The novel finite element formulations fall into the category of geometrically exact Kirchhoff-Love beams. A prominent characteristic of this category is that the absence of shear deformation is strongly enforced by removing two degrees of freedom. Further, the corresponding beam theories exhibit not only translational but also rotational degrees of freedom and their configurations thus form a non-additive and non-commutative space. Sophisticated interpolation schemes are required that need to be tested not only for locking, spatial convergence behavior, and energy conservation, but also for observer invariance and path-independence. For the three novel beam element formulations all these properties are analytically and numerically studied and confirmed, if applicable. Two different rotation parameterization strategies are employed based on the well-known geodesic interpolation used in many Simo-Reissner beams and the lesser known split into the so-called \textit{smallest rotation} and a torsional part. Application of the former parameterization results in a mixed finite element formulation intrinsically free of locking phenomena. Additionally, the first geometrically exact Kirchhoff-Love beam element is presented, which strongly enforces inextensibility by removing another degree of freedom. Furthermore, the numerical efficiency of the new beam formulations is compared to other beam elements that allow for or suppress shear deformation. When modeling very slender beams, the new elements offer distinct numerical advantages.
Standard molecular dynamics simulations, which are commonly used to study polymers, suffer from a lack of a careful mathematical basis and the use of an expensive explicit time integration scheme. To circumvent these shortcomings and to be able to simulate stretching experiments on relevant time scales, the problem is described by a stochastic partial differential equation, which can be solved using the finite element method with a backward Euler temporal discretization. In detail, the polymer is represented by a Kirchhoff-Love beam with a linear elastic constitutive model. Inertial and electrostatic forces are neglected. It is deformed by a distributed load mimicking collisions with molecules of the surrounding fluid. Naturally, this load heavily fluctuates over time and space and mean values need to be computed in a Monte Carlo manner. To vastly speed up the fitting process to experimental data in a Bayesian framework, a surrogate model based on a Gaussian process is set up, which directly computes the mean values for given material parameters. The uncertainties and correlations of the material parameters are studied and compared to the literature.
Author(s): Matthias C. Schulz
Series: Mechanics and Adaptronics
Publisher: Springer
Year: 2022
Language: English
Pages: 149
City: Cham
Preface
Zusammenfassung
Summary
Contents
List of Figures
List of Tables
1 Introduction
1.1 Motivation
1.2 Focus of This Thesis
1.3 Relevance of the Present Work
1.4 Outline
2 Modeling of Slender Bodies
2.1 A General Model for Physical Space
2.2 Definition of Slender Bodies
2.3 The Special Orthogonal Group SO(3)
2.3.1 The Tangent Space of SO(3) and the Corresponding Exponential Map
2.3.2 Derivatives and Variations of Orthogonal Transformations
2.3.3 Parameterization of Orthogonal Rotations
2.4 Simo-Reissner Beam Formulation
2.4.1 Kinematic Assumptions and Description
2.4.2 Balance of Linear and Angular Momentum
2.4.3 Weak Form and Resulting Deformation Measures
2.4.4 Constitutive Relations
2.5 Kirchhoff-Love Beam Formulations
2.5.1 Advantages of Kirchhoff-Love Beam Formulations
2.5.2 Kirchhoff-Love Beam Formulation Based on Constraint Translation
2.5.3 Kirchhoff-Love Beam Formulation Based on Constraint Rotation
2.5.4 Kirchhoff-Love Beam Formulations Based on Weak Enforcement
2.5.5 Reduced Kirchhoff-Love Beam Formulations
2.5.6 Analytical Formulation of Kirchhoff-Love Beams
3 Finite-Element Formulation of Slender Bodies Modeled by Geometrically Exact Beams
3.1 Discretization in Time
3.1.1 Identification of Primary Fields
3.1.2 Generalized-alphaα Method for Elements of double struck upper RmathbbR and double struck upper R cubedmathbbR3
3.1.3 Generalized-alphaα Method for Elements of bold upper S upper O bold left parenthesis bold 3 bold right parenthesisSO(3)
3.1.4 Order of Temporal and Spatial Discretization
3.2 Discretization in Space
3.2.1 Essentials of the Finite-Element Method
3.2.2 Discretization of Elements of double struck upper RmathbbR and double struck upper R cubedmathbbR3
3.2.3 Discretization of Elements of upper S upper O left parenthesis 3 right parenthesisSO(3)
3.3 Simo-Reissner Beam Element
3.4 Kirchhoff-Love Beam Elements
3.4.1 Kirchhoff-Love Beam Elements Based on Constraint Translation
3.4.2 Inextensible Kirchhoff-Love Beam Element
3.4.3 Kirchhoff-Love Beam Elements Based on Weak Enforcement
3.4.4 Nested Assembly Processes
3.4.5 Dirichlet Boundary Conditions and Joints
3.5 Requirements of Beam Formulations
3.5.1 Differentiability of Spatially Discretized Fields
3.5.2 Objectivity and Path-Independence
3.5.3 Numerical Locking
3.5.4 Conservation Properties
3.5.5 Optimal Convergence Order
3.6 Numerical Examples
3.6.1 Objectivity
3.6.2 Path-Independence
3.6.3 Locking and Convergence Order
3.6.4 Performance
3.6.5 Free Oscillation
4 Modeling the Mechanics of Single Polymer Chains in the Finite-Element Framework
4.1 Basic Model Assumptions
4.2 Probabilistic Basics
4.3 Viscous and Temperature Effects
4.3.1 Viscous Forces and Moments
4.3.2 Stochastic Forces and Moments
4.4 Constitutive Parameters and Discretization
4.5 Surrogate Modeling
4.5.1 Linear Model for Regression
4.5.2 Gaussian Process Regression
4.6 Numerical Results
4.6.1 Distributions for Different Boundary Conditions and Loads
4.6.2 Tracking the Accuracy of Monte Carlo Methods
4.6.3 Obtaining Force-Extension Curves
4.6.4 Surrogate Modeling and Validation
4.6.5 Fit to Experimental Data in a Bayesian Framework
5 Conclusion
5.1 Novel Geometrically Exact Kirchhoff-Love Beams
5.2 Finite-Element Brownian Dynamics Simulations for Bayesian Inference
Bibliography
