Parallel anisotropic 3D mesh adaptation by mesh modification

Abstract

Improvements to a local modification-based anisotropic mesh adaptation procedure are presented. The first improvement focuses on control of the local operations that modify the mesh to satisfy the given anisotropic mesh metric field. The second is the parallelization of the mesh modification procedures to support effective parallel adaptive analysis. The resulting procedures are demonstrated on general curved 3D domains where the anisotropic mesh size field is defined by either an analytic expression or by an adaptive correction indicator as part of a flow solution process.

Appendix: Analytical size field definition

In this section, the four analytical size fields used in Sect. 4 are described.

A planar shock

We consider the following analytical size

$$ h_{1} = h_{{\rm max}} |1 - \hbox{e}^{-|x-0.5|}| + 0.003$$

with h _max=0.2 and the analytical metric is given by:

$${\mathcal{M}} = {\mathcal{R}} \Lambda {\mathcal{R}}^{-1},\quad\hbox{with}\; \Lambda = \left(\begin{array}{*{20}c} h_{1}^{-2} & 0 & 0 \\ 0 & h_{{\rm max}}^{-2}&0 \\ 0 & 0 & h_{{\rm max}}^{-2}\\ \end{array} \right)\quad\hbox{and}\; {\mathcal{R}} = \left(\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right). $$

A cylindrical shock

The analytical size field representing a cylindrical shock of radius R=1 and axis z=0 is defined by:

$$h_{1} = h_{{\rm max}} |1 - \hbox{e}^{-2 |x^{2} + y^{2} - R^{2}|}| + 0.0015$$

with h _max=0.14 and the analytical metric is given by:

$${\mathcal{M}} = {\mathcal{R}}\Lambda {\mathcal{R}}^{-1},\quad\hbox{with}\;\Lambda = \left(\begin{array}{*{20}c} h_{1}^{-2} & 0 & 0 \\ 0 & h_{max}^{-2}&0 \\ 0 & 0 & h_{max}^{-2}\\ \end{array}\right)\quad\hbox{and}\; {\mathcal{R}} = \left(\begin{array}{*{20}c} x/r & -y/r & 0 \\ y/r & x/r & 0 \\ 0 & 0 & 1\\ \end{array} \right) .$$

where $r=\sqrt{x^{2} + y^{2}}.$

Two spherical shocks

We consider an analytical size field representing two spherical shocks of radius R=1 and centers (0, 0, 0) and (1, 0, 0), defined by:

$$\begin{aligned} h_{1} =& h_{{\rm max}} |1 - \hbox{e}^{-3|x^{2} + y^{2} + z^{2} - R^{2}|}| + 0.0015,\\ h_{2} =& h_{{\rm max}} |1 - \hbox{e}^{-3|(x-1)^{2} + y^{2} + z^{2} - R^{2}|}| + 0.0015\\ \end{aligned}$$

with h _max=0.14 and the analytical metric is given by:

$${\mathcal{M}} = {\mathcal{R}} \Lambda {\mathcal{R}}^{-1},\quad \hbox{with}\; \Lambda = \left(\begin{array}{*{20}c} h_{1}^{-2} & 0 & 0 \\ 0 & h_{2}^{-2} & 0 \\ 0 & 0 & h_{max}^{-2}\\ \end{array} \right)\quad\hbox{and}\; {\mathcal{R}} = \left(\begin{array}{*{20}c} \vdots & \vdots & \vdots\\ v_{1} & v_{2} &v_{3}\\ \vdots & \vdots & \vdots\\ \end{array}\right), $$

where v ₁=(x/r ₁, y/r ₁, z/r ₁) with $r_{1} = \sqrt{x^{2} + y^{2} + z^{2}},$ v ₂ =((x−1)/r ₂, y/r ₂, z/r ₂) with $r_{2} = \sqrt{(x-1)^{2} + y^{2} + z^{2}}$ and v ₃=v ₁ ∧ v ₂.

In parallel, we define the same metric field with the following size:

$$\begin{aligned} h_{1} =& h_{{\rm max}} |1 - \hbox{e}^{-3|x^{2} + y^{2} + z^{2} - R^{2}|}| + 0.004 ,\\ h_{2} =& h_{{\rm max}} |1 - \hbox{e}^{-3|(x-1)^{2} + y^{2} + z^{2} - R^{2}|}| + 0.001\\ \end{aligned}$$

with h _max=0.14.