Error Analysis on Numerical Integration Algorithms in a Hypoelasticity Framework

Abstract

This report determines local truncation errors for common stress integration algorithms used in explicit finite element codes with hypoelastic material models. The hypoelastic integration algorithms in question utilize an operator splitting procedure in a rotation neutralized configuration, where the stress response is determined from de- coupling the total deformation into rotational and strain dependent components. This document analyzes the error in evolving the stress given a one-step time increment Δt and compares the errors associated with both the rotational and strain components of the operator splitting method. A slight modification to a traditional algorithm is proposed and studied, where the rate of deformation is appropriately rotated from the midstep configuration at t n+1/2 to the end step configuration at t n+1 before the constitutive evaluation. The proposed modification either completely eliminates the error associated with the rotation rate or is of the same order of magnitude as the original algorithm for the three test cases considered in this report. These cases consist of an unaxial stretch with a constant true strain rate with a rigid body rotation, an uniaxial stretch with a constant engineering strain rate with a rigid body rotation, and a simple shear deformation. All three cases are compared to a closed form solution, and in almost every test case the alternative algorithm yields the most accurate one-step local truncation error.