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Mathematical Methods for Mechanics - A Handbook with MATLAB Experiments-Springer (2008).pdf

Contents
Chapter 1. Mathematical Auxiliaries .........................1
1.1. Matrix Computations ............................................1
Vector and Matrix Products – Determinants and Cofactors – Eigenvalues and Eigenvectors – Decompositions of a Matrix – Linear Systems
of Equations – Projectors and Reflectors – The QR-Algorithm – The
Moore-Penrose Inverse – Over- and Underdetermined Linear Systems – Rotations in R3 – Matrices with Definite Real Part
1.2. Brief on Vector Analysis ....................................... 17
Notations and Definitions – Differential Rules – Integral Rules –
Coordinate-Free Definitions – Potentials and Vector Fields
1.3. Curves in R3 .....................................................25
Curvature and Torsion – Frenet’s Formulas
1.4. Linear Differential Equations .................................. 27
Homogenous Linear Differential Equations with Constant Coefficients –
Inhomogenous Linear Differential Equations with Constant Coefficients
and Special Right Sides – The General Solution – Example
1.5. Linear Differential Systems of First Order ................... 31
Autonomous Homogenous Systems with Diagonable Matrix – Autonomous Homogenous Systems with Undiagonable Matrix – On Stability – General Linear Systems – Special Right Sides – Boundary Value
Problems – Periodic Solutions
1.6. The Flux Integral and its Vector Field . . . . . . . . . . . . . . . . . . . . . . . 38
The Flux Integral – Stationary Vector Fields – Straightening of Vector
Fields – Invariants – Transformation – Examples
1.7. Vector Spaces .................................................. 45
Spaces of Continuous Functions – Banach Spaces – Linear Mappings –
Linear Functionals and Hyperplanes – Dual Spaces – Hilbert Spaces
– Sobolev Spaces – On Boundary Values – Poperties of Hs
0(Ω) and
Hs(Ω) – Equivalent Norms on Hs
0(Ω) and Hs(Ω)
X Contents
1.8. Derivatives .......................................................52
Gateaux and Frechet ´ Derivative – Properties – Examples
1.9. Mappings in Banach Spaces ................................... 57
Linear Operators – Projectors – Implicit Functions
1.10. Convex Sets and Functions ....................................61
Convex Sets and Cones – Separation Theorems – Cone Properties –
Convex Functions
1.11. Quadratic Functionals ......................................... 70
The Energy Functional – Operators in Hilbert Space – Projectors in
Hilbert Space – Properties of the Energy Functional – Ritz Approximation
Chapter 2. Numerical Methods ...............................79
2.1. Interpolation and Approximation ............................. 80
The General Interpolation Problem – Interpolating Polynomials – Interpolation after Lagrange – Interpolation after Newton – Interpolation
of Derivatives – Approximation by Bezier´ Polynomials – Interpolating
Splines
2.2. Orthogonal Polynomials ........................................90
Construction – The Formulas of Rodriguez – Minimum Property of
Chebyshev Polynomials
2.3. Numerical Integration ..........................................94
Integration Rules of Lagrange – Composite Integration Rules – Gauß
Integration – Suboptimal Integration Rules – Barycentric Coordinates
in Triangle – Domain Integrals
2.4. Initial Value Problems ........................................ 105
Euler’s Method – General One-Step Methods – Asymptotic Expansion
and Extrapolation – Runge-Kutta Methods – Multistep Methods –
Stability – Stiff Differential Systems – Further Examples – Full Implicit
Runge-Kutta Methods
2.5. Boundary Value Problems .................................... 128
The Linear Problem – Nonlinear Case – Boundary Value Problems with
Parameter – Example
2.6. Periodic Problems ............................................. 133
Problems with Known Period – Problems with Unknown Period – Examples
2.7. Differential Algebraic Problems .............................. 136
Formulation of the Problem – Runge-Kutta Methods – Regular Matrix Pencils – Differential Index – Semi-Explicit Runge-Kutta Methods
2.8. Hints to the MATLAB Programs ............................ 141
Contents XI
Chapter 3. Optimization .......................................143
