The different types of meshes allow the representation of a larger geometric domain to be optimized using smaller discrete cells. Meshes are commonly used to compute solutions to partial differential equations and in computer graphics applications, as well as to analyze geographic and cartographic data. A mesh divides space into elements (also called cells or zones) over which equations can be solved, which then allows the solution to be approximated over a larger domain. Element borders can be constrained to fit internal or external boundaries within a model. Elements of higher quality (better shape) have more suitable numerical properties, although the goodness of these elements depends on the type of equations being worked with and the properties of the particular solution sought.
Common cell shapes
two-dimensional space
There are two types of two-dimensional cell shapes commonly used: the triangle and the quadrilateral.
The least computationally suitable elements are those with some very acute interior angle, very short sides, or both characteristics.
This cell shape consists of 3 sides and is one of the simplest mesh types. A triangular surface mesh is always quick and easy to create. It is more common in unstructured lattices.
It is the basic shape of the 4-sided cell, as shown in the figure. It is most common in structured grids.
Quadrilateral elements generally cannot be or become concave.
Three-dimensional space
The basic three-dimensional elements are tetrahedra, pyramids "Pyramid (geometry)"), triangular prisms and hexahedrons. They all have triangular and quadrilateral faces.
Extruded two-dimensional models can be represented entirely by prisms and hexahedrons such as extruded triangles and quadrilaterals.
In general, quadrilateral faces in 3 dimensions may not be perfectly flat. A non-planar quadrilateral face can be considered a thin tetrahedral volume shared by two neighboring elements.
Tights
Introduction
The different types of meshes allow the representation of a larger geometric domain to be optimized using smaller discrete cells. Meshes are commonly used to compute solutions to partial differential equations and in computer graphics applications, as well as to analyze geographic and cartographic data. A mesh divides space into elements (also called cells or zones) over which equations can be solved, which then allows the solution to be approximated over a larger domain. Element borders can be constrained to fit internal or external boundaries within a model. Elements of higher quality (better shape) have more suitable numerical properties, although the goodness of these elements depends on the type of equations being worked with and the properties of the particular solution sought.
Common cell shapes
two-dimensional space
There are two types of two-dimensional cell shapes commonly used: the triangle and the quadrilateral.
The least computationally suitable elements are those with some very acute interior angle, very short sides, or both characteristics.
This cell shape consists of 3 sides and is one of the simplest mesh types. A triangular surface mesh is always quick and easy to create. It is more common in unstructured lattices.
It is the basic shape of the 4-sided cell, as shown in the figure. It is most common in structured grids.
Quadrilateral elements generally cannot be or become concave.
Three-dimensional space
The basic three-dimensional elements are tetrahedra, pyramids "Pyramid (geometry)"), triangular prisms and hexahedrons. They all have triangular and quadrilateral faces.
Extruded two-dimensional models can be represented entirely by prisms and hexahedrons such as extruded triangles and quadrilaterals.
A tetrahedron has 4 vertices, 6 edges and is bounded by 4 triangular faces. In most cases a tetrahedral volume mesh can be automatically generated.
A pyramid with a quadrilateral base has 5 vertices and 8 edges; delimited by 4 triangular faces and 1 quadrilateral. They are effectively used as transition elements between square and triangular face elements and others in hybrid meshes and lattices.
A triangular prism has 6 vertices and 9 edges; delimited by 2 triangular faces and 3 quadrilaterals. The advantage of this type of element is that it resolves the boundary layer efficiently.
A cuboid, topologically equivalent to a cube, has 8 vertices and 12 edges; delimited by 6 quadrilateral faces. It is also called hexahedron or brick.[1] For the same number of cells, the precision of solutions in hexahedral meshes is the highest.
The pyramid and triangular prism zones can be considered computationally as degenerate hexahedrons, where some edges have been reduced to zero. Other degenerate shapes can also be represented from a hexahedron.
A polyhedral (dual) element has any number of vertices, edges and faces. It generally requires more computing operations per cell due to the number of neighboring elements (typically 10),[2] although this drawback is offset by the precision of the calculation.
Reticle Classification
Structured lattices
Structured lattices are identified by their regular connectivity. Possible element options are 2D quadrilateral and 3D hexahedron. This model is very space efficient, since the neighborhood relationships are defined by the storage layout. Some other advantages of structured network over unstructured network are better convergence and higher resolution.[3][4][5].
