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Finite element approach to solution of multidimensional quasi-harmonic field functions
Abstract
This paper focuses on the methodical approach for the solution of field
problems whose function can be expressed as derivatives and integrated functional or on solution of quasi-harmonic functions whose physical behaviors are governed by a general quasi-harmonic differential equation that can be treated as a quadratic functional that can be minimized over a region. The functional of a stress field function was established using mixed methods analogous to variational principle, minimum total potential principle and finite element method. The functional of function, Φ(x,y) was formed using Euler equivalent integral and finite element shape function for a function expressed in derivative form. The minimization of the functional gave the stationary values of the function which minimize the functional. The solution of the functional gave the minimum value of the function. Possible solutions of states that minimize the functional was achieved by finite element solution procedure while the minimum values of the stationary states were solved by solving the functional. The functional obtained for each finite element is minimized with respect to associated degrees of freedom of the element and assembly method applied to all elements minimization equation to obtain system of equations equal to unconstrained nodes in the region .The element equations are assembled and solved by substitution to obtain the values of the function at discrete points. The values of the function at the discrete points did not vary significantly with boundary points values. The minimum value of the function representing the critical or the functional of the function is evaluated as 24MPa.
problems whose function can be expressed as derivatives and integrated functional or on solution of quasi-harmonic functions whose physical behaviors are governed by a general quasi-harmonic differential equation that can be treated as a quadratic functional that can be minimized over a region. The functional of a stress field function was established using mixed methods analogous to variational principle, minimum total potential principle and finite element method. The functional of function, Φ(x,y) was formed using Euler equivalent integral and finite element shape function for a function expressed in derivative form. The minimization of the functional gave the stationary values of the function which minimize the functional. The solution of the functional gave the minimum value of the function. Possible solutions of states that minimize the functional was achieved by finite element solution procedure while the minimum values of the stationary states were solved by solving the functional. The functional obtained for each finite element is minimized with respect to associated degrees of freedom of the element and assembly method applied to all elements minimization equation to obtain system of equations equal to unconstrained nodes in the region .The element equations are assembled and solved by substitution to obtain the values of the function at discrete points. The values of the function at the discrete points did not vary significantly with boundary points values. The minimum value of the function representing the critical or the functional of the function is evaluated as 24MPa.
