Convergence behavior of 3D finite elements for Neo‐Hookean material

Abstract

Purpose-The purpose of this paper is to examine quadratic convergence of finite element analysis for hyperelastic material at finite strains via Abaqus-UMAT as well as classification of the rates of convergence for iterative solutions in regular cases. Design/methodology/approach-Different formulations for stiffness-Hessian form of the free energy functionals-are systematically given for getting the rate-independent analytical tangent and the numerical tangent as well as rate-dependent tangents using the objective Jaumann rate of Kirchoff stress tensor as used in Abaqus. The convergence rates for available element types in Abaqus are computed and compared for simple but significant nonlinear elastic problems, such as using the 8-node linear brick (B-bar) element-also with hybrid pressure formulation and with incompatible modesfurther the 20-node quadratic brick element with corresponding modifications as well as the 6-node linear triangular prism element and 4-node linear tetrahedral element with modifications. Findings-By using the Jaumann rate of Kirchoff stress tensor for both, rate dependent and rate independent problems, quadratic or nearly quadratic convergence is achieved for most of the used elements using Abaqus-UMAT interface. But in case of using rate independent analytical tangent for rate independent problems, even convergence at all is not assured for all elements and the considered problems. Originality/value-First time the convergence properties of 3D finite elements available in Abaqus sre systematically treated for elastic material at finite strain via Abaqus-UMAT.

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications

“…The derivation of an analytical elasticity tensor is not an easy task, which is the reason why the approximate elasticity tensors are often used, although they worsen the rate of convergence and accuracy of analysis' results. Stein and Sagar [23] have found for the Neo-Hooke hyperelastic model that only an analytically derived elasticity tensor assures a quadratic rate of convergence 1 . For the approximate elasticity tensors, not only the convergence rates are not quadratic but even convergence at all is not assured for all elements and considered problems.…”

“…The incremental constitutive equation 14which is formulated in the reference configuration can be transformed into the current configuration 4 . It takes the following form [2], [23]:…”

“…where e k (k = 1, 2, 3) denotes a unit vector of a Cartesian basis. It should be noted that the elasticity tensor which should be implemented into the subroutine UMAT is slightly different from (23) and takes the form:…”

“…Subsequent use of (17), the transformation rule C τc i jkl = F ip F jq F kr F ls C pqrs , (23) and (24) leads to the the following elasticity tensor expressed by means of Eulerian variables 6 :…”

Finite element implementation of slightly compressible and incompressible first invariant-based hyperelasticity: theory, coding, exemplary problems

“…The approximate material Jacobians always worsen the rate of convergence of the numerical calculations. It was demonstrated by Stein and Sagar (2008) that for the neo-Hooke hyperelastic model, the quadratic rate of convergence 2 is obtained only when the analytical material Jacobian is used. The utilization of the approximate material Jacobians resulted in worsening the convergence rate and, in the case of some of the considered problems and finite element types, it caused lack of convergence.…”