Geometric Optimization of Radiative Enclosures Through Nonlinear Programming

2001 ◽  
Author(s):  
K. J. Daun ◽  
J. R. Howell ◽  
D. P. Morton
Keyword(s):  
Nonlinear Programming ◽  
Newton’S Method ◽  
Newton's Method ◽  
Design Procedure ◽  
Computational Effort ◽  
Geometric Optimization ◽  
Spline Curves ◽  
Analysis Technique ◽  
Gradient Based ◽  
Optimum Solution

Abstract Optimization routines have become a popular tool in the design of thermal systems. These routines dramatically reduce the time and computational effort needed to find the optimal design by reducing the number of iterations required by the forward design procedure. Also, these methods often find solutions that are not intuitive to the designer. Although these routines have been applied to solve conduction and convection problems, they have not been used to design radiant enclosures. This paper introduces a methodology for applying nonlinear programming to design 2-D radiant enclosures. The process is facilitated by representing the enclosure surface parametrically with B-spline curves, and an infinitesimal-area analysis technique is then used to solve the radiosity distribution within the enclosure. The enclosure geometry is repeatedly adjusted with a gradient-based minimization algorithm, until the optimum solution is found. This technique is demonstrated by optimizing the geometry of a 2-D radiant enclosure, with the objective to obtain a specified radiosity distribution over a portion of the enclosure surface. The steepest descent, Newton’s method, and quasi-Newton’s method are used to find the optimum enclosure geometry, and the performance of these methods is compared.

Application of the Monomial Method to Shape Optimization

1994 ◽  
Vol 116 (4) ◽  
pp. 1013-1018 ◽  
Author(s):  
S. A. Burns
Keyword(s):  
Shape Optimization ◽  
Newton’S Method ◽  
Structural Design ◽  
Newton's Method ◽  
Computational Effort ◽  
Algebraic Equations ◽  
Structural Shape ◽  
Continuum Structures ◽  
Structural Shape Optimization ◽  
Nonlinear Algebraic Equations

The monomial method is an alternative to Newton’s method for solving systems of nonlinear algebraic equations. It possesses several properties not shared by Newton’s method that enhance performance, yet does not require substantial computational effort beyond that required for Newton’s method. Previous work has demonstrated that the monomial method treats problems in structural design very effectively. This paper combines the monomial method with the method of generalized geometric programming to treat the problem of structural shape optimization of continuum structures modeled by finite elements.

Application of the Monomial Method to Shape Optimization

1993 ◽  
Author(s):  
Scott A. Burns
Keyword(s):  
Shape Optimization ◽  
Newton’S Method ◽  
Structural Design ◽  
Newton's Method ◽  
Computational Effort ◽  
Algebraic Equations ◽  
Structural Shape ◽  
Continuum Structures ◽  
Structural Shape Optimization ◽  
Nonlinear Algebraic Equations

Abstract The monomial method is an alternative to Newton’s method for solving systems of nonlinear algebraic equations. It possesses several properties not shared by Newton’s method that enhance performance, yet does not require substantial computational effort beyond that required for Newton’s method. Previous work has demonstrated that the monomial method treats problems in structural design very effectively. This paper combines the monomial method with the method of generalized geometric programming to treat the problem of structural shape optimization of continuum structures modeled by finite elements.

scholarly journals A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ababu Teklemariam Tiruneh ◽  
W. N. Ndlela ◽  
S. J. Nkambule
Keyword(s):  
Newton Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Computational Effort ◽  
Secant Methods ◽  
Iterative Solutions ◽  
Solutions Of Equations ◽  
Modified Formula

An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method.

scholarly journals Extending the Applicability of Stirling’s Method

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 35 ◽  
Author(s):  
Cristina Amorós ◽  
Ioannis K. Argyros ◽  
Á. Alberto Magreñán ◽  
Samundra Regmi ◽  
Rubén González ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Newton’S Method ◽  
Newton's Method ◽  
Computational Effort ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
New Findings

Stirling’s method is considered as an alternative to Newton’s method when the latter fails to converge to a solution of a nonlinear equation. Both methods converge quadratically under similar convergence criteria and require the same computational effort. However, Stirling’s method has shortcomings too. In particular, contractive conditions are assumed to show convergence. However, these conditions limit its applicability. The novelty of our paper lies in the fact that our convergence criteria do not require contractive conditions. Hence, we extend its applicability of Stirling’s method. Numerical examples illustrate our new findings.

Optimal Power Flow Using Newton’s Method

2012 ◽  
Vol 3 (2) ◽  
pp. 167-169
Author(s):  
F.M.PATEL F.M.PATEL ◽  
◽  
N. B. PANCHAL N. B. PANCHAL
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Optimal Power
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