Unstructured lattices
An unstructured lattice is identified by irregular connectivity. It cannot be easily expressed as a two-dimensional or three-dimensional "Vector (computing)") array in a computer's memory. This allows for any possible element that a solver can use. Compared to structured meshes, for which neighborhood relationships are implicit, this model can be very spatially inefficient, as it requires explicit storage of neighborhood relationships. The storage requirements of a structured network and an unstructured network are within a constant factor. These grids usually use 2D triangles and 3D tetrahedra.[6].
Hybrid reticles
A hybrid grid contains a mixture of structured portions and unstructured portions. It integrates structured meshes and unstructured meshes efficiently. Those parts of geometry that are regular may have structured grids and those that are complex may have unstructured grids. These grids can be non-conforming, meaning that the grid lines do not need to coincide at block boundaries.[7].
Quality of a mesh
Contenido
Se considera que una malla tiene mayor calidad si permite calcular más rápidamente una solución más precisa. La precisión y la velocidad están en contraposición. Disminuir el tamaño de la malla siempre aumenta la precisión, pero también aumenta el costo computacional.
La precisión depende tanto del error de discretización como del error admisible de la solución. Una malla dada es una aproximación discreta del espacio y, por lo tanto, solo puede proporcionar una solución aproximada, incluso cuando las ecuaciones se resuelvan exactamente. En los gráficos por computadora mediante trazado de rayos, el número de rayos disparados es otra fuente de error de discretización. Para el error de solución, para las PDE se requieren muchas iteraciones sobre toda la malla. El cálculo finaliza antes de que las ecuaciones se resuelvan exactamente. La elección del tipo de elemento de malla afecta tanto a la discretización como al error de solución.
La precisión depende tanto del número total de elementos como de la forma de los elementos individuales. La velocidad de cada iteración crece (linealmente) con la cantidad de elementos, y la cantidad de iteraciones necesarias depende del valor de la solución local y del gradiente en comparación con la forma y el tamaño de los elementos locales.
Solution Accuracy
A coarse mesh can provide an accurate solution if the solution is constant, so the accuracy depends on the particular case of the problem.
You can selectively refine the mesh in areas where solution gradients are high, thus increasing fidelity there. Precision, including interpolated values within an element, depends on the type and shape of the element.
Convergence rate
Each iteration reduces the error between the calculated solution and the true one. A faster rate of convergence means lower error with fewer iterations.
A lower quality mesh may omit important features, such as the boundary layer for fluid flow. The discretization error will be large and the convergence rate will be impaired; the solution may not converge at all.
Mesh independence
A solution is considered grid independent if the discretization and error of the solution are small enough given enough iterations. This is essential to know to obtain comparative results. A mesh convergence study consists of refining elements and comparing the refined solutions with the coarse solutions. If further refinement (or other changes) does not significantly change the solution, the mesh is an independent grid.
Choice of mesh type
Si la precisión es lo más importante, entonces la malla hexaédrica es la más preferible. Se requiere que la densidad de la malla sea lo suficientemente alta para capturar todas las características del flujo, pero del mismo modo, no debe ser tan alta como para capturar detalles superfluos, lo que sobrecargaría la CPU y haría perder más tiempo. Siempre que hay una pared, la malla adyacente a la pared es lo suficientemente fina como para resolver el flujo de la capa límite y, en general, se prefieren las celdas cuadrangulares, hexagonales y prismáticas, frente a triángulos, tetraedros y pirámides. Las celdas Quad y Hex se pueden estirar donde el flujo está completamente desarrollado y es unidimensional.
En función de la asimetría, la suavidad y la relación de aspecto, se puede decidir la idoneidad de una malla.[8].
Asymmetry
The asymmetry of a grid is a good indicator of the quality and suitability of the mesh. A large asymmetry compromises the accuracy of the interpolated regions. There are three methods to determine the asymmetry of a grid.
This method is applicable only to triangles and tetrahedral elements, and is the default method.
This method applies to all cell and face shapes, and is almost always used for prisms and pyramids.
Another common measure of quality is based on equiangular inclination.
where:.
A skewness of 0 is the best possible and a skewness of one is almost never preferred. For hexagonal and quadrangular cells, the skewness should not exceed 0.85 to obtain a fairly accurate solution.
For triangular cells, the skewness should not exceed 0.85 and for quadrilateral cells, the skewness should not exceed 0.9.
