An isogeometric analysis for elliptic homogenization problems

TL;DR

This paper introduces an isogeometric analysis-based multiscale method for elliptic homogenization problems, offering improved accuracy and efficiency over traditional finite element approaches by utilizing NURBS basis functions.

Contribution

It develops the IGA-HMM framework, integrating NURBS for macro and micro discretizations, with proven error estimates and adaptive refinement strategies, enhancing the solution of highly oscillatory homogenization problems.

Findings

Demonstrates superior accuracy of IGA-HMM over standard methods

Provides optimal micro refinement strategies for NURBS basis functions

Numerical results confirm excellent performance and efficiency

Abstract

A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard Finite Element Heterogeneous Multiscale Method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rational B-splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Beside the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. Numerical results show the excellent performance of the proposed method.