Non-standard Discretisation Methods in Solid Mechanics (Lecture Notes in Applied and Computational Mechanics, 98)

✍ Scribed by Jörg Schröder (editor), Peter Wriggers (editor)

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 561
• Category: Library

✦ Synopsis

This edited volume summarizes research being pursued within the DFG Priority Programme 1748: "Reliable Simulation Methods in Solid Mechanics. Development of non-standard discretisation methods, mechanical and mathematical analysis", the aim of which was to develop novel discretisation methods based e.g. on mixed finite element methods, isogeometric approaches as well as discontinuous Galerkin formulations, including a sound mathematical analysis for geometrically as well as physically nonlinear problems. The Priority Programme has established an international framework for mechanical and applied mathematical research to pursue open challenges on an inter-disciplinary level. The compiled results can be understood as state of the art in the research field and show promising ways of further research in the respective areas. The book is intended for doctoral and post-doctoral students in civil engineering, mechanical engineering, applied mathematics and physics, as well as industrial researchers interested in the field.

✦ Table of Contents

Preface
Contents
Hybrid Discretizations in Solid Mechanics for Non-linear and Non-smooth Problems
1 Introduction
2 Continuous, Discontinuous and Hybrid Discretizations
2.1 A Weakly Conforming Method
2.2 Incomplete Interior Penalty Galerkin Method
2.3 Hybrid Discontinuous Galerkin Method
2.4 Symmetric Hybrid Discontinuous Galerkin Method
2.5 Cohesive Discontinuous Galerkin Method
3 Numerical Evaluation of the Numerical Schemes
3.1 An Illustrating Smooth Example in 2D
3.2 A Corner Singularity in 3D
3.3 Cook's Membrane: A Benchmark Problem
3.4 Elasto-Plastic Deformation of an Annulus
3.5 Material Discontinuities at Interfaces: A Ring with Different Materials
3.6 A Fiber Composite with Nearly Incompressible Inclusions
3.7 A Benchmark Configuration for Thin Structures
3.8 An Inelastic Model Combining Plasticity and Damage
3.9 A Hybrid Approximation of a Contact Problem
4 Conclusion
References
Novel Finite Elements - Mixed, Hybrid and Virtual Element Formulations at Finite Strains for 3D Applications
1 Introduction and State of the Art
2 Brief Continuum-Mechanical Background
3 Mixed FE Technology for Large Deformations
3.1 Consistent Stabilization for Displacement-Pressure Elements
3.2 Hellinger-Reissner Principle for Large Deformations
4 Virtual Element Technology for Large Deformations
4.1 Displacement VEM Space and Projector Operators
4.2 Construction of Displacement Based and Two-Field Mixed VEM Approximation
4.3 Numerical Example
5 Conclusion and Outlook
References
Robust and Efficient Finite Element Discretizations for Higher-Order Gradient Formulations
1 Introduction
1.1 Definitions
2 Formulation for Finite Strain Gradient Elasticity
2.1 Gradient Elasticity Fundamentals
2.2 Rot-Free Finite Element Formulation
2.3 Finite Element Discretization
2.4 Numerical Examples
3 Gradient Enhanced Damage at Finite Strains
3.1 Continuum Damage Mechanics with Gradient Enhancement
3.2 Finite Elements for Gradient Damage
3.3 Numerical Examples
4 Conclusion
References
Stress Equilibration for Hyperelastic Models
1 Introduction
2 Hyperelasticity and Stress Equilibration
3 Localized Stress Equilibration
4 Error Estimation
5 Computational Experiments
References
Adaptive Least-Squares, Discontinuous Petrov-Galerkin, and Hybrid High-Order Methods
1 Introduction
1.1 Motivation
1.2 Three Nonstandard Discretizations
1.3 Adaptive Mesh-Refinement
1.4 Outline of the Presentation
2 Notation
