Finite Element Analysis for Engineers	4
	Preface	6
	1 Introduction	16
	2 The Basic Procedure	20
	3 Some Elasticity Theory	38
	3.1 Displacements and Strains	38
	3.1.1 For the Truss	38
	3.1.2 For Plane Stress	40
	3.1.3 In Space	46
	3.1.4 For the Plate	47
	3.2 Stress-Strain Relations	49
	3.3 Basics of Thermomechanical Loading	59
	3.4 Basic Principles of Natural Vibration	62
	3.5 Basic Principles of Non-linear Calculations	65
	4 Finite Elements and Element Matrices	78
	4.1 Basics of Element Stiffness Matrices	80
	4.2 Constitutive Matrices	84
	4.3 B Matrix	85
	4.4 Shape Functions	86
	4.5 Integration	96
	4.6 The Application of Loads, Load Vectors	103
	4.6.1 The Basic Procedure	103
	4.6.2 Plate Elements	106
	4.6.3 Volume Elements	108
	4.6.4 Plane and Axial-Symmetrical State of Stress	119
	4.6.5 Distributed Loads for Beams	121
	4.6.6 Gerber Joints for Beams	123
	4.7 A complete Element Stiffness Routine	127
	4.8 Some Remarks on Modelling	136
	4.8.1 Choice of Element Types	136
	4.8.2 Polymers and Material Laws	144
	4.8.3 Structural Optimization	145
	4.9 Some Remarks on Shells	149
	4.10 Element Matrices for Heat Transfer	163
	4.11 Element Matrices for Vibration	165
	4.12 Element Matrices of the Non-linear Finite Element Analysis	167
	5 Compilation, Storage Schemes and Boundary Conditions	178
	5.1 Compilation	178
	5.2 Storage Schemes	189
	5.2.1 Band Width Storage Scheme	191
	5.2.2 The Skyline Storage Scheme	195
	5.2.3 The Jennings Storage Scheme	197
	5.2.4 The Non-Zero Storage Scheme	205
	5.2.5 Summary of the Storage Schemes	211
	5.3 Boundary Conditions	212
	5.3.1 Single Forces and Single Displacements	212
	5.3.2 Distributed Loads with Plates	215
	5.3.3 Fixture of plates	217
	5.3.4 Boundary Conditions in Temperature Analyses	218
	5.3.5 Boundary Conditions with Vibration	221
	5.3.6 Boundary Conditions in the Non-linear Finite Element Analysis	222
	6 Solvers	224
	6.1 Direct Solvers	225
	6.1.1 The Cholesky Solver	227
	6.2 Condition and Scaling	229
	6.3 Iterative Solvers	238
	6.3.1 The Jacobi Method	240
	6.3.2 The Gauss-Seidel Method	241
	6.3.3 The SOR Method and the JOR Method	241
	6.3.4 The basic CG Solver	242
	6.3.5 The CG Solver with Pre-conditioning	244
	6.4 Solver for Thermomechanical Problems	259
	6.5 Solver for Vibration Problems	259
	6.6 Solver for the Non-linear Finite Element Analysis	269
	7 Stresses and Nodal Forces	272
	7.1 Stresses	272
	7.2 Reduced Stresses	279
	7.3 Nodal Forces	286
	8 Mesh Generation of Curvilinear Finite Elements	290
	8.1 Basis Considerations of the Procedure	290
	8.2 Mathematical Foundations	292
	8.3 Description of a Simple Mapped Mesher	296
	9 Z88: The Basics	304
	9.1 General Information	304
	9.1.1 Summary of the Z88 Element Library	305
	9.2 The Open Source FE Program Z88	317
	9.2.1 Overview of the Z88 Program Modules	317
	9.2.2 Dynamic Memory Z88	320
	9.2.3 The Input and Output of Z88:	323
	9.3 The Freeware FE Program Z88Aurora	327
