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# 数学代写|有限元方法代写finite differences method代考|MS-E1653 Properties of the FEM

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## 数学代写|有限元代写Finite Element Method代考|Properties of the FEM

Using the FEM, one can usually expect only an approximated solution. In Example 4.1, however, we obtained the exact solution. Why? This is because the exact solution of the deformation for the bar is a first order polynomial (see Eq. (4.46)). The shape functions used in our FEM analysis are also first order polynomials that are constructed using complete monomials up to the first order. Therefore, the exact solution of the problem is included in the set of assumed displacements in FEM shape functions. In Chapter 3, we understand that the FEM based on Hamilton’s principle guarantees to choose the best possible solution that can be produced by the shape functions. In Example 4.1, the best possible solution that can be produced by the shape function is the exact solution, due to the reproduction property of the shape functions, and the FEM has indeed reproduced it exactly. We therefore confirmed the reproduction property of the FEM that if the exact solution can be formed by the basis functions used to construct the FEM shape function, the FEM will always produce the exact solution, provided there is no numerical error involved in computation of the FEM solution.
Making use of this property, one may try to add in basis functions that form the exact solution or part of the exact solution, if that is possible, so as to achieve better accuracy in the FEM solution.

## 数学代写|有限元代写Finite Element Method代考|Convergence property of the FEM

For complex problems, the solution cannot be written in the form of a combination of monomials. Therefore, the FEM using polynomial shape functions will not produce the exact solution for such a problem. The question now is, how can one ensure that the FEM can produce a good approximation of the solution of a complex problem? The insurance is given by the convergence property of the FEM, which states that the FEM solution will converge to the exact solution that is continuous at arbitrary accuracy when the element size becomes infinitely small, and as long as the complete linear polynomial basis is included in the basis to form the FEM shape functions. The theoretical background for this convergence feature of the FEM is due to the fact that any continuous function can always be approximated by a first order polynomial with a second order of refinement error. This fact can be revealed by using the local Taylor expansion, based on which a continuous (displacement) function $u(x)$ can always be approximated using the following equation:
$$u=u_i+\left.\frac{\partial u}{\partial x}\right|_i\left(x-x_i\right)+O\left(h^2\right)$$
where $h$ is the characteristic size that relates to $\left(x-x_i\right)$, or the size of the element.

## 数学代写|有限元代写Finite Element Method代考|Convergence property of the FEM

$$u=u_i+\left.\frac{\partial u}{\partial x}\right|_i\left(x-x_i\right)+O\left(h^2\right)$$

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