We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.

中文翻译：

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子流形上积分的二阶定向形状导数

我们使用形状微分的变体计算平滑子流形上积分的一阶和二阶形状敏感性。结果是根据一个扰动向量场的二次形式，产生扰动泛函的二阶二次模型。我们讨论了这个导数的结构，在一般几何框架中推导出域表达式和哈达玛形式，并给出了对出现的项的详细几何解释。