Reduced quadrature for Finite Element and Isogeometric methods in nonlinear solids

We extend the recently proposed framework using reduced quadrature in the Finite Element and Isogeometric methods for solid mechanics to the nonlinear realm. The proposed approach makes use of the governing equations in the updated Lagrangian formulation in combination with the rate form of the constitutive laws. The key ingredient in the framework is the careful development and use of the Taylor series expansion in the integrands of the internal work terms. The resulting formulation relies on the evaluation of stress gradients, for which the evolution equations and update algorithms are developed. The versatility of the proposed approach is demonstrated on an extensive set of numerical examples employing a variety of constitutive models. The resulting formulations are especially effective in alleviating volumetric locking for the cases of nearly-incompressible and plastic deformations.