Variational Methods for Engineers with Matlab

<p>Introduction xi</p>
<p>Chapter 1. Integrals 1</p>
<p>1.1. Riemann integrals 3</p>
<p>1.2. Lebesgue integrals 6</p>
<p>1.3. Matlab&reg; classes for a Riemann integral by trapezoidal integration 10</p>
<p>1.4. Matlab&reg; classes for Lebesgue s integral 17</p>
<p>1.5. Matlab&reg; classes for evaluation of the integrals when is defined by a subprogram 33</p>
<p>1.6. Matlab&reg; classes for partitions including the evaluation of the integrals 38</p>
<p>Chapter 2. Variational Methods for Algebraic Equations 51</p>
<p>2.1. Linear systems 52</p>
<p>2.2. Algebraic equations depending upon a parameter 62</p>
<p>2.2.1. Approximation of the solution by collocation 63</p>
<p>2.2.2. Variational approximation of the solution 65</p>
<p>2.2.3. Linear equations 66</p>
<p>2.2.4. Connection to orthogonal projections 67</p>
<p>2.2.5. Numerical determination of the orthogonal projections 69</p>
<p>2.2.6. Matlab&reg; classes for a numerical solution 70</p>
<p>2.3. Exercises 98</p>
<p>Chapter 3. Hilbert Spaces for Engineers 103</p>
<p>3.1. Vector spaces 107</p>
<p>3.2. Distance, norm and scalar product 109</p>
<p>3.2.1. Distance 110</p>
<p>3.2.2. Norm 111</p>
<p>3.2.3. Scalar product 112</p>
<p>3.2.4. Cartesian products of vector spaces 117</p>
<p>3.2.5. A Matlab&reg; class for scalar products and norms 118</p>
<p>3.2.6. A Matlab&reg; class for Gram Schmidt orthonormalization 125</p>
<p>3.3. Continuous maps 132</p>
<p>3.4. Sequences and convergence 134</p>
<p>3.4.1. Sequences 134</p>
<p>3.4.2. Convergence (or strong convergence) 134</p>
<p>3.4.3. Weak convergence 138</p>
<p>3.4.4. Compactness 139</p>
<p>3.5. Hilbert spaces and completeness 141</p>
<p>3.5.1. Fixed points 142</p>
<p>3.6. Open and closed sets 144</p>
<p>3.6.1. Closure of a set 144</p>
<p>3.6.2. Open and closed sets 145</p>
<p>3.6.3. Dense subspaces 146</p>
<p>3.7. Orthogonal projection 147</p>
<p>3.7.1. Orthogonal projection on a subspace 147</p>
<p>3.7.2. Orthogonal projection on a convex subset 150</p>
<p>3.7.3. Orthogonal projection on an affine subspace 151</p>
<p>3.7.4. Matlab&reg; determination of orthogonal projections 153</p>
<p>3.8. Series and separable spaces 157</p>
<p>3.8.1. Series 158</p>
<p>3.8.2. Separable spaces and Fourier series 159</p>
<p>3.9. Duality 161</p>
<p>3.9.1. Linear functionals 161</p>
<p>3.9.2. Kernel of a linear functional 164</p>
<p>3.9.3. Riesz s theorem 166</p>
<p>3.10. Generating a Hilbert basis 167</p>
<p>3.10.1. 1D situations 168</p>
<p>3.10.2. 2D situations 169</p>
<p>3.10.3. 3D situations 172</p>
<p>3.10.4. Using a sequence of finite families 175</p>
<p>3.11. Exercises 175</p>
<p>Chapter 4. Functional Spaces for Engineers 185</p>
<p>4.1. The L2 ( ) space 186</p>
<p>4.2. Weak derivatives 189</p>
<p>4.2.1. Second–order weak derivatives 191</p>
<p>4.2.2. Gradient, divergence, Laplacian 192</p>
<p>4.2.3. Higher–order weak derivatives 195</p>
<p>4.2.4. Matlab&reg; determination of weak derivatives 195</p>
<p>4.3. Sobolev spaces 199</p>
<p>4.3.1. Point values 201</p>
<p>4.4. Variational equations involving elements of a functional space 203</p>
<p>4.5. Reducing multiple indexes to a single one 205</p>
<p>4.6. Existence and uniqueness of the solution of a variational equation 207</p>
<p>4.7. Linear variational equations in separable spaces 210</p>
<p>4.8. Parametric variational equations 211</p>
<p>4.9. A Matlab&reg; class for variational equations 213</p>
<p>4.10. Exercises 216</p>
