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TWO-DIMENSIONAL LAGRANGIAN HYDRODYNAMIC DIFFERENCE EQUATIONS＆William D．Schulz 1

Ⅰ．Introduction 1

Ⅱ．Definitions,Notation,and Transformation Relations 2

Ⅲ．Conservation Equations 6

Ⅳ．Tensor Artificial Viscosity 8

Ⅴ．Stability of the Differential Equations 15

Ⅳ．Hydrodynamic Difference Equations 21

Ⅶ．Dt Control 32

Ⅷ．Boundary Treatment 39

Ⅸ．Sample Problem 40

Reference 45

MIXED EULERIAN-LAGRANGIAN METHOD＆R．M．Frank and R．B．Lazarus 47

Ⅰ．Introduction 47

Ⅱ．Cell Equations 48

Ⅲ．Euler-Lagrange Mesh 49

Ⅳ．Choice of Variables 50

Ⅴ．Space Differencing 51

Ⅵ．Time Differencing 55

Ⅶ．Artificial Forces 55

Ⅷ．Courant Stability 56

Ⅸ．Test Calculations 57

Ⅹ．Details of the Machine Program 58

Ⅺ．Heat Flow 66

THE STRIP CODE AND THE JETTING OF GAS BETWEEN PLATES＆John G．Trulio 69

Ⅰ．Introduction 69

Ⅱ．Simple Theory for the Jetting of Gas between Two Plates 71

Ⅲ． The Strip Code 75

Ⅳ． Comments on the Convergence of Numerical Methods in Hydrodynamics 87

Ⅴ． Application of the Strip Code to the Problem of the Jetting of Gas 92

Ⅵ． Conclusion 100

Ⅶ． Appendices 103

References 115

CEL:A TIME-DEPENDENT,TWO-SPACE-DIMENSIONAL,COUPLED EULERIAN-LAGRANGE CODE＆W．F．Noh 117

Ⅰ．Introduction 117

Ⅱ．A General Discussion of Hydrodynamical Calculations 121

Ⅲ．Description of the CEL Code 125

Ⅳ．Generalization of the Difference Approximation for Partial Derivatives over an Arbitrary Region 130

Ⅴ．Eulerian Difference Equations for Interior Cells of the Mesh 136

Ⅵ．The Conservation Form of the Hydrodynamical Equations When the Space Variables x and y May Have an Arbitrary Velocity Relative to the Fluid Velocity 140

Ⅶ．General Considerations for the Boundary Zone Calculations 143

Ⅷ．Boundary Zones and Boundary Sequences 144

Ⅸ．Boundary Zone Difference Equations for Variables p,e,etc．,Which Are Centered in Boundary Zones Cn(k,l) 148

Ⅹ．Difference Equations for the Components of Momentum(m=pu,n=pv) Which Are Centered in the Boundary Zones Cn(k+1/2,l+1/2) 151

Ⅺ．Lagrange Difference Equations 153

Ⅻ．Discussion and Graphical Results of Several CEL Calculations 156

References 179

THE TENSOR CODE＆G．Maenchen and S．Sack 181

Ⅰ．Introduction 181

Ⅱ．Definitions and Notation 184

Ⅲ．Momentum Equation 187

Ⅳ．Boundary Conditions 191

Ⅴ．Volume and Strain Rate 192

Ⅵ．Plastic Yield 196

Ⅶ．Fracture 198

Ⅷ．Damping 202

Ⅸ．Energy Equation and Stability Control 205

Ⅹ．Examples 206

References 210

CALCULATION OF ELASTIC-PLASTIC FLOW＆Mark L．Wilkins 211

Ⅰ．Introduction 211

Ⅱ．Equations of State 212

Ⅲ．One-Dimensional Elastic-Plastic Flow 223

Ⅳ．Two-Dimensional Elastic-Plastic Flow 226

Appendix Ⅰ．Finite Difference Equations for the Equations of Section Ⅲ 241

Appendix Ⅱ．Finite Difference Equations for the Equations of Section Ⅳ 244

References 262

SOLUTION BY CHARACTERISTICS OF THE EQUATIONS OF ONE-DIMENSIONAL UNSTEADY FLOW＆N．E．Hoskin 265

Ⅰ．Introduction 265

Ⅱ．Equations of Motion 268

Ⅲ．Structure of a Computer Program 274

Ⅳ．Calculation by Characteristics at Fixed Time Intervals 288

Ⅴ．Conclusions 293

References 293

THE SOLUTION OF TWO-DIMENSIONAL HYDRODYNAMIC EQUATIONS BY THE METHOD OF CHARACTERISTICS＆D．J．Richardson 295

Ⅰ．Introduction 295

Ⅱ．The Equations of Hydrodynamic Flow 296

Ⅲ．Characteristic Surfaces and Bi-Characteristic Curves 296

Ⅳ．Relations Holding along Bi-Characteristics 298

Ⅴ．The General Bi-Characteristic Method 300

Ⅵ．Shock Surfaces 302

Ⅶ．Computational Aspects of the Two-Dimensional Characteristics Method 303

Ⅷ．Topological Aspects of the Computer Program 306

Ⅸ．Appendices 310

References 318

THE PARTICLE-IN-CELL COMPUTING METHOD FOR FLUID DYNAMICS＆Francis H．Harlow 319

Ⅰ．Introduction 319

Ⅱ．The Basic Computing Method 322

Ⅲ．Modified Forms of the Method 331

Ⅳ．Properties of the PIC Method 333

Ⅴ．Applicability and Limitations of the PIC Method 339

Ⅵ．Application to a Specific Problem 342

References 343

THE TIME DEPENDENT FLOW OF AN INCOMPRESSIBLE VISCOUS FLUID＆Jacob Fromm 345

Ⅰ．Introduction 346

Ⅱ．The Numerical Method 349

Ⅲ．Initial and Boundary Conditions 355

Ⅳ．Modification of the Vorticity Equation 362

Ⅴ．Functionals of the Motion;A Test of the Method 371

Ⅵ．Applications 376

References 382

AUTHOR INDEX 383

SUBJECT INDEX 384