Finite Elements Approaches in the Solution of Field Functions in Multidimensional Space: A Case of Boundary Value Problems
Abstract: An idealized two dimensional continuum region of GRP composite was used to develop an efficient method for solving continuum problems formulated for space domains. The continuum problem is solved by minimization of a functional formulated through a finite element procedure employing triangular elements and assumption of linear approximation polynomial. The assemblage of elements functional derivatives system of equations through FEM assembly procedure made possible the definition of a unique and parametrically defined model from which the solution of continuum configuration with an arbitrary number of scales is solved. The finite element method(FEM )developed is recommended to be applied in the evaluation of the function of functions in irregular shaped continuum whose boundary conditions are specified such as in the evaluation of displacement in structures and solid mechanics problems, evaluation of temperature distribution in heat conduction problems, evaluation of displacement potential in acoustic fluids evaluation of pressure in potential flows, evaluation of velocity in general flows, evaluation of electric potential in electrostatics, evaluation of magnetic potential in magnetostatics and in the solution of time dependent field problems. A unified computational model with standard error of 0.15 and correlation coefficient of 0.72 was developed to aid analysis and easy prediction of regional function with which the continuum function was successfully modeled and optimized through gradient search and Lagrange multipliers approach. Above all the optimization schemes of gradient search and Lagrangian multiplier confirmed local minimum of function as 0.006-0.00847 to confirm the predictions of FEM and constraint conditions.
Cite this paper: C. Ihueze, O. Christian and E. Onyemaechi, "Finite Elements Approaches in the Solution of Field Functions in Multidimensional Space: A Case of Boundary Value Problems," Journal of Minerals and Materials Characterization and Engineering, Vol. 9 No. 10, 2010, pp. 929-959. doi: 10.4236/jmmce.2010.910068.
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