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Variational Methods for Engineers with Matlab®
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Variational Methods for Engineers with Matlab®
by Eduardo Souza de Cursi
Variational Methods for Engineers with Matlab
Cover
Title
Copyright
Introduction
Chapter 1. Integrals
1.1 Riemann integrals
1.2 Lebesgue integrals
1.3 Matlab® classes for a Riemann integral by trapezoidal integration
1.4 Matlab® classes for Lebesgue’s integral
1.5 Matlab® classes for evaluation of the integrals when/is defined by a subprogram
1.6 Matlab® classes for partitions including the evaluation of the integrals
Chapter 2. Variational Methods for Algebraic Equations
2.1 Linear systems
2.2 Algebraic equations depending upon a parameter
2.3 Exercises
Chapter 3. Hilbert Spaces for Engineers
3.1 Vector spaces
3.2 Distance, norm and scalar product
3.3 Continuous maps
3.4 Sequences and convergence
3.5 Hilbert spaces and completeness
3.6 Open and closed sets
3.7 Orthogonal projection
3.8 Series and separable spaces
3.9 Duality
3.10 Generating a Hilbert basis
3.11 Exercises
Chapter 4. Functional Spaces for Engineers
4.1 The L2 (Ω) space (Ω) space
4.2 Weak derivatives
4.3 Sobolev spaces
4.4 Variational equations involving elements of a functional space
4.5 Reducing multiple indexes to a single one
4.6 Existence and uniqueness of the solution of a variational equation
4.7 Linear variational equations in separable spaces
4.8 Parametric variational equations
4.9 A Matlab® class for variational equations
4.10 Exercises
Chapter 5. Variational Methods for Differential Equations
5.1 A simple situation: the oscillator with one degree of freedom
5.2 Connection between differential equations and variational equations
5.3 Variational approximation of differential equations
5.4 Evolution partial differential equations
5.5 Exercises
Chapter 6. Dirac’s Delta
6.1 A simple example
6.2 Functional definition of Dirac’s delta
6.3 Approximations of Dirac’s delta
6.4 Smoothed particle approximations of Dirac’s delta
6.5 Derivation using Dirac’s delta approximations
6.6 A Matlab® class for smoothed particle approximations
6.7 Green’s functions
Chapter 7. Functionals and Calculus of Variations
7.1 Differentials
7.2 Gâteaux derivatives of functionals
7.3 Convex functionals
7.4 Standard methods for the determination of Gâteaux derivatives
7.5 Numerical evaluation and use of Gâteaux differentials
7.6 Minimum of the energy
7.7 Lagrange’s multipliers
7.8 Primal and dual problems
7.9 Matlab® determination of minimum energy solutions
7.10 First-order control problems
7.11 Second-order control problems
7.12 A variational approach for multiobjective optimization
7.13 Matlab® implementation of the variational approach for biobjective optimization
7.14 Exercises
Bibliography
Index
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Title
Table of Contents
Cover
Title
Copyright
Introduction
Chapter 1. Integrals
1.1 Riemann integrals
1.2 Lebesgue integrals
1.3 Matlab® classes for a Riemann integral by trapezoidal integration
1.4 Matlab® classes for Lebesgue’s integral
1.5 Matlab® classes for evaluation of the integrals when/is defined by a subprogram
1.6 Matlab® classes for partitions including the evaluation of the integrals
Chapter 2. Variational Methods for Algebraic Equations
2.1 Linear systems
2.2 Algebraic equations depending upon a parameter
2.3 Exercises
Chapter 3. Hilbert Spaces for Engineers
3.1 Vector spaces
3.2 Distance, norm and scalar product
3.3 Continuous maps
3.4 Sequences and convergence
3.5 Hilbert spaces and completeness
3.6 Open and closed sets
3.7 Orthogonal projection
3.8 Series and separable spaces
3.9 Duality
3.10 Generating a Hilbert basis
3.11 Exercises
Chapter 4. Functional Spaces for Engineers
4.1 The L
2
(Ω) space
4.2 Weak derivatives
4.3 Sobolev spaces
4.4 Variational equations involving elements of a functional space
4.5 Reducing multiple indexes to a single one
4.6 Existence and uniqueness of the solution of a variational equation
4.7 Linear variational equations in separable spaces
4.8 Parametric variational equations
4.9 A Matlab® class for variational equations
4.10 Exercises
Chapter 5. Variational Methods for Differential Equations
5.1 A simple situation: the oscillator with one degree of freedom
5.2 Connection between differential equations and variational equations
5.3 Variational approximation of differential equations
5.4 Evolution partial differential equations
5.5 Exercises
Chapter 6. Dirac’s Delta
6.1 A simple example
6.2 Functional definition of Dirac’s delta
6.3 Approximations of Dirac’s delta
6.4 Smoothed particle approximations of Dirac’s delta
6.5 Derivation using Dirac’s delta approximations
6.6 A Matlab® class for smoothed particle approximations
6.7 Green’s functions
Chapter 7. Functionals and Calculus of Variations
7.1 Differentials
7.2 Gâteaux derivatives of functionals
7.3 Convex functionals
7.4 Standard methods for the determination of Gâteaux derivatives
7.5 Numerical evaluation and use of Gâteaux differentials
7.6 Minimum of the energy
7.7 Lagrange’s multipliers
7.8 Primal and dual problems
7.9 Matlab® determination of minimum energy solutions
7.10 First-order control problems
7.11 Second-order control problems
7.12 A variational approach for multiobjective optimization
7.13 Matlab® implementation of the variational approach for biobjective optimization
7.14 Exercises
Bibliography
Index
Guide
Cover
Table of Contents
Begin Reading
List of Illustrations
Chapter 1: Integrals
Figure 1.1.
