Table 8 coefficients used to capture the solution with the finite element approximation for the quadratic patch test
Sukumar [14] | Moës [27] | Belytchko [28] | |
|---|---|---|---|
Linear \(\phi (x)\) and linear F(x) | The analytical solution cannot be represented. Incompatibility for the central element. | ||
Quadratic \(\phi (x)\) and quadratic F(x) | The analytical solution cannot be represented. Incompatible with blending elements. | No solution. Incompatibility for the central element. | The analytical solution cannot be represented. Incompatible with blending elements. |
Quadratic \(\phi (x)\) and linear F(x) | The analytical solution cannot be represented. Incompatible with blending elements. | \(\left\{ \begin{aligned} a_1&= 0\\ a_2&= \frac{f}{4E_1}\left( \frac{5}{2}x_0 L - x_0^2\right) + \\&\frac{f}{4E_2}\left( \frac{3}{2}L^2 - \frac{5}{2}x_0 L + x_0^2\right) \\ a_3&= \frac{f}{2E_1}x_0(2L - x_0) + \frac{f}{2E_2}(x_0 - L)^2\\ b_1&= \frac{f}{4}\left( \frac{1}{E_1} - \frac{1}{E_2}\right) (2L - x_0)\\ b_2&= \frac{f}{8}\left( \frac{1}{E_1} - \frac{1}{E_2}\right) (3L - 2x_0)\\ b_3&= \frac{f}{4}\left( \frac{1}{E_1} - \frac{1}{E_2}\right) (L - x_0) \end{aligned}\right. \) | The analytical solution cannot be represented. Incompatible with blending elements. |