3.1. Minimization of a Function ................................... 144
Descent Methods – Negative Examples – Convergence – Efficient Choice
of Descent Direction – Newton’s Method
3.2. Extrema with Constraints .................................... 149
Formulation of the Problem – Multiplier Rule – Kuhn-Tucker Points –
Example
3.3. Linear Programming ........................................... 154
Examples – Formulation of the Problem – Projection Method – Optimality Condition – Optimal Step Length – Change of Basis – Algorithm – Degenerated Extremal Points – Multiple Solutions – Equality
Constraints – Sensitivity – The Dual Problem – The Tableau – Example
3.4. Linear-Quadratic Problems ................................... 164
Primal Projection Method – The Algorithm plqp.m – Dual Projection
Method – The Algorithm dlqp.m – Examples for the Dual Method
3.5. Nonlinear Optimization ....................................... 169
Gradient Projection Method – Typical Iteration Step – Restoration –
Penalty Methods – The Algorithm sqp.m – Supplements – Examples
3.6. A Brief on Lagrange Theory ..................................177
Formulation of the Problem – Lagrange Problem – Saddlepoint Problems – Primal and Dual Problems – Geometrical Interpretation – Lokal
Lagrange Theory – Examples
3.7. Hints to the MATLAB Programs ............................ 191
Chapter 4. Variation and Control ...........................193
4.1. Variation ........................................................194
Extremal Problem, Variational Problem and Boundary Value Problem –
Modified Problems – Variable Terminal Point – Legendre Transformation – Lagrange Function and Hamilton Function – Examples
4.2. Control Problems without Constraints ...................... 211
Formulation of the Problem – Free Terminal Time – The Free Lagrange Multipliers – The Costate – Maximum Principle – The State
Regulator Problem
4.3. Control Problems with Constraints ..........................220
Formulation of the Problem – Necessary Conditions – On the Maximum
Principle
4.4. Examples ....................................................... 226
Numerical Approach – Examples
4.5. On the Re-Entry Problem .................................... 236
4.6. Hints to the MATLAB Programs ............................ 240
XII Contents
Chapter 5. The Road as Goal ................................ 241
5.1. Bifurcation Problems ..........................................242
Fredholm Operators – Formulation of the Problem – LjapunovSchmidt Reduction – The Branching Equation – Some Further Results –
Examples – Symmetry – Examples with Symmetry
5.2. Scaling .......................................................... 257
Modified Ljapunov-Schmidt Reduction – Homogenous Problems –
Nonlinear Eigenvalue Problem – Perturbated Eigenvalue Problem –
General Branching Points
5.3. Calculation of Singular Points ................................264
Classification – Turning Points – Calculation of Simple Branching Points
5.4. Ordinary Differential Systems ................................ 268
Linear Boundary Value Problem – Adjoint Boundary Value Problem –
Nonlinear Boundary Value Problems – Examples
5.5. Hopf Bifurcation ...............................................275
Formulation of the Problem – Simple Examples – Transformation to
Uniform Period – An Eigenvalue Problem – Scaled Problem – Discretization – Numerical Solution – Examples
5.6. Numerical Bifurcation .........................................288
Two Algorithms – A Classic Example
5.7. Continuation ................................................... 295
Formulation of the Problem – Predictor Step – Corrector Step – Examples
5.8. Hints to the MATLAB Programs ............................ 300
Chapter 6. Mass Points and Rigid Bodies ................301
6.1. The Force and its Moment ....................................301
6.2. Dynamics of a Mass Point .................................... 303
Equations of Motion – Energy – Hamilton’s Principle – Systems with
one Degree of Freedom – Rigid Rotation
6.3. Mass Point in Central Field .................................. 310
Equation of Motion – Total Energy – Shape of the Orbit – Kepler’s
Problem – Examples
6.4. Systems of Mass Points ....................................... 319
Equations of Motion – Potential and Kinetic Energy – Mass Points with
