The proper generalized decomposition for advanced numerical simulations ch03 ex Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom. Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab© software used to generate these examples.
Chapter 3: WEAK AND VARIATIONAL FORMS OF THE POISSON’S EQUATION 3–16 Homework Exercises for Chapter Solutions EXERCISE 3.1 = TPE [T ] k (∇T )2 d V + V = h T dV + V ∂T ∂ x1 k V ∂T ∂ x2 + qT ˆ dS ST + (E3.3) ∂T ∂ x3 dV + k T dV + V qT ˆ d S ST EXERCISE 3.2 (a) Let stTPE denote the standard functional for heat conduction with prescribed temperature and flux BC on Su and Sq , derived previously Take the convective flux residual qnu − χ (u − u ) of the RBC, multiply by the variation of the conjugate quantity δu and integrate over the convection surface Sr Upon integration by parts one gets δ TPE =δ st TPE − χ (u − u ) δu d S (E3.4) Sr The last term is the first variation of − TPE = ρ u· u dV + V (b) χ (u Sr − u )2 d S with respect to u Consequently s u dV − V qˆ u d S − Sq χ (u − u )2 d S (E3.5) Sr Figure E3.1 shows a possible graphical representation out of many possibilities In the figure u, ρ and σ have been replaced by T , −k and −h, respectively, for better fit with the thermal problem ^ T ^ T=T on ST h T div q + h = in V g = grad T in V χ, Tc g q = −k g in V q qn = q.n = q^ on Sq qn = χ (T−Tc ) on Sr Figure E3.1 A Strong Form Tonti diagram for the thermal conduction problem with convection BC (E3.1) 3–16 q^ 3–17 Solutions to Exercises EXERCISE 3.3 An exact linearization is possible: σ (T − Tr4 ) = σ (T − Tr )(T + T Tr + T Tr2 + Tr3 ) = σ (T − Tr )h r (E3.6) where h r = T + T Tr + T Tr2 + Tr3 If h r is “frozen” during variation, the previous exercise shows that the appropriate surface term to add to the functional is σ h r (T − Tr )2 − 12 (E3.7) Sr EXERCISE 3.4 The PBC residual is u − uˆ over Su Multiply by the variation of the conjugate quantity: δqn , and integrate over Su Upon integration by parts and passing to the functional one obtains the given term with the − sign 3–17 ... variation, the previous exercise shows that the appropriate surface term to add to the functional is σ h r (T − Tr )2 − 12 (E3.7) Sr EXERCISE 3.4 The PBC residual is u − uˆ over Su Multiply by the. .. Multiply by the variation of the conjugate quantity: δqn , and integrate over Su Upon integration by parts and passing to the functional one obtains the given term with the − sign 3–17 ...3–17 Solutions to Exercises EXERCISE 3.3 An exact linearization is possible: σ (T − Tr4 ) = σ (T − Tr )(T + T Tr + T Tr2 + Tr3