Advanced topics in computational partial differential equations: numerical methods and Diffpack programming
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| Format: | Book |
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| Language: | German |
| Published: |
Berlin [u.a.]
Springer
2003
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| Series: | Lecture notes in computational science and engineering
33 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 658 S. graph. Darst. |
| ISBN: | 3540014381 |
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| 245 | 1 | 0 | |a Advanced topics in computational partial differential equations |b numerical methods and Diffpack programming |c Hans Petter Langtangen ... ed. |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
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| 490 | 1 | |a Lecture notes in computational science and engineering |v 33 | |
| 630 | 0 | 4 | |a Diffpack (Computer file) |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions |x Data processing | |
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| adam_text | IMAGE 1
HANS PETTER LANGTANGEN
ASLAK TVEITO EDITORS
ADVANCED TOPICS
IN COMPUTATIONAL PARTIAL DIFFERENTIAL EQUATIONS
NUMERICAL METHODS AND DIFFPACK PROGRAMMING
JBN SPRINGER
IMAGE 2
TABLE OF CONTENTS
1 PARALLEL COMPUTING 1
X. CAI, E. ACKLAM, H. P. LANGTANGEN, A. TVEITO 1.1 INTRODUCTION TO
PARALLEL COMPUTING 1
1.1.1 DIFFERENT HARDWARE ARCHITECTURES 2
1.1.2 THE MESSAGE-PASSING PROGRAMMING MODEL 2
1.1.3 A MULTICOMPUTER MODEL 3
1.1.4 TWO EXAMPLES OF PARALLEL COMPUTING 3
1.1.5 PERFORMANCE MODELING 5
1.1.6 PERFORMANCE ANALYSIS OF THE TWO EXAMPLES 6
1.2 A DIFFERENT PERFORMANCE MODEL 7
1.2.1 THE 2D CASE 8
1.2.2 THE 3D CASE 10
1.2.3 A GENERAL MODEL 11
1.3 THE FIRST MPI ENCOUNTER 11
1.3.1 HELLO WORLD IN PARALLEL 11
1.3.2 COMPUTING THE NORM OF A VECTOR 12
1.4 BASIC PARALLEL PROGRAMMING WITH DIFFPACK 14
1.4.1 DIFFPACK S BASIC INTERFACE TO MPI 14
1.4.2 EXAMPLE: THE NORM OF A VECTOR 17
1.5 PARALLELIZING EXPLICIT FD SCHEMES 18
1.5.1 PARTITIONING LATTICE GRIDS 18
1.5.2 PROBLEM FORMULATION 20
1.5.3 HIDING THE COMMUNICATION COST 22
1.5.4 EXAMPLE: THE WAVE EQUATION IN 1D 23
1.5.5 HOW TO CREATE PARALLEL FD SOLVERS IN DIFFPACK 26
1.5.6 TOOLS FOR PARALLEL FINITE DIFFERENCE METHODS 27
1.5.7 EXAMPLE: HEAT CONDUCTION IN 2D 30
1.5.8 THE SEQUENTIAL HEAT CONDUCTION SOLVER 32
1.5.9 THE PARALLEL HEAT CONDUCTION SOLVER 34
1.6 PARALLELIZING FE COMPUTATIONS ON UNSTRUCTURED GRIDS 37
1.6.1 PARALLEL LINEAR ALGEBRA OPERATIONS 38
1.6.2 OBJECT-ORIENTED DESIGN OF A PARALLEL TOOLBOX 42
1.6.3 MAJOR MEMBER FUNCTIONS OF GRIDPARTADM 46
1.6.4 A PARALLELIZATION EXAMPLE 49
1.6.5 STORAGE OF COMPUTATION RESULTS AND VISUALIZATION 51
1.6.6 QUESTIONS AND ANSWERS 52
REFERENCES 55
IMAGE 3
X TABLE OF CONTENTS
2 OVERLAPPING DOMAIN DECOMPOSITION M E T H O DS 57
X. CAI
2.1 INTRODUCTION 57
2.2 THE MATHEMATICAL FORMULATIONS 58
