论文标题
用于计算形状导数的抽象拉格朗日框架
An abstract lagrangian framework for computing shape derivatives
论文作者
论文摘要
在本文中，我们研究了一个抽象框架，用于计算受PDE约束的功能的形状衍生物。我们使用隐式函数定理在为应用程序量身定制的抽象设置中重新访问Lagrangian方法，以塑造优化。该抽象框架产生了实用公式，以计算形状功能的衍生物，状态的材料衍生物和伴随状态。此外，它允许洞悉状态的材料衍生物与伴随状态之间的双重性。我们向分布式形状衍生物的计算显示了涉及线性椭圆形，非线性椭圆，抛物线PDE和分布的问题。我们将我们的方法与计算形状导数（包括材料衍生方法和平均伴随方法）的其他技术进行比较。
In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of our main result to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.