3 Least-Squares Finite Element Methods in Computational Mechanics
3.1 Least-Squares Finite Element Methods
3.2 Natural Adaptive Mesh-Refinement
3.3 Alternative A posteriori Error Control
3.4 Axioms of Adaptivity
4 Least-Squares Finite Element Methods in Nonlinear Computational Mechanics
4.1 Convex Energy Minimization
4.2 Least-Squares Formulation
4.3 Numerical Experiments
4.4 Comments
5 Discontinuous Petrov-Galerkin
5.1 Optimal Test Functions
5.2 Breaking Spaces and Forms
5.3 Adaptive Mesh-Refinement
5.4 Axioms of Adaptivity
6 Discontinuous Petrov-Galerkin in Nonlinear Computational Mechanics
6.1 Nonlinear Discontinuous Petrov-Galerkin
6.2 Alternative Formulations
6.3 Existence and Uniqueness of Discrete Solutions
6.4 Numerical Experiments
7 Hybrid High-Order Method
7.1 Discrete Ansatz Space
7.2 Reconstruction Operators and Stabilization
7.3 HHO in Computational Mechanics
7.4 Reliable and Efficient Error Control
7.5 Numerical Experiment on L-Shaped Domain with Corner Singularity
8 HHO in Nonlinear Computational Mechanics
8.1 A Class of Degenerate Convex Minimization Problems
8.2 The Unstabilized HHO Method
8.3 A priori Analysis
8.4 A posteriori Analysis
8.5 A Topology Optimization Problem: Optimal Design
References
Least-Squares Finite Element Formulation for Finite Strain Elasto-Plasticity
1 Introduction
2 Elasto-Plasticity for the Framework of Finite Strains
3 Least-Squares Finite Element Formulation for Finite Strain Elasto-Plasticity
4 The Least-Squares Functional as an Error Estimator
5 Numerical Analysis
6 Conclusion
References
Hybrid Mixed Finite Element Formulations Based on a Least-Squares Approach
1 Introduction
2 Continuous Least-Squares Finite Element Formulation
3 Hybrid Mixed Finite Element Based on a Least-Squares Approach
3.1 Weak Form and Linearization of the Hybrid Mixed Formulation
3.2 Discretization and Implementation Aspects
4 Numerical Analysis for Hybrid Mixed Formulations
4.1 Cook's Membrane Problem
4.2 Quartered Plate Example
5 Conclusion
References
Adaptive and Pressure-Robust Discretization of Incompressible Pressure-Driven Phase-Field Fracture
1 Introduction
2 Notation and Equations
2.1 Pressurized Phase-Field Fracture in a Displacement Formulation
2.2 Pressurized Phase-Field Fracture in a Mixed Formulation
3 Discrete Formulation
4 Residual-Type a Posteriori Error Estimator
5 Numerical Tests
5.1 Sneddon-Inspired Test Cases (Example 1)
5.2 Incompressible Material Surrounded with a Compressible Layer (Example 2)
5.3 Nonhomogeneous Pressure Test Case with a Compressible Layer (Example 3)
6 Conclusions
References
A Phase-Field Approach to Pneumatic Fracture
1 Introduction
2 Phase-Field Model
2.1 Linear Elasticity
2.2 Finite Elasticity
2.3 Discretization
3 Multilevel Solution Strategies
4 Discussions and Extensions of the Phase-Field Model
4.1 Influencing Parameters
4.2 Externally Driven Fracture
5 Numerical Examples
5.1 Conchoidal Fracture
5.2 Pressure Driven Crack Growth
6 Summary
References
Adaptive Isogeometric Phase-Field Modeling of Weak and Strong Discontinuities
1 Introduction
2 Local Mesh Refinement
2.1 Truncated Hierarchical B-Splines
2.2 T-Splines
2.3 Unstructured T-Splines
3 Spline-Based Analysis
3.1 Spectral Superiority of Splines
3.2 Adapted Heterogeneous Spline Spaces
4 Adaptive Isogeometric Analysis
4.1 THB-Splines or T-Splines – A Computational Comparison
4.2 Mesh Adaptivity for Incremental Solution Schemes
5 Weak and Strong Discontinuities in Solid Mechanics
5.1 Embedded Material Interfaces in Linear Elasticity
5.2 Brittle and Ductile Fracture in Homogeneous and Heterogeneous Materials
6 Conclusion
References
Phase Field Modeling of Brittle and Ductile Fracture
1 Phase Field Model of Brittle Fracture
2 Exponential Shape Functions
