Mathematical Analysis Of Continuum Mechanics And Industrial Applications: Proceedings Of The International Conference Comfos15 (mathematics For Industry)

This book focuses on mathematical theory and numerical simulation related to various aspects of continuum mechanics, such as fracture mechanics, elasticity, plasticity, pattern dynamics, inverse problems, optimal shape design, material design, and disaster estimation related to earthquakes. Because these problems have become more important in engineering and industry, further development of mathematical study of them is required for future applications. Leading researchers with profound knowledge of mathematical analysis from the fields of applied mathematics, physics, seismology, engineering, and industry provide the contents of this book. They help readers to understand that mathematical theory can be applied not only to different types of industry, but also to a broad range of industrial problems including materials, processes, and products. Preface 6 Contents 8 Part I Fracture Mechanics 10 Strong but Slippery Adhesion of Mushroom-Shaped Polysaccharide Gels 11 1 Introduction 11 2 Materials and Methods 13 2.1 Gel Preparation 13 2.2 Molding of Mushroom-Shaped Gels 13 2.3 Mechanical Tests 13 2.4 Visualization of Stress Distribution at Interface 14 2.5 FEM Analysis 14 3 Results 15 3.1 Mechanical Measurement 15 3.2 Visualizing Distribution of Stress at Interface 16 3.3 FEM Analysis 17 4 Discussion 17 References 20 Bridging the Scales Between Discrete and Continuum Dislocation Models 22 1 Disagreeing Plasticity Models 22 2 Upscaling of Dislocation Walls 23 2.1 Pile-Up of Dislocation Walls 23 2.2 Discrete-to-continuum Limit 25 3 Upscaling of Mixed Positive and Negative Dislocations 27 3.1 Setting 28 3.2 Γ-Convergence 28 3.3 Convergence of Gradient Flows 30 4 Implications for Plasticity Models 31 References 31 Phase Field Crack Growth Model with Hydrogen Embrittlement 33 1 Phase Field Approach for Crack Propagation 33 2 Phase Field Crack Growth Model with Hydrogen Embrittlement 34 3 Numerical Results 36 4 Estimation of Hydrogen Effects 38 5 Concluding Remarks 39 References 39 On Singularities in 2D Linearized Elasticity 41 1 Introduction 41 1.1 Domain 42 1.2 The Linearized Elasticity Equation 42 2 Problems 43 2.1 Case 1 43 2.2 Case 2 43 2.3 Case 3 44 3 Convergent Series Expansions 45 3.1 Case 1 45 3.2 Case 2 46 3.3 Case 3 51 4 Conclusion 52 References 53 Part II Shape Optimization 54 Two-Parameter Topological Expansion of Helmholtz Problems with Inhomogeneity 55 1 Introduction 55 2 Forward Helmholtz Problem in Inhomogeneous Medium 56 2.1 Inner Asymptotic Expansion in the Near-Field 61 2.2 Outer Asymptotic Expansion in the Far-Field 65 2.3 Uniform Asymptotic Expansion of the Solution 71 3 Inverse Helmholtz Problem in Inhomogeneous Medium 73 3.1 High-Order Topological Expansion of the Objective 75 3.2 Zero-Order Necessary Optimality Condition 79 3.3 The Analytic Solution in One Dimension 81 3.4 The Numerical Solution in Two Dimensions 82 References 83 Solution of Shape Optimization Problem and Its Application to Product Design 86 1 Introduction 86 2 Outline of Shape Optimization Problem and Its Solution 88 3 Shape Optimization for a Link Mechanism 94 4 Shape Optimization for Suppressing Brake Squeal 95 5 Method of Designing Beads in Shell Structure 96 6 Shape Optimization of Flow Field to Improve Hydrodynamic Stability 97 7 Shape Optimization of an Electrostatic Capacitive Sensor 97 8 Summary 99 References 100 Shape Optimization by GJ-Integral: Localization Method for Composite Material 102 1 Introduction 102 2 GJ-Integral 104 3 Shape Sensitivity of Composite Materials 105 4 Shape Optimization 107 5 Finite Element Analysis 108 References 112 Shape Optimization Approach by Traction Method to Inverse Free Boundary Problems 113 1 Introduction 113 2 Free Boundary Problem and Shape Optimization 114 3 Variation Formula of the Cost Functional 115 4 Traction Method 118 5 Numerical Examples 120 6 Conclusions 122 References 124 Part III Earthquakes and Inverse Problems 126 Synthesis of Seismic Wave Envelopes Based on the Markov Approximation 127 1 Introduction 127 2 Random Media 129 3 The Born Approximation 129 4 The Markov Approximation 131 5 Extension of the Markov Approximation 135 5.1 The Hybrid Method 135 5.2 An Analytical Solution for von Karman Type Random Media 136 5.3 Velocity Discontinuity in the Markov Approximation 136 6 Application of the Markov Approximation 139 7 Summary 140 References 140 Propagation Velocity of Pulse-Like Rupture Along Earthquake Faults 143 1 Introduction 143 2 Formulation 145 2.1 PDE and Boundary Conditions 145 2.2 Fundamental Solution and Integral Equation 146 3 Friction Modeling 147 3.1 RSP Modeling Revisited 147 3.2 Modeling the Microscopic Structure of Fault Planes 148 3.3 Determination