A new global optimization technique for solving partial differential equations

2006 ◽  
pp. 394-398
Author(s):  
I. Karpouzas ◽  
Y. Cherruault
Keyword(s):  
Global Optimization ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimization Technique ◽  
Partial Differential ◽  
Global Optimization Technique

Explicit Solutions for Linear Partial Differential Equations Using Bezier Functions

2006 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimization Technique ◽  
Nonlinear Boundary Value Problems ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Standard Design ◽  
Set Up ◽  
The One ◽  
Domain Discretization

Solutions in basic polynomial form are obtained for linear partial differential equations through the use of Bezier functions. The procedure is a direct extension of a similar technique employed for nonlinear boundary value problems defined by systems of ordinary differential equations. The Bezier functions define Bezier surfaces that are generated using a bipolynomial Bernstein basis function. The solution is identified through a standard design optimization technique. The set up is direct and involves minimizing the error in the residuals of the differential equations over the domain. No domain discretization is necessary. The procedure is not problem dependent and is adaptive through the selection of the order of the Bezier functions. Three examples: (1) the Poisson equation; (2) the one dimensional heat equation; and (3) the slender two-dimensional cantilever beam are solved. The Bezier solutions compare excellently with the analytical solutions.

Solution of some Engineering Partial Differential Equations Governed by the Minimal of a Functional by Global Optimization Method

2013 ◽  
Vol 29 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Y. M. Cheng ◽  
D. Z. Li ◽  
N. Li ◽  
Y. Y Lee ◽  
S. K. Au
Keyword(s):  
Finite Element ◽  
Global Optimization ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Optimization Method ◽  
Global Optimization Method ◽  
Partial Differential ◽  
Special Cases

AbstractMany engineering problems are governed by partial differential equations which can be solved by analytical as well as numerical methods, and examples include the plasticity problem of a geotechnical system, seepage problem and elasticity problem. Although the governing differential equations can be solved by either iterative finite difference method or finite element, there are however limitations to these methods in some special cases which will be discussed in the present paper. The solutions of these governing differential equations can all be viewed as the stationary value of a functional. Using an approximate solution as the initial solution, the stationary value of the functional can be obtained easily by modern global optimization method. Through the comparisons between analytical solutions and fine mesh finite element analysis, the use of global optimization method will be demonstrated to be equivalent to the solutions of the governing partial differential equations. The use of global optimization method can be an alternative to the finite difference/ finite element method in solving an engineering problem, and it is particularly attractive when an approximate solution is available or can be estimated easily.

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