Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity

Abstract

International audienceMany models in polymer processing and composites manufacturing are defined in degenerated three-dimensional domains, involving plate or shell geometries. The reduction of models from 3D to 2D is not obvious when complex physics are involved. The hypotheses to be introduced for reaching this dimensionality reduction are sometimes unclear, and most of possible proposals will have a narrow interval of validity. The only getaway is to explore new discretization strategies able to circumvent or at least alleviate the drawbacks related to mesh-based discretizations of fully 3D models defined in plate or shell domains. An in-plane–out-of-plane separated representation of the involved fields within the context of the Proper Generalized Decomposition allows solving the fully 3D model by keeping a 2D characteristic computational complexity. Moreover the PGD features allow the introduction of many extra-coordinates, as for example the orientation of the different laminate plies, without affecting the solvability of the resulting multidimensional model

Models defined in stratified-plate domains. A review on efficient 3D PGD-based solution procedures

“…With the weak form of the elastic problem defined in the reference domain Ξ = Ω × I, with (ξ, η) ∈ Ω and ζ ∈ I, the situation is quite similar to the one encountered in the analysis of elastic problems in plate geometries, that was addressed in Bognet et al (2012).…”

“…Thus, the resulting complexity scales as N · N 1 · N 2 . By comparing both complexities we can notice that as soon as N 3 N the use of separated representations leads to impressive computing time savings, making possible the solution of models never until now solved, and even using light computing platforms (Bognet et al, 2012).…”

“…As discussed in Bognet et al (2012) this separated representation allows computing 3D solutions while keeping a computational complexity characteristic of 2D solution procedures. If we consider a hexahedral domain discretized using a regular structured grid with N 1 , N 2 and N 3 nodes in the x 1 , x 2 and x 3 directions respectively, usual mesh-based discretization strategies imply a challenging issue because the number of nodes involved in the model scales with N 1 · N 2 · N 3 , however, by using the separated representation and assuming that the solution involves N modes, one must solve about N 2D problems related to the functions involving the in-plane coordinates x and the same number of 1D problems related to the functions involving the thickness coordinate x 3 .…”

“…We proposed in Bognet et al (2012) and original in-plane-out-of-plane decomposition of the 3D elastic solution in a plate geometry. The elastic problem was defined in a plate domain Ξ = Ω × I with x = (x 1 , x 2 ) ∈ Ω, Ω ⊂ R 2 and x 3 ∈ I, I = [0, H] ⊂ R, being H the plate thickness.…”

“…Thus an in-plane-out-of-plane decomposition was proposed for solving flow problems in laminates (Chinesta, Ammar et al, 2011;Ghnatios et al, 2013), thermal problems in extruded geometries and laminates , elasticity problems in plates (Bognet et al, 2012) and shells geometries . In general the 3D solution was obtained from the solution of a sequence of 2D problems (the ones involving the in-plane coordinates) and 1D problems (the ones involving the coordinate related to the plate thickness).…”