Riemann sums: partitions of the horizontal axis
Figure 1.2.
Adding supplementary points increases
and decreases
Figure 1.3.
Lebesgue’s approach to integration: partition of the vertical axis
Figure 1.4.
Lebesgue measure obtained in example 1.3
Figure 1.5.
Results for example 1.4
Figure 1.6.
Lebesgue measure in example 1.5
Figure 1.7.
Results for Lebesgue measure in example 1.6
Chapter 2: Variational Methods for Algebraic Equations
Figure 2.1.
A physical interpretation of the method
Figure 2.2.
Orthogonal projection in a simple noisy situation
Figure 2.3.
Derivative in a simple noisy situation
Figure 2.4.
Orthogonal projection in a simple noisy situation
Figure 2.5.
Derivative in a simple noisy situation
Figure 2.6.
Solutions obtained for a polynomial of degree 6
Figure 2.7.
Solutions obtained for a polynomial of degree 5 and “mean” method
Figure 2.8.
Solutions obtained for a polynomial of degree 5 and “integral” method
Figure 2.9.
Solutions obtained for a polynomial of degree 6
Figure 2.10.
Solutions obtained for a polynomial of degree 5
Figure 2.11.
Solutions obtained by collocation using different basis
Figure 2.12.
Solutions obtained by variational approach (mean) using different basis
Chapter 3: Hilbert Spaces for Engineers
Figure 3.1.
Results for example 3.4 (using sp0 and ‘subprograms’)
Figure 3.2.
Results for example 3.4 (using sp0 and ‘tables’)
Figure 3.3.
Results for example 3.4 using sp1 and ‘subprograms’
Figure 3.4.
Results for example 3.4 using sp1 and ‘tables’
Figure 3.5.
Geometrical interpretation of the orthogonal projection on a vector subspace
Figure 3.6.
Geometrical interpretation of the orthogonal projection on a convex subset
Figure 3.7.
Geometrical interpretation of the orthogonal projection on an affine subspace
Figure 3.8.
A second interpretation of the orthogonal projection on an affine subspace
Figure 3.9.
Orthogonal projection of x
5
onto a polynomial subspace (degree <= 3)
Figure 3.10.
Orthogonal projection of sin(x) onto a polynomial subspace (degree <= 3)
Chapter 4: Functional Spaces for Engineers
Figure 4.1.
Evaluation of the weak derivative of x
4
using subprograms and tables
Figure 4.2.
Evaluation of the weak derivative of |x| using subprograms and tables
Figure 4.3.
Solution of the nonlinear variational equation in example 4.8
Figure 4.4.
Solution of the linear variational equation in example 4.9
Chapter 5: Variational Methods for Differential Equations
Figure 5.1.
A simple harmonic oscillator
Figure 5.2.
Examples of solutions for beam under flexion. For a color version of the figure, see www.iste.co.uk/souzadecursi/variational.zip
Figure 5.3.
Examples of solutions for beam under flexion (linear variation of a)
Figure 5.4.
Examples of modes furnished by the trigonometric family. For a color version of the figure, see www.iste.co.uk/souzadecursi/variational.zip
Figure 5.5.
