Fast and multiscale formation of isogeometric matrices of microstructured geometric models

Abstract

The matrix formation associated to high-order discretizations is known to be numerically demanding. Based on the existing procedure of interpolation and lookup, we design a multiscale assembly procedure to reduce the exorbitant assembly time in the context of isogeometric linear elasticity of complex microstructured geometries modeled via spline compositions. The developed isogeometric approach involves a polynomial approximation occurring at the macro-scale and the use of lookup tables with pre-computed integrals incorporating the micro-scale information. We provide theoretical insights and numerical examples to investigate the performance of the procedure. The strategy turns out to be of great interest not only to form finite element operators but also to compute other quantities in a fast manner as for instance sensitivity analyses commonly used in design optimization.

A fully non-invasive hybrid IGA/FEM scheme for the analysis of localized non-linear phenomena

“…In this sense, our method is consistent with the current development of open-source IG libraries (see, e.g., Nutils (http://www.nutils.org) or pyiga (https://github.com/c-f-h/pyiga) as Python libraries, tIGAr [40] for a FENICS-based implementation, and YETI (https://lamcosplm.insa-lyon.fr/projects/yeti/) for a Fortran-Python implementation). Even more interestingly, since the IG code is only used at the global scale, where standard elasticity may be sufficient, our strategy also appears totally relevant with the newly developed fast assembly and solution procedures for IGA in the linear regime (see, e.g., sum factorization [58], use of look-up tables [59,60], weighted quadrature [61], and domain decomposition solvers [62,63]).…”

Reduced order modeling based inexact FETI‐DP solver for lattice structures

“…This is done via functional composition, meaning that the subdomains are the images of the composed mappings: $$$$ {\Omega}^{(s)}=\left({G}^{(s)}\circ T\right)\left[\tilde{\Omega}\right]. $$$$ More information regarding this multiscale geometric representation can be found in References 31,46 for instance.…”

“…They are generic in terms of applications but they may not appear optimal in the specific context of lattice structures. In particular, they do not make use of the geometric similarity among different cells in the numerical solution, what allows a drastic reduction in terms of memory and computational cost, as demonstrated for the assembly of the numerical operators in Reference 31. The objective of this work is to bring forward this idea of benefiting from the repetitive character of the lattice geometries to build a dedicated HPC solver.…”

“…In addition, it is fully consistent with IsoGeometric Analysis (IGA), 47,48 thus allowing to directly analyze such models (see, e.g., References 49,50) while avoiding possibly delicate meshing procedures. A first step toward the efficient analysis of lattices modeled by spline composition has been recently performed with the development of a fast multiscale assembly procedure for the IGA operators (see previous contribution 31 ). The strategy makes use of look‐up tables with precomputed integrals associated to the cell pattern and macro‐fields that encode the mechanical behavior related to the macro‐shape.…”

Topology Optimization and Fatigue Life Estimation of Sustainable Medical Waste Shredder Blade

“…Various structural topology optimization methods have been adopted by researchers to improve the material distribution of engineering structures [20][21][22][23]. While some researchers have used internal microstructure-based material optimization methods and proposed various microstructures to improve material efficiency [24][25][26][27][28]. However, to the author's best knowledge, there is no study found in the literature that has focused on implementing any such methodology for the optimization of any shredding system.…”