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rohithsrinivaas / CFD-Body-Fitted-Grid-Generation-for-Elliptic-domain

This repo contains the code to generate body fitted grid for elliptic domain

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Body-Fitted-Grid-Generation-for-Elliptic-domain

Introduction

This is a 2D elliptic mesh (grid) generator which works by solving the partial differential equations. This algorithm works by calculating curve slopes using a tilted parabola tangent line fitter. The mesh generator is packaged as a Java program which can be compiled and executed via Eclipse Java IDE.The program allows one to the following parameters for the fluid domain: major axis length, minor axis length, x-resolution, y-resolution, center of the puncture circle and the radius of the puncture circle.

Mathematical Framework

The algorithms implemented in this project are mainly founded upon the principles of differential geometry and tensor calculus. The most fundamental concept behind the mathematics governing this project is the transition between coordinate systems in order to obtain the solution to partial differential equations in the most efficient manner possible. More specifically, in the context of this project, this implies transforming a set of equations written in Cartesian coordinates to curvilinear coordinates. This concept can be extended to any number of spatial dimensions, which will later be shown. However, a two-dimensional solution was developed in order to demonstrate the feasibility of generating smooth elliptic grids with the prescribed specifications.\

Consider a system of n dimensions which can be represented by the set of Cartesian coordinates

where each of the partial derivatives comes from the definition of the covariant base vectors. Each of these base vectors describe how one coordinate system changes with respect to another, when any particular coordinate is held fixed. For our two-dimensional problem, we can expand these sums to yield the following expressions for the metric tensors.\

Elliptic Mesh Generation Algorithm

To construct an initial mesh, the Transfinite Interpolation algorithm is applied to the given domain constrained by the specified boundary conditions. This algorithm is implemented by mapping each point within the domain (regardless of the boundaries) to a new domain existing within the boundaries. This algorithm works by iteratively solving the parametric vector equation.

where and represent parameters in the original domain and $\vec{l}$, $\vec{t}$, $\vec{b}$ and $\vec{r}$ represent the curves defining the left, top, bottom and right boundaries. $P_{ij}$ represents the point of intersection between curve and $\vec{i}$, $\vec{j}$\

At the heart of the solver is the mesh smoothing algorithm, which at a high level, works by solving the pair of Laplace equations\

where $\xi$ and $\eta$ represent the x and y coordinates of every point in the target domain, mapped to a transformed, computational space using the change of variables method. This renders the calculations simpler and faster to compute. However, we wish to solve the inverse problem, where we transition from the computational space to the curvilinear solution space. Using tensor mathematics, it can be shown that this problem entails solving the equations.\

where $g_{ij}$ is the covariant metric tensor at entry (i,j) within the matrix of covariant tensor components defining the mapping of the computational space coordinates $(\xi,\eta)$ onto the physical solution space coordinates (x,y). In this model, x and y are computed as functions of $\xi$ and $\eta$ .

This set of equations are the elliptic PDEs known as the Winslow equations. These are applied to the mesh using the method of mixed-order finite differences on the partial derivatives (and tensor coefficients, as they are a function of these derivatives), thereby resulting in the equations (for a single node):

where i and j are the coordinates of a node in the mesh in computational space. Here $\Delta \xi$and $\Delta \eta$ are equal increments in $\xi$ and $\eta$ respectively.

The coefficients for these equations can be generated for each point to form a system of linear equations, which is then modeled in matrix representation, resulting in a tri-diagonal matrix. This matrix is then solved iteratively using the Thomas Tri-Diagonal Matrix Algorithm line-by-line by traversing from the bottom up.

Mesh Quality Analysis

In order to determine the quality of the resulting mesh, it was necessary to construct an objective means of quality measurement. Therefore, several statistical procedures were implemented in the program to produce a meaningful mesh quality analysis report. The metrics which are presented are divided into the following categories:

  • Orthogonality Metrics

    • Standard deviation of angles

    • Mean angle

    • Maximum deviation from 90 degrees

    • Percentage of angles within x degrees from 90 degrees (x can be set as a constant in the code)

  • Cell Quality Metrics

    • Average aspect ratio of all cells

    • Standard deviation of all aspect ratios

Here, “angle” refers to the angle of intersection of grid lines at a node and “aspect ratio” refers to the skewness of a grid cell measured as a ratio of the cell’s longest side to its shortest side.

Code structure

The following shows the code structure in the Eclipse Java IDE.The program is written in a modular way, allowing it to be altered for new boundary cases.

  • EllipticMeshGenerator2D.java

    • This is the main program.

  • MeshHelper.java

    • Class containing helper methods for setting up the visualized meshes at different stages throughout the generation process

  • MeshSolver.java

    • Class containing methods which perform the main algorithmic computations throughout the mesh generation process

  • MeshStatistics.java

    • Class for analyzing the quality of the generated meshes and reporting key statistics

  • TiltedParabolaFitter.java

    • Class which curve fits a set of three points with a parabola at an angle theta such that the middle point

  • GifSequenceWriter.java

    • Creates a new GifSequenceWriter

Mathematical libraries like JMathPlot(as org.jar here) as Referenced libraries for plotting and for numerical computation.

Results and Discussions

Here as the given domain was horizontally symmetric, only the top half of the domain is concerned for solving the problem. Since only three smooth boundaries are present, a split point has to be chosen, so as to create the four boundaries. This point is generally at $x = a/2$ where a is the semi-major axis length.

