Nodally Integrated Finite Element Formulation for Mindlin-Reissner Plates

This work describes a nodally integrated finite element formulation for plates under the Mindlin-Reissner theory. The formulation makes use of the weighted residual method and nodal integration to derive the assumed strain relations. An element formulation for four-node quadrilateral elements is implemented in the nonlinear finite element solver Abaqus using the UEL user element subroutine. Numerical tests are carried out on the new element and the results are presented. Index Terms— Abaqus, CAE, element formulation, finite element analysis, four-node quadrilateral element, nodal integration, UEL —————————— a —————————— 1 I N nodal integration, the numerical integration is carried out at the corner nodes of the element, rather than at quadrature points inside the element. Research on finite element formu- lations emerged in the beginning of the 1970s, while nodal integration in FEA is a relatively recent development, with the first works beginning in the year 2000 describing its applica- tion to tetrahedral elements. Wolff (1) discussed nodally inte- grated finite elements, focussing on solid elements. It was shown that nodally integrated elements show good conver- gence compared to full integration when applied to incom- pressible media. Nodal integration is also attractive because of its applicability to meshfree methods as described by Quak et al (2), who compared gauss integration and nodal integration for meshless analyses. This is because, as described in this formulation, the method focuses on a nodal patch, rather than on an elemental area, as conventional elements do. In large deformation processes, for example in extrusion and injection moulding, finite elements can suffer from excessive mesh de- formation. Meshless methods are well suited to avoid these problems. However, the obstacle that must be overcome in developing nodally integrated elements is that the quantities at nodal points are not continuous, and the nodes are shared among multiple elements. These elements also suffer from shear locking, being based on the Mindlin-Reissner theory for thick plates. Wang and Chen (3) described a Mindlin-Reissn er plate formulation with nodal integration. Castelazzi and Krysl (4) introduced Reissner-Mindlin plate elements with nodal inte- gration in which the nodal integration is derived from the a priori satisfaction of the weighted residuals. They have ap- plied the same formulation, called the NIPE technique to a nine-node quadrilateral plate element (5). However, in this case, a variational energy method is used instead of the weighted residual method in their previous work. The result- ing element is found to be an improvement over the earlier one. Giner, E. et al (6) and Park, K. et al (7), have implemented special elements for fracture mechanics in Abaqus UEL. Abaqus is an extremely capable nonlinear solver with a large element library that allows analysis of even the most complex structural problems. The default element formula- tions in Abaqus are accurate, robust and reliable enough, hav- ing been extensively tested. However, situations arise in which the Abaqus element library would not serve the pur- pose, and the user would like to define their own elements, such as:  Modelling non-structural physical processes that are coupled to structural behaviour  Applying solution-dependent loads  Modelling active control mechanisms  Studying the behaviour of proposed formulations For example, elements can be developed to function as con- trol or feedback mechanisms in an analysis that consists of regular elements. Moreover, it is easier to maintain and port a subroutine than to do the same for a complete finite element program.