Isogeometric analysis (IGA) is a numerical simulation method which is directly based on the NURBS-based representation of CAD models. It exploits the tensor-product structure of 2- or 3-dimensional NURBS objects to parameterize the physical domain. Hence the physical domain is parameterized with respect to a rectangle or to a cube. Consequently, singularly parameterized NURBS surfaces and NURBS volumes are needed in order to represent non-quadrangular or non-hexahedral domains without splitting, thereby producing a very compact and convenient representation.

The Galerkin projection introduces finite-dimensional spaces of test functions in the weak formulation of partial differential equations. In particular, the test functions used in isogeometric analysis are obtained by composing the inverse of the domain parameterization with the NURBS basis functions. In the case of singular parameterizations, however, some of the resulting test functions do not necessarily fulfill the required regularity properties. Consequently, numerical methods for the solution of partial differential equations cannot be applied properly.

We discuss the regularity properties of the test functions. For one- and two-dimensional domains we consider several important classes of singularities of NURBS parameterizations. For specific cases we derive additional conditions which guarantee the regularity of the test functions. In addition we present a modification scheme for the discretized function space in case of insufficient regularity. It is also shown how these results can be applied for computational domains in higher dimensions that can be parameterized via sweeping.

等几何分析 (IGA) 是一种直接基于基于 NURBS 的 CAD 模型表示的数值模拟方法。它利用 2 维或 3 维 NURBS 对象的张量积结构来参数化物理域。因此，物理域是相对于矩形或立方体进行参数化的。因此，需要单独参数化的 NURBS 曲面和 NURBS 体积，以便在不分裂的情况下表示非四边形或非六面体域，从而产生非常紧凑和方便的表示。

Galerkin 投影在偏微分方程的弱公式中引入了测试函数的有限维空间。特别是，等几何分析中使用的测试函数是通过将域参数化的逆与 NURBS 基函数组合而获得的。然而，在奇异参数化的情况下，一些结果测试函数不一定满足所需的正则性。因此，不能正确应用求解偏微分方程的数值方法。

我们讨论测试函数的正则性。对于一维和二维域，我们考虑 NURBS 参数化的几个重要类别的奇点。对于特定情况，我们推导出额外的条件来保证测试函数的规律性。此外，我们提出了一种在规则性不足的情况下对离散化函数空间的修改方案。还展示了如何将这些结果应用于可以通过扫描参数化的更高维度的计算域。