Examples of modes furnished by the polynomial family. For a color version of the figure, see www.iste.co.uk/souzadecursi/variational.zip
Figure 5.6.
Examples of modes furnished by the Q2 family. For a color version of the figure, see www.iste.co.uk/souzadecursi/variational.zip
Figure 5.7.
Examples of modes furnished by the T1 family. For a color version of the figure, see www.iste.co.uk/souzadecursi/variational.zip
Figure 5.8.
Comparison between the mean along the vertical axis and a one-dimensional mode
(
α
=
0.1)
Figure 5.9.
Solution for a uniform monochromatic force
Figure 5.10.
Solution for a uniform multichromatic force
Figure 5.11.
Solution for a uniform monochromatic force
Figure 5.12.
Solution for a uniform multichromatic force
Chapter 6: Dirac’s Delta
Figure 6.1.
Velocity of a mass “kicked” at time t
= 0
Figure 6.2.
An example of smoothed particle approximation
Figure 6.3.
An example of numerical derivation using smoothed particle approximation
Figure 6.4.
Approximation of the set Ω by Ω
δ
Figure 6.5.
Approximation of the set Ω
in the non-smooth case
Figure 6.6.
The discretization used in example 6.5
Figure 6.7.
Example of numerical solution by Green’s function (example 6.5)
Chapter 7: Functionals and Calculus of Variations
Figure 7.1.
Differential of a functional
Figure 7.2.
A convex functional
Figure 7.3.
A simple system
Figure 7.4.
Determination of a Gâteaux derivative for two different scalar products
Figure 7.5.
Situation of example 7.4
Figure 7.6.
Situation of example 7.5
Figure 7.7.
Analysis of the situation of example 7.5
Figure 7.8.
Determination of the minimum of a functional
Figure 7.9.
Determination of the unconstrained minimum of a functional
Figure 7.10.
Determination of the constrained minimum of a functional by Uzawa’s method
Figure 7.11.
Determination of a first-order control
Figure 7.12.
Determination of a first-order control
Figure 7.13.
Second-order control: a linear oscillator
Figure 7.14.
Variational approach for multiobjective optimization (2 objectives)
Figure 7.15.
Variational approach for a simple biobjective optimization problem
Figure 7.16.
Variational approach for a biobjective optimization problem
Chapter 1: Integrals
Program 1.1.
A class for the evaluation of Riemann integrals
Program 1.2.
A class for the creation of the tables F
Program 1.3.
A class for the evaluation of Lebesgue integrals
Program 1.4.
A class for the integration of functions defined by subprograms
Program 1.5.
An example of evaluation
Program 1.6.
Evaluation of a 3D integral
Program 1.7.
Evaluation of a 3d integral with a discontinuous function
Program 1.8.
Definition of a class for partition of intervals
Program 1.9.
A class for partitions of rectangles
Program 1.10.
A class for three-dimensional partitions
Program 2.1.
A class for the determination of partial solutions of linear systems
Chapter 2: Variational Methods for Algebraic Equations
Program 2.2.
Polynomial basis
Program 2.3.
Approximation
Program 2.4.
Variational solution of parametric algebraic equations
Program 2.5.
Trigonometrical basis
Program 2.6.
P1 Finite Element basis
Chapter 3: Hilbert Spaces for Engineers
Program 3.1.
A class for the determination of partial solutions of linear systems
Program 3.2.
Definition of u and v in example 3.2
The code:
Program 3.3.
Definition of u and v in example 3.3
Program 3.4.
A class for the Gram-Schmidt orthogonalization
Program 3.5.
Definition of the family G in example 3.4
Program 3.6.
A class for orthogonal projection
Chapter 4: Functional Spaces for Engineers
Program 4.1.
A class for weak derivatives of functions of one variable
Program 4.2.
A class for one-dimensional variational equations
Program 4.3.
Definition of
a
and tin example 4.8
Chapter 6: Dirac’s Delta
Program 6.1.
A class for kernels
Program 6.2.
A class for kernels
Chapter 7: Functionals and Calculus of Variations
Program 7.1.
A class for periodic trigonometrical basis
List of Tables
Chapter 1: Integrals
Table 1.1.
Intrinsic functions for the evaluation of integrals
Table 1.2.
Intrinsic function for older versions of Matlab®0
Chapter 7: Functionals and Calculus of Variations
Table 7.1.
Lagrange’s Multipliers and penalties
Table 7.2.
Ascent directions for Uzawa’s method
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