The splitting of separatrices for a standard family of area-preserving transformations

This paper compares the system of equations underlying Alonso's theory of movement with that of Wilson's standard family of spatial-interaction models. It is shown that the Alonso model is equivalent to one of Wilson's four standard models depending on the assumption at the outset about which of the total outflows and/or inflows are known. This result turns out to supersede earlier findings—inconsistent only in appearance—which were derived independently by Wilson and Ledent. In addition to this, an original contribution of this paper—obtained as a byproduct of the process leading to the aforementioned result—is to provide an exact methodology permitting one to solve the Alonso model for each possible choice of the input data.

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On an area-preserving inverse curvature flow of convex closed plane curves

Journal of Functional Analysis ◽

10.1016/j.jfa.2021.108931 ◽

2021 ◽

Vol 280 (8) ◽

pp. 108931

Author(s):

Laiyuan Gao ◽

Shengliang Pan ◽

Dong-Ho Tsai

Keyword(s):

Curvature Flow ◽

Plane Curves ◽

Area Preserving ◽

Closed Plane

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Area-Preserving Surface Flattening Using Lie Advection

Lecture Notes in Computer Science - Medical Image Computing and Computer-Assisted Intervention – MICCAI 2011 ◽

10.1007/978-3-642-23629-7_41 ◽

2011 ◽

pp. 335-342 ◽

Cited By ~ 2

Author(s):

Guangyu Zou ◽

Jiaxi Hu ◽

Xianfeng Gu ◽

Jing Hua

Keyword(s):

Surface Flattening ◽

Area Preserving

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Airfoil Optimization Using Area-Preserving Free-Form Deformation

Volume 1: Advances in Aerospace Technology ◽

10.1115/imece2015-50904 ◽

2015 ◽

Author(s):

Stavros N. Leloudas ◽

Giorgos A. Strofylas ◽

Ioannis K. Nikolos

Keyword(s):

Cross Sectional Area ◽

Equality Constraint ◽

Optimization Process ◽

Free Form ◽

Sectional Area ◽

Cross Sectional ◽

Airfoil Optimization ◽

Free Form Deformation ◽

Correction Problem ◽

Area Preserving

Given the importance of structural integrity of aerodynamic shapes, the necessity of including a cross-sectional area equality constraint among other geometrical and aerodynamic ones arises during the optimization process of an airfoil. In this work an airfoil optimization scheme is presented, based on Area-Preserving Free-Form Deformation (AP FFD), which serves as an alternative technique for the fulfillment of a cross-sectional area equality constraint. The AP FFD is based on the idea of solving an area correction problem, where a minimum possible offset is applied on all free-to-move control points of the FFD lattice, subject to the area preservation constraint. Due to the linearity of the area constraint in each axis, the extraction of an inexpensive closed-form solution to the area preservation problem is possible by using Lagrange Multipliers. A parallel Differential Evolution (DE) algorithm serves as the optimizer, assisted by two Artificial Neural Networks as surrogates. The use of multiple surrogate models, in conjunction with the inexpensive solution to the area correction problem, render the optimization process time efficient. The application of the proposed methodology for wind turbine airfoil optimization demonstrates its applicability and effectiveness.

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