2.1 Quadratic Shape Functions
2.2 Exponential Shape Functions
2.3 3d Exponential Shape Functions
2.4 Numerical Test
2.5 Adaptive Numerical Integration
2.6 Blending Elements
2.7 Adaptive Orientation 2d
3 Phase Field Model for Ductile Fracture
3.1 Phase Field Modeling of Ductile Fracture
3.2 Analysis of a 1D-Bar Problem
3.3 Plane Strain Simulations
3.4 3D Simulations
4 Concluding Remarks
References
Adaptive Quadrature and Remeshing Strategies for the Finite Cell Method at Large Deformations
1 Introduction
2 The Finite Cell Method
3 Adaptive Quadrature Based on the Moment Fitting
3.1 Moment Fitting
3.2 Adaptive Moment Fitting
3.3 Numerical Examples
4 A Remeshing Strategy for the Finite Cell Method
4.1 Kinematics
4.2 Remeshing Procedure
4.3 Numerical Examples
5 Conclusion
References
The Finite Cell Method for Simulation of Additive Manufacturing
1 Introduction
2 The Finite Cell Method
3 The Finite Cell Method Combined with Locally-Refined p-FEM
4 The Finite Cell Method Combined with Locally-Refined IGA
5 Simulation of Additive Manufacturing
5.1 The Thermal Model
5.2 Experimental Validation
5.3 The Finite Cell Method in SLM Process Simulations
5.4 Thermo-Mechanical Part-Scale Simulation
6 Credits and Permissions
References
Error Control and Adaptivity for the Finite Cell Method
1 Introduction
2 The Finite Cell Method for the Poisson Problem
3 Basis Functions for Finite Cell Meshes with Hanging Nodes
4 Residual-Based Error Estimation for the Finite Cell Method
4.1 Reliability
4.2 Numerical Example
5 Dual Weighted Residual Error Estimation for the Finite Cell Method
5.1 Error Identity
5.2 Refinement Strategy
5.3 Numerical Example
6 Conclusion and Outlook
References
Frontiers in Mortar Methods for Isogeometric Analysis
1 Introduction
2 Coupled Simulations with Mortar Methods in HPC
3 Basic Equations and Isogeometric Analysis
4 Mortar Techniques for Isogeometric Analysis
4.1 Biorthogonal Splines for Isogeometric Analysis
4.2 Multi-patch Analysis for Kirchhoff–Love Shells
4.3 Weak Cn Coupling for Solids
4.4 Crosspoint Modification
4.5 Hybrid Approaches for Higher-Order Continuity Constraints
5 Mortar Contact Formulations for Isogeometric Analysis
5.1 Biorthogonal Basis Functions Applied to Contact Mechanics
5.2 Thermomechanical Contact Problems
6 Multi-dimensional Coupling
7 Conclusions
References
Collocation Methods and Beyond in Non-linear Mechanics
1 Introduction
1.1 Collocation and Isogeometric Analysis
1.2 Beyond Collocation in Uncertainty Quantification
2 Isogeometric Collocation for Linear and Non-linear Mechanics
2.1 Introduction to Isogeometric Collocation
2.2 Variational Collocation and a New Reduced Quadrature Rule
2.3 Isogeometric Collocation for Hyperelasticity
2.4 Mixed Isogeometric Collocation for Nearly Incompressible Elasticity and Elastoplasticity
2.5 Contact Approaches for Isogeometric Collocation
2.6 Isogeometric Collocation for Geometrically Non-linear Structural Elements
3 Beyond Stochastic Collocation
3.1 Bayesian Numerics
3.2 Multilevel Monte Carlo Method
3.3 Stochastic Upscaling
3.4 Nonlinear Identification
4 Conclusion
References
Approximation Schemes for Materials with Discontinuities
1 Introduction
2 Mathematical Formulations to Handle Discontinuities in Time
3 Finite Element Approximation for Total Variation Regularized Problems
3.1 Model Problem and Analytical Properties
3.2 Notation in Finite Element Spaces
3.3 Finite Element Discretization
3.4 Iterative Solution Methods
3.5 Fully Discrete Approximation of Rate-Independent Damage Processes
4 Fully Discrete Approximation of Dynamic Phase-Field Fracture by Viscous Regularization
4.1 Basic Assumptions and Main Result
4.2 Proof of Proposition12
4.3 Outline of the Proof of Convergence Theorem3
References