The Best Methods for Mesh Interpolation
Key Takeaways

Mesh interpolation is used to approximate curvature between points in a 2D or 3D grid.

Point grids are used to represent real objects and interpolation is used to match a curve model to the curvature of a real object’s surface.

The curvature described in a mesh interpolation curve will be used in a numerical approximation of the main CFD equations.
CFD simulations, computer graphics, surface predictions in GIS, and structural dynamics all rely on a common technique to approximate surface curvature. This technique is mesh interpolation, which is used to predict curvature between points in a 2D or 3D point grid. Interpolation methods and the curvature model used in interpolation will be a major determinant of simulation accuracy and computation time. Simulation designers should have an idea of which mesh interpolation methods they should choose in order to balance accuracy and computing power requirements.
Why should we even consider mesh interpolation and how does it relate to simulation model generation? Although we would like curved surfaces on real objects to be easily approximated using analytical functions, this is not possible, and some method of interpolation must be used to extract the trend between data points in a 2D or 3D grid. Numerical methods will use the point relationships defined in these generated curves to solve the primary CFD equations.
Mesh Interpolation Overview
Interpolation is a curve fitting process, where a set of points is used to fit the shape of a curve such that the curve closely passes through the relevant set of points. In effect, it is a process of determining parameters in a model; for CFD simulations, these curve parameters define the curvature of surfaces along which a fluid will flow. Many interpolation methods exist for different types of curves, all of which are intended to balance the resolution and computational power required to converge to an accurate mesh.
All mesh interpolation methods are used to approximate surface curvature in a 3D model. This involves the following process:
 Grid generation: Mesh generation begins with grid generation, as the grid points defining shapes and surfaces in a real system will be used to create a mesh for CFD simulations.
 Curve selection: The simulation designer will need to select an appropriate curve that sufficiently approximates the shape of surfaces in the system, but does not require excessive computation to solve CFD equations.
 Apply discretization: The main fluid dynamics equations (either NavierStokes, Euler, or a reduced model) are discretized using the geometric definition developed with the interpolation procedure.
Grid generation is simple and effectively involves sampling points along surfaces and within volumetric regions in a system in order to define the grid. The grid points can be generated adaptively, as described in the section below. The next step in mesh interpolation requires selecting a curve that will be used to approximate flow contours along surfaces in a simulation model.
Comparison of a structured grid and curved mesh determined with an interpolation procedure for a heart model.
Curve Selection
Technically, any curve, including a straight line, can be used to approximate the shape of a real surface. Whether the approximated curve accurately conforms to the true curvature of the surface is another matter. Mesh interpolation procedures have been developed for several classes of curves:

Bspline interpolation: Bsplines are a general set of curves that link multiple polynomials together at their endpoints (called knots). This can be seen as a form of adaptive mesh interpolation that allows reduced order polynomials to be generated in regions where high curvature is not needed.

Polynomial interpolation: This is the most generic interpolation method, as it involves fitting a polynomial to a set of grid points to generate a mesh. Technically, polynomials of any order can be used for mesh interpolation, including linear interpolation used in simple meshes or higher order polynomials with highly accurate results.

Parametric interpolation: This type of interpolation is simple, as it relies on a parameter to define curvature in 3D. The same parameter is then used in discretization via a functional transformation.
Relation to Adaptive Meshing
Adaptive meshing steadily applies finer grid resolution to the system in order to ensure simulation results have higher accuracy in regions of the system with high gradients. Interpolation is involved in adaptive meshing in two possible ways, both of which are intended to steadily increase the resolution and simulation accuracy in the presence of high flow rate gradients.

Increasing mesh density: The standard method used in adaptive meshing is to increase the grid density in the neighborhood of high flow gradients. This will steadily increase the simulation resolution only in areas where high accuracy is required.

Increasing mesh order: The alternative method is to increase the polynomial degree when highorder meshing is used while keeping the mesh density constant. This can help increase accuracy in high flow gradient regions without a significant increase in computational cost as long as the interpolation procedure is not too involved.
Adaptive meshing schemes used in commercial and opensource CFD applications will use the first method, as it is the simplest to implement and does not require developing an additional numerical scheme to handle higherorder polynomial interpolation in high flow gradient regions. Adaptive mesh order methods, known as pmethods, have long been the subject of applied mathematics research and are discussed extensively in the literature. One seminal paper can be found below:
Advanced CFD simulation suites with meshing utilities will include builtin procedures for mesh interpolation. The complete set of fluid dynamics analysis and simulation tools in Pointwise from Cadence are ideal for defining and running CFD simulations with modern numerical approaches, ranging from full NavierStokes simulations to reduced turbulence models in complex systems.
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