Numerical Methods in Engineering, 45 (1999)
Polynomial reproducibility of the approximation C 0 PoU (conventional tent for C b= p +1 FEM shape function) Continuous partition of unity with C k -GFEM C k PoU for C b= p Defining an approximation subspace p = degree of polynomial enrichment Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Mendonça, Barcellos and Torres, Robust Ck/C0 generalized FEM approximations for higher-order Concluding remarks conformity requirements: application to Reddy’s HSDT model for anisotropic laminated plates . Composite Structures, 96 (2013) Mendonça, Barcellos and Torres, Analysis of anisotropic Mindlin plate model by continuous and non- continuous GFEM . Finite Element in Analysis and Design, 47 (2011) Barcellos, Mendonça and Duarte, A Ck continuous generalized finite element formulation applied to laminated Kirchhoff plate model . Computational Mechanics, 44 (2009) 7
Geometric enrichment pattern Continuous partition of unity with C k -GFEM Defining an approximation subspace Quality assessment through global measures R1 > R2 Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks branch functions on orange nodes and inside the circles, 8 and uniform polynomial enrichment
Convergence in terms of global values mode I opening crack loading C 0 Continuous partition of unity with C k -GFEM Defining an geometric pattern of approximation singular enrichment subspace p – convergence ! Quality assessment through global + measures Configurational forces method uniform polynomial Quality assessment enrichment through local measures C k Smoothness, enrichments Concluding remarks 9
Configurational mechanics Continuous partition of unity with C k -GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment , on Ω , on Ω through local measures Smoothness, enrichments Concluding remarks Eshelby, The force on an elastic singularity . Philosophical Transactions of the Royal Society A: mathematical, physical and engineering sciences, 244 (1951) Kienzler and Herrmann, Mechanics in material space with applications to defect and fracture mechanics . Springer, 2000 Ruter and Stein, On the duality of global finite element discretization error control in small strain 10 Newtonial and Eshelbian mechanics. Technische Mechanik, 23 (2003)
Variational balance of material linear momentum strong-form , where Eshelby tensor inhomogeneity force Continuous partition of unity with C k -GFEM and defining Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks weak-form 11
Post-processing of nodal configurational forces Continuous partition of unity with C k -GFEM Defining an approximation subspace Eshelby tensor Quality assessment through global measures Configurational forces ∴ method Quality assessment through local measures where Smoothness, enrichments thus, defining Concluding remarks nodal configurational 12 force
Local measure using configurational forces Continuous partition of unity with C k -GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local Eshelby, The force on an elastic singularity . Philosophical Transactions of the Royal Society A: measures mathematical, physical and engineering sciences, 244 (1951) Smoothness, enrichments Kienzler and Herrmann, Mechanics in material space with applications to defect and fracture mechanics . Springer, 2000 Concluding remarks Mueller and Maugin, On material forces and finite element discretizations . Computational Mechanics, 29 (2002) Glaser and Steinmann, On material forces within the extended finite element method . Proceedings of the sixth European Solid Mechanics Conference (2006) Häusler, Lindhorst and Horst, Combination of the material force concept and the extended finite element method for mixed mode crack growth simulations . International Journal for Numerical Methods 13 in Engineering, 85 (2011)
Topologic enrichment pattern Continuous partition of unity with C k -GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks - Branch functions on orange nodes; - Uniform p -enrichment 14 - Local p -enrichment around the crack tip
Mixed mode loading K I = 1.0, K II = 1.0 Continuous partition of unity with C k -GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks 15
Convergence in global values p = 0 p = 1 Continuous partition of p = 2 unity with C k -GFEM p = 3 Defining an uniform polynomial Topologic pattern of approximation enrichment, singular enrichment subspace p = 1 b = 1 ... Quality assessment p = 4 through global + measures Configurational forces method Localized polynomial Quality assessment enrichment, p p = 1 through local measures p = 2 Smoothness, p = 3 enrichments Concluding remarks uniform polynomial p = 2 enrichment, p = 3 b = 2 p = 4 16
Convergence of J-integral Continuous partition of unity with C k -GFEM Defining an Topologic pattern of approximation singular enrichment subspace uniform polynomial Quality assessment enrichment, through global + b = 1 measures Configurational forces method Localized polynomial Quality assessment enrichment through local measures Smoothness, enrichments Concluding remarks uniform polynomial enrichment, b = 2 17
Exact error dispersion: y -stress Uniform polynomial Continuous partition of Topologic pattern of unity with C k -GFEM enrichment, singular enrichment Defining an b = 1 approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks 18 C ∞, 451 DOFs C0, 191 DOFs
Exact error dispersion: y -stress uniform polynomial enrichment, b = 1 Continuous partition of topologic pattern of unity with C k -GFEM + singular enrichment Defining an localized polynomial approximation subspace enrichment, Quality assessment p = 2 through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks 19 C ∞, 469 DOFs C0, 203 DOFs
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