Area-Preserving Transformations: Cavalieri in 2D

2020 ◽  
Vol 113 (1) ◽  
pp. 53-60
Author(s):  
Nicholas H. Wasserman ◽  
Keith Weber ◽  
Timothy Fukawa-Connelly ◽  
Juan Pablo Mejía-Ramos
Keyword(s):  
Riemann Integral ◽  
Area Preserving ◽  
Cavalieri’S Principle

A 2D version of Cavalieri's Principle is productive for the teaching of area. In this manuscript, we consider an area-preserving transformation, “segment-skewing,” which provides alternative justification methods for area formulas, conceptual insights into statements about area, and foreshadows transitions about area in calculus via the Riemann integral.

The Riemann Integral and Area-Preserving Transformations

2021 ◽  
pp. 177-190
Author(s):  
Nicholas H. Wasserman ◽  
Timothy Fukawa-Connelly ◽  
Keith Weber ◽  
Juan Pablo Mejia-Ramos ◽  
Stephen Abbott
Keyword(s):  
Riemann Integral ◽  
Area Preserving

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

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

Airfoil Optimization Using Area-Preserving Free-Form Deformation

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.

Area-preserving diffeomorphisms and the stability of the atmosphere

1990 ◽  
Vol 7 (12) ◽  
pp. 2361-2365 ◽  
Author(s):  
J S Dowker ◽  
Wei Mo-zheng
Keyword(s):  
The Stability ◽  
Area Preserving

Invariant curves for area-preserving twist maps far from integrable

1991 ◽  
Vol 65 (3-4) ◽  
pp. 617-643 ◽  
Author(s):  
Alessandra Celletti ◽  
Luigi Chierchia
Keyword(s):  
Invariant Curves ◽  
Twist Maps ◽  
Area Preserving
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