The proper generalized decomposition for advanced numerical simulations ch04 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.
4–15 Solutions to Exercises Homework Exercises for Chapter Solutions EXERCISE 4.1 Transforming the weak-BE integral by parts as before: L (M κ ) δw d x = M κ δw d x − M κ δθ w B − V κ δw , (E4.5) B and adding to the other weak relations we get L L q δw d x − Mˆ δθ w (M w δκ + M κ δw − M κ δκ) d x − L The integral B L (M w δκ + M κ δw ) d x is the variation of is the variation of − 12 L 0 − Vˆ δw , (E4.6) B M w κ d x whereas the integral L −M κ δκ d x M κ κ d x Therefore the required functional is L L w [w, κ] = (M − Mκ) κ dx − q w d x − Mˆ θ w B − Vˆ w (E4.7) B to which the one given in the Exercise statement can be contracted If the PBC is made weak the following additional boundary term appears: (wˆ − w) V κ A + (θˆ − θ w ) M κ A (E4.8) EXERCISE 4.2 Integrating U by parts: L U= L E I (w )2 d x = L Mw d x = M w d x + 12 Mw 0 − 12 M w B B (E4.9) At equilibrium M = q, Mˆ = M| B and Vˆ = −M | B , which substituted in the above gives L U= ˆ qw d x + 12 Mw B + 12 Vˆ w B = 12 W (E4.10) EXERCISE 4.3 To be done (not assigned so far) L L Proof: δ M w κ d x = δ (M w δκ + δ M w κ )d x = tricky part is M w κ = E I w κ = E I κw = M κ κ w L (M w δκ + M κ δκ w )d x = 4–15 L (M w δκ + M κ δw )d x The 4–16 Chapter 4: THE BERNOULLI-EULER BEAM EXERCISE 4.4 Transforming the weak-BE integral by parts: L (M κ ) δw d x = M κ δw d x − M κ δθ w B − V κ δw , (E4.11) B and adding to the other weak relations we get L L L (M w − M κ ) δκ d x + q δw d x − Mˆ δθ w M κ δw d x − 0 B − Vˆ δw , (E4.12) B The first integral is the variation of L (M κ κ − 12 M κ κ) d x, (E4.13) whereas the second is the variation of L L Mκ w dx = L E Iκ w dx = L M w κ d x E I w κ dx = (E4.14) Therefore the required functional is L [w, κ] = L Mκκ dx + L (M w − M κ ) κ d x − q w d x − Mˆ θ w B − Vˆ w , B (E4.15) as in the Exercise statement This may be further contracted to L L (M w − 12 M κ ) κ d x − [w, κ] = q w d x − Mˆ θ w B − Vˆ w B (E4.16) If the PBC is made weak the following additional boundary term appears: (wˆ − w) V κ A + (θˆ − θ w ) M κ A (E4.17) The derivation is very similar to that of Exercise 4.1 and gives L L Mκκ [w, κ, M] = + M(κ w − κ) d x − 0 ˆ w qw d x − Mθ B − Vˆ w B (E4.18) Note: there is an infinite number of three-field functionals and associated weak forms The one shown above leads to the historically important Hu-Washizu functional If you start from a different diagram you may arrive at another three-field functional 4–16 ... (E4.12) B The first integral is the variation of L (M κ κ − 12 M κ κ) d x, (E4.13) whereas the second is the variation of L L Mκ w dx = L E Iκ w dx = L M w κ d x E I w κ dx = (E4.14) Therefore the required...4–16 Chapter 4: THE BERNOULLI-EULER BEAM EXERCISE 4.4 Transforming the weak-BE integral by parts: L (M κ ) δw d x = M κ δw d x − M κ δθ w B − V κ δw , (E4.11) B and adding to the other weak relations... , B (E4.15) as in the Exercise statement This may be further contracted to L L (M w − 12 M κ ) κ d x − [w, κ] = q w d x − Mˆ θ w B − Vˆ w B (E4.16) If the PBC is made weak the following additional