Variational Methods for Engineers with Matlab

Eduardo Souza de Cursi, Professor (INSA-Rouen, France); Dean of International Affairs (INSA-Rouen, France); Director LOFIMS (INSA-Rouen, France).

• Introduction xiChapter 1. Integrals 11.1. Riemann integrals 31.2. Lebesgue integrals 61.3. Matlab® classes for a Riemann integral by trapezoidal integration 101.4. Matlab® classes for Lebesgue’s integral 171.5. Matlab® classes for evaluation of the integrals when is defined by a subprogram 331.6. Matlab® classes for partitions including the evaluation of the integrals 38Chapter 2. Variational Methods for Algebraic Equations 512.1. Linear systems 522.2. Algebraic equations depending upon a parameter 622.2.1. Approximation of the solution by collocation 632.2.2. Variational approximation of the solution 652.2.3. Linear equations 662.2.4. Connection to orthogonal projections 672.2.5. Numerical determination of the orthogonal projections 692.2.6. Matlab® classes for a numerical solution 702.3. Exercises 98Chapter 3. Hilbert Spaces for Engineers 1033.1. Vector spaces 1073.2. Distance, norm and scalar product 1093.2.1. Distance 1103.2.2. Norm 1113.2.3. Scalar product 1123.2.4. Cartesian products of vector spaces 1173.2.5. A Matlab® class for scalar products and norms 1183.2.6. A Matlab® class for Gram–Schmidt orthonormalization 1253.3. Continuous maps 1323.4. Sequences and convergence 1343.4.1. Sequences 1343.4.2. Convergence (or strong convergence) 1343.4.3. Weak convergence 1383.4.4. Compactness 1393.5. Hilbert spaces and completeness 1413.5.1. Fixed points 1423.6. Open and closed sets 1443.6.1. Closure of a set 1443.6.2. Open and closed sets 1453.6.3. Dense subspaces 1463.7. Orthogonal projection 1473.7.1. Orthogonal projection on a subspace 1473.7.2. Orthogonal projection on a convex subset 1503.7.3. Orthogonal projection on an affine subspace 1513.7.4. Matlab® determination of orthogonal projections 1533.8. Series and separable spaces 1573.8.1. Series 1583.8.2. Separable spaces and Fourier series 1593.9. Duality 1613.9.1. Linear functionals 1613.9.2. Kernel of a linear functional 1643.9.3. Riesz’s theorem 1663.10. Generating a Hilbert basis 1673.10.1. 1D situations 1683.10.2. 2D situations 1693.10.3. 3D situations 1723.10.4. Using a sequence of finite families 1753.11. Exercises 175Chapter 4. Functional Spaces for Engineers 1854.1. The L2 () space 1864.2. Weak derivatives 1894.2.1. Second-order weak derivatives 1914.2.2. Gradient, divergence, Laplacian 1924.2.3. Higher-order weak derivatives 1954.2.4. Matlab® determination of weak derivatives 1954.3. Sobolev spaces 1994.3.1. Point values 2014.4. Variational equations involving elements of a functional space 2034.5. Reducing multiple indexes to a single one 2054.6. Existence and uniqueness of the solution of a variational equation 2074.7. Linear variational equations in separable spaces 2104.8. Parametric variational equations 2114.9. A Matlab® class for variational equations 2134.10. Exercises 216Chapter 5. Variational Methods for Differential Equations 2215.1. A simple situation: the oscillator with one degree of freedom 2245.1.1. Newton’s equation of motion 2255.1.2. Boundary value problem and initial condition problem 2265.1.3. Generating a variational formulation 2265.1.4. Generating an approximation of a variational equation 2305.1.5. Application to the first variational formulation of the BVP 2305.1.6. Application to the second variational formulation of the BVP 2315.1.7. Application to the first variational formulation of the ICP 2325.1.8. Application to the second variational formulation of the ICP 2325.2. Connection between differential equations and variational equations 2335.2.1. From a variational equation to a differential equation 2335.2.2. From a differential equation to a variational equation 2365.3. Variational approximation of differential equations 2435.4. Evolution partial differential equations 2535.4.1. Linear equations 2535.4.2. Nonlinear equations 2555.4.3. Motion equations 2565.4.4. Motion of a bar 2645.4.5. Motion of a beam under flexion 2685.5. Exercises 272Chapter 6. Dirac’s Delta 2796.1. A simple example 2836.2. Functional definition of Dirac’s delta 2856.2.1. Compact support functions 2856.2.2. Infinitely differentiable functions having a compact support 2866.2.3. Formal definition of Dirac’s delta 2876.2.4. Dirac’s delta as a probability 2876.3. Approximations of Dirac’s delta 2886.4. Smoothed particle approximations of Dirac’s delta 2896.5. Derivation using Dirac’s delta approximations 2916.6. A Matlab® class for smoothed particle approximations 2926.7. Green’s functions 2986.7.1. Adjoint operators 2996.7.2. Green’s functions 3016.7.3. Using fundamental solutions to solve differential equations 302Chapter 7. Functionals and Calculus of Variations 3197.1. Differentials 3207.2. Gâteaux derivatives of functionals 3217.3. Convex functionals 3247.4. Standard methods for the determination of Gâteaux derivatives 3267.4.1. Methods for practical calculation 3267.4.2. Gâteaux derivatives and equations of the motion of a system 3297.4.3. Gâteaux derivatives of Lagrangians 3327.4.4. Gâteaux derivatives of fields 3337.5. Numerical evaluation and use of Gâteaux differentials 3347.5.1. Numerical evaluation of a functional 3357.5.2. Determination of a Gâteaux derivative 3367.5.3. Determination of the derivatives with respect to the Hilbertian coefficients 3397.5.4. Solving an equation involving the Gâteaux differential 3437.5.5. Determining an optimal point 3457.6. Minimum of the energy 3477.7. Lagrange’s multipliers 3497.8. Primal and dual problems 3527.9. Matlab® determination of minimum energy solutions 3547.10. First-order control problems 3667.11. Second-order control problems 3717.12. A variational approach for multiobjective optimization 3747.13. Matlab® implementation of the variational approach for biobjective optimization 3847.14. Exercises 388Bibliography 393Index 411