The MITC3+ shell element in geometric nonlinear analysis

Highlights

• We present the MITC3+ shell finite element for geometric nonlinear analysis.

• The Lagrangian formulation is employed allowing for large displacement kinematics.

• An excellent performance of the element is demonstrated in nonlinear example solutions.

Abstract

In this paper, we present the MITC3+ shell finite element for geometric nonlinear analysis and demonstrate its performance. The MITC3+ shell element, recently proposed for linear analysis [1], represents a further development of the MITC3 shell element. The total Lagrangian formulation is employed allowing for large displacements and large rotations. Considering several analysis problems, the nonlinear solutions using the MITC3+ shell element are compared with those obtained using the MITC3 and MITC4 shell elements. We conclude that the MITC3+ shell element shows, in the problems considered, the same excellent performance in geometric nonlinear analysis as already observed in linear analysis.

Introduction

Due to significant efforts over the past decades, the finite element method has become a powerful tool for the linear and nonlinear analyses of shell structures [2]. The available capabilities have been continuously improved in reliability and effectiveness but there are still important research and development tasks to be accomplished. One such task is the development of an optimal 3-node shell element.

When modeling complex shell structures, an effective mesh of triangular shell elements is relatively easy to generate provided general element shapes are allowed, that is, the element employed must be effective even when used in general triangular shapes. Lee et al. proposed recently the 3-node MITC3+ shell element for linear analysis that was shown to perform even well when highly distorted elements are used [1]. Hence this element is a good candidate for use in general meshes. Of course, a shell surface can also be meshed rather easily using 4-node quadrilateral elements, but then in general practical analysis quite distorted elements might be present that show low predictive capabilities. Namely, quadrilateral elements generally do not perform well when highly distorted.

It is difficult to develop effective shell finite elements that give reliable and efficient solutions for general shell problems, when considering the various shell geometries, boundary and loading conditions, and mesh patterns used [2], [3], [4]. The difficulty is basically due to the highly sensitive and complex behavior of shell structures (categorized as bending dominated, membrane dominated and mixed behaviors), in particular, when the shell thickness is small [3], [4]. Then a shell finite element discretization frequently gives too stiff solutions. This phenomenon is called “locking”, which must be alleviated for reliable shell finite element analysis. Among various schemes to alleviate the locking, the MITC (Mixed Interpolation of Tensorial Components) scheme has been used very successfully in the development of general shell elements [1], [2], [3], [4], [5], [6], [7].

The MITC3+ shell element is based on the concepts of the MITC3 shell element developed by Lee and Bathe [6] with an enrichment by a cubic bubble function for the rotations. The cubic bubble function provides a higher-order interpolation inside the element to enrich the element behavior while maintaining the linear interpolation along the element edges. The degrees of freedom corresponding to the bubble function can be statically condensed out on the element level. To alleviate shear locking, a newly developed assumed transverse shear strain field is used. The shell element passes the basic numerical tests – the isotropy, zero energy mode and patch tests – and shows an excellent convergence behavior in linear analysis.

A particular strength of the MITC approach is that a formulation achieved for linear analysis can be directly extended to nonlinear analysis by “simply” using the appropriate stress and strain measures [2], although the performance in nonlinear solutions must then of course still be studied.

Our objective in this paper is to present the formulation of the MITC3+ shell element in geometric nonlinear analysis. The standard total Lagrangian formulation is employed allowing for large displacements and large rotations [2]. Solving various shell problems, the performance of the MITC3+ shell element is evaluated by comparison of the solution accuracies obtained with the MITC3+, MITC3 and MITC4 shell elements. Of course, the MITC4 quadrilateral shell element has been, and is, widely employed in engineering practice due to its superior performance in both linear and nonlinear analyses. Our study reveals that the performance of the MITC3+ shell element in nonlinear analysis is as good as the performance of the MITC4 shell element, even when highly distorted meshes are used.

Next, in Section 2, the linear formulation of the MITC3+ shell element is reviewed and, in Section 3, we present the geometric nonlinear formulation. In Section 4, we examine the performance of the MITC3+ shell element in geometric nonlinear analysis through the solutions of various shell problems.