Proper Generalized Decompositions

“This book provides a brief introduction to Proper Generalized Decompositions (PGD), with strong emphasis on computational aspects. The book discusses the implementation of PGD for the Poisson problem, parameter-dependent problems, linear-elasticity, and dynamical problems. For every problem, matrix assembly is developed and Matlab routines are presented.” (Dante Kalise, Mathematical Reviews, July, 2017)

• Introduction.- 2 To begin with: PGD for Poisson problems.- 2.1 Introduction.- 2.2 The Poisson problem.- 2.3 Matrix structure of the problem.- 2.4 Matlab code for the Poisson problem.- 3 Parametric problems.- 3.1 A particularly challenging problem: a moving load as a parameter.- 3.2 The problem under the PGD formalism.- 3.2.1 Computation of S(s) assuming R(x) is known.- 3.2.2 Computation of R(x) assuming S(s) is known.- 3.3 Matrix structure of the problem.- 3.4 Matlab code for the influence line problem.- 4 PGD for non-linear problems.- 4.1 Hyperelasticity.- 4.2 Matrix structure of the problem.- 4.2.1 Matrix form of the term T2.- 4.2.2 Matrix form of the term T4.- 4.2.3 Matrix form of the term T6.- 4.2.4 Matrix form for the term T8.- 4.2.5 Matrix form of the term T9.- 4.2.6 Matrix form of the term T10.- 4.2.7 Final comments.- 4.3 Matlab code.- 5 PGD for dynamical problems.- 5.1 Taking initial conditions as parameters.- 5.2 Developing the weak form of the problem.- 5.3 Matrix form of the problem.- 5.3.1 Time integration of the equations of motion.- 5.3.2 Computing a reduced-order basis for the field of initial conditions.- 5.3.3 Projection of the equations onto a reduced, parametric basis.- 5.4 Matlab code.- References.- Index.