# Higher Order Panel Method / BEM Current State Overview

Discussion in 'Hydrodynamics and Aerodynamics' started by DCockey, Mar 12, 2011.

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### DCockeySenior Member

Any recommendations for an overview of the current state of higher order panel methods / boundary element methods, particularly those used for potential flow aerodynamics and hydrodynamics? I'm particularly interested in any which use the nodes at the corners of panels/elements as the collocation points, and any which use higher order (higher than constant potential / constant normal velocity) on "flat" or bilinear panels/elements.

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### DCockeySenior Member

Also interested in any methods which use a velocity rather than potential formulation.

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### BYDEJunior Member

I carry out quite a lot of research and development on panel methods (hydrodynamics and free surface mainly), so I could be helpful to you.

I don't really understand what you mean with ''use the nodes at the corners of panels/elements as the collocation points''. To my knowledge, normally you'd use the centroid of the panel as the collocation point, because near the edges of the panel you'd face numerical difficulties

Higher order methods (for both potential and geometry description) are a mature technique since several years. However imo simpler methods are often preferred because of their better stability and lower computational time

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### Leo LazauskasSenior Member

It is possible to use nodes at the corners of panels, but then some form of de-singularisation is often used. The choice of the locations of the singularities in these methods can sometimes be a bit tricky.

Depending on the specific application, better results are often achievable by not using the centroid of each panel. For example, my simple Lifting Surface Program (LSP) uses Chebyshev (i.e. cosine-spaced) grids and collocation points. There are a variety of non-uniform spacings that can be found in the literature for Vortex Lattice Methods for planar lifting surfaces, e.g. Lan's method, Stark's method, and ad hoc techniques such as those used by De Jarnette and others.

Leo.

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### BYDEJunior Member

I don't think he meant the spacing, which of course can be non uniform (I usually cluster the grid towards the ends of the hull using a linear variation).

I think that he says that, given the 4 corners of each quadrilateral panel, the collocation point is placed on one of them. Never heard about it honestly.
Probably I'm reading it wrong anyway

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### Leo LazauskasSenior Member

That would be very advisable in many applications.
I have found it much faster (in my ship codes) to use equally-spaced stations and waterlines and to use a different quadrature technique for each integral. This can speed up calculations (over brute force methods) by factors of 10,000 or more. Without that treatment non-linear optimisation techniques (e.g. memetic algorithms) would be far too slow.

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### DCockeySenior Member

Yes, I'm talking about putting the collocation points at the corners of the panels, and yes, I'm very aware of the problems that occur with the usual formulations when this is done. I did some research into a variety of 2D panel methods for unsteady subsonic and supersonic aerodynamics thirty years ago.

About twenty five years ago I developed a velocity based method for 2D which uses linear panels with the collocation points at the nodes on the corners of the panels. I used a very simple idea, which apparently is unique, to eliminate "analytical" singularities at the corners (other than sharp corners in the geometry). I still had to properly evaluate principal value integrals. The integral equations can be a Fredholm equation of the first type, of the second type, or a combination. Mixed boundary conditions do not cause any difficulties. It worked very well and was less sensitive to the arrangement of the panels than lower order methods. I didn't publish anything and the code and results were misplaced during a job change and house move.

More recently I've come up with several ways to extend the method to 3D using a quadratic potential, linear velocity formulation and bi-linear triangular or quadrilateral panels. Before I spend considerable time with further development I'd like to determine if someone else has come up with a similar formulation.

I should add that the work I've done is for "thick" bodies including those with wakes and lift. I have not extended it to thin lifting surfaces to data. The original impetus for the development was the intention to use it as the outer part of a hybrid scheme for bluff body and automobile external aerodynamics where a RANSE solve would be used for the near-field solution and the potential panel method would be used for the far field solution. We only did one preliminary test of that concept but it look promising.

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### BYDEJunior Member

I see, that's interesting.
I think it's confined to your specific method though, as I guess it uses Green's identity or something else based on boundary integral equations.
Instead I use a non-linear rankine panel method. It seems quite accurate on a rather wide range of hull, but also slow and sometimes unstable. (still under development though)

Do you have experience with memetic algorithms? Are they much slower than genetic algorithms?
I don't know much about them actually, I have a weakness for Particle Swarm Optimization

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### Leo LazauskasSenior Member

Many people have had that problem with non-linear codes. If a wave breaks anywhere in the field of interest many methods fail.
(One of the nice features of linear theory and of field methods is that they might produce unrealistic wave heights in some small part of the wave domain, but they never fail.)

I dream in terms of artificial worlds and evolutionary algorithms
Memetic algorithms are really just genetic algorithms that incorporate local search techniques.

They are useful for some optimisation problems but, like many other similar techniques, they are hopeless when there are many non-linear interactions between variables.

I do, however, love the whole field of Alife. Most of my work in aero and hydro is to find fast methods of calculating objective functions I find interesting so I can then embed them in artificial worlds.

Cheers,
Leo.

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### BYDEJunior Member

What is the advantage of placing the coll.point at the corner?
For a better representation of the geometry, you can project it to the actual surface according to the normal vector (basically coll. point and control point don't coincide anymore). But usually it makes a small difference overall.

Many de-singularization techniques have been suggested in the last 20 years, so it is difficult to state if your idea has been already used more recently without knowing the details of your method.
Moreover I'm mainly on potential-based methods for free surface flows. I have little references on velocity-based methods. Perhaps Leo can provide a better help.

But the RANSE code would still give a solution in all the domain, and better than the panel method.
Do you mean to use the results of the panel method to initialize the RANS computation? This is done quite commonly (at least in ship hydrodynamics)

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### DCockeySenior Member

Everything is defined and solved for at the nodes. The geometry is defined by the coordinates and normal vectors of the surface at the nodes. The normal vectors are those of the surface to be solved for, not the panels. The nodes define the corners of the panels. The boundary conditions are specified at the nodes, and can be mixed. The unknown velocity potential and velocity is solved for at the nodes.
The de-singularization techniques I'm aware of move the panels off of the surface with the collocation points on the surface. My understanding is they can be sensitive to the space between the panels and surface.

My method has the panel corners and collocation points directly on the surface. Convergence does not depend on panel arrangement.

They are potential flow solvers. The 2D method were formulated in terms of the velocity at the nodes. The 3D methods I'm considering are formulated in terms of the potential and the gradient of the potential (velocity) at the nodes. Using velocity alone in 3D raises the complication if insuring the results are irrotational (outside of wake sheets).

It was a zonal method. The inner zone used the RANSE code, the outer zone used the potential panel method which implicity includes the far-field boundary conditions. Other attempts at such methods up to that time usually