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

等距有限元公式有助于和教唆设计和分析之间更紧密的联系。利用设计系统的几何描述及其基本函数进行分析。目前的CAD软件是面向曲面的，使用NURBS(非均匀有理B-Splines)来描述几何图形。薄壁结构，如自由曲面，仅由其参考曲面使用NURBS曲面来定义。NURBS在有限元方法中的使用提供了全新的可能性。除几何外，在每个离散化过程中，每个点上的空间导数和曲面法线都被精确定义。通过升阶和插入结点，可以构造元素间具有期望连续性的逼近空间及其适当的基函数。将保持精确的几何图形。考虑到这些特性，应发展一种高效、可靠和广泛适用的薄壁结构分析等几何壳式。壳体的定义应以CAD系统的几何描述为参考面。因此，可以避免复杂的到其他几何描述的转换，例如NURBS体积。所采用的非线性运动学应允许大变形和有限旋转。必须考虑横向剪切应变。厚度方向的几何图形的精确描述和运动学的一致近似对于确保阶数升高的变形收敛尤为重要。具有复杂交叉点的几何形状的计算既不需要稳定钻井旋转，也不需要人工干预。有效和广泛应用的壳体公式的基本成分是消除锁定效应的有效方法以及纳入一般三维本构关系。应有利地利用NURBS的特性来消除锁定效应。具有适当的变形和旋转基函数的自适应近似空间避免了横向剪切锁定。具有独立变形、应变和应力的混合变分公式显著减少了膜锁定以及其他锁定效应。解空间的选择允许在单元水平上进行静态凝聚，以确保数值效率。适当地选择应变和应力的ansatz，可以使用一般的三维本构关系，而不需要局部迭代。最后，应提供一套全面的基准实例，以显示所建议方法的能力。

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.