Constraints – D’Alembert’s Principle – Examples
6.5. The Three-Body Problem .....................................328
Formulation of Problem – Two-Body Problem – Restricted Three-Body
Problem – Periodic Solutions
6.6. Rotating Frames ............................................... 334
Rotation of a Body – Two Rotations – Motion in Rotating System –
Coriolis Force – Example
Contents XIII
6.7. Inertia Tensor and Top ........................................339
Inertia Tensor – Rigid Body with Stationary Point – Rotors – Top without External Forces – Symmetric Top without External Forces – Leaded
Symmetric Top – Kinematic Euler Equations – Heavy Symmetric
6.8. On Multibody Problems ...................................... 349
6.9. On Some Principles of Mechanics ............................353
Energy Principle – Extremal Principle – D’Alembert and Lagrange
– Hamilton’s Principle – Jacobi’s Principle
6.10. Hints to the MATLAB Programs ........................... 357
Chapter 7. Rods and Beams ..................................359
7.1. Bending Beam ................................................. 359
Tension Rod – Bending Beam– Total Energy – Variational Problem and
Boundary Value Problem – Balance of Moments – Further Boundary
Conditions – Existence of Solution
7.2. Eigenvalue Problems .......................................... 367
Generalized Eigenvalue Problem – Buckling of a Beam – Oscillating
Beam
7.3. Numerical Approximation .................................... 373
Tension Rod – Bending Beam – Examples
7.4. Frameworks of Rods ...........................................376
Tension Rod in General Position – Plane and Spacial Frameworks –
Support Conditions – Support Loads – Examples
7.5. Frameworks of Beams ......................................... 382
Torsion – Total Energy – Beam with Bending and Torsion – Numerical
Approximation
7.6. Hints to the MATLAB Programs ............................ 386
Chapter 8. Continuum Theory ...............................387
8.1. Deformations ...................................................387
Deformation – Derivation of the Gradient – Material Derivatives (Substantial Derivatives) – Piola Transformation – Pull Back of Divergence
Theorem
8.2. The Three Transport Theorems .............................393
8.3. Conservation Laws .............................................396
Conservation Law of Mass, Momentum, Angular Momentum and Energy – Conservation Laws in Differential Form – Second Law of Thermodynamics
8.4. Material Forms .................................................403
Conservation Laws – Variational Problem – Extremal Problem –
Hamilton’s Principle
Top – Energy – Examples
XIV Contents
8.5. Linear Elasticity Theory ...................................... 408
Strain- and Stress Tensor – Extremal Problem and Variational Problem – Boundary Value Problem – St.Venant-Kirchhoff Material –
Elasticity and Compliance Matrix
8.6. Discs ............................................................ 413
Plane Stress – Plane Strain
8.7. Kirchhoff ’s Plate ...............................................415
Extremal Problem and Variational Problem – Transformation – Boundary Value Problem – Babuska Paradoxon – Example
8.8. Von Karman’s Plate and the Membrane .................... 421
Strain Energy – Airy’s Stress Function – Von Karman’s Equations
8.9. On Fluids and Gases .......................................... 424
Conservation Laws – Notations – Conservation Laws of Viscous Fluids –
Homogenous Fluids
8.10. Navier-Stokes Equations ..................................... 427
Velocity-Pressure Form – Boundary Value Problem – Non-Dimensional
System – Stream-Function Vorticity Form – Connection with the Plate
Equation – Calculation of Pressure
Chapter 9. Finite Elements ................................... 435
9.1. Elliptic Boundary Value Problems ........................... 435
Extremal Problem – Weak Form – Boundary Value Problem – Existence
of Solutions
9.2. From Formula to Figure, Example ........................... 439
Formulation of the Problem – Approximation – Linear Triangular Elements – Implementation of Dirichlet Boundary Conditions – Implementation of Cauchy Boundary Conditions
9.3. Constructing Finite Elements ................................ 445
Formulation of the Problem – Integration by Shape Functions – Reduction to Unit Triangle – Examples
