The Shapes of Things - A Practical Guide to Differential Geometry and the Shape Derivative
The Shapes of Things - A Practical Guide to Differential Geometry and the Shape Derivative
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事物的形状 - 微分几何和形状导数实用指南
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
A. Aschoff
中科院分区：
文献类型：
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作者：
J. Ostwald;A. Aschoff
我们周围的许多东西都有依赖于它们形状的特性，例如，流中刚体的阻力特性。这个微分几何的独立概述解释了如何区分函数（在微积分意义上）相对于形状变量。这种方法对于理解包含几何偏微分方程（PDE）的数学模型很有用，它允许读者获得比微分几何文本中通常提供的更清晰的几何量（如曲率）的公式。读者将学习如何通过开发表面上的基本微积分工具并将其与变分法相结合来计算几何灵敏度。几个应用程序，利用形状衍生物和许多插图，帮助建立直觉包括在内。观众：这本书是各种形状导数公式的方便参考，对于任何对表面几何和形状优化感兴趣的人都应该有价值。研究生可以用它来快速掌握形状微分的机器。研究连续介质力学、流体力学、数值分析和偏微分方程的科学家会发现这本书对表面几何是关键的和/或几何随时间演变的问题很有帮助。那些想要学习形状差异化的基础知识的人也会发现它很有用。
Many things around us have properties that depend on their shape-for example, the drag characteristics of a rigid body in a flow. This self-contained overview of differential geometry explains how to differentiate a function (in the calculus sense) with respect to a shape variable. This approach, which is useful for understanding mathematical models containing geometric partial differential equations (PDEs), allows readers to obtain formulas for geometric quantities (such as curvature) that are clearer than those usually offered in differential geometry texts. Readers will learn how to compute sensitivities with respect to geometry by developing basic calculus tools on surfaces and combining them with the calculus of variations. Several applications that utilize shape derivatives and many illustrations that help build intuition are included. Audience: This book is a convenient reference for various shape derivative formulas and should be of value to anyone interested in surface geometry and shape optimization. Graduate students can use it to quickly get up to speed on the machinery of shape differential calculus. Scientists studying continuum mechanics, fluid mechanics, numerical analysis, and PDEs will find the book helpful for problems in which surface geometry is critical and/or geometry evolves in time. Those who want to learn the basics of shape differentiation will also find it useful.