<p>Chapter 5. Variational Methods for Differential Equations 221</p>
<p>5.1. A simple situation: the oscillator with one degree of freedom 224</p>
<p>5.1.1. Newton s equation of motion 225</p>
<p>5.1.2. Boundary value problem and initial condition problem 226</p>
<p>5.1.3. Generating a variational formulation 226</p>
<p>5.1.4. Generating an approximation of a variational equation 230</p>
<p>5.1.5. Application to the first variational formulation of the BVP 230</p>
<p>5.1.6. Application to the second variational formulation of the BVP 231</p>
<p>5.1.7. Application to the first variational formulation of the ICP 232</p>
<p>5.1.8. Application to the second variational formulation of the ICP 232</p>
<p>5.2. Connection between differential equations and variational equations 233</p>
<p>5.2.1. From a variational equation to a differential equation 233</p>
<p>5.2.2. From a differential equation to a variational equation 236</p>
<p>5.3. Variational approximation of differential equations 243</p>
<p>5.4. Evolution partial differential equations 253</p>
<p>5.4.1. Linear equations 253</p>
<p>5.4.2. Nonlinear equations 255</p>
<p>5.4.3. Motion equations 256</p>
<p>5.4.4. Motion of a bar 264</p>
<p>5.4.5. Motion of a beam under flexion 268</p>
<p>5.5. Exercises 272</p>
<p>Chapter 6. Dirac s Delta 279</p>
<p>6.1. A simple example 283</p>
<p>6.2. Functional definition of Dirac s delta 285</p>
<p>6.2.1. Compact support functions 285</p>
<p>6.2.2. Infinitely differentiable functions having a compact support 286</p>
<p>6.2.3. Formal definition of Dirac s delta 287</p>
<p>6.2.4. Dirac s delta as a probability 287</p>
<p>6.3. Approximations of Dirac s delta 288</p>
<p>6.4. Smoothed particle approximations of Dirac s delta 289</p>
<p>6.5. Derivation using Dirac s delta approximations 291</p>
<p>6.6. A Matlab&reg; class for smoothed particle approximations 292</p>
<p>6.7. Green s functions 298</p>
<p>6.7.1. Adjoint operators 299</p>
<p>6.7.2. Green s functions 301</p>
<p>6.7.3. Using fundamental solutions to solve differential equations 302</p>
<p>Chapter 7. Functionals and Calculus of Variations 319</p>
<p>7.1. Differentials 320</p>
<p>7.2. G&acirc;teaux derivatives of functionals 321</p>
<p>7.3. Convex functionals 324</p>
<p>7.4. Standard methods for the determination of G&acirc;teaux derivatives 326</p>
<p>7.4.1. Methods for practical calculation 326</p>
<p>7.4.2. G&acirc;teaux derivatives and equations of the motion of a system 329</p>
<p>7.4.3. G&acirc;teaux derivatives of Lagrangians 332</p>
<p>7.4.4. G&acirc;teaux derivatives of fields 333</p>
<p>7.5. Numerical evaluation and use of G&acirc;teaux differentials 334</p>
<p>7.5.1. Numerical evaluation of a functional 335</p>
<p>7.5.2. Determination of a G&acirc;teaux derivative 336</p>
<p>7.5.3. Determination of the derivatives with respect to the Hilbertian coefficients 339</p>
<p>7.5.4. Solving an equation involving the G&acirc;teaux differential 343</p>
<p>7.5.5. Determining an optimal point 345</p>
<p>7.6. Minimum of the energy 347</p>
<p>7.7. Lagrange s multipliers 349</p>
<p>7.8. Primal and dual problems 352</p>
<p>7.9. Matlab&reg; determination of minimum energy solutions 354</p>
<p>7.10. First–order control problems 366</p>
<p>7.11. Second–order control problems 371</p>
<p>7.12. A variational approach for multiobjective optimization 374</p>
<p>7.13. Matlab&reg; implementation of the variational approach for biobjective optimization 384</p>
<p>7.14. Exercises 388</p>
<p>Bibliography 393</p>
<p>Index 411</p>