Smoothness
The size change should also be smooth. There should be no sudden jumps in cell size, because this can cause erroneous results on nearby nodes.
aspect ratio
It is the relationship between the longest and shortest side of a cell. Ideally it should be equal to 1 to ensure best results. For multidimensional flow"), it should be close to one. Additionally, local variations in cell sizes should be minimal, that is, the sizes of adjacent cells should not vary by more than 20%. Having a large aspect ratio "Aspect Ratio (Geometry)") can cause an interpolation error of unacceptable magnitude.
Mesh generation and improvement
In two dimensions, flipping and smoothing are powerful tools for adapting a poor mesh to a good mesh. Flipping involves combining two triangles to form a quadrilateral and then dividing the quadrilateral in the other direction to produce two new triangles. Flipping is used to improve quality measures of a triangle, such as asymmetry. Smoothing a mesh improves the shapes of elements and their overall quality by adjusting the linking of their vertices. In smoothing a mesh, core features, such as the non-zero pattern of a linear system, are preserved, since the topology of the mesh remains invariant. Laplacian smoothing") is the most used technique.
[6] ↑ Mavriplis, D.J. (1996), «Mesh Generation and adaptivity for complex geometries and flows», Handbook of Computational Fluid Mechanics .
[7] ↑ Bern, Marshall; Plassmann, Paul (2000), «Mesh Generation», Handbook of Computational Geometry. Elsevier Science .
[8] ↑ «Meshing, Lecture 7». Andre Bakker. Consultado el 10 de noviembre de 2012.: http://www.bakker.org
In general, quadrilateral faces in 3 dimensions may not be perfectly flat. A non-planar quadrilateral face can be considered a thin tetrahedral volume shared by two neighboring elements.
A tetrahedron has 4 vertices, 6 edges and is bounded by 4 triangular faces. In most cases a tetrahedral volume mesh can be automatically generated.
A pyramid with a quadrilateral base has 5 vertices and 8 edges; delimited by 4 triangular faces and 1 quadrilateral. They are effectively used as transition elements between square and triangular face elements and others in hybrid meshes and lattices.
A triangular prism has 6 vertices and 9 edges; delimited by 2 triangular faces and 3 quadrilaterals. The advantage of this type of element is that it resolves the boundary layer efficiently.
A cuboid, topologically equivalent to a cube, has 8 vertices and 12 edges; delimited by 6 quadrilateral faces. It is also called hexahedron or brick.[1] For the same number of cells, the precision of solutions in hexahedral meshes is the highest.
The pyramid and triangular prism zones can be considered computationally as degenerate hexahedrons, where some edges have been reduced to zero. Other degenerate shapes can also be represented from a hexahedron.
A polyhedral (dual) element has any number of vertices, edges and faces. It generally requires more computing operations per cell due to the number of neighboring elements (typically 10),[2] although this drawback is offset by the precision of the calculation.
Reticle Classification
Structured lattices
Structured lattices are identified by their regular connectivity. Possible element options are 2D quadrilateral and 3D hexahedron. This model is very space efficient, since the neighborhood relationships are defined by the storage layout. Some other advantages of structured network over unstructured network are better convergence and higher resolution.[3][4][5].
Unstructured lattices
An unstructured lattice is identified by irregular connectivity. It cannot be easily expressed as a two-dimensional or three-dimensional "Vector (computing)") array in a computer's memory. This allows for any possible element that a solver can use. Compared to structured meshes, for which neighborhood relationships are implicit, this model can be very spatially inefficient, as it requires explicit storage of neighborhood relationships. The storage requirements of a structured network and an unstructured network are within a constant factor. These grids usually use 2D triangles and 3D tetrahedra.[6].
Hybrid reticles
A hybrid grid contains a mixture of structured portions and unstructured portions. It integrates structured meshes and unstructured meshes efficiently. Those parts of geometry that are regular may have structured grids and those that are complex may have unstructured grids. These grids can be non-conforming, meaning that the grid lines do not need to coincide at block boundaries.[7].
Quality of a mesh
Contenido
Se considera que una malla tiene mayor calidad si permite calcular más rápidamente una solución más precisa. La precisión y la velocidad están en contraposición. Disminuir el tamaño de la malla siempre aumenta la precisión, pero también aumenta el costo computacional.
