FEM at polar coordinate origin

Jonas Holdeman · October 28, 2019, 21:54

Can anyone send me accessible links to a discussion of how to handle the polar coordinate origin using the finite element method? Physically it is an ordinary point, but is singular in the polar coordinate system. All of the nodes which share the origin are of course correlated. My DOFs are the stream function and incompressible fluid velocities. I can do this when the origin is on the problem boundary (and specified as a Dirichlet BC) as in the semicircular driven cavity. I have attached results using quadrilateral elements, shown in polar and Cartesian coordinates for this example. Thanks.

FMDenaro · October 29, 2019, 04:35

I don't know if the problem can be seen in a similar way but using FVM (weak problem), there is no problem with the 1/r term. No terms are written on the r=0 point.

Jonas Holdeman · October 29, 2019, 10:29

The problem is not with 1/r term. I am assembling the matrix on an element by element basis in MATLAB. Using Gauss quadrature, the functions are never evaluated at r=0. When developing a code, when a section of code does not do what you expect it to do, the problem may be with the coding, or it may be a conceptual problem, even with a wrong expectation.

In the example figure, 28 nodes in polar coordinates corresponding to r=0 share and contribute to the same physical point, but only one is independent (actually none are independent in the semicircular driven cavity as they are all specified by BCs, but I want to consider the circular driven cavity). For the circular cavity, the number would be reduced by one because of periodic boundary conditions. The remaining 26 nodes must yield the same physical velocity and stream function. So there would be one independent node and 27 nodal/DOF constraints. Like Dirichlet BCs, these 27 are removed from the solution set (though they do contribute to the 28th equation) and restored to their proper constrained values after solution of the independent equations.

At least that is my concept of what should be happening. But I would like to see what others have done.

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October 29, 2019, 04:35		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 Rep Power: 73	I don't know if the problem can be seen in a similar way but using FVM (weak problem), there is no problem with the 1/r term. No terms are written on the r=0 point.

October 29, 2019, 10:29		#3
Jonas Holdeman Senior Member Jonas T. Holdeman, Jr. Join Date: Mar 2009 Location: Knoxville, Tennessee Posts: 128 Rep Power: 18	The problem is not with 1/r term. I am assembling the matrix on an element by element basis in MATLAB. Using Gauss quadrature, the functions are never evaluated at r=0. When developing a code, when a section of code does not do what you expect it to do, the problem may be with the coding, or it may be a conceptual problem, even with a wrong expectation. In the example figure, 28 nodes in polar coordinates corresponding to r=0 share and contribute to the same physical point, but only one is independent (actually none are independent in the semicircular driven cavity as they are all specified by BCs, but I want to consider the circular driven cavity). For the circular cavity, the number would be reduced by one because of periodic boundary conditions. The remaining 26 nodes must yield the same physical velocity and stream function. So there would be one independent node and 27 nodal/DOF constraints. Like Dirichlet BCs, these 27 are removed from the solution set (though they do contribute to the 28th equation) and restored to their proper constrained values after solution of the independent equations. At least that is my concept of what should be happening. But I would like to see what others have done.