	9.3.1 Overview of the Z88Aurora Modules	327
	9.3.2 Memory Requirement in Z88Aurora	330
	9.3.3 The Input and Output of Z88Aurora	331
	10 Z88: The Modules	334
	10.1 The Linear Solver Z88R	334
	10.1.1 Z88R: The Cholesky Solver	335
	10.1.2 Z88R: The Sparse Matrix Solvers SICCG and SORCG	336
	10.1.3 Z88R: The Sparse Matrix multi-core Solver PARDISO	338
	10.1.4 Which Solver to choose?	339
	10.1.5 Explanations for Stress Calculations	339
	10.1.6 Explanations for Nodal Force Calculations	340
	10.2 The Mapped Mesher Z88N	340
	10.3 The Advanced Mapped Mesher in Z88Aurora	343
	10.3.1 The Use of Z88N in Z88Aurora	343
	10.3.2 Tetrahedron Refiner Z88MTV	344
	10.3.3 The 2D Shell Thickener Z88MVS	346
	10.4 The OpenGL Plot Program Z88O in Z88 V14 OS or the Post-Processor of Z88Aurora	346
	10.5 The DXF Converter Z88X	350
	10.6 The 3D Converter Z88G	359
	10.7 The Ansys Converter Z88ASY in Z88Aurora	362
	10.8The Abaqus Converter Z88INP in Z88Aurora	364
	10.9 Das Cuthill-McKee Program Z88H	365
	10.10 The STEP Import Z88GEOCON (STEP) in Z88Aurora	367
	10.11 The STL Converter Z88GEOCON (STL) in Z88Aurora	369
	10.12 The Tetrahedron Mesher in Z88Aurora	370
	10.13 The Picking Module of Z88Aurora	371
	10.14 The Material Data Base of Z88Aurora	373
	10.15 Applying Boundary Conditions in Z88Aurora	373
	10.16 The User Support with Spider in Z88Aurora	374
	10.17 The Thermomechanical Solver in Z88Aurora	375
	10.18 The free Vibration Solver in Z88Aurora	378
	10.19 The Non-linear Solver Z88NL of Z88Aurora	381
	11 Generating Input Files	386
	11.1 General Information	386
	11.2 General Structure Data File Z88I1.TXT	388
	11.3 Boundary Condition File Z88I2.TXT	389
	11.4 Surface and Pressure Loads File Z88I5.TXT	392
	11.5 Material Parameters File Z88MAT.TXT	397
	11.6 Material Data File *.TXT	398
	11.7 Element Parameters File Z88ELP.TXT	398
	11.8 Integration Order File Z88INT.TXT	400
	11.9 Mapped Mesher Input File Z88NI.TXT	401
	11.10 Solver Parameters File Z88MAN.TXT	405
	11.11 Comparison of the different Z88 Data File Formats	408
	12 The Finite Elements of Z88 and Z88Aurora	410
	12.1 Hexahedron No..1 with 8 Nodes	410
	12.2 Beam No..2 with 2 Nodes in Space	413
	12.3 Plane Stress Element No..3 with 6 Nodes	415
	12.4 Truss No..4 in Space	416
	12.5 Shaft No..5 with 2 Nodes	417
	12.6 Torus No..6 with 3 Nodes	419
	12.7 Plane Stress Element No..7 with 8 Nodes	420
	12.8 Torus No..8 with 8 Nodes	422
	12.9 Truss No. 9 in the Plane	424
	12.10 Hexahedron No..10 with 20 Nodes	426
	12.11 Plane Stress Element No..11 with 12 Nodes	429
	12.12 Torus No..12 with 12 Nodes	431
	12.13 Beam No..13 in the Plane	433
	12.14 Plane Stress Element No..14 with 6 Nodes	434
	12.15 Torus No..15 with 6 Nodes	436
	12.16 Tetrahedron No.16 with 10 Nodes	439
	12.17 Tetrahedron No..17 with 4 Nodes	442
	12.18 Plate No..18 with 6 Nodes	444
	12.19 Plate No..19 with 16 Nodes	446
	12.20 Plate No..20 with 8 Nodes	449
	12.21 Shell No..21 with 16 Nodes	451