Mesh Quality report for 4 X 4 resolution

  • Initial (Transfinite interpolation)

    • Boundary Angles

      • The standard deviation of all boundary angles is: 2.412145192252833 deg

      • The mean boundary angle is: 94.69510733620636 deg

      • The maximum deviation of all angles from 90 deg on the boundary is: 56.867308795456815 deg

      • The corresponding angle is: 33.132691204543185 deg and is located at position: [0, 8]

      • The percent of angles within 10 deg of 90 deg on the boundary is: 56.25 %

    • Interior angles

      • The standard deviation of all interior angles is: 1.0413714606748834 deg

      • The mean interior angle is: 92.12785308218719 deg

      • The maximum deviation of all angles from 90 deg on the interior is: 55.89652205783994 deg

      • The corresponding angle is: 34.10347794216006 deg and is located at position: [1, 8]

      • The percent of angles within 10 deg of 90 deg on the interior is: 48.4375 %

    • Aspect ratio

      • The average aspect ratio of all cells is: 1.4045298830932862

      • The standard deviation of all aspect ratios is: 0.39380432798738363

  • Final( Elliptic PDE solution )

    Completed in 54 iterations

    • Boundary Angles

      • The standard deviation of all boundary angles is: 2.578732547131888 deg

      • The mean boundary angle is: 93.07141464775451 deg

      • The maximum deviation of all angles from 90 deg on the boundary is: 53.15052220555345 deg

      • The corresponding angle is: 36.84947779444655 deg and is located at position: [0, 8] The percent of angles within 10 deg of 90 deg on the boundary is: 56.25 %

    • Interior Angles

      • The standard deviation of all interior angles is: 0.8789846429198418 deg

      • The mean interior angle is: 95.4981186288393 deg

      • The maximum deviation of all angles from 90 deg on the interior is: 36.63218456011564 deg

      • The corresponding angle is: 53.36781543988436 deg and is located at position: [1, 8] The percent of angles within 10 deg of 90 deg on the interior is: 55.729166666666664 %

    • Aspect ratios

      • The average aspect ratio of all cells is: 1.423311852960875

      • The standard deviation of all aspect ratios is: 0.3527840284371575

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Mesh Quality report for 2 X 2 resolution

  • Initial (Transfinite interpolation)

    • Boundary Angles

      • The standard deviation of all boundary angles is: 1.6913003493993906 deg

      • The mean boundary angle is: 95.47273957497313 deg

      • The maximum deviation of all angles from 90 deg on the boundary is: 63.2461081175267 deg

      • The corresponding angle is: 26.753891882473297 deg and is located at position: [0,16]

      • The percent of angles within 10 deg of 90 deg on the boundary is: 57.6530612244898 %

    • Interior angles

      • The standard deviation of all interior angles is: 0.5182833312075722 deg

      • The mean interior angle is: 93.08632549460636 deg

      • The maximum deviation of all angles from 90 deg on the interior is: 62.829524417793834 deg

      • The corresponding angle is: 27.17047558220617 deg and is located at position: [1,16]

      • The percent of angles within 10 deg of 90 deg on the interior is: 51.770095793419415 %

    • Aspect ratio

      • The average aspect ratio of all cells is: 1.3872556063719226

      • The standard deviation of all aspect ratios is: 0.5090135522422773

  • Final( Elliptic PDE solution )

    Completed in 54 iterations

    • Boundary Angles

      • The standard deviation of all boundary angles is: 2.578732547131888 deg

      • The mean boundary angle is: 93.07141464775451 deg

      • The maximum deviation of all angles from 90 deg on the boundary is: 53.15052220555345 deg

      • The corresponding angle is: 36.84947779444655 deg and is located at position: [0, 8] The percent of angles within 10 deg of 90 deg on the boundary is: 56.25 %

    • Interior Angles

      • The standard deviation of all interior angles is: 0.8789846429198418 deg

      • The mean interior angle is: 95.4981186288393 deg

      • The maximum deviation of all angles from 90 deg on the interior is: 36.63218456011564 deg

      • The corresponding angle is: 53.36781543988436 deg and is located at position: [1, 8] The percent of angles within 10 deg of 90 deg on the interior is: 55.729166666666664 %

    • Aspect ratios

      • The average aspect ratio of all cells is: 1.423311852960875

      • The standard deviation of all aspect ratios is: 0.3527840284371575

Conclusion

The overall code takes less than 60 to 90 seconds to complete for a grid size of 100 X 100. The code is available in https://drive.google.com/drive/folders/1U3yX2MTPRi4IPsgqF_tQLkUXxwFiET2J?usp=sharing

Acknowledgements

During the course of this project , https://github.com/Cvarier/2D-Elliptic-Mesh-Generator was highly helpful.

References

  • Sreenivas Jayanthi Computational Fluid Dynamics for Engineers and Scientists: Springer.

  • Farrashkhalvat, M., and J. P. Miles. Basic structured grid generation with an introduction to unstructured grid generation. Oxford: Butterworth Heinemann, 2003. Print.

  • Versteeg, H.K, and W. Malalasekera. An introduction to computational fluid dynamics: the finite volume method. Harlow: Pearson Education, 2011. Print.

  • Knupp, Patrick M., and Stanly Steinberg. Fundamentals of grid generation. Boca Raton: CRC Press, 1994. Print.

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This repo contains the code to generate body fitted grid for elliptic domain

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