Improving the robustness of the finite cell method for problems with large deformations
Zusammenfassung
Zahlreiche Ingenieuranwendungen mit komplexen Strukturen und mikrostrukturierten Materialien wie Schaumstoffen oder 3D-gedruckten Objekten stellen eine Herausforderung für die traditionelle Finite-Elemente-Methode (FEM) dar. Die Finite-Cell-Methode (FCM) kombiniert den fiktiven Gebietsansatz mit hierarchischen Ansatzfunktionen, um den Netzgenerierungsaufwand zu reduzieren. Eine nicht-negative Moment-Fitting-Quadratur-Regel wird entwickelt, um die Integration von gebrochenen Zellen zu verbessern und die NewtonRaphson-Methode in der nichtlinearen Berechnung zu stabilisieren. Zusätzlich wird eine Eigenwert-Stabilisierungs-Technik entwickelt, um die Konditionszahl der Schnittzellen zu reduzieren. Ein Remeshing-Verfahren wird entwickelt, um die FCM-Robustheit zu erhöhen und größere Lastschritte zu ermöglichen. Numerische Beispiele veranschaulichen die Wirksamkeit dieser Ansätze. Engineering applications often involve complex structures and microstructured materials, such as foams or 3D printed objects, posing challenges for traditional finite element methods due to the complex geometry. The finite cell method (FCM) is employed to eliminate mesh creation overhead, combining the fictit...
Schlagworte
- I–VIII Titelei/Inhaltsverzeichnis I–VIII
- 1–5 1 Introduction 1–5
- 1.1 Motivation
- 1.2 Aims and scope of this work
- 1.3 Outline of the thesis
- 6–16 2 Basic equations of continuum mechanics 6–16
- 2.1 Kinematics
- 2.1.1 Motion and deformation gradient
- 2.1.2 Strain measures
- 2.2 Equilibrium and stress measures
- 2.2.1 Equilibrium
- 2.2.2 Stress measures
- 2.3 Constitutive equations
- 2.3.1 Linear elasticity
- 2.3.2 Hyperelasticity
- 2.3.3 Finite J2 elastoplasticity
- 2.4 Weak form of equilibrium
- 2.5 Weak form of equilibrium in the initial configuration
- 2.6 Weak form of equilibrium in the current configuration
- 2.7 Linearization of the weak form
- 2.7.1 Linearization of the weak form in the initial configuration
- 2.7.2 Linearization of the weak form in the current configuration
- 17–26 3 The finite cell method 17–26
- 3.1 The weak forms in the extended domain
- 3.2 Spatial discretization
- 3.2.1 Displacement field and mapping functions
- 3.2.2 Discretization of the weak forms
- 3.2.3 Boundary conditions
- 3.3 Numerical integration
- 3.3.1 Gaussian quadrature
- 3.3.2 Adaptive Gaussian quadratures
- 27–58 4 Non-negative moment fitting quadrature for cut finite elements and cells 27–58
- 4.1 Moment fitting
- 4.2 Adaptive moment fitting
- 4.3 Reduced adaptive moment fitting
- 4.4 Non-negative moment fitting
- 4.5 Non-negative least square solver
- 4.6 Preliminary investigations
- 4.6.1 Recovery of the Gauss-Legendre quadrature
- 4.6.2 Cell cut by a sphere with radius r = 1.55 mm
- 4.6.3 Cell cut by a sphere with radius r = 0.3 mm
- 4.7 Stabilization of the solution in the fictitious domain
- 4.8 Application to problems in hyperelasticity
- 4.8.1 Plate with a cylindrical hole
- 4.8.2 Single pore of a foam
- 4.9 Application to problems in elastoplasticity
- 4.9.1 Plate with a cylindrical hole
- 4.9.2 Single pore of a foam
- 59–88 5 Eigenvalue stabilization technique to increase the robustness of the FCM 59–88
- 5.1 Material stabilization (α-method)
- 5.2 Eigenvalue stabilization technique
- 5.2.1 Excluding the rigid body modes from the stabilization
- 5.2.2 Compute the rigid body modes analytically
- 5.3 Preliminary investigations
- 5.3.1 Plate with a circular inclusion
- 5.3.2 Plate with a cylindrical hole
- 5.4 Nonlinear applications in finite strains
- 5.4.1 Combining the eigenvalue stabilization with the α-method
- 5.4.2 Single cube connector
- 5.4.3 Complex cube connector
- 5.4.4 Single pore of a foam
- 89–129 6 Remeshing strategy for large deformations in the FCM 89–129
- 6.1 Kinematic relations
- 6.2 A procedure for performing the remeshing technique
- 6.3 Mesh generation and geometry description on any configuration state
- 6.4 Remeshing criteria
- 6.4.1 Ratio of Jacobians
- 6.4.2 Orthogonality
- 6.4.3 Aspect ratio
- 6.4.4 Investigation of the remeshing criteria on a single element
- 6.5 Data transfer
- 6.5.1 Radial basis function interpolation
- 6.5.2 Inverse distance weighting interpolation
- 6.6 Numerical investigations in finite strain hyperelasticity
- 6.6.1 Plate with a cylindrical hole
- 6.6.2 Single cube connector
- 6.6.3 Complex cube connector
- 6.6.4 Foam
- 6.7 Numerical investigations in finite strain elastoplasticity
- 6.7.1 Plate with a cylindrical hole
- 6.7.2 Single cube connector
- 6.7.3 Complex cube connector
- 6.7.4 Single pore of a foam
- 130–133 7 Summary and outlook 130–133
- 134–134 Publications resulting from this project 134–134
- 135–152 Bibliography 135–152
