By Francisco Chinesta, Roland Keunings, Adrien Leygue
ISBN-10: 3319028642
ISBN-13: 9783319028644
ISBN-10: 3319028650
ISBN-13: 9783319028651
Many difficulties in clinical computing are intractable with classical numerical thoughts. those fail, for instance, within the resolution of high-dimensional types because of the exponential elevate of the variety of levels of freedom.
Recently, the authors of this publication and their collaborators have built a unique strategy, known as right Generalized Decomposition (PGD) that has confirmed to be an important leap forward. The PGD builds via a successive enrichment technique a numerical approximation of the unknown fields in a separated shape. even if first brought and effectively confirmed within the context of high-dimensional difficulties, the PGD enables a totally new strategy for addressing extra normal difficulties in technology and engineering. certainly, many not easy difficulties will be successfully solid right into a multi-dimensional framework, hence commencing totally new resolution options within the PGD framework. for example, the fabric parameters and boundary stipulations showing in a specific mathematical version will be considered as extra-coordinates of the matter as well as the standard coordinates corresponding to area and time. within the PGD framework, this enriched version is solved just once to yield a parametric answer that incorporates all specific ideas for particular values of the parameters.
The PGD has now attracted the eye of a giant variety of study teams all over the world. the current textual content is the 1st on hand e-book describing the PGD. It offers a truly readable and functional advent that permits the reader to speedy seize the most gains of the tactic. through the e-book, the PGD is utilized to difficulties of accelerating complexity, and the method is illustrated by way of rigorously chosen numerical examples. furthermore, the reader has unfastened entry to the Matlab© software program used to generate those examples.
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Extra info for The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer
Sample text
1) on a two-dimensional rectangular domain = x × y = (0, 2) × (0, 1) for which we seek a solution using the procedure described above. The source term f is set to f = 1, in which case we obtain analytically the following exact solution for u(x, y): u ex (x, y) = 64 mπx · sin (nπy) . 28) The unknown functions X i (x) and Yi (y) are sought on a uniform grid with M points. All one-dimensional differential problems arising in the solution procedure are solved using second-order finite differences, while the integrals are evaluated numerically using the trapezoidal rule.
This is due to the fact that the original Poisson equation involves a second-order x-derivative of the unknown field u. With either the weighted residual or strong formulations, the homogeneous Dirichp p let boundary conditions X n (x = 0) = X n (x = L) = 0 are readily specified. p Having thus computed X n (x), we are now ready to proceed with the second step of iteration p. p p • Calculating Yn (y) from the just-computed X n (x) The procedure exactly mirrors what we have done above. Indeed, we simply exchange the roles played by all relevant functions of x and y.
Metivier, M. Normandin, F. Chinesta, Non-incremental transient solution of the Rayleigh-Bnard convection model using the PGD. J. Nonnewton. Fluid Mech. 200, 65–78 (2013). 007 (2013) 32. B. Mokdad, A. Ammar, M. Normandin, F. R. Clermont, A fully deterministic micromacro simulation of complex flows involving reversible network fluid models. Math. Comput. Simul. 80, 1936–1961 (2010) 33. E. Pruliere, A. Ammar, N. El Kissi, F. Chinesta, Recirculating flows involving short fiber suspensions: numerical difficulties and efficient advanced micro-macro solvers.
The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer by Francisco Chinesta, Roland Keunings, Adrien Leygue
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