What it means when a mesh is described as “high-order.”
Why mesh curving is perhaps more important than elevating its order.
The benefits of using a high-order mesh relative to a linear mesh.
How a high-order mesh is created from a linear mesh.
This hybrid mesh for two turbine blades includes a mix of linear mesh elements of different shapes (hexahedra, tetrahedra, etc.). High-order meshing can be used to reduce the mesh density around critical surfaces without sacrificing accuracy.
Anytime a numerical simulation needs to be performed in a complex system, some discretization scheme is needed in both the governing equations and the geometry. In CFD simulations, meshing is used to discretize the system geometry and create a set of points where the solution of the governing equations will be computed. One of the challenges in modern CFD is to balance the need for high accuracy, high resolution, and low computational effort in simulations. To that end, many meshing methods have been developed such that they can accommodate very complex geometries without increasing the computational complexity.
Among the various meshing methods used in CFD simulations, using a high-order is one powerful technique that helps balance accuracy, resolution, and computational cost. The goal in high-order meshing is to take advantage of high degree polynomial curves to build a mesh for a CFD problem, thereby providing much higher accuracy than linear meshes in complex systems. In this article, we’ll explore high-order meshes, how they are generated, and how they stack up to linear meshes in terms of accuracy and computational complexity.
What Is Mesh Order?
A high-order mesh connects adjacent mesh points with a curve of polynomial degree higher than one (i.e. linear). The easiest way to understand high-order meshing is to compare a high-order mesh with a linear mesh. In a linear mesh, volume cells are constructed from a set of straight lines connecting grid points. A high-order mesh connects grid points with a non-linear polynomial function (e.g. second degree quadratic), thus the technique is called “mesh curving.”
CFD meshing software that uses mesh curving or high-order mesh generation can typically use degree 2 (quadratic) to degree 4 (quartic) polynomials. If the technique were applied with a degree 1 polynomial, we would arrive back at a linear mesh, thus mesh curving is a more generalized mesh generation method. There are several geometric and mathematical advantages to using a curved mesh, but the primary advantage is computational.
Linear vs. High-Order Meshes
In the system shown below, we have a turbine blade where a linear mesh has been applied along the surface and along the boundary region. As can be seen near the bottom boundary of the turbine blade, the mesh is very dense as the base is approached. This is required to accurately approximate the shape of the curved surface as well as the gradient in the curved boundary layer flow along the surface. Since the gradient will be larger in a linear coordinate system as the surface is approached, we would expect the mesh density to increase along with the magnitude of the flow gradient.
This example linear mesh could be refined with a high-order mesh generation technique.
By applying mesh curving, we can produce a grid that more closely follows the curve along the surface of the turbine blade without increasing the node density. In a linear mesh, surfaces with higher curvature will require higher node density in order to provide the desired accuracy. This then requires greater computational time to solve the problem at hand because the number of arithmetic calculations in a numerical algorithm will scale linearly with grid density.
High-Order Mesh Generation From a Linear Mesh
A high-order mesh can be constructed from an existing linear mesh through interpolation. Regression is used to determine the coefficients in a polynomial model or in a roughly equivalent spline model. Interpolation can then be used to fill in the data between endpoints, giving a set of points for a curved mesh that follows the desired polynomial model. High-order mesh generation requires applying similar processes to a linear mesh (either structured or hybrid) in order to extract a polynomial curve relating successive points in the mesh.
Consider the example below, where a linear mesh is applied to a curved turbine blade. After imposing the boundary conditions on the linear mesh, an algorithm is used to fit the linear mesh points to a polynomial curve. The simulation designer has the freedom to choose the mesh order (polynomial) that might be best for their simulation. For some surfaces with specific polynomial curvature, the resulting polynomial mesh could very closely follow the surface, but without requiring the same node density as in a linear mesh.
Once polynomial curves are defined along the relevant surfaces, interpolation can be used to effectively develop a mesh with any density that might be desired. The solution accuracy can then be tuned by carefully adjusting the interpolated mesh density and using different interpolation techniques. An example of some interpolated high-order meshes is shown below (left). In the right panel, we can see that some interpolation methods might produce artefacts in the resulting interpolated mesh, so it is important to select the right interpolation technique to produce an accurate mesh result.
Interpolated polynomial mesh and an apparent linear mesh resulting from artefacts in the interpolation procedure.
The mesh generation tools in Pointwise from Cadence can help you create accurate simulations with a high-order mesh in complex geometries without a significant increase in computational complexity.