Definitions

A shape function is most of the time a polynomial that is used to interpolate between the computed values at nodes in an element in order to find the value of any point in this element.

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We need as many shape function as there is nodes making up the element

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For example given a shape function $N_i$ and values $v_i$ at every given point, we can state that the value at a point $x$ is:

$$ v(x) = \sum_{i=0}^n N_i(x)v_i $$

Properties

• At any node, the associated shape function is 1
• Summation: $\forall x \in \Omega, \sum_{i=0}^n N_i(x) = 1$
• Continuity: Within the element boundary, the polynomial approximation for u is C∞ continuous. The continuity across the element boundaries depends on the polynomial approximation for the variable. For example, the variable has only C0 continuity across the linear and quadratic element boundaries (i.e., its first derivative is piecewise continuous).
• Completeness: This is an important requirement for shape functions to ensure the convergence of finite element approximate solution to the exact solution of governing equation.