Newton's aerodynamic problem in media of chaotically moving particles

Hessian measures in the aerodynamic Newton problem

Simple natural proofs of all known results regarding the aerodynamic Newton problem are obtained. Additional new theorems and new promising formulas in terms of Hessian measures are found. We would like to express our deep gratitude to Gerd Wachsmuth for his very important comment which turned one result upside down. Please ignore version 1 of the article.

The analytical solution to Newton's aerodynamic problem in the class of bodies with vertical plane of symmetry and developable side boundary

The method of Hessian measures is used to find the differential equation that defines the optimal shape of nonrotationally symmetric bodies with minimal resistance moving in a rare medium. The synthesis of optimal solutions is described. A theorem on the optimality of the obtained solutions is proved.

Newton空气动力学问题中的局部极小值与偏导数范数之间的不等式

Local minima in Newton's aerodynamical problem and inequalities between norms of partial derivatives

针对Newton最小阻力曲面问题，本文将平面曲面的局部极小性归结为边界消失二元凹函数偏导数$L_2$范数的不等式，并给出依赖区域几何的完备判据。

Full-Space Approach to Aerodynamic Shape Optimization

Aerodynamic shape optimization (ASO) involves finding an optimal surface while constraining a set of nonlinear partial differential equations (PDE). The conventional approaches use quasi-Newton methods operating in the reduced-space, where the PDE constraints are eliminated at each design step by decoupling the flow solver from the optimizer. Conversely, the full-space Lagrange-Newton-Krylov-Schur (LNKS) approach couples the design and flow iteration by simultaneously minimizing the objective function and improving feasibility of the PDE constraints, which requires less iterations of the forwa…

拟芬斯勒度量的等周问题与最小空气动力学阻力的形状

Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance

该研究针对牛顿空气动力学中固定面积二维凸体的最小阻力问题，将其归约为拟芬斯勒度量等周问题，引入振荡微小性准则，揭示最优物体边界奇点条件并构造均匀振荡下的精确最优形状。

The problem of the body of revolution of minimal resistance

Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of {\it convex} axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but {\it generally nonconvex} bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the valu…