论文标题
NGSOLVE中的完全和半自动化的形状分化
Fully and Semi-Automated Shape Differentiation in NGSolve
论文作者
论文摘要
在本文中，我们介绍了有限元软件NGSOLVE中自动形状差异化的框架。我们的方法结合了数学拉格朗日方法，用于区分PDE限制的形状函数和NGSOLVE的自动分化功能。用户可以确定需要哪种自动化程度，从而允许该软件的更类似于自定义的或更类似黑色的盒子。 我们讨论了无约束模型问题的一阶和二阶形状衍生物的自动生成，以及受不同类型的部分微分方程约束的更现实的问题。我们认为线性以及非线性问题以及在表面上构成的问题。在数值实验中，我们通过泰勒测试验证了计算的衍生物的准确性。最后，我们提出了第一阶和二阶形状优化算法，并为几个数值优化示例提供了说明，从非线性弹性到麦克斯韦的方程。
In this paper we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE constrained shape functions with the automated differentiation capabilities of NGSolve. The user can decide which degree of automatisation is required and thus allows for either a more custom-like or black-box-like behaviour of the software. We discuss the automatic generation of first and second order shape derivatives for unconstrained model problems as well as for more realistic problems that are constrained by different types of partial differential equations. We consider linear as well as nonlinear problems and also problems which are posed on surfaces. In numerical experiments we verify the accuracy of the computed derivatives via a Taylor test. Finally we present first and second order shape optimisation algorithms and illustrate them for several numerical optimisation examples ranging from nonlinear elasticity to Maxwell's equations.
