Two-dimensional Mesh Generator in Generalized Coordinates Implemented in Python

The Cartesian system $(x,y)$ is called the physical domain and the generalized system $(\xi,\eta)$ is called the transformed domain or computational domain.

The transformation from a non-trivial geometry described in a Cartesian coordinate system to a generalized coordinate system involves transformation metrics, or mathematical relations, that can accurately describe the transformed data.

The mapping of irregular or regular geometries written in Cartesian coordinates $(x,y)$, Fig. 1a, is performed numerically for regular geometries written in generalized coordinate system $(\xi,\eta)$, Fig. 1b.

Since the transformed domain is regular, for convenience, unitary normalization of elementary volumes is assumed, that is, $\Delta\xi=\Delta\eta=1$. In this way, even if in the physical plane the coordinated lines assume arbitrary spacing, in the computational plane the dimensions are fixed to the unit.

To obtain the $(\xi,\eta)$ system, the following generating equations are used

The transformation metrics, based on differentials of the Eqs. (1) and (2), are

Starting from the assumption that it is possible to find a representation in a Cartesian coordinate system for a model described in generalized coordinates, then we admit the existence of the inverse of the equations Eqs. (1) and (2), so

Replacing Eq. (6) in Eq. (10), we get $d^{f}=BAd^{f}$, so that $BA=I$, or equivalent $A=B^{-1}$. Thus, the $A$ matrix becomes

In this work, we opted for the use of elliptical partial differential equations (EPDE) as a method of generating two-dimensional meshes, since their solutions do not generate null Jacobian, and the lines $\xi$ or $\eta$ never intersect (THOMPSON, 1985; MALISKA, 2004). Thus, the governing equations for the generation of two-dimensional meshes in a domain, for example the domain illustrated in Fig. 2a, are

Solving the equations (12) and (13), in relation to the $(x,y)$ coordinates, using the transformation metrics of the $(\xi,\eta)$ system, then the coordinated lines can be generated in the directions $\xi$ and $\eta$, inside the computational mesh, Fig. 2b, as in any other geometry, through the equations

The numerical solution of elliptical EDPs (14) and (15), subject to initial and boundary conditions, provide the lines $\xi$ and $\eta$, which generate the computational mesh (DE BORTOLI, 2000, MALISKA, 2004; CIRILO et al., 2006; FORTUNA, 2012). For convenience, Eqs. (14) and (15) can be written using a generic $\phi$ variable as

To approximate the derivatives in Eq. (19), the finite difference method is used. The mesh nodes are labeled by the cardinal points $P$ (center), $E$ (east), $W$ (west), $N$ (north), $S$ (south), $NW$ (northwest ), $SW$ (southwest), $SE$ (southeast) and $NE$ (northeast), as in Fig. 3.

Approximating the derivative terms of Eq. ( Ref eq27) by means of central differences, around $P$, we obtain

Thus, the numerical solution of the two-dimensional mesh generation equations, Eqs. (14) and (15), written according to the generic $\phi$ variable, Eq. (21), is given by

Next, the mesh generator in generalized coordinates is presented.

For the definition and creation of a geometry mapped in generalized coordinates, it starts from a initial set of points that describe the border, that is, the physical contour. In this work, the Parametric Linear Spline method was chosen to interpolate the set of border points under study. Parametric Linear Spline equations (BURDEN et al., 2015) are

$\forall t\in[t_{i-1},t_{i}]$, where $s$ interpolating spline curve and $t$ is the interpolated variable.

Having obtained the interpolating lines of the border, Eqs. (29) and (30), the number of partitions desired for each border must be defined.

Then, points of the Splines curves are selected, which define the new border of the approximate geometry. To do this, an approach is used to position the points by weighted average of the components of the border, ie

Finally, on the set of ordered pairs, the resolution of the equation (22) is applied.

In summary, the implemented algorithm can be illustrated in this way:

The next two sections are intended to exemplify and test the algorithm. In the first applications, monoblock meshes are built (3 applications). Following a multiblock mesh is obtained. The multiblock technique consists of dividing the physical domain into parts and, on each of the parts, applying the generation of meshes in generalized coordinates. Such a procedure allows to better capture the complexity of the geometry compared to the strategy employed on a single block (CIRILO, 2006; PARDO et al., 2012, SAITA et al., 2017).

As a first application of the mesh generation algorithm, the set of input data presented in Table 1 was used. The weighted values for $\Gamma_{j}$, $j=$ 1 and 3 are omitted, since they are known for complementarity, that is, $p^{\Gamma_{1}\Gamma_{3}}_{x}+p^{\Gamma_{2}\Gamma_{4}}_{x}=1$ e $p^{\Gamma_{1}\Gamma_{3}}_{y}+p^{\Gamma_{2}\Gamma_{4}}_{y}=1$.

Given the set of predefined points, Fig. 4a, the number of partitions and the weights at the borders, Table 1, then the border geometry is obtained by Parametric Spline Linear interpolation. In sequence, the algorithm calculates the new border points, using the given refinement. In this way, the algorithm starts to solve the transformation metrics, finally generating the mesh in generalized coordinates. Figs. 4b and 4c present the meshes in generalized coordinates, according to the refinements described in Table 1.

As a second application, a mesh in a doubly-connected domain was generated. The set of input data is shown in Table 2.

In this application, the left/right border corresponds to the line segment that joins the two circles, while the upper/lower border corresponds to the two circles, as shown in Fig 5a. From the set of points given in Fig. 5a and the input data in Table 2, the algorithm constructs the meshes shown in Figs. 5b and 5c, according to the refinements described in Table 2.

The latest generation of monoblock mesh is shaped like a bottle, as illustrated in Fig. 6. The set of input data is explained in Table 3.

Using the set of predefined points, Fig. 6a, and the input data in Table 3, the meshes in generalized coordinates are generated, as shown in Fig. 6b and Fig. 6c, according to the refinements described in Table 3.

automated mesh generator In the applications presented for monoblock meshes, the automated mesh generator precisely built the geometry in the physical domain.

In order to show a more complex application, points from the contour of Lake Igapó I, located in Londrina, Paraná, Brazil, were collected. The points were obtained from satellite images offered by the Google Earth software (GOOGLE, 2020). The generated mesh has the same scale as the physical domain.

Unlike previous applications, in addition to generating the monoblock meshes (now called monobloc sub-meshes), in this application the sub-meshes must be connected. The multiblock mesh algorithm uses the monoblock mesh generator as a subroutine of the multiblock algorithm.

In summary, the multiblock algorithm can be illustrated in this way:

For the geometry of Lago Igapo, it is considered that it consists of three blocks, Lake Igapó I (main block) and its two tributaries (two other blocks). The set of predefined points are given in Fig. 7a. The input data are presented in Table 4. As for the weighted average, the percentage weights ($p^{\Gamma_{1}\Gamma_{3}}$ e $p^{\Gamma_{2}\Gamma_{4}}$), in $x$ and $y$, were all chosen as $0.5$.

After executing the multiblock mesh algorithm, the meshes were obtained in generalized coordinates as shown in Figs. 7b and 7c, according to the refinements of Table 4.

Again, the automated mesh generator accurately represented the physical domain of Igapó I lake, located in Londrina, Paraná, Brazil.