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Institute
A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition
(2024)
A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.
In isogeometric analysis (IGA), which was founded by Hughes et al. [1], the geometry representation is used for the analysis as well. Hence, due to the exact description of the geometry, analysis results can be improved [1, 2]. Therefore, different kinds of splines, like non-uniform rational B-splines (NURBS) [3], are used as shape functions for the discretizations. In linear elasticity problems, for standard formulations, shear locking phenomena can occur due to the different orders of the derivatives in the unbalanced strain-displacement relation. This effect can be reduced using shape functions of higher order, causing an increased computational effort. For low-order formulations, the degrees of shape functions can be adapted accordingly in order to counteract this effect. For an isogeometric displacement-stress mixed Reissner-Mindlin shell formulation such adapted approximation spaces were investigated in [4], for instance. In [5], additionally to a proper choice of shape function spaces, convective coordinates are employed in the derivation of isogeometric shell formulations.
In this contribution, a convective displacement-based isogeometric formulation is introduced, wherein the displacements in the different surface directions are approximated independently using appropriate approximation orders. Therefore, two different meshes are generated from the NURBSdescribed geometry representation, employing order elevation solely for one of the two surface directions. Thus, the order is elevated in opposite directions for these two meshes. The two different possibilities of order elevation are investigated. Furthermore, the use of different convective basis systems is studied. This comprises convective basis systems in each control point, computed according to [6], as well as convective basis systems determined from the local geometry direction in each integration point. The achieved results are compared to whose of a two-dimensional displacementstress mixed formulation presented in [7].
Isogeometric analysis and mixed finite element methods offer promising opportunities to enhance analysis results for complex problems like incompressible elasticity and are able to cope with different locking phenomena. In this contribution, a mixed two-field isogeometric formulation with independent approximations for displacements and stresses is derived, and its ability to counteract different types of locking is investigated using two examples. Furthermore, the influence of the continuity of the stress shape functions on the accurancy of results and convergence behaviour is shown.
In order to overcome locking effects that especially occur for lower order finite element formulations, different methods can be employed. This can be conducted using mixed formulations or adapted approximation orders, for instance. Hence, in order to tackle shear locking that is caused by non-matching interpolation degrees in the shear strain equation, an irreducible and a mixed Reissner-Mindlin plate formulation with accordingly adapted conforming discretizations are derived within the scope of this contribution. In addition, non-uniform rational B-splines (NURBS) are employed therefore, in order to benefit from the properties and refinement strategies offered by isogeometric analysis (IGA) and to achieve more accurate results. The effect of various combinations of interpolation orders on the convergence behavior and the ability to alleviate locking is investigated for both the irreducible and the mixed isogeometric plate formulation and examined for a benchmark example. This is also supplemented by investigations on the stability of the considered variants, tested by the existence of the correct number of zero-energy modes.