9.4. Further Topics ................................................. 452
Hermitian Elements – Normal Derivatives – Argyris’ Triangle – A
Triangular Element with Curvilinear Edges – Finite Elements for Discs
– On the Patch Test – A Cubic Triangular Element for Plates
9.5. On Singular Elements ......................................... 467
Transition to Polar Coordinates – Laplace Equation – Example
9.6. Navier-Stokes Equations ...................................... 471
Incompressible Stationary Equations – Convective Term – TaylorHood Element – Stream-Function Vorticity Form – Coupled Stationary
System – Boundary Conditions for Stream-Function Vorticity Form
Contents XV
9.7. Mixed Applications ............................................482
Heat Conduction – Convection – Mass Transport – Shallow Water Problems
9.8. Examples ....................................................... 489
Navier-Stokes Equations – Convection – Shallow Water Problems –
Discs and Plates
9.9. Hints to the MATLAB Programs ............................ 498
Chapter 10. A Survey on Tensor Calculus ................503
10.1. Tensor Algebra ................................................ 503
Transformation of Basis and Components – Scalar Product Spaces –
Identifying V and Vd – General Tensors – Representation and Transformation of Tensors – Tensor Product – Vector Space of Tensors –
Representation of General Tensors – Transformation of General Tensors – Contraction – Scalar Product of Tensors – Raising and Lowering
of Indices – Examples
10.2. Algebra of Alternating Tensors ..............................520
Alternating Tensors – Alternating Part of Tensors – Exterior Product of
Tensors – Basis – Representation of Alternating Tensors – Basis Transformation – Scalar Product of Alternating Tensors
10.3. Differential Forms in Rn ...................................... 525
The Abstract Tangential Space and Pfaffian Forms – Differential
Forms – Exterior Derivatives – Closed and Exact Forms – Hodge Star
Operator and Integral Theorems – Transformations – Push Forward
10.4. Tensor Analysis ............................................... 537
Euklidian Manifolds – Natural Coordinate Systems – Representation
and Transformation – Christoffel Symbols – Divergence of Gradient
of a Scalar Field – Gradient of a Tensor – Divergence of a Tensor Field
– Rotation of a Vector Field
10.5. Examples .......................................................550
Brief Recapitulation – Orthogonal Natural Coordinate Systems – Divergence and Rotation
10.6. Transformation Groups .......................................555
Notations and Definitions – Examples – One-Parametric Transformation Groups – Generator of a Group
Chapter 11. Case Studies ......................................561
11.1. An Example of Gas Dynamics ............................... 561
11.2. The Reissner-Mindlin Plate ..................................563
11.3. Examples of Multibody Problems ...........................565
Double Pendulum – Seven-Body Problem (Andrew’s Squeezer) –
Roboter after Schiehlen
XVI Contents
11.4. Dancing Discs ................................................. 568
General Discs – Cogwheels – Gears with Zero-Gearing
11.5. Buckling of a Circular Plate ................................. 574
Chapter 12. Appendix ..........................................577
12.1. Notations and Tables ......................................... 577
Notations – Measure Units and Physical Constants – Shape Functions
of Complete Cubic Triangular Element – Argyris’ Element
12.2. Matrix Zoo .................................................... 581
12.3. Translation and Rotation .....................................583
12.4. Trigonometric Interpolation ................................. 585
Fourier Series – Discrete Fourier Transformation – Trigonometric
Interpolation
12.5. Further Properties of Vector Spaces ........................ 591
Sets of Measure Zero – Functions of Bounded Variation – The Dual
Space of C[a, b] – Examples
12.6. Cycloids ........................................................593
12.7. Quaternions and Rotations ...................................596
Complex Numbers – Quaternions – Composed Rotations
References ........................................................... 599
Index ................................................................. 611