Even when the simulation is performed multiple times, same results prevail. Why is this so?
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This is governed by the principle of minimization of energy. It states that when a boundary condition like displacement or force is applied, of the numerous possible configurations that the body can take, only that configuration where the total energy is minimum is the one that is chosen.
However, the earliest mathematical papers on FEM can be found in the works of Schellback  and Courant . FEM was independently developed by engineers to address structural mechanics problems related to aerospace and civil engineering. An interesting review of these historical developments can be found in Oden . Finite element method is in itself a semester course. In this article, a concise description of the mechanism of FEM is described.
Consider a simple 1-D problem to depict the various stages involved in FEA. The PDE or differential form is known as the strong form and the integral form is known as the weak form.
Consider the simple PDE as shown below. The equation is multiplied by a trial function v x on both sides and integrated with the domain [0,1]. Now, using integration of parts, the LHS of the above equation can be reduced to. As it can be seen, the order of continuity required for the unknown function u x is reduced by one. The earlier differential equation required u x to be differentiable at least twice while the integral equation requires it to be differentiable only once.
The same is true for multi-dimensional functions, but the derivatives are replaced by gradients and divergence. Without going into the mathematics, the Riesz representation theorem can prove that there is a unique solution for u x for the integral and hence the differential form. In addition, if f x is smooth, it also ensures that u x is smooth. Once the integral or weak form has been set up, the next step is the discretization of the weak form.
The integral form needs to be solved numerically and hence the integration is converted to a summation that can be calculated numerically. In addition, one of the primary goals of discretization is also to convert the integral form to a set of matrix equations that can be solved using well-known theories of matrix algebra.
As shown in Fig. The unknown functional u x are calculated at the nodal points. Interpolation functions are defined for each element to interpolate, for values inside the element, using nodal values. These interpolation functions are also often referred to as shape or ansatz functions.www.balterrainternacional.com/wp-content/2020-01-15/786.php
NAFEMS - International Association Engineering Modelling
Thus the unknown functional u x can be reduced to. Similarly, interpolation can be used for the other functions v x and f x present in the weak form, so that the weak form can be rewritten as. The summation schemes can be transformed into matrix products and can be rewritten as. Note above that the earlier trial function v x that had been multiplied does not exist anymore in the resulting matrix equation.
Further on, using numerical integration schemes, like Gauss or Newton-Cotes quadrature, the integrations in the weak form that forms the tangent stiffness and residual vector are also handled easily. Once the matrix equations have been established, the equations are passed on to a solver to solve the system of equations. Depending on the type of problem, direct or iterative solvers are generally used.
As discussed earlier, traditional FEM technology has demonstrated shortcomings in modeling problems related to fluid mechanics and wave propagation. Several improvements have been made recently to improve the solution process and extend the applicability of finite element analysis to a wide range of problems. Some of the important ones still being used include:. Bubnov-Galerkin method requires continuity of displacement across elements. Although problems like contact, fracture, and damage involve discontinuities and jumps that cannot be directly handled by the finite element method.
XFEM works through the expansion of the shape functions with Heaviside step functions. Extra degrees of freedom are assigned to the nodes around the point of discontinuity so that the jumps can be considered. It combines the features of the traditional FEM and meshless methods. One of the advantages of GFEM is the prevention of re-meshing around singularities. In several problems, like contact or incompressibility, constraints are imposed using Lagrange multipliers.
These extra degrees of freedom arising from Lagrange multipliers are solved independently. The system of equations is solved like a coupled system of equations. This is not the same as doing h- and p- refinements separately. When automatic hp-refinement is used, and an element is divided into smaller elements h-refinement , each element can have different polynomial orders as well. DG-FEM has shown significant promise for utilizing the idea of finite elements to solve hyperbolic equations, where traditional finite element methods have been weak.
In addition, it has also shown improvements in bending and incompressible problems which are typically observed in most material processes. Here, additional constraints are added to the weak form that includes a penalty parameter to prevent interpenetration and terms for other equilibrium of stresses between the elements.
We hope this article has covered the answers to your most important questions regarding what is the finite element method. To discover all the features provided by the SimScale cloud-based simulation platform, download this overview or watch the recording of one of our webinars.
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