Hi,

I come across the Jacobian very frequently in the Grid generation book. However, I'm not quite understand the application of this Jacobian. It seems like a derivative of the physical coord with respect to the cartersian coord. Can anyone kind enough to briefly point out its physical meaning or significance in grid generation? I looked at several books and they all showed me a full lot of equations without much explanation.

Thank you for your time.

The Jacobian is used to transform from one coordinate system to another. In the case of general grid, which may be in general curvilinear and nonorthogonal, some operations (e.g., calculation of derivatives wrt this sytem or a cell volume) may be more conveinetly done in this system and then transformed to the Cartesian system, or vice versa.

Thanks for your advice.

Can Jacobian be used as a measure of grid quality? Each grid point should have its unique Jacobian, right? If so, will there be any effect if the Jacobian of a grid point is negative/positive or too large/small?

Sorry for the long question here. I tried to understand how this can be applied in the CFD. I read thru some books about the simple algorithm used by most commercial code to solve the flow problem. The equation used are all expressed in terms of the catersian coord.

So what you are saying here is we have to first transform the grid points from physical coord into the cartesian coord before solving the flow equations on these grid points, right?

The significance of the Jacobian is that it represents the validity of the mapping from physical space to computational space. In order for a mapping to be valid the Jacobian should have the same sign throughout the domain, i.e. for a transformation from a right-handed coordinate system to a right-handed coordinate system the Jacobian should be greater than zero. Where the Jacobian goes to zero, the mapping fails to be one-to-one and the inverse transformation no longer exists. In practical terms the Jacobian can be used as a measure of grid quality - since it also can be thought of as the volume of a grid cell the Jacobian distribution should be smoothly varying in the domain.

These remarks are in the context of the finite element method, in particular for incompressible flow using primative variables, but similar considerations apply in other methods.

To find the flow field, one needs to solve a set of equations, which in matrix equation form involves assembling a global stiffness matrix involving integrals over each element of the grid. These integrals are usually evaluated by transforming from the global coordinates to a standard reference element (square or triangle in 2D or cube or tet in 3D) and using Gaussian quadrature. It can easily be demonstrated using the chain rule that solenoidal fields transform as V(x,y)=J/D*V(q,r), where J is the Jacobian matrix of the transformation and D is the determinant of J (hint: write V as the curl of a stream function in 2D). (x,y) are the global coordinates and (q,r) are the local coordinates. This relation is still true in 3D, but the proof is a bit more difficult to demonstrate. If one uses divergence-free elements, the field on each element must transform as above. To evaluate the needed integrals, one needs to know the Jacobian at each integration point. Further, the global velocities at the nodes must be transformed using the inverse or the above transformation. If one knows the Jacobian at the corner nodes, then the values at the integration points can be found by interpolation. Of course if one is using rectangular elements and scalar functions for the vector components (enforcing mass conservation with a seperate equation), the transformations commute through the shape functions and cancel, and the Jacobian is contained implicitly in the values of h_x and h_y. That is why you don't see much discussion of the material above.

But you can also generate curved meshes using Hermite geometric functions at the corner nodes of blocks. The derivative DOFs of the Hermite functions are just the Jacobian. This is an attractive alternative to using the coordinates at midside nodes. With curved meshes or meshes where opposite sides are not parallel, the full formalism above is important.

Similar considerations apply to irrotational fields, but the transformation is different (curls transform differently from gradients).

[Crank-Shaft]

Hey everyone,

I am currently involved in the development of a research CFD code for compressible flows, which will be eventually used to implement hybrid RANS/LES models.

I have been trying to find a robust, direct way to calculate cell volumes using the Jacobian matrix. This code accepts grids in structured, curvilinear format and hence a coordinate transformation from the physical to the computational domain would be represented by J = d(\xi, \eta, \zeta)/d(x,y,z) however, does the reciprocal of this give me the cell volume.

I am trying to find a representative cell size definition and wanted to use a definition such as \sqrt{|V|} based on each cell volume V.

I look forward to your comments. Thanks in advance.