The hierarchical finite cell method for nonlinear problems: Moment fitting quadratures, basis function removel, and remeshing

Produktbeschreibung

In this thesis, several approaches are discussed in order to further enhance the performance of the finite cell method (FCM). Thereby, novel moment fitting quadrature schemes are introduced that allow to reduce the effort of the numerical integration process significantly. Further, a basis function removal scheme is proposed to improve the conditioning behavior of the resulting equation system. Finally, an innovative remeshing strategy is presented that overcomes the problem of severely distorted elements for simulations with large deformations.

Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Goal and scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Basic elements of continuum mechanics 6
2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Motion and deformation . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Strain measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Equilibrium and stress measures . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Stress measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 Small strain elastoplasticity . . . . . . . . . . . . . . . . . . . . . . 16
2.3.4 Finite strain plasticity . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Strong and weak form of equilibrium . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Strong and weak form in the initial configuration . . . . . . . . . . 19
2.4.2 Strong and weak form in the current configuration . . . . . . . . . . 20
2.5 Linearization of the weak form . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.1 Linearized weak form in the initial configuration . . . . . . . . . . . 20
2.5.2 Linearized weak form in the current configuration . . . . . . . . . . 22
3 The finite cell method 23
3.1 Fictitious domain approach . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Weak forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Linearized weak forms . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Discretization of the weak forms . . . . . . . . . . . . . . . . . . . . 27
3.3 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Gaussian quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Adaptive Gaussian quadrature scheme . . . . . . . . . . . . . . . . 33
4 Moment fitting quadratures 36
4.1 Moment fitting approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.2 Point distribution schemes . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.3 Computation of the moments . . . . . . . . . . . . . . . . . . . . . 42
4.1.4 Computation of the weights . . . . . . . . . . . . . . . . . . . . . . 43
4.1.5 Optimized points and weights . . . . . . . . . . . . . . . . . . . . . 43
4.1.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.6.1 Cell cut by a sphere . . . . . . . . . . . . . . . . . . . . . 45
4.1.6.2 Recovery of the Gauss-Legendre quadrature . . . . . . . . 54
4.2 Adaptive moment fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Moment fitting without solving an equation system . . . . . . . . . 57
4.2.2 Computation of the moment fitting weights . . . . . . . . . . . . . 59
4.3 Applications to the finite cell method . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Hydrostatic sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2 Porous material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.2.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.2.2 Small strain elastoplasticity . . . . . . . . . . . . . . . . . 69
4.3.3 Cube with a cylindrical hole . . . . . . . . . . . . . . . . . . . . . . 72
4.3.4 Thick-walled plate with a circular hole . . . . . . . . . . . . . . . . 75
5 Basis function removal for the FCM 79
5.1 A simple function removal strategy for the hierarchical basis . . . . . . . . 82
5.1.1 Affected and nonaffected modes of the hierarchical basis . . . . . . 83
5.1.2 Removal criterion of affected modes . . . . . . . . . . . . . . . . . . 84
5.1.3 Implementation scheme . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Benchmark problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.2 Small strain elastoplasticity . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Finite strain problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1 Single cube connector under pressure . . . . . . . . . . . . . . . . . 98
5.3.2 Complex cube connector under pressure . . . . . . . . . . . . . . . 107
5.3.3 Single pore of a foam-like structure under pressure . . . . . . . . . . 116
6 A remeshing strategy for the FCM 124
6.1 Kinematic relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2 Remeshing procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2.1 Remeshing criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2.1.1 Ratio of Jacobians . . . . . . . . . . . . . . . . . . . . . . 129
6.2.1.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2.1.3 Inverse aspect ratio . . . . . . . . . . . . . . . . . . . . . . 130
6.2.1.4 Performance of the suggested remeshing criteria . . . . . . 130
6.2.2 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2.3 Data transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 Finite strain problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.1 Plate with a circular hole . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.2 Single cube connector . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3.3 Complex cube connector . . . . . . . . . . . . . . . . . . . . . . . . 146
7 Summary and outlook 150
Bibliography 154

Keywords: Finite cell method, Fictitious domain approach, High-order finite element methods, Numerical integration, Moment fitting quadratures, Basis function removal, Remeshing, Data transfer, Nonlinear problems, Finite strain problems, Finite cell method, Fictitious domain approach, High-order finite element methods, Numerical integration, Moment fitting quadratures, Basis function removal, Remeshing, Data transfer, Nonlinear problems, Finite strain problems