of Kinematic Parameters 149 4 Discussion 149 4.1 Analysis with Huge Uncertainties of Parameters 149 4.2 Applicability to Interfacial Problem 151 5 Conclusions 151 References 151 Inverse Source Problem for aWave Equation with Final Observation Data 153 1 Introduction 153 2 Generic Well-Posedness for Problem 1 and the Proof 154 3 Iterative Thresholding Algorithm 157 4 Numerical Experiments 160 4.1 One-Dimensional Examples 161 4.2 Two-Dimensional Examples 162 References 163 Part IV Fluid Mechanics and Interface Dynamics 165 The Contribution of Kawada to the Analytical Solution for the Velocity Induced by a Helical Vortex Filament and Modern Applications of Helical Vortices 166 1 Introduction 166 2 Bibliography of Sandi Kawada 23 168 3 Kawada's Potential Theory for a Helical Vortex Filament 170 4 von Kármán's Recognition 172 References 172 A New Model for Fungal Hyphae Growth Using the Thin Viscous Sheet Equations 174 1 Introduction 174 2 The New Fungal Hyphae Growth Model 175 2.1 Steady Tip Growth 176 2.2 The Cell Wall as a Thin Viscous Sheet 176 2.3 The Spitzenkörper as a Ballistic Point Source 177 2.4 Age Function 178 3 Governing Equations: A First Order 5-Dimensional ODE 179 4 Numerical Results 180 4.1 Asymptotic Expansions at the Apex and Base 181 4.2 Connecting the Asymptotic Expansions 181 5 Conclusion and Future Work 183 References 183 On Boundary Conditions for Hele-Shaw Problem 184 1 Introduction 184 2 Formulations 186 2.1 Hele-Shaw Problem 186 2.2 Boundary Conditions 187 3 Weakly Nonlinear Stability Analysis 189 3.1 Analytical Results 189 4 Concluding Remarks 193 References 193 Part V Industrial Applications 194 Computer Simulation of the Phase Separation of Polymeric Materials for Industrial Applications 195 1 Microphase Separation of Block Copolymers for the Study of Thermoplastic Elastomers 195 1.1 Introduction 195 1.2 Calculation Method 196 1.3 Results 197 2 Non-solvent Induced Phase Separation (NIPS) for the Study of Hollow Fibers 200 2.1 Introduction 200 2.2 Calculation Model 200 2.3 Results 202 References 203 Highly Parallel Computation of Generalized Eigenvalue Problem in Vibration for Automatic Transmission of Vehicles Using the Sakurai--Sugiura Method and Supercomputers 205 1 Introduction 206 2 Eigenvalue Problem in Automatic Transmission of Vehicles 207 3 Sakurai--Sugiura Method 208 4 Numerical Experiments 210 4.1 Finite Element Model 210 4.2 Computational Environments 211 4.3 Parallel Performance of MUMPS Linear Solver 211 4.4 Scalability of the Sakurai--Sugiura Method on the K Computer 213 4.5 Performance Comparison 213 5 Conclusions 215 References 215 Mathematical Analysis of Synchronization from the Perspective of Network Science 217 1 Introduction 217 2 Functional Equations Based on Kuramoto Models 218 3 Existing Arguments and Remaining Issues 219 4 Function Spaces 219 5 Local and Global-in-Time Solvability of (1) 221 6 Vanishing Diffusion Limit 223 7 Conclusion and Further Studies 225 References 225 Index 227 Front Matter....Pages i-viii Front Matter....Pages 1-1 Strong but Slippery Adhesion of Mushroom-Shaped Polysaccharide Gels....Pages 3-13 Bridging the Scales Between Discrete and Continuum Dislocation Models....Pages 15-25 Phase Field Crack Growth Model with Hydrogen Embrittlement....Pages 27-34 On Singularities in 2D Linearized Elasticity....Pages 35-47 Front Matter....Pages 49-49 Two-Parameter Topological Expansion of Helmholtz Problems with Inhomogeneity....Pages 51-81 Solution of Shape Optimization Problem and Its Application to Product Design....Pages 83-98 Shape Optimization by GJ-Integral: Localization Method for Composite Material....Pages 99-109 Shape Optimization Approach by Traction Method to Inverse Free Boundary Problems....Pages 111-123 Front Matter....Pages 125-125 Synthesis of Seismic Wave Envelopes Based on the Markov Approximation....Pages 127-142 Propagation Velocity of Pulse-Like Rupture Along Earthquake Faults....Pages 143-152 Inverse Source Problem for a Wave Equation with Final Observation Data....Pages 153-164 Front Matter....Pages 165-165 The Contribution of Kawada to the Analytical Solution for the Velocity Induced by a Helical Vortex Filament and Modern Applications of Helical Vortices....Pages 167-174 A New Model for Fungal Hyphae Growth Using the Thin Viscous Sheet Equations....Pages 175-184 On Boundary Conditions for Hele-Shaw Problem....Pages 185-194 Front Matter....Pages 195-195 Computer Simulation of the Phase Separation of Polymeric Materials for Industrial Applications....Pages 197-206 Highly Parallel Computation of Generalized Eigenvalue Problem in Vibration for Automatic Transmission of Vehicles Using the Sakurai–Sugiura Method and Supercomputers....Pages 207-218 Mathematical Analysis of Synchronization from the Perspective of Network Science....Pages 219-228 Back Matter....Pages 229-231