Donea & Huerta 2D box geometry benchmark#

This section was contributed by Cedric Thieulot.

This benchmark is taken from Donea and Huerta’s book (Donea and Huerta 2003). The domain is a unit square and the viscosity and density are set to 1. The components of the gravity vector \(\mathbf g\) are prescribed as $\(\begin{aligned} g_x &=& (12 - 24y) x^4 + (-24 + 48y) x^3 + (-48y + 72y^2 - 48 y^3 + 12) x^2 \nonumber\\ && + (-2 + 24y -72y^2+48y^3)x + 1-4y + 12y^2-8y^3 \nonumber\\ g_y &=& (8 - 48y + 48 y^2) x^3 + (-12 + 72y - 72y^2) x^2 \nonumber\\ && + (4 - 24y + 48y^2 - 48y^3 + 24y^4) x - 12y^2 + 24y^3 - 12y^4.\end{aligned}\)\( The exact solution can then be chosen as follows, if one prescribes Dirichlet boundary values for the velocity using the same formula: \)\(\begin{aligned} u(x,y) &=& x^2(1- x)^2 (2y - 6y^2 + 4y^3) \nonumber\\ v(x,y) &=& -y^2 (1 - y)^2 (2x - 6x^2 + 4x^3) \nonumber\\ p(x,y) &=& x(1 -x) -1/6.\end{aligned}\)\( Note that the pressure satisfies \)\int_{\Omega} p ; \text{d}x = 0$. The gravity, pressure and velocity fields are shown in Fig. [fig:doneahuerta-benchmark].

The convergence of the numerical error of this benchmark has been analyzed by changing the mesh refinement level in the input file, and results show that the velocity shows cubic error convergence, while the pressure shows quadratic convergence in the \(L_2\) norm, as expected when using the \(Q_2\times Q_1\) element.

   

Donea, J., and A. Huerta. 2003. Finite Element Methods for Flow Problems. John Wiley & Sons, Ltd.