Convergence analysis of configurational forces for brittle cracks - - PowerPoint PPT Presentation

▶
convergence analysis of configurational forces for
SMART_READER_LITE
LIVE PREVIEW

Convergence analysis of configurational forces for brittle cracks - - PowerPoint PPT Presentation

Mar 13, 2023 129 likes •432 views

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

slide-1
SLIDE 1

Convergence analysis of configurational forces for brittle cracks modeled through Ck-Generalized FEM

Diego Amadeu F. Torres

Department of Mechanical Engineering

Federal University of Santa Catarina Brazil Clovis S. de Barcellos Paulo de Tarso R. Mendonça

  • 11th. World Congress on

Computational Mechanics

slide-2
SLIDE 2

Presentation topics

  • Construction of continuous Partition of Unity at C

C k -GFEM

  • 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
slide-3
SLIDE 3
  • Regularity;
  • Polynomial reproducibility;
  • Efficient enrichment patterns;
  • Flat-top property;
  • Integration cost;
  • Integration of C0 and Ck-GFEM.

GFEM/XFEM versus Ck-GFEM

3

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments

slide-4
SLIDE 4

C C k partition of unity – convex clouds

Edwards, C∞ finite element basis functions, Report 45, Institute for Computational Engineering and Sciences – The University of Texas at Austin, 1996

Arbitrary patch shape; Arbitrary element shape; free of coordinate mapping; k = p -1 k = ∞

Barcellos, Mendonça and Duarte, A Ck continuous generalized finite element formulation applied to laminated Kirchhoff plate model. Computational Mechanics, 44 (2009) Duarte, Kim and Quaresma, Arbitrarily smooth generalized finite element approximations. Computer Methods in Applied Mechanics and Engineering, 196 (2006)

4

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-5
SLIDE 5

C C ∞partition of unity – convex clouds

α Mα= 6 5

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Some improvements beyond...

slide-6
SLIDE 6

Galerkin aproximation

e.g. for reducing mesh dependences if p=3

Belytschko and Black, Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45 (1999)

6

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-7
SLIDE 7

Polynomial reproducibility of the approximation

Mendonça, Barcellos and Torres, Analysis of anisotropic Mindlin plate model by continuous and non- continuous GFEM. Finite Element in Analysis and Design, 47 (2011) Mendonça, Barcellos and Torres, Robust Ck/C0 generalized FEM approximations for higher-order conformity requirements: application to Reddy’s HSDT model for anisotropic laminated plates. Composite Structures, 96 (2013)

b=p+1 for C C 0 PoU (conventional tent FEM shape function) p = degree of polynomial enrichment b=p for C C k PoU

Barcellos, Mendonça and Duarte, A Ck continuous generalized finite element formulation applied to laminated Kirchhoff plate model. Computational Mechanics, 44 (2009)

7

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-8
SLIDE 8

Geometric enrichment pattern

>

R1 R2

branch functions on orange nodes and inside the circles, and uniform polynomial enrichment 8

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-9
SLIDE 9

Convergence in terms of global values

p – convergence !

geometric pattern of singular enrichment uniform polynomial enrichment

+

mode I opening crack loading

9

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

Ck C0

slide-10
SLIDE 10

Configurational mechanics

Kienzler and Herrmann, Mechanics in material space with applications to defect and fracture

  • mechanics. Springer, 2000

Eshelby, The force on an elastic singularity. Philosophical Transactions of the Royal Society A: mathematical, physical and engineering sciences, 244 (1951)

10

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

Ruter and Stein, On the duality of global finite element discretization error control in small strain Newtonial and Eshelbian mechanics. Technische Mechanik, 23 (2003)

, on Ω , on Ω

slide-11
SLIDE 11

Variational balance of material linear momentum

inhomogeneity force Eshelby tensor 11

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

strong-form weak-form , where and defining

slide-12
SLIDE 12

Post-processing of nodal configurational forces

12

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

where

∴

thus, defining nodal configurational force Eshelby tensor

slide-13
SLIDE 13

Local measure using configurational forces

Kienzler and Herrmann, Mechanics in material space with applications to defect and fracture

