# Interpolation between two surface meshes

• Starbug
In summary, the conversation discusses a problem involving a surface represented as a Bezier patch and the need to obtain load values for points in a trimesh. The proposed solution is to use splining to interpolate the data and extrapolate it onto the mesh. The use of Akima splines is suggested as a potential method for this problem.

#### Starbug

Hi,

I've been given a problem to solve and I was hoping for some pointers.

To start with we have a surface which is originally represented as a Bezier patch, a peicewise set of bi-cubic bezier surfaces. It might be important to mention that the surface is "fairly two dimensional", the surface is single-valued for each X-Z coordinate, there have been a few problems where it was convenient just to work in the X-Z plane and then just project to a height value.

To perform different types of analysis on the surface we have created two meshes. The first mesh has points that form quadrangles, as if a grid had been laid over the surface. One of the analysis results is a set of load values on the surface, each node in the mesh has 3d vector for the force at that point. We also have another mesh in which the elements are triangles, this is made in a more sophisticated way and there is no necessary connection between points in the two meshes.

So the problem, what is the best way to get load values for the points in the trimesh? Obviously some interpolation scheme is needed. I've been googling around on the subject but I'm not sure how to proceed. For each point on the trimesh the starting point would be to find the quadrangle containing it in the quadmesh. The interpolation methods I've looked at for 2d or 3d problems generally seem to require a regular grid or lattice as input, whereas here we have a set of 3d points but on a 2d surface. Also another question, when doing vector valued interpolation in this sort of context is it usual to just treat each component independently or are there problems with this. (I've seen some contexts mentioned in which it is problematic, normal vectors in graphics calculations particularly.)

(I'm not sure if this is the best forum to post this in, perhaps this counts as a mechanical engineering problem? I don't have a engineering background myself.)

You could possibly look at splining. You can spline your data (there are 2D and 3D splining routines) and then interpolate the triangular points from the spline.

We have similar problems where we have inflow boundary conditions that are the result of experimental runs. There are not enough data points for the grid that we use, so we spline the data and then extrapolate it onto our mesh. We use a lot of Akima splines for data sets like that, I would look into it, they seem to work quite well; although, a simple B-spline might work well enough.

I would approach this problem by first understanding the underlying mathematical concepts and principles involved in surface interpolation. In this case, it seems like the goal is to interpolate load values from a quadrilateral mesh to a triangular mesh.

One possible approach would be to use a barycentric interpolation method, which is commonly used for interpolation between triangular meshes. This method involves finding the barycentric coordinates of a point within a triangle, and then using those coordinates to interpolate the value from the surrounding points in the quadrilateral mesh. This approach can work well if the quadrilateral mesh is relatively regular and the points are evenly distributed.

If the quadrilateral mesh is irregular or the points are not evenly distributed, a more complex interpolation method may be needed. One option could be to use a spline interpolation, which involves fitting a smooth curve or surface through the given points. This approach can work well for irregularly distributed points, but may require more computational resources.

In terms of treating vector-valued interpolation, it is important to consider the context and purpose of the analysis. If the vector values represent physical forces, it may be more appropriate to treat each component independently. However, if the vector values represent a single physical quantity (such as velocity), it may be more appropriate to interpolate the vector as a whole.

In conclusion, the best approach for interpolating load values from a quadrilateral mesh to a triangular mesh will depend on the specific characteristics of the meshes and the context of the analysis. Further research and experimentation may be needed to determine the most suitable method for your particular problem. It may also be helpful to consult with experts in the field of surface interpolation or mechanical engineering for additional insights and guidance.

## 1. What is interpolation between two surface meshes?

Interpolation between two surface meshes is a method used to fill in the gaps or missing data points between two existing surface meshes. It is a way to create a continuous surface between the two meshes, resulting in a more uniform and complete representation of the data.

## 2. How does interpolation between two surface meshes work?

Interpolation between two surface meshes works by using mathematical algorithms to estimate the values of the missing data points based on the values of the surrounding known data points. This can be done using various interpolation methods such as linear, polynomial, or spline interpolation.

## 3. In what applications is interpolation between two surface meshes commonly used?

Interpolation between two surface meshes is commonly used in computer graphics, 3D modeling, and animation to create smooth and realistic surfaces between two existing mesh objects. It is also used in scientific and engineering fields to fill in missing data in 3D scans or simulations.

## 4. What are the advantages of using interpolation between two surface meshes?

The main advantage of interpolation between two surface meshes is that it allows for a more accurate and complete representation of the data. It also helps to reduce noise and errors in the data as it fills in the gaps between the meshes. Additionally, interpolation can help to create a smoother and more visually appealing surface between the two meshes.

## 5. Are there any limitations to using interpolation between two surface meshes?

Yes, there are some limitations to using interpolation between two surface meshes. The accuracy of the interpolated data points is highly dependent on the quality and density of the existing meshes. If the meshes have a low resolution or contain a lot of noise, the interpolated data may not be as accurate. Additionally, interpolation may not work well for complex or irregularly shaped meshes, as the best-fit surface may not accurately represent the data.