Finite Elements for Large Strains - A double mixed (M 2 ) Formulation

Transcription

1 Finite Elements for Large Strains - A double mixed (M 2 ) Formulation

2 Motivation Development of user friendly elements robustness simple treatment of incompressible materials complex geometries geometrical and material nonlinearities coarse and non-uniform meshes efficiency simple implementation

3 Numerical simulation of incompressible materials is still challenging within the context of large deformations Challenges: Locking in incompressibility Large deformations Non-convex element shapes Robustness

4 Deformation Measures F (X) = Grad ϕ(x) (F, cof F, detf ) dx = F dx, n da = cof[f ] N da and dv =det[f ] dv cof [F ] = F 22 F 33 F 32 F 23 F 31 F 23 F 21 F 33 F 21 F 32 F 31 F 22 F 32 F 13 F 12 F 33 F 11 F 33 F 31 F 13 F 31 F 12 F 11 F 32 F 12 F 23 F 22 F 13 F 21 F 13 F 11 F 23 F 11 F 22 F 21 F 12 C = F T F

5 Polyconvex Material Potential Hu-Washizu Functional

6 Hu-Washizu Functional

7 Hu-Washizu Functional Variation

8 Hu-Washizu Functional Choice of Strain Energy Function

9 Hu-Washizu Functional Choice of Strain Energy Function Linearization, Definitions

10 Linearization of Hu-Washizu Functional

11 Finite Element Approximation

12 Finite Element Approximation

13 Finite Element Approximation

14 Newton Solution Procedure

15 Mixed finite element formulation Displacement field

16 Mixed finite element formulation Displacement field Pressure Terms constant

17 Mixed finite element formulation Approximation of co factor Approximation of Lagrange Multiplier

18 Mixed finite element formulation

19 Mixed finite element formulation

20 Mixed finite element formulation Block structure of

21 Mixed finite element formulations Three-dimensional tetraeder

22 Local Algorithm at element level

23 Local algorithm at element level

24 Local algorithm at element level

25 Examples: isotropic strain energy function Mixed form

26 Examples: isotropic strain energy function Mixed form Stress free inital configuration

27 Examples: isotropic strain energy function Mixed form Stress free inital configuration

28 Mesh distortion sensitivity: Cantilever beam

29 Mesh distortion sensitivity: Cantilever beam

30 Mesh distortion sensitivity: Cantilever beam

31 Cook s membran problem

32 Cook s membran problem

33 Cook s membran problem

34 Cook s membran problem

35 Anisotropic strain energy function Schröder et al. 2009

36 Anisotropic strain energy function Mixed form Schröder et al. 2009

37 Anisotropic strain energy function Mixed form Schröder et al. 2009

38 Anisotropic strain energy function Mixed form Schröder et al. 2009

39 Cook s membran problem

40 Cook s membran problem

41 Simulation of arterial walls

42 Simulation of arterial walls Isotropic material model for calcification and extracellular lipid Transversly isotropic material model for media and adventitia Complete strain energy function for media and adventitia Isotropic material model for plaque Schröder & Neff 2004, Balzani 2006, Brands et al. 2008

43 Simulation of arterial walls

44 Simulation of arterial walls

45 36 Development of the EI9 element Motivation Split of the strain energy function - Different Treatment for homogenous and inhomogenous part Variational formulation Ansatz and Implementation Numerical Tests - non-uniformly meshed beam - incompressible block - surface buckling Conclusion and outlook

46 37 Split of the Strain Energy Function Additive split into a homogeneous and an inhomogeneous part (Nadler & Rubin, 2003) Split of the deformation gradient: homogeneous part

47 38 Homogeneous Part Compressible Neo-Hooke Material with

48 39 Inhomogeneous Part (1) Linear Elastic Material: with (Nadler & Rubin, 2003)

49 40 Inhomogeneous Part (2) Multiplicative Split of the inhomogeneous part of the displacement gradient leading to with Ansatz for inhomogeneous part of to leading (Wilson, Taylor, Doherty & Ghaboussi, 1973)

50 41 Potential and Variation Hu-Washizu

51 41 Potential and Variation Hu-Washizu

52 41 Potential and Variation Hu-Washizu with

53 42 Ansatz and Implementation

54

55 with weak form

56 45 Residual and tangent stiffness matrix Equation system on element level (Simo, Armero & Taylor, 1993)

57 Numerical Example: Objectivity Test

58 47 Numerical Example: Beam (1) System Deformed System

59 48 Numerical Example: Beam (2)

60 Numerical Example: Thin Plate Ring

61 Numerical Example: Thin Plate Ring (2)

62 Numerical Example: Spherical Shell

63 52 Numerical Example: Block (1) System Deformed System

64 53 Numerical Example: Block (2)

65 Numerical Example: Surface Buckling 54 Q1 Q2 Q1P0 Q1/EI9 (b) QM1/E12 (c)

66 55 Conclusion and Outlook Advantages of Q1/EI9 robust, no hourglassing no adjustment of any parameters works for coarse and unaligned meshes handles incompressible material has good bending properties Outlook Inplementation of inelastic constitutive models (more complex than for standard elements) Solution by TSCG12 element

67 TSCG12 Element for plasticity using 12 enhanced modes and an expansion of shape function derivatives using Taylor series Korelc, Wriggers, Soric ()!