  • mechanics. Springer, 2000

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 in Engineering, 85 (2011) Eshelby, The force on an elastic singularity. Philosophical Transactions of the Royal Society A: mathematical, physical and engineering sciences, 244 (1951)

13

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-14
SLIDE 14

Topologic enrichment pattern

  • Branch functions on orange nodes;
  • Uniform p-enrichment
  • Local p-enrichment around the crack tip

14

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-15
SLIDE 15

Mixed mode loading

KI = 1.0, KII = 1.0

15

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-16
SLIDE 16

Convergence in global values

Topologic pattern of singular enrichment uniform polynomial enrichment, b = 1 uniform polynomial enrichment, b = 2 Localized polynomial enrichment, p

+

p = 1 p = 2 p = 1 p = 0 p = 1 p = 2 p = 3 16

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

p = 4 ... p = 3 p = 2 p = 3 p = 4

slide-17
SLIDE 17

Convergence of J-integral

uniform polynomial enrichment, b = 1 uniform polynomial enrichment, b = 2 Topologic pattern of singular enrichment Localized polynomial enrichment

+

17

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-18
SLIDE 18

Exact error dispersion: y-stress

C∞, 451 DOFs C0, 191 DOFs Topologic pattern of singular enrichment Uniform polynomial enrichment, b = 1

18

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-19
SLIDE 19

Exact error dispersion: y-stress

uniform polynomial enrichment, b = 1 localized polynomial enrichment, p = 2 C0, 203 DOFs C∞, 469 DOFs topologic pattern of singular enrichment

+

19

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-20
SLIDE 20

Exact error dispersion: y-stress

topologic pattern of singular enrichment uniform polynomial enrichment, b = 2 C0, 451 DOFs C∞, 841 DOFs

20

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-21
SLIDE 21

Collecting data of the three cases...

21

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

b=2, 451 DOFs, - 85x10-3 b=1, p=2, 203 DOFs, 130x10-3 b=1, 191 DOFs, 145x10-3 b=2, 841 DOFs, - 19x10-3 b=1, p=2, 469 DOFs, 62x10-3 b=1, 451 DOFs, 62x10-3

C C 0 C C ∞

slide-22
SLIDE 22

Angle of crack growth: θADV

KI = 1.0, KII = 1.0

θADV 22

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

slide-23
SLIDE 23

Angle of crack advance: θADV

Configurational forces are restorative forces !

23

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

R

2

slide-24
SLIDE 24

Angle of crack advance: θADV, b = 1

Topologic pattern of singular enrichment Uniform polynomial enrichment, b = 1

Crack tip enrichment Nodes for the configurational force

~3% ~8% ~1% 24

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

C0 Ck

slide-25
SLIDE 25

Topologic pattern of singular enrichment Uniform polynomial enrichment, b = 2

25

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments Concluding remarks

C0 Ck Angle of crack advance: θADV, b = 2

slide-26
SLIDE 26

Joining Ck-GFEM and C0-GFEM

Shepard equation -> Application of smooth PoUs only where they are convenient ! C 0 and C k weighting functions Partition of unity

26

Continuous partition of unity with Ck-GFEM Defining an approximation subspace Quality assessment through global measures Configurational forces method Quality assessment through local measures Smoothness, enrichments and conditioning Some improvements Concluding remarks

Straightforward!!!

slide-27
SLIDE 27

Concluding remarks

Disavantages: - Lower polynomial reproducibility than C C 0 -GFEM /XFEM;

  • Higher integration cost;

Solution: - Apply smooth PoUs only at convenient zones of the model ! C k -GFEM shows better estimates of J and θADV than C 0 -GFEM:

  • Probably due to the well determined flat-top;
  • Smoothness seems to reduce transition effects around singular enrichment;
  • Lower dependence with enrichment pattern;
  • Lower dependence on the size of the region used to compute configurational

forces;

  • In general, C k -GFEM combines:
  • Higher regularity;
  • Flat-top property;
  • Definition on global coordinates -> admits extremely distorted meshes;
  • Compact support.
slide-28
SLIDE 28

Acknowledgements

National Council for Scientific and Technological Development

Ministry of Science, Technology and Innovation

  • f Brazil
slide-29
SLIDE 29

Thank you!

mendonca@grante.ufsc.br