Gears are one of the most widely used transmission components. Their operation relies on the contact between mating gear teeth flanks for the transmission of power. Accurate prediction of the contact stresses at these regions, is crucial for the design and dimensioning of these systems. Gear design is centered around highly smooth involute curves that greatly influence their contact behaviour. In this paper, a fully adaptive isogeometric contact modelling scheme, based on hierarchical splines, is presented and applied to the simulation of gear contact problems. In particular, isogeometric simulation is performed for the modelling of mating pair of gear teeth, regarded as linearly elastic bodies. A boundary fitted B-Spline representation of the teeth is automatically generated from engineering design parameters and is used to define the initial discretisation basis. The numerical integration over the contact region is addressed using the so called, Gauss-Point to Surface formulation and a closest point projection procedure. Truncated hierarchical B-Splines are used to capture the highly localised nature of contact, while effectively reducing the number of degrees of freedom. The adaptivity is driven by the strain energy density gradient, which allows to automatically localise the mesh without a priori knowledge of the contact region between the teeth flanks. In our experiments we justify the choices made in different steps of our algorithm and we assess the performance of our adaptive solver with respect to classical tensor product B-Splines.

中文翻译：

---

自适应等几何齿轮接触分析：几何生成、截断分层 B 样条细化和验证


齿轮是使用最广泛的传动部件之一。它们的运行依赖于配合的齿轮齿面之间的接触来传递动力。准确预测这些区域的接触应力对于这些系统的设计和尺寸确定至关重要。齿轮设计以高度平滑的渐开线曲线为中心，这极大地影响了它们的接触行为。在本文中，提出了一种基于分层样条的完全自适应等几何接触建模方案，并将其应用于齿轮接触问题的仿真。特别是，执行等几何模拟用于对配对轮齿的建模，这些轮齿被视为线性弹性体。齿的边界拟合 B 样条表示是根据工程设计参数自动生成的，用于定义初始离散化基础。接触区域的数值积分使用所谓的高斯点到表面公式和最近点投影程序来解决。截断的分层 B 样条用于捕获接触的高度局部性质，同时有效减少自由度的数量。自适应性由应变能密度梯度驱动，这允许自动定位网格，而无需先验了解齿面之间的接触区域。在我们的实验中，我们证明了在算法的不同步骤中所做的选择是合理的，并评估了自适应求解器相对于经典张量积 B 样条的性能。