Proper Generalized Decomposition with time adaptive space separation for transient wave propagation problems in separable domains

Exploring space separation techniques for 3D elastic waves simulations

“…As an alternative, Quaranta et al [23] proposed perform ing a space separation, either with ( 18) or (19), at each time step to compute a space separated form of the solution within a PGD-based time incremental solver. In our previous work [21], we evidenced that this time incremental procedure is in fact well adapted to wave propagation problems. Indeed, the propagation of a planar wave is mathematically described with a phase variable of the form k • x -w t , with w the wave frequency and k the propagation vector [37].…”

“…plate, cylinder). This paper is a continuation of our previous work [21] validated on two dimensional configurations.…”

“…A seemingly simple discretization grid with 100 nodes in each direction results in 10 6 unknowns at each time step (for each component of the displacement field), which already is a challenging numerical problem. In the proposed framework, however, the unknowns are the nodal values of the 2D or 1D fonctions involved in the separated represen tations (20) or (21). This problem is solved with a PGD algorithm as described in the following.…”

“…In our previous work [21], we used low-order finite elements to salve 2D wave propagation problems with the time adap tive space separation. However, our numerical experiments showed unsatisfactory results for some problems, as pre sented in Sect.…”

“…Fol lowing the guideline provided in the previous section, the enrichment criterion is selected equal to 10-4 . A maximum number of 10 iterations is implemented in the stagnation loop as recommended in our previous work [21]. The central dif ference method is employed for the time integration (a = 0) and a uniform time step of 34.6ms is used.…”