Finite element method and its procedure
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Basic Steps of the Finite Element Method
As stated in the introduction, the finite element method is a numerical procedure for obtaining solutions to boundary-value problems. The principle of the method is to replace an entire continuous domain by a number of subdomains in which the unknown function is represented by simple interpolation functions with unknown coefficients. Thus, the original boundary-value problem with an infinit number of degrees of freedom is converted into a problem with a finite number of degrees of freedom, or in other words, the solution of the whole system is approximated by a finite number of unknown coefficients. Therefore, a finite element analysis of a boundary-value problem should include the following basic steps:Discretization or subdivision of the domainSelection of the interpolation functions (to provide an approximation of the unknown solution within an element)Formulation of the system of equations ( also the major step in FEM. The typical Ritz variational and Galerkin methods can be used.)Solution of the system of equations (Once we have solved the system of equations, we can then compute the desired parameters and display the result in form of curves, plots, or color pictures, which are more meaningful and interpretable.)The first step and the last step are most relevant with the visualization process. These 2 steps are described in greater detail below.
Domain Discretization
As stated in the introduction, the finite element method is a numerical procedure for obtaining solutions to boundary-value problems. The principle of the method is to replace an entire continuous domain by a number of subdomains in which the unknown function is represented by simple interpolation functions with unknown coefficients. Thus, the original boundary-value problem with an infinit number of degrees of freedom is converted into a problem with a finite number of degrees of freedom, or in other words, the solution of the whole system is approximated by a finite number of unknown coefficients. Therefore, a finite element analysis of a boundary-value problem should include the following basic steps:Discretization or subdivision of the domainSelection of the interpolation functions (to provide an approximation of the unknown solution within an element)Formulation of the system of equations ( also the major step in FEM. The typical Ritz variational and Galerkin methods can be used.)Solution of the system of equations (Once we have solved the system of equations, we can then compute the desired parameters and display the result in form of curves, plots, or color pictures, which are more meaningful and interpretable.)The first step and the last step are most relevant with the visualization process. These 2 steps are described in greater detail below.
Domain Discretization
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