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Quadratic NURBS-based discretizations of the Galerkin method suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements have been recently introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we generalize CAS elements to vanquish membrane locking in quadratic NURBS-based discretizations of linear Kirchhoff–Love shells. CAS elements bilinearly interpolate the membrane strains at the four corners of each element. Thus, the assumed strains have C 0 continuity across element boundaries. To the best of the authors' knowledge, CAS elements are the first assumed-strain treatment to effectively overcome membrane locking in quadratic NURBS-based discretizations of Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional degrees of freedom are added, (2) No additional systems of algebraic equations need to be solved, (3) No matrix multiplications or matrix inversions are needed to obtain the stiffness matrix, and (4) The nonzero pattern of the stiffness matrix is preserved. The benchmark problems show that CAS elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are an effective locking treatment since this element type results in more accurate displacements for coarse meshes and excises the spurious oscillations of the membrane forces. The benchmark problems also show that CAS elements outperform state-of-the-art element types based on Lagrange polynomials equipped with either assumed-strain or reduced-integration locking treatments. [ABSTRACT FROM AUTHOR]

NURBS-based discretizations of the Galerkin method suffer from membrane locking when applied to primal formulations of curved thin-walled structures. We consider linear plane curved Kirchhoff rods as a model problem to study how to remove membrane locking from NURBS-based discretizations. In this work, we propose continuous-assumed-strain (CAS) elements, an assumed strain treatment that removes membrane locking from quadratic NURBS for an ample range of slenderness ratios. CAS elements take advantage of the C 1 inter-element continuity of the displacement vector given by quadratic NURBS to interpolate the membrane strain using linear Lagrange polynomials while preserving the C 0 inter-element continuity of the membrane strain. To the authors' knowledge, CAS elements are the first NURBS-based element type able to remove membrane locking for a broad range of slenderness ratios that combines the following distinctive characteristics: (1) No additional degrees of freedom are added, (2) No additional systems of algebraic equations need to be solved, and (3) The nonzero pattern of the stiffness matrix is preserved. Since the only additional computations required by the proposed element type are to evaluate the derivatives of the basis functions and the unit tangent vector at the knots, the proposed scheme barely increases the computational cost with respect to the locking-prone NURBS-based discretization of the primal formulation. The benchmark problems show that the convergence of CAS elements is independent of the slenderness ratio up to 1 0 4 while the convergence of quadratic NURBS elements with full and reduced integration, local B ̄ elements, and local ANS elements depends heavily on the slenderness ratio and the error can even increase as the mesh is refined. The numerical examples also show how CAS elements remove the spurious oscillations in stress resultants caused by membrane locking while quadratic NURBS elements with full and reduced integration, local B ̄ elements, and local ANS elements suffer from large-amplitude spurious oscillations in stress resultants. In short, CAS elements are an accurate, robust, and computationally efficient numerical scheme to overcome membrane locking in quadratic NURBS-based discretizations. [ABSTRACT FROM AUTHOR]

G-splines are a generalization of B-splines that deals with extraordinary points by imposing G 1 constraints across their spoke edges, thus obtaining a continuous tangent plane throughout the surface. Using the isoparametric concept and the Bubnov–Galerkin method to solve partial differential equations with G-splines results in discretizations with global C 1 continuity in physical space. Extraordinary points (EPs) are required to represent manifold surfaces with arbitrary topological genus. In this work, we allow both interior and boundary EPs and there are no limitations regarding how close EPs can be from each other. Reaching this level of flexibility is necessary so that splines with EPs can become mainstream in the design-through-analysis cycle of the complex thin-walled structures that appear in engineering applications. To the authors' knowledge, the two EP constructions based on imposing G 1 constraints proposed in this work are the first two EP constructions used in isogeometric analysis (IGA) that combine the following distinctive characteristics: (1) Only vertex-based control points are used and they behave as geometric shape handles, (2) any control point of the control net can potentially be an EP, (3) global C 1 continuity in physical space is obtained without introducing singularities, (4) faces around EPs are not split into multiple elements, i.e., Bézier meshes with uniform element size are obtained, and (5) good surface quality is attained. The studies of convergence and surface quality performed in this paper suggest that G-splines are more suitable for IGA than EP constructions based on the D-patch framework. Finally, we have represented the stiffener, the inner part, and the outer part of a B-pillar with G-spline surfaces and solved eigenvalue problems using both Kirchhoff–Love and Reissner–Mindlin shell theories. The results are compared with bilinear quadrilateral meshes and excellent agreement is found between G-splines and conventional finite elements. In summary, G-splines are a viable alternative to design and analyze thin-walled structures using the same geometric representation so as to streamline the design-through-analysis cycle. [ABSTRACT FROM AUTHOR]

Analysis-suitable T-splines (AST-splines) are a promising candidate to achieve a seamless integration between the design and the analysis of thin-walled structures in industrial settings. In this work, we allow multiple extraordinary points per face, i.e., we remove the restriction of preceding works that required extraordinary points to be at least four rings apart from each other. We do so by mathematically showing that AST-splines with multiple extraordinary points per face are linearly independent and their polynomial basis functions form a non-negative partition of unity. This extension of the subset of AST-splines drastically increases the flexibility to build geometries using AST-splines; e.g., much coarser meshes can be constructed around small holes. The AST-spline spaces detailed in this work have C 1 inter-element continuity near extraordinary points and C 2 inter-element continuity elsewhere. For the convergence studies performed in this paper involving second- and fourth-order linear elliptic problems with manufactured solutions, we have not found any drawback caused by allowing multiple EPs per face in either the first refinement levels or the asymptotic behavior. To illustrate a possible isogeometric framework that is already available, we design the B-pillar and the side outer panel of a car using T-splines with the commercial software Autodesk Fusion360, import the control nets into our in-house code to build AST-splines, and import the Bézier extraction information into the commercial software LS-DYNA to solve eigenvalue problems. The results are compared with conventional finite elements and good agreement is found between AST-splines and conventional finite elements. • Smooth splines with multiple extraordinary points per face are studied. • A mathematical proof of linear independence is provided. • Excellent convergence from level 0 is obtained. • Geometries with arbitrary topological genus are built. • Comparisons with the commercial software LS-DYNA are included. [ABSTRACT FROM AUTHOR]