Reduced quadrature for FEM, IGA and meshfree methods

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.