Balanced approximation spaces and mixed variational principles to eliminate locking effects in isogeometric shell analysis

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Isogeometric finite element formulations aid and abet a tighter linkage between design and analysis. The geometry description of the design system together with its basis functions is used for analysis. Current CAD software is surface-oriented and uses NURBS (Non-Uniform Rational B-Splines) to describe the geometry. Thin-walled structures, e.g. free form surfaces, are defined merely by their reference surface using NURBS surfaces.The usage of NURBS in the context of the finite element method offers completely new possibilities. Besides geometry also spatial derivatives and the surface normal are defined exactly in every point for every discretization. Approximation spaces and their appropriate basis functions with the desired continuity between elements can be constructed by order elevation and knot insertion. The exact geometry is maintained. Considering these properties, an efficient, reliable and widely applicable isogeometric shell formulation for the analysis of thin-walled structures shall be developed. The geometry description of the CAD system shall be used as reference surface to define the shell body. Thus, an elaborate conversion into other geometry descriptions, e.g. NURBS volumes, can be avoided. The employed nonlinear kinematic shall allow large deformations and finite rotations. Transverse shear strains have to be considered. An exact description of the geometry in the thickness direction together with a consistent approximation of the kinematic is especially important to ensure convergence of deformations for order elevation. The computation of geometries with complex intersections shall require neither drilling rotation stabilization nor a manual intervention. Essential ingredients of an efficient and widely applicable shell formulation are efficacious methods for the elimination of locking effects as well as the incorporation of general three dimensional constitutive laws. The properties of NURBS shall be used advantageously to eliminate locking effects. Transverse shear locking is precluded by adapted approximation spaces with appropriate basis functions for deformations and rotations. A mixed variational formulation with independent deformations, strains and stresses significantly reduces membrane locking, as well as other locking effects. The choice of the solution spaces allows a statical condensation on element level to ensure numerical efficacy. An appropriate choice of the ansatz for strains and stresses allows the usage of general three dimensional constitutive laws without local iterations. Finally, a comprehensive set of benchmark examples shall be provided in order to show the capability of the proposed methods.