In this paper, we perform a rigourous version of shape and topological derivatives for optimizations problems under constraint Helmoltz problems. A shape and topological optimization problem is formulated by introducing cost functional. We derive first by considering the lagradian method the shape derivative of the functional. It is also proven a topological derivative with the same approach. An application to several unconstrained shape functions arising from differential geometry are also given.

翻译：在本文中,我们在赫尔莫尔茨问题的限制下,对优化问题的形状和表层衍生物进行严格化的版本,通过引入成本功能来形成形状和表层优化问题,我们首先从函数的形状衍生物拉格迪亚法中得出,还证明这是一种具有相同方法的表层衍生物,还给出了不同几何法产生的若干不受限制的形状函数的应用。