육면체 지배적 유한 요소 모델의 해석을 위한 다면체 요소의 개발 및 활용
Development of a Polyhedral Element for the Analysis of Hexahedral-Dominant Finite Element Models and Its Applications
- 주제(키워드) Hexahedral meshes , MLS shape functions , Nonlinear Analysis , Shape Optimization
- 발행기관 서울과학기술대학교
- 지도교수 김현규
- 발행년도 2018
- 학위수여년월 2019. 2
- 학위명 박사
- 학과 일반대학원 기계공학과
- 실제URI http://www.dcollection.net/handler/snut/200000178214
- 본문언어 영어
초록/요약
본 연구는 육면체 지배적 유한요소 모델을 위한 효과적 다면체 요소에 대해 나타낸다. 육면체 지배적 유한요소망은 박스 형태의 균일한 배경 육면체 요소망을 CAD 모델의 형상 또는 level set함수의 0-등위면으로 분할함으로써 간단히 구성된다. 육면체 지배적 유한요소망의 내부는 균일한 육면체 요소로 구성되고, 모델의 경계면에서 Marching Cubes 알고리즘에 의해 직선의 모서리와 곡면을 가지는 절단 육면체 형상의 다면체 요소로 구성된다. 또한 Marching Cubes 알고리즘의 면 구성의 모호성에 기인한 다면체 요소의 불일치면을 해소하기 위한 간단한 방법이 제시된다. 다면체 요소는 중심점 기반 분할을 통해 사면체 하위영역으로 분할되며, 이에 기반한 이동최소제곱근사(Moving Least Square approximation)를 통해 다면체 요소의 형상함수를 구성한다. 본 연구의 다면체 형상함수는 기존의 유한요소 형상함수와 유사하게 요소 내부에서의 연속성 및 완전성, 요소 간 경계면에서의 적합성, 절점에서의 Kronecker-delta 조건을 만족한다. 본 연구에서 제시한 육면체 지배적 유한요소망은 초탄성 및 탄소성 재료의 대변형 거동 문제와 level set 함수 기반 형상최적화 문제에 적용된다.
more초록/요약
This study presents an efficient polyhedral element for the analysis of a hexahedral-dominant finite element model and its application. A hexahedral-dominant finite element mesh can be easily constructed by splitting a regular background hexahedral mesh in a simple block with a geometric model representing by CAD surfaces or the zero-isosurface of a level set function. Based on the marching cubes algorithm, polyhedral elements with straight edges but possibly non-planar faces are generated at the domain boundaries, while regular hexahedral elements remain in the interior region. A simple method is proposed to handle nonconforming polyhedral elements at the interface due to the face ambiguity arising in the marching cubes algorithm. Shape functions for polyhedral elements are derived from moving least square approximation based on a tetrahedral subdivision of polyhedral domains by a centroid-based subdivision technique. The polyhedral shape functions in this study have similar properties to conventional finite element shape functions in terms of continuity and completeness within elements, compatibility across inter-element boundaries and the Kronecker-delta property. The present approach using hexahedral-dominant meshes with polyhedral elements at domain boundaries is successfully applied to solve large deformation problems of hyper-elastic and elastic-plastic materials and level set based shape optimization problems.
more목차
Summary ⅰ
List of Tables v
List of Figures vi
List of Acronyms and Abbreviations xi
List of Symbols xii
1. Introduction 1
2. Hexahedral-Dominant Meshes and Polyhedral Elements 7
2.1. Generation of Hexahedral-Dominant Meshes by Using CAD Surfaces 7
2.2. Generation of Hexahedral-Dominant Meshes by Using the Zero-Isosurface of a Level Set Functions 10
2.2.1 Generation of Trimmed Hexahedral Meshes 12
2.2.2 Treatment of Nonconforming Meshes at The Interface 18
2.2.3. Quality Improvement of Polyhedral Elements 23
3. Shape Functions for Polyhedral Elements Using MLS Approximation 25
3.1. Introduction 25
3.2. MLS Shape Functions for Polyhedral Elements with Non-Planar Faces 25
3.2.1 Subdivision of Polyhedral Domains into Tetrahedral Sub-Domains 28
3.2.2 Support Points and Ten-Point Tetrahedral Sub-Domains 31
3.2.3 MLS Weight and Shape Functions 34
3.2.4 Numerical Integration 39
3.3. A Patch Test with Polyhedral and Hexahedral Elements 42
4. Nonlinear Solid Mechanics Problems Using Hexahedral-Dominant Meshes 45
4.1. Incremental FE Formulations 45
4.2. Hyperelastic and Elastic-Plastic Materials 48
4.3. Numerical Examples 54
4.3.1 A Spiral Beam of a Saint-Venant Kirchhoff Hyperelastic Material 56
4.3.2 A Rubber Bushing of a Compressible Mooney-Rivlin Material 60
4.3.3 A Rectangular Block with Arbitrary Voids of an Elastic-Plastic Material 63
5. Level-Set Based Shape Optimization Problems Using Hexahedral-Dominant Meshes 66
5.1. Introduction 66
5.2. Shape Optimization Problems Based on Level Set Method 74
5.2.1 Optimization Problem Formulation 75
5.2.2 Level Set Based Shape Optimization 78
5.2.3 Global Stress Measure Using a P-Norm Aggregation Function 79
5.2.4 Sensitivity Analysis 81
5.2.4.1 The Compliance Minimization under a Volume Constraint 81
5.2.4.2 The Volume Minimization under Stress Constraints 84
5.3. Explicit Representation for the Solid Domain Using Trimmed H8 Meshes 89
5.3.1 A Comparison of the Stress Distribution between the Present Method Using a Trimmed H8 Mesh and the Density-Based Methods 91
5.3.2 Extension and Regularization of the Normal Velocity Field 93
5.3.2.1 Extension of the Normal Velocity Field 93
5.3.2.2 Regularization of the Normal Velocity Field 96
5.4. Numerical Examples 97
5.4.1 A Short Cantilever Beam 99
5.4.2 An L-shaped Beam 101
5.4.3 A Chair Problem under Multiple Loads 103
6. Conclusions 105
Related Publications 107
References 108
초록 121
Acknowledgements 122