La precisión depende tanto del error de discretización como del error admisible de la solución. Una malla dada es una aproximación discreta del espacio y, por lo tanto, solo puede proporcionar una solución aproximada, incluso cuando las ecuaciones se resuelvan exactamente. En los gráficos por computadora mediante trazado de rayos, el número de rayos disparados es otra fuente de error de discretización. Para el error de solución, para las PDE se requieren muchas iteraciones sobre toda la malla. El cálculo finaliza antes de que las ecuaciones se resuelvan exactamente. La elección del tipo de elemento de malla afecta tanto a la discretización como al error de solución.
La precisión depende tanto del número total de elementos como de la forma de los elementos individuales. La velocidad de cada iteración crece (linealmente) con la cantidad de elementos, y la cantidad de iteraciones necesarias depende del valor de la solución local y del gradiente en comparación con la forma y el tamaño de los elementos locales.
Solution Accuracy
A coarse mesh can provide an accurate solution if the solution is constant, so the accuracy depends on the particular case of the problem.
You can selectively refine the mesh in areas where solution gradients are high, thus increasing fidelity there. Precision, including interpolated values within an element, depends on the type and shape of the element.
Convergence rate
Each iteration reduces the error between the calculated solution and the true one. A faster rate of convergence means lower error with fewer iterations.
A lower quality mesh may omit important features, such as the boundary layer for fluid flow. The discretization error will be large and the convergence rate will be impaired; the solution may not converge at all.
Mesh independence
A solution is considered grid independent if the discretization and error of the solution are small enough given enough iterations. This is essential to know to obtain comparative results. A mesh convergence study consists of refining elements and comparing the refined solutions with the coarse solutions. If further refinement (or other changes) does not significantly change the solution, the mesh is an independent grid.
Choice of mesh type
Si la precisión es lo más importante, entonces la malla hexaédrica es la más preferible. Se requiere que la densidad de la malla sea lo suficientemente alta para capturar todas las características del flujo, pero del mismo modo, no debe ser tan alta como para capturar detalles superfluos, lo que sobrecargaría la CPU y haría perder más tiempo. Siempre que hay una pared, la malla adyacente a la pared es lo suficientemente fina como para resolver el flujo de la capa límite y, en general, se prefieren las celdas cuadrangulares, hexagonales y prismáticas, frente a triángulos, tetraedros y pirámides. Las celdas Quad y Hex se pueden estirar donde el flujo está completamente desarrollado y es unidimensional.
En función de la asimetría, la suavidad y la relación de aspecto, se puede decidir la idoneidad de una malla.[8].
Asymmetry
The asymmetry of a grid is a good indicator of the quality and suitability of the mesh. A large asymmetry compromises the accuracy of the interpolated regions. There are three methods to determine the asymmetry of a grid.
This method is applicable only to triangles and tetrahedral elements, and is the default method.
This method applies to all cell and face shapes, and is almost always used for prisms and pyramids.
Another common measure of quality is based on equiangular inclination.
where:.
A skewness of 0 is the best possible and a skewness of one is almost never preferred. For hexagonal and quadrangular cells, the skewness should not exceed 0.85 to obtain a fairly accurate solution.
For triangular cells, the skewness should not exceed 0.85 and for quadrilateral cells, the skewness should not exceed 0.9.
Smoothness
The size change should also be smooth. There should be no sudden jumps in cell size, because this can cause erroneous results on nearby nodes.
aspect ratio
It is the relationship between the longest and shortest side of a cell. Ideally it should be equal to 1 to ensure best results. For multidimensional flow"), it should be close to one. Additionally, local variations in cell sizes should be minimal, that is, the sizes of adjacent cells should not vary by more than 20%. Having a large aspect ratio "Aspect Ratio (Geometry)") can cause an interpolation error of unacceptable magnitude.
Mesh generation and improvement
In two dimensions, flipping and smoothing are powerful tools for adapting a poor mesh to a good mesh. Flipping involves combining two triangles to form a quadrilateral and then dividing the quadrilateral in the other direction to produce two new triangles. Flipping is used to improve quality measures of a triangle, such as asymmetry. Smoothing a mesh improves the shapes of elements and their overall quality by adjusting the linking of their vertices. In smoothing a mesh, core features, such as the non-zero pattern of a linear system, are preserved, since the topology of the mesh remains invariant. Laplacian smoothing") is the most used technique.