2.2.1 THE CLASSICAL ALTERNATING SCHWARZ METHOD 58
2.2.2 THE MULTIPLICATIVE SCHWARZ METHOD 59
2.2.3 THE ADDITIVE SCHWARZ METHOD 62
2.2.4 FORMULATION IN THE RESIDUAL FORM 62
2.2.5 OVERLAPPING DD AS PRECONDITIONER 63
2.3 A 1D EXAMPLE 64
2.4 SOME IMPORTANT ISSUES 67
2.4.1 DOMAIN PARTITIONING 67
2.4.2 SUBDOMAIN DISCRETIZATIONS IN MATRIX FORM 67
2.4.3 INEXACT SUBDOMAIN SOLVER 67
2.4.4 COARSE GRID CORRECTIONS 68
2.4.5 LINEAR SYSTEM SOLVERS AT DIFFERENT LEVELS 68
2.5 COMPONENTS OF OVERLAPPING DD METHODS 69
2.6 A GENERIC IMPLEMENTATION FRAMEWORK 70
2.6.1 THE SIMULATOR-PARALLEL APPROACH 70
2.6.2 OVERALL DESIGN OF THE IMPLEMENTATION FRAMEWORK 71
2.6.3 SUBDOMAIN SOLVERS 71
2.6.4 THE COMMUNICATION PART 76
2.6.5 THE GLOBAL ADMINISTRATOR 76
2.6.6 SUMMARY OF THE IMPLEMENTATION FRAMEWORK 80
2.7 PARALLEL OVERLAPPING DD METHODS 80
2.8 TWO APPLICATION EXAMPLES 82
2.8.1 THE POISSONL EXAMPLE 83
2.8.2 THE HEATL EXAMPLE 88
2.8.3 SOME PROGRAMMING RULES 93
REFERENCES 95
3 SOFTWARE TOOLS FOR MULTIGRID M E T H O DS 97
K.-A. MARDAL, G. W. ZUMBUSCH, H. P. LANGTANGEN 3.1 INTRODUCTION 97
3.2 SKETCH OF HOW MULTILEVEL METHODS ARE IMPLEMENTED IN DIFFPACK 99 3.3
IMPLEMENTING MULTIGRID METHODS 100
3.3.1 EXTENDING AN EXISTING APPLICATION WITH MULTIGRID 100
3.3.2 MAKING GRID HIERARCHIES 104
3.3.3 A PROPER TEST EXAMPLE 105
3.4 SETTING UP AN INPUT FILE 105
3.4.1 RUNNING A PIAIN GAUSSIAN ELIMINATION 105
3.4.2 FILLING OUT MGTOOLS MENUE ITEMS 106
3.5 PLAYING AROUND WITH MULTIGRID 107
3.5.1 NUMBER OF GRIDS AND NUMBER OF ITERATIONS 108
3.5.2 CONVERGENCE MONITORS 109
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TABLE OF CONTENTS XI
3.5.3 SMOOTHER 110
3.5.4 W CYCLE AND NESTED ITERATION 113
3.5.5 COARSE-GRID SOLVER 116
3.5.6 MULTIGRID AS A PRECONDITIONER 117
3.5.7 ADDITIVE PRECONDITIONER 119
3.6 EQUIPPING THE POISSON2 SOLVER WITH MULTIGRID 121
3.6.1 MULTIGRID AND NEUMANN OR ROBIN BOUNDARY CONDITIONS 122 3.6.2
DEBUGGING MULTIGRID 124
3.6.3 DOMAINS WITH GEOMETRIE SINGULARITIES 128
3.6.4 JUMPING COEFRCIENTS 129
3.6.5 ANISOTROPIE PROBLEMS AND (R)ILU SMOOTHING 133
3.7 SYSTEMS OF EQUATIONS, LINEAR ELASTICITY 133
3.7.1 EXTENDING ELASTICITYL WITH MULTIGRID 134
3.7.2 THE A-DEPENDENCY 136
3.8 NONLINEAR PROBLEMS 136
3.8.1 THE NONLINEAR MULTIGRID METHOD 138
3.8.2 THE PULL APPROXIMATION SCHEME 139
3.8.3 SOFTWARE TOOLS FOR NONLINEAR MULTIGRID 141
3.8.4 DEFAULT IMPLEMENTATION OF SOME COMPONENTS 144
3.8.5 EXPERIMENTS WITH NONLINEAR MULTIGRID 145
3.8.6 NONLINEAR MULTIGRID FOR A LINEAR PROBLEM 147
3.8.7 NONLINEAR MULTIGRID APPLIED TO A NON-LINEAR PROBLEM . . 148
REFERENCES 150
4 MIXED FINITE ELEMENTS 153
K.-A. MARDAL, H. P. LANGTANGEN 4.1 INTRODUCTION 153
4.2 MODEL PROBLEMS 154
4.2.1 THE STOKES PROBLEM 154
4.2.2 THE MIXED POISSON PROBLEM 154
4.3 MIXED FORMULATION 155
4.3.1 WEIGHTED RESIDUAL METHODS 155
4.3.2 MIXED ELEMENTS, DO WE REALLY NEED THEM? 158
4.3.3 FUNCTION SPACE FORMULATION 159
4.3.4 THE BABUSKA-BREZZI CONDITIONS 161
4.4 SOME BASIC CONCEPTS OF A FINITE ELEMENT 163
4.4.1 A GENERAL FINITE ELEMENT 163
4.4.2 EXAMPLES OF FINITE ELEMENT SPACES 165
4.4.3 DIFFPACK IMPLEMENTATION 175
4.4.4 NUMBERING STRATEGIES 177
4.5 SOME CODE EXAMPLES 181
4.5.1 A DEMO PROGRAM FOR SPECIAL VTRSUS GENERAL NUMBERING 181 4.6
PROGRAMMING WITH MIXED FINITE ELEMENTB IN A SIMULATOR 183
4.6.1 A STANDARD SOLVER WITH SOME NEW UTILITIES 183
4.6.2 A SIMULATOR FOR THE STOKES PROBLEM 186
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XII TABLE OF CONTENTS
4.6.3 NUMERICAL EXPERIMENTS FOR THE STOKES SIMULATOR 191
4.6.4 A SIMULATOR FOR THE MIXED POISSON PROBLEM 192
4.6.5 NUMERICAL EXPERIMENTS FOR THE MIXED POISSON SIMULATOR 195
REFERENCES 196
5 SYSTEMS OF P D ES AND BLOCK PRECONDITIONING 199
K.-A. MARDAL, J. SUNDNES, H. P. LANGTANGEN, A. TVEITO 5.1 INTRODUCTION
199
5.2 BLOCK PRECONDITIONERS IN GENERAL 200
5.2.1 OPERATOR SPLITTING TECHNIQUES 201
5.2.2 CONJUGATE GRADIENT-LIKE METHODS 203
5.2.3 SOFTWARE TOOLS FOR BLOCK MATRICES AND NUMBERING STRATEGIES 205
5.2.4 SOFTWARE TOOLS FOR BLOCK STRUCTURED PRECONDITIONERS.... 208 5.3
THE BIDOMAIN EQUATIONS 209
5.3.1 THE MATHEMATICAL MODEL 209
5.3.2 NUMERICAL METHOD 211
5.3.3 SOLUTION OF THE LINEAR SYSTEM 212
5.3.4 A SIMULATOR FOR THE BIDOMAIN MODEL 213
5.3.5 A SIMULATOR WITH MULTIGRID 220
5.3.6 A SOLVER BASED ON OPERATOR SPLITTING 222
5.4 TWO SADDLE POINT PROBLEMS 226
5.4.1 STOKES PROBLEM 227
5.4.2 MIXED POISSON PROBLEM 231
REFERENCES 235
6 FULLY IMPLICIT M E T H O DS FOR SYSTEMS OF P D ES 237
AE. 0DEGAERD, H. P. LANGTANGEN, A. TVEITO 6.1 INTRODUCTION 237
6.2 IMPLEMENTATION OF SOLVERS FOR PDE SYSTEMS IN DIFFPACK 238
6.2.1 HANDLING SYSTEMS OF PDES 238
6.2.2 A SPECIFIC 2 X2 SYSTEM 239
6.3 PROBLEM WITH THE GAUSS-SEIDEL METHOD, BY EXAMPLE 240
6.3.1 A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS 240
6.4 FULLY IMPLICIT IMPLEMENTATION 243
6.4.1 A SYSTEM OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS 244
6.4.2 IMPLEMENTATION 244
6.4.3 SYSTEMS OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS.. . 247 6.5
APPLICATIONS 248
6.5.1 A SYSTEM OF PDES MODELLING PIPEFLOW 248
6.5.2 COMPARISON OF THE GAUSS-SEIDEL AND THE FULLY IMPLICIT METHOD 250
6.5.3 TWO-PHASE POROUS MEDIA FLOW 251
6.5.4 SOLVER 254
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TABLE OF CONTENTS XIII
6.5.5 COMPARISON OF THE GAUSS-SEIDEL METHOD AND THE FULLY IMPLICIT
METHOD 254
6.6 CONCLUSION 255
REFERENCES 256
7 STOCHASTIC P A R T I AL DIFFERENTIAL EQUATIONS 257
H. P. LANGTANGEN, H. OSNES 7.1 INTRODUCTION 257
7.2 SOME SIMPLE EXAMPLES 259
7.2.1 BENDING OF A BEAM 260
7.2.2 VISCOUS DRAG FORCES 264
7.2.3 HEAT CONDUCTION IN THE EARTH S CRUST 266
7.2.4 TRANSPORT PHENOMENA 268
7.2.5 GENERIC PROBLEMS 269
7.3 SOLUTION METHODS 271
7.3.1 MONTE CARLO SIMULATION 272
7.3.2 PERTURBATION METHODS 275
7.4 QUICK OVERVIEW OF DIFFPACK TOOLS 277
7.5 TOOLS FOR RANDOM VARIABLES 279
7.5.1 RANDOM NUMBER GENERATION 279
7.5.2 DESCRIPTION OF PROBABILITY DISTRIBUTIONS 280
7.5.3 ESTIMATION OF STATISTICS OF RANDOM VARIABLES 282
7.5.4 EXAMPLE: SIMULATION OF A STOCHASTIC BEAM 284
7.5.5 SUGGESTED DESIGN OF STOCHASTIC PDE SIMULATORS 287
7.5.6 EXAMPLE: STOCHASTIC HEAT CONDUCTION 290
7.6 DIFFPACK TOOLS FOR RANDOM FIELDS 298
7.6.1 GENERATION OF RANDOM FIELDS 298
7.6.2 STATISTICAL PROPERTIES 303
7.6.3 EXAMPLE: A STOCHASTIC POISSON EQUATION 304
7.7 SUMMARY 307
7.A TRANSFORMATION OF RANDOM VARIABLES 308
7.A.1 TRANSFORMATION OF A SINGLE RANDOM VARIABLE 308
7.A.2 TRANSFORMATION OF NORMAL AND LOGNORMAL RANDOM VARIABLES 311
7.A.3 OTHER VARIABLE TRANSFORMATIONS 312
7.A.4 PARTIAL AND APPROXIMATE ANALYSIS OF RANDOM TRANSFORMATIONS 314
7.B IMPLEMENTING A NEW DISTRIBUTION 316
REFERENCES 318
8 USING DIFFPACK FROM P Y T H ON SCRIPTS 321
H. P. LANGTANGEN, K.-A. MARDAL 8.1 INTRODUCTION 321
8.1.1 THE ADVANTAGES OF HIGH-LEVEL LANGUAGES 322
8.1.2 PYTHON SCRIPTING AND DIFFPACK 323
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XIV
TABLE OF CONTENTS
8.1.3 EXAMPLE: RUNNING A DIFFPACK SIMULATOR FROM PYTHON .. 324 8.2
DEVELOPING PYTHON INTERFACES TO C / C ++ FUNCTIONS 325
8.2.1 WRAPPER FUNCTIONS 326
8.2.2 CREATING AND USING A MODULE 328
8.2.3 HOW SWIG SIMPLIFIES WRITING WRAPPER CODE 328
8.2.4 WRITING PYTHON INTERFACES TO DIFFPACK RELIES ON SWIG . 329 8.3
COMPILING AND LINKING WRAPPER CODE WITH DIFFPACK 329
8.3.1 MAKEFILES 330
8.3.2 SUMMARY OF CREATING THE INTERFACE 332
8.3.3 A TRIVIAL TEST MODULE 333
8.3.4 COMMON ERRORS IN THE LINKING PROCESS 334
8.3.5 PROBLEMS WITH INTERFACING A SIMULATOR CLASS 335
8.3.6 PYTHON INTERFACE, VERSION 1 335
8.3.7 VERSION 2; SETTING INPUT PARAMETERS 337
8.3.8 VERSION 3; USING THE C PREPROCESSOR TO EXPAND MACROS 340 8.3.9
VERSION 4; EXTENDING THE PYTHON INTERFACE WITH AUXILIARY FUNCTIONS 342
8.3.10 SWIG POINTERS 343
8.4 CONVERTING DATA BETWEEN DIFFPACK AND PYTHON 346
8.4.1 A CLASS FOR DATA CONVERSION 346
8.4.2 CONVERSION BETWEEN VEC AND NUMPY ARRAYS 346
8.4.3 CREATING A PYTHON INTERFACE TO THE CONVERSION CLASS . .. 348 8.4.4
EXAMPLES ON USING THE CONVERSION CLASS INTERFACE 349
8.4.5 A STRING TYPEMAP 350
8.5 BUILDING AN INTERFACE TO A MORE ADVANCED SIMULATOR 351
8.5.1 COMPUTING EMPIRICAL CONVERGENCE ESTIMATES 351
8.5.2 VISUALIZATION WITH VTK 355
8.6 INSTALLING PYTHON, SWIG ETC 356
8.7 CONCLUDING REMARKS 359
REFERENCES 359
9 PERFORMANCE MODELING OF P DE SOLVERS 361
X. CAI, A. M. BRUASET, H. P. LANGTANGEN, G. T. LINES, K. SAMUELSSON, W.
SHEN, A. TVEITO, G. ZUMBUSCH 9.1 INTRODUCTION 361
9.2 MODEL PROBLEMS 363
9.2.1 THE ELLIPTIC BOUNDARY-VALUE PROBLEM 363
9.2.2 THE LINEAR ELASTICITY PROBLEM 365
9.2.3 THE PARABOLIC PROBLEM 366
9.2.4 THE NONLINEAR WATER WAVE PROBLEM 366
9.2.5 THE TWO-PHASE FLOW PROBLEM IN 2D 368
9.2.6 THE HEART-TORSO COUPLED SIMULATIONS 369
9.2.7 THE SPECIES TRANSPORT PROBLEM 370
9.2.8 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS 371
9.3 NUMERICAL METHODS 371
IMAGE 8
TABLE OF CONTENTS XV
9.3.1 THE ELLIPTIC PROBLEMS 371
9.3.2 THE HC2 PROBLEM 372
9.3.3 THE NONLINEAR WATER WAVE PROBLEM 372
9.3.4 THE T F2 PROBLEM 372
9.3.5 THE H T2 AND H T3 SIMULATIONS 373
9.3.6 THE A D3 PROBLEM 373
9.3.7 THE NS2 PROBLEM 374
9.3.8 SOLUTION METHODS FOR LINEAR SYSTEMS 374
9.3.9 DIFFPACK IMPLEMENTATION AND NOTATION 376
9.4 TOTAL CPU TIME CONSUMPTION 377
9.4.1 ESTABLISHING PERFORMANCE MODELS 378
9.4.2 THE SOFTWARE AND HARDWARE SPECIFLCATION 379
9.4.3 THE BEST PERFORMANCE MODEL FOR EACH TEST PROBLEM . .. 380 9.4.4
THE CPU MEASUREMENTS AND PERFORMANCE MODELS 380
9.4.5 SOME REMARKS ABOUT THE MEASUREMENTS 385
9.4.6 MEASUREMENTS OBTAINED ON ANOTHER PLATFORM 385
9.5 SOLUTION OF LINEAR SYSTEMS 386
9.5.1 SUMMARY 387
9.5.2 MEASUREMENTS 387
9.5.3 EMCIENCY OF THE LINEAR ALGEBRA TOOLS IN DIFFPACK 392
9.6 CONSTRUCTION OF LINEAR SYSTEMS 392
9.6.1 THE PROCESS 392
9.6.2 SOME GUIDELINES 394
9.6.3 THE MAPPED LAPLACE EQUATION IN THE W A3 PROBLEM . .. 395 9.6.4
PARABOLIC PROBLEMS 396
9.7 CONCLUDING REMARKS 397
REFERENCES 398
10 ELECTRICAL ACTIVITY IN T HE H U M AN HEART 401
J. SUNDNES, G.T. LINES, P. GR0TTUM, A. TVEITO 10.1 THE BASIC PHYSIOLOGY
401
10.2 OUTLINE OF A MATHEMATICAL MODEL 402
10.3 THE BIDOMAIN MODEL 404
10.3.1 A CONTINUOUS MODEL FOR THE HEART TISSUE 404
10.3.2 DERIVATION OF THE BIDOMAIN EQUATIONS 405
10.4 A COMPLETE MATHEMATICAL MODEL 407
10.4.1 BOUNDARY CONDITIONS ON THE HEART-TORSO INTERFACE 407
10.4.2 BOUNDARY CONDITIONS ON THE SURFACE OF THE BODY 408
10.4.3 SUMMARY OF THE MATHEMATICAL PROBLEM 408
10.5 PHYSIOLOGY OF THE HEART MUSCLE TISSUE 409
10.5.1 PHYSIOLOGY OF THE CELL MEMBRANE 409
10.5.2 THE NERNST POTENTIAL 410
10.5.3 MODELS FOR THE IONIC CURRENT 412
10.5.4 ELECTRIC CIRCUIT MODEL FOR THE MEMBRANE 412
10.5.5 CHANNEL GATING 413
IMAGE 9
XVI TABLE OF CONTENTS
10.5.6 THE HODGKIN-HUXLEY MODEL 415
10.5.7 THE BEELER-REUTER MODEL 415
10.5.8 THE LUO-RUDY MODEL 416
10.5.9 A DIFFERENT MODEL FOR CALCIUM DYNAMICS 420
10.5.10 STRUCTURE OF THE HEART TISSUE 425
10.6 THE NUMERICAL METHOD 427
10.6.1 SIMPLIFICATIONS DUE TO BOUNDARY CONDITIONS ON DH . . .. 427
10.6.2 TIME DISCRETIZATION 428
10.6.3 DISCRETIZATION IN SPACE 429
10.6.4 CALCULATION OF THE CONDUCTIVITY TENSORS 431
10.6.5 SOLUTION OF THE ODES 431
10.7 IMPLEMENTATION 433
10.7.1 DESIGN 433
10.7.2 THE COMMONREL CLASS 434
10.7.3 THE CELLS CLASS 436
10.7.4 THE CELLMODEL CLASS 436
10.7.5 THE PARABOLIC CLASS 437
10.7.6 THE ELLIPTIC CLASS 437
10.8 OPTIMIZATION OF THE SIMULATOR 438
10.8.1 AN IMPLICIT ODE SOLVER 438
10.8.2 LOCAL MAKESYSTEM ROUTINES 440
10.8.3 ADAPTIVITY 440
10.9 SIMULATION RESULTS 442
10.9.1 TEST RESULTS ON A SIMPLE 3D GEOMETRY 443
10.9.2 PHYSIOLOGICALLY CORRECT GEOMETRIES 445
10.10 CONCLUDING REMARKS 447
REFERENCES 448
11 M A T H E M A T I C AL MODELS OF FINANCIAL DERIVATIVES ... 451 O.
SKAVHAUG, B. F. NIELSEN, A. TVEITO 11.1 INTRODUCTION 451
11.2 BASIC ASSUMPTIONS 453
11.2.1 ARBITRAGE-FREE MARKETS 453
11.2.2 THE EFFICIENT MARKET HYPOTHESIS 454
11.2.3 OTHER ASSUMPTIONS 454
11.3 FORWARDS AND FUTURES 454
11.3.1 THE FORWARD CONTRACT 455
11.3.2 THE FORWARD PRICE 4!
11.3.3 THE FUTURE CONTRACT 4;
11.4 THE BLACK-SCHOLES ANALYSIS 4!
11.4.1 ITO S LEMMA 4!
11.4.2 ELIMINATION OF RANDOMNESS 4(
11.4.3 THE BLACK-SCHOLES EQUATION 4F
11.5 EUROPEAN CALL AND PUT OPTIONS 4F
11.5.1 EUROPEAN CALL OPTIONS 40^
IMAGE 10
TABLE OF CONTENTS XVII
11.5.2 EUROPEAN PUT OPTIONS 464
11.5.3 PUT-CALL PARITY 465
11.5.4 ANALYTICAL SOLUTIONS 465
11.6 AMERICAN OPTIONS 466
11.6.1 AMERICAN PUT OPTIONS 467
11.7 EXOTIC OPTIONS 468
11.7.1 CORRELATION-DEPENDENT OPTIONS 468
11.7.2 PATH-DEPENDENT OPTIONS 471
11.8 HEDGING 474
11.8.1 THE DELTA GREEK 474
11.8.2 THE GAMMA GREEK 476
11.8.3 THE THETA GREEK 478
11.8.4 THE VEGA GREEK 479
11.8.5 THE RHO GREEK 480
11.9 REMARKS 481
REFERENCES 481
12 NUMERICAL M E T H O DS FOR FINANCIAL DERIVATIVES 483
0. SKAVHAUG, B. F. NIELSEN, A. TVEITO 12.1 INTRODUCTION 483
12.2 MODEL SUMMARY 484
12.3 MONTE-CARLO METHODS 488
12.4 LATTICE METHODS 490
12.5 FINITE DIFFERENCE METHODS 493
12.5.1 EUROPEAN OPTIONS 494
12.5.2 AMERICAN OPTIONS 496
12.6 FINITE ELEMENT METHODS 497
12.6.1 IMPLEMENTING A FEM SOLVER 498
12.6.2 EXTENSIONS 504
REFERENCES 505
13 FINITE ELEMENT MODELING OF ELASTIC STRUCTURES ... 507 T. THORVALDSEN,
H. P. LANGTANGEN, H. OSNES 13.1 INTRODUCTION 507
13.1.1 TWO VERSIONS OF THE FINITE ELEMENT METHOD 507
13.1.2 TWO ELEMENT CONCEPTS 509
13.2 AN INTRODUCTORY EXAMPLE; BAR ELEMENTS 510
13.2.1 DIFFERENTIAL EQUATION FORMULATION 511
13.2.2 ENERGY FORMULATION 516
13.3 ANOTHER EXAMPLE; BEAM ELEMENTS 518
13.3.1 DIFFERENTIAL EQUATION FORMULATION 519
13.3.2 ENERGY FORMULATION 521
13.4 GENERAL THREE-DIMENSIONAL ELASTICITY 522
13.4.1 DIFFERENTIAL EQUATION FORMULATION 522
13.4.2 ENERGY FORMULATION 525
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XVIII TABLE OF CONTENTS
13.5 DEGREES OF PREEDOM AND BASIS FUNCTIONS 527
13.5.1 BAR ELEMENTS 528
13.5.2 BEAM ELEMENTS 530
13.5.3 FRAME ELEMENTS 531
13.5.4 DKT PLATE ELEMENTS 532
13.6 MATERIAL TYPES AND ELASTICITY MATRICES 534
13.6.1 LINEAR ELASTIC, ISOTROPIE MATERIAL 535
13.6.2 ORTHOTROPIC MATERIAL; FIBER COMPOSITE 535
13.7 ELEMENT MATRICES IN LOCAL COORDINATES 537
13.7.1 ELEMENTS WITH LINEAR, ISOTROPIE MATERIAL PROPERTIES . . .. 537
13.7.2 ELEMENTS WITH ORTHOTROPIC MATERIAL PROPERTIES 539
13.7.3 DKT PLATE ELEMENTS 539
13.8 ELEMENT LOAD VECTORS IN LOCAL COORDINATES 539
13.8.1 BAR ELEMENTS 540
13.8.2 BEAM ELEMENTS 542
13.8.3 FRAME ELEMENTS 544
13.8.4 DKT PLATE ELEMENTS 545
13.9 ELEMENT MATRICES AND VECTORS IN GLOBAL COORDINATES 546
13.9.1 BAR, BEAM, AND FRAME ELEMENTS 546
13.9.2 DKT PLATE ELEMENTS 546
13.10 ELEMENT FORCES, STRESSES, AND STRAINS 549
13.10.1 BAR, BEAM, AND FRAME ELEMENTS 549
13.10.2 DKT PLATE ELEMENTS 550
13.11 IMPLEMENTATION OF STRUCTURAL ELEMENTS 551
13.11.1 CLASS STRUCTELMDEF 552
13.11.2 CLASS BARELM 553
13.11.3 CLASS BEAMELM 553
13.11.4 CLASS ISODKTELM 554
13.11.5 CLASS ORTHODKTELM 554
13.11.6 CLASS STRUCTELMS 554
13.11.7 HOW TO IMPLEMENT NEW STRUCTURAL ELEMENTS 555
13.12 SOME EXAMPLE PROGRAMS 556
13.12.1 THE BAR ELEMENT SIMULATOR 557
13.12.2 INDICATORS IN THE ANSYS SIMULATORS 560
13.13 TEST PROBLEMS 561
13.13.1 BAR ELEMENTS IN 1D 561
13.13.2 BAR ELEMENTS IN 2D 562
13.13.3 BEAM ELEMENTS IN 2D; THREE-STOREY FRAMEWORK 567
13.13.4 BAR AND FRAME ELEMENTS IN 2D 569
13.13.5 TWISTING OF A SQUARE PLATE; ISOTROPIE MATERIAL 570
13.13.6 SIMPLY SUPPORTED FIBER COMPOSITE PLATE 572
13.13.7 TEST PROBLEMS USING ANSYS INPUT FILES 573
13.13.8 TEST PROBLEMS SUMMARY 574
13.14 SUMMARY 574
REFERENCES 576
IMAGE 12
TABLE OF CONTENTS XIX
14 SIMULATION OF ALUMINUM EXTRUSION 577
K. M. OKSTAD, T. KVAMSDAL 14.1 INTRODUCTION 577
14.2 MATHEMATICAL FORMULATION 579
14.2.1 BASIC DEFINITIONS 579
14.2.2 GOVERNING EQUATIONS 580
14.2.3 BOUNDARY CONDITIONS 581
14.2.4 VARIATIONAL FORMULATION 583
14.3 FINITE ELEMENT IMPLEMENTATION 585
14.3.1 TIME DISCRETIZATION 585
14.3.2 SPATIAL DISCRETIZATION 586
14.3.3 GLOBAL SOLUTION PROCEDURE AND MESH MOVEMENT 586
14.4 OBJECT-ORIENTED IMPLEMENTATION 587
14.4.1 INTRODUCTION 587
14.4.2 CLASS HIERARCHY FOR THE PROBLEM-DEPENDENT DATA 588
14.4.3 CLASS HIERARCHY FOR THE NUMERICAL SOLVERS 593
14.5 NUMERICAL EXPERIMENTS 600
14.5.1 THE JEFFERY-HAMEL FLOW PROBLEM 600
14.5.2 THE EXTRUSION PROBLEM 603
14.5.3 SIMULATIONS OF THE TEMPERATURE 604
14.6 CONCLUDING REMARKS 604
REFERENCES 608
15 SIMULATION OF SEDIMENTARY BASINS 611
A. KJELDSTAD, H. P. LANGTANGEN, J. SKOGSEID, K. BJ0RLYKKE 15.1
INTRODUCTION 611
15.2 THE GEOMECHANICAL AND MATHEMATICAL PROBLEM 613
15.2.1 ELASTIC DEFORMATIONS 613
15.2.2 FLUID FLOW 616
15.2.3 HEAT TRANSFER 619
15.2.4 INITIAL CONDITIONS 620
15.2.5 BOUNDARY CONDITIONS 622
15.3 NUMERICAL METHODS 622
15.3.1 DISCRETIZATION TECHNIQUE 623
15.3.2 NONLINEAR SOLUTION TECHNIQUE 623
15.3.3 THE LINEAR SYSTEM 624
15.4 IMPLEMENTING A SOLVER FOR A SYSTEM OF PDES 628
15.5 VERIFICATION 630
15.5.1 CYLINDER WITH CONCENTRIC CIRCULAR HOLE 630
15.5.2 ONE-DIMENSIONAL CONSOLIDATION 633
15.6 A MAGMATIC SILL INTRUSION CASE STUDY 635
15.6.1 CASE DEFINITION 641
15.6.2 RESULTS AND DISCUSSION 643
15.7 CONCLUDING REMARKS 656
REFERENCES 657
|
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| id | DE-604.BV017304440 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:16:22Z |
| institution | BVB |
| isbn | 3540014381 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010430147 |
| oclc_num | 53178519 |
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| physical | XIX, 658 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
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| publisher | Springer |
| record_format | marc |
| series | Lecture notes in computational science and engineering |
| series2 | Lecture notes in computational science and engineering |
| spelling | Advanced topics in computational partial differential equations numerical methods and Diffpack programming Hans Petter Langtangen ... ed. Berlin [u.a.] Springer 2003 XIX, 658 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in computational science and engineering 33 Diffpack (Computer file) Datenverarbeitung Differential equations, Partial Numerical solutions Data processing C++ (DE-588)4193909-8 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Programmierumgebung (DE-588)4134837-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s C++ (DE-588)4193909-8 s Programmierumgebung (DE-588)4134837-0 s DE-604 Langtangen, Hans Petter 1962- Sonstige (DE-588)1019109599 oth Lecture notes in computational science and engineering 33 (DE-604)BV011386476 33 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010430147&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Advanced topics in computational partial differential equations numerical methods and Diffpack programming Lecture notes in computational science and engineering Diffpack (Computer file) Datenverarbeitung Differential equations, Partial Numerical solutions Data processing C++ (DE-588)4193909-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Programmierumgebung (DE-588)4134837-0 gnd |
| subject_GND | (DE-588)4193909-8 (DE-588)4128130-5 (DE-588)4044779-0 (DE-588)4134837-0 |
| title | Advanced topics in computational partial differential equations numerical methods and Diffpack programming |
| title_auth | Advanced topics in computational partial differential equations numerical methods and Diffpack programming |
| title_exact_search | Advanced topics in computational partial differential equations numerical methods and Diffpack programming |
| title_full | Advanced topics in computational partial differential equations numerical methods and Diffpack programming Hans Petter Langtangen ... ed. |
| title_fullStr | Advanced topics in computational partial differential equations numerical methods and Diffpack programming Hans Petter Langtangen ... ed. |
| title_full_unstemmed | Advanced topics in computational partial differential equations numerical methods and Diffpack programming Hans Petter Langtangen ... ed. |
| title_short | Advanced topics in computational partial differential equations |
| title_sort | advanced topics in computational partial differential equations numerical methods and diffpack programming |
| title_sub | numerical methods and Diffpack programming |
| topic | Diffpack (Computer file) Datenverarbeitung Differential equations, Partial Numerical solutions Data processing C++ (DE-588)4193909-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Programmierumgebung (DE-588)4134837-0 gnd |
| topic_facet | Diffpack (Computer file) Datenverarbeitung Differential equations, Partial Numerical solutions Data processing C++ Numerisches Verfahren Partielle Differentialgleichung Programmierumgebung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010430147&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011386476 |
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