An immersed boundary shell element formulation based on Kirchhoff-Love shell theory is presented here that uses a background mesh consisting of uniform 3D B-spline elements within which the geometry and the boundaries are embedded. In this theory, rotation is assumed to be identical to slope which introduces second derivatives of deflection in the weak form. As a result, elements that provide tangent continuous approximation of the displacement are needed. A 3D shell element is formulated under small deformation assumptions that has three translational degrees of freedom (DOF) per node and uses quadratic B-Spline shape functions. The displacement field is transformed to a local coordinate system attached to the shell surface and defined such that the shell theory assumptions are satisfied. Essential boundary conditions are applied using the step boundary method where approximate step functions are utilized to construct trial and test solutions that satisfy the boundary conditions. Benchmark problems that are commonly used to test shell elements are used to validate the 3D B-spline thin shell element presented here.

本文介绍了一种基于Kirchhoff-Love壳理论的浸入式边界壳元素公式，该公式使用了由均匀3D B样条元素组成的背景网格，其中嵌入了几何和边界。在该理论中，假定旋转与坡度相同，坡度以弱形式引入了挠度的二阶导数。结果，需要提供位移的切线连​​续近似的元素。3D壳单元是在较小变形假设下制定的，每个节点具有三个平移自由度（DOF），并使用二次B样条曲线形状函数。位移场被转换为连接到壳表面的局部坐标系，并定义为满足壳理论假设。使用分步边界法应用基本边界条件，其中近似分步函数用于构建满足边界条件的试验和测试解决方案。通常用于测试壳单元的基准问题用于验证此处介绍的3D B样条薄壳单元。