数学代写|有限元方法代写Finite Element Method代考|AMCS329

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数学代写|有限元方法代写Finite Element Method代考|Approximation of Geometry

We wish to use the Gauss quadrature to numerically evaluate all integrals in finite element method. The Gauss quadrature requires us to express the integral in terms of $\xi$ over the interval -1 to +1 . We assume a relation (or transformation) between the problem coordinate $x$ and natural coordinate $\xi$ in the form
$$x=f(\xi)$$
where $f$ is assumed to be a one-to-one transformation. An example of $f(\xi)$ is provided by Eq. (8.2.7), where $n=2$ :
$$f(\xi)=x_a^e+\frac{1}{2} h_e(1+\xi)$$
In this case, $f(\xi)$ is a linear function of $\xi$. Hence, a straight line is transformed into a straight line.

It is natural to think of approximating the geometry in the same way as we approximated a dependent variable. In other words, the transformation $x=$ $f(\xi)$ can be written as
$$x=\sum_{i=1}^m x_i^e \hat{\psi}i^e(\xi)$$ where $x_i^e$ is the global coordinate of the $i$ th node of the element $\Omega_e$ and $\hat{\psi}_i^e$ are the Lagrange interpolation functions of degree $m-1$. When $m=2$ we have a linear transformation and Eq. (8.2.11) is exactly the same as Eq. (8.2.7). When $m=3$, Eq. (8.2.11) expresses a quadratic relation between $x$ and $\xi$. The functions $\hat{\psi}_i^e$ are called shape functions because they are used to express the geometry or shape of the element. When the element is a straight line, the mapping is linear because the two end points, $x_1^e$ and $x_n^e$ are sufficient to define a line. The transformation in Eq. (8.2.11) allows us to rewrite integrals involving $x$ as those in terms of $\xi$ : $$\int{x_a^e}^{x_b^e} F(x) d x=\int_{-1}^1 \hat{F}(\xi) d \xi, \quad \hat{F}(\xi) d \xi=F(x(\xi)) d x$$
so that the Gauss quadrature can be used to evaluate the integral over $[-1,1]$. The differential element $d x$ in the global coordinate system $x$ is related to the differential element $d \xi$ in the natural coordinate system $\xi$ by
$$d x=\frac{d x}{d \xi} d \xi=J_e d \xi$$
where $J_e$ is called the Jacobian of the transformation. We have
$$J_e=\frac{d x}{d \xi}=\frac{d}{d \xi}\left(\sum_{i=1}^m x_i^e \hat{\psi}i^e\right)=\sum{i=1}^m x_i^e \frac{d \hat{\psi}_i^e}{d \xi}$$

数学代写|有限元方法代写Finite Element Method代考|Parametric Formulations

Recall that a dependent variable $u$ is approximated in an element $\Omega_e$ by expressions of the form
$$u(x)=\sum_{j=1}^n u_j^e \psi_j^e(x)$$
In general, the degree of approximation used to describe the coordinate transformation in Eq. (8.2.11) is not equal to the degree of approximation in Eq. (8.2.16) used to represent a dependable variable, $\hat{\psi}_i^e \neq \psi_i^e$. In other words, two independent meshes of elements may be used in the finite element formulation of a problem: one for the approximation of the geometry $x$ and the other for the interpolation of the dependent variable $u$. Depending on the relationship between the degree of approximation used for the coordinate transformation and that used for the dependent variable, the finite element formulations are classified into three categories:

Subparametric formulations: $m<n$

Isoparametric formulations: $m=n$

Superparametric formulations: $m>n$
In subparametric formulations, the geometry is represented by lower-order elements than those used to approximate the dependent variable. An example of this category is provided by the Euler-Bernoulli beam element, where the Hermite cubic functions are used to approximate the deflection $w(x)$ and linear interpolation can be used, when straight beams are analyzed, to represent the geometry. In isoparametric formulations (which are the most common in practice), the same element is used to approximate the geometry as well as the dependent unknowns: $\psi_i^e(x)=\hat{\psi}_i^e(\xi)$. In the superparametric formulations, the geometry is represented with higher-order elements than those used to approximate the dependent variables. The superparametric formulation is seldom used in practice. It is not correct to say “isoparametric element” because an element is what it is (i.e., linear, quadratic, and so on).

有限元方法代考

数学代写|有限元方法代写Finite Element Method代考|Approximation of Geometry

$$x=f(\xi)$$

$$f(\xi)=x_a^e+\frac{1}{2} h_e(1+\xi)$$

有限元方法代写

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