We present a consistent and efficient approach to the formulation of geometric nonlinear finite elements for isogeometric analysis (IGA) and isogeometric B-Rep analysis (IBRA) based on the adjoint method. IGA elements are computationally expensive, especially for high polynomial degrees. Using the method presented here enables us to reduce this disadvantage and develop a methodical framework for the efficient implementation of IGA elements. The elements are consistently derived from energy functionals. The load vector and stiffness matrix are obtained from the first and second order derivatives of the energy. Starting from the functional, we apply the concept of algorithmic or automatic differentiation to compute the precise derivatives. Here, we compare the direct (forward) and adjoint (reversed) methods. Analysis of the computational graph allows us to optimize the computation and identify recurring modules. It turns out that using the adjoint method leads to a core-congruential formulation, which enables a clean separation between the mechanical behavior and the geometric description. This is particularly useful in CAD-integrated analysis, where mechanical properties are applied to different geometry types. The adjoint method produces the same results but requires significantly fewer operations and fewer intermediate results. Moreover, the number of intermediate results is no longer dependent on the polynomial degree of the NURBS. This is important for implementation efficiency and computation speed.

The procedure can be applied to arbitrary element formulations and coupling conditions based on energy functionals. For demonstration purposes, we present the proposed approach specifically for use with geometrically nonlinear trusses, beams, membranes, shells, and coupling conditions based on the penalty method.

我们提出一种一致有效的方法，用于基于伴随方法的等几何分析（IGA）和等几何B-Rep分析（IBRA ）的几何非线性有限元公式化。IGA元素在计算上很昂贵，尤其是对于高多项式而言。使用此处介绍的方法使我们能够减少此缺点，并开发出一种用于有效实施IGA元素的方法框架。这些元素始终来自能量功能。载荷矢量和刚度矩阵是从能量的一阶和二阶导数获得的。从函数开始，我们应用算法或自动微分以计算精确的导数。在这里，我们比较直接（正向）和伴随（反向）方法。对计算图的分析使我们能够优化计算并确定重复模块。事实证明，使用伴随方法会导致核心一致的表述，可以在机械行为和几何描述之间进行清晰的分离。这在将机械属性应用于不同几何类型的CAD集成分析中特别有用。伴随方法产生相同的结果，但是所需的操作少得多，中间结果也少得多。此外，中间结果的数量不再取决于NURBS的多项式。这对于实现效率和计算速度很重要。

该程序可以应用于基于能量功能的任意元素配方和耦合条件。出于演示目的，我们基于惩罚方法，提出了专门针对几何非线性桁架，梁，膜，壳体和耦合条件使用的拟议方法。