Reduced quadrature for FEM, IGA and meshfree methods

• Published in: Computer methods in applied mechanics and engineering 2021-01, Vol.373, p.113521, Article 113521
• Main Authors: Moutsanidis, Georgios, Li, Weican, Bazilevs, Yuri
• Format: Article
• Language: English
• Subjects: Computational fluid dynamics; Finite element method; Hourglass control; Isogeometric analysis; Material point method; Meshfree methods; Meshless methods; One-point quadrature; Quadratures; Reduced integration; Stiffness matrix; Taylor series
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.113521

We propose a framework to improve one-point quadrature and, more generally, reduced integration for finite element methods (FEM), Isogeometric Analysis (IGA), and immersed methods. The framework makes use of first- and higher-order Taylor expansion of the integrands involved in the principle of virtual work, and the analytical integration of the resulting correction terms. Explicit forms of the correction terms, which eliminate rank deficiency of the resulting stiffness matrices and ensure optimal convergence of the discrete formulation, are derived for C0-continuous linear and quadratic FEM, and C1-continuous quadratic Non-Uniform Rational B-Splines (NURBS). The proposed methodology naturally extends to immersed methods, such as the Material-Point Method (MPM), provided the background discretization is sufficiently smooth. It also naturally lends itself to handle nearly incompressible materials, where the resulting correction is applied only to the deviatoric part of the internal virtual work term. •Proposed reduced quadrature for FEM, IGA, and Meshfree methods.•Used higher-order Taylor series expansion of the integrands in the weak form.•Developed a technique to treat near-incompressibility.•Achieved same accuracy as full quadrature at a fraction of the cost.•Reduced the integration error in material point method-like techniques.