	12.22 Shell No..22 with 12 Nodes	453
	12.23 Shell No..23 with 8 Nodes	455
	12.24 Shell No..24 with 6 Nodes	457
	12.25 Element/Solver Overview Z88Aurora V2	459
	13	13
	460	460
	13.1 Flat Wrench (Plate No..7)	467
	13.1.1 With Z88 V14	468
	13.1.2With Z88Aurora V2	476
	13.2 Crane Girder made of Trusses No..4	486
	13.2.1 With Z88 V14	487
	13.2.2 With Z88Aurora V2	492
	13.3 Gear Shaft with Shaft No..5	497
	13.3.1 With Z88 V14	499
	13.3.2 With Z88Aurora V2	502
	13.4 Bending Girder with Beam No..13	506
	13.4.1 With Z88 V14	507
	13.4.2 With Z88Aurora V2	511
	13.5 Plate Segment of Hexahedrons No..1 and No..10	515
	13.5.1 With Z88 V14	516
	13.5.2 With Z88Aurora V2	522
	13.6 Pipe under Internal Pressure, Plain Stress Element No..7	525
	13.6.1 With Z88 V14	526
	13.6.2 With Z88Aurora V2	533
	13.7 Pipe under Internal Pressure, Torus No..8	535
	13.7.1 With Z88 V14	536
	13.7.2 With Z88Aurora V2	542
	13.8 Two-Stroke Engine Piston	544
	13.8.1 With Z88 V14	545
	13.8.2 With Z88Aurora V2	549
	13.9 RINGSPANN Spring and Belleville Spring	554
	13.9.1 With Z88 V14	556
	13.9.2 With Z88Aurora V2	559
	13.10 Liquid Gas Tank	561
	13.10.1 With Z88 V14	561
	13.10.2 With Z88Aurora V2	565
	13.11 Motorcycle Crankshaft	567
	13.11.1 With Z88 V14	569
	13.11.2 With Z88Aurora V2	574
	13.12 Torque-measuring hub	578
	13.12.1 With Z88 V14	579
	13.12.2 With Z88Aurora V2	580
	13.13 Plane Frameworks	581
	13.13.1 With Z88 V14	582
	13.13.2 With Z88Aurora V2	602
	13.14 Gearwheel	604
	13.14.1 With Z88 V14	605
	13.14.2 With Z88AuroraV2	610
	13.15 3D Wrench	614
	13.15.1 With Z88 V14	614
	13.15.2 with Z88Aurora V2	626
	13.16 Force Measuring Element, Plane Stress Elements No..7	628
	13.16.1 With Z88 V14	628
	13.16.2 With Z88Aurora V2	638
	13.17 Circular Plate, Plates No..20	639
	13.17.1 With Z88 V14	641
	13.17.2 With Z88Aurora V2	645
	13.18 Rectangular Plate with 16 Nodes Plates No..19	646
	13.18.1 With Z88 V14	646
	13.18.2 With Z88Aurora V2	653
	13.19 Four-stroke Engine Pistons with Tetrahedrons No..16	654
	13.19.1 With Z88 V14	655
	13.19.2 With Z88Aurora V2	659
	13.20 Motorcar Fan Wheel	662
	13.20.1 With Z88 V14	664
	13.20.2 With Z88Aurora V2	665
	13.21 Diesel Piston	668
	13.21.1 With Z88 V14	669
	13.21.2 With Z88Aurora V2	671
	13.22 Calculation of a Stress Concentration Factor	672
	13.22.1 With Z88 V14	673
	13.22.2 With Z88Aurora V2	678
	13.23 Gear Root Stress	679
	13.23.1 With Z88 V14	681
	13.23.2 With Z88Aurora V2	683
	13.24 Square Pipe, Shell No..24	685
	13.24.1 With Z88 V14	686
	13.24.2 With Z88Aurora V2	688
	13.25 Submarine made of Shells No..22	692
	13.26 Gear Wheel out of Tetrahedrons No..17	697
	13.27 Oscillating Drum	700
	13.28 Modal Analysis Crankshaft	704
	13.29 Thermo-mechanical Analysis of a Spoon	707
	13.30 Thermal Analysis of a four-stroke Engine Piston	713
	13.31 Non-linear Calculation of a Belleville Spring	717
	13.32 Non-linear Calculation of a Hinge	721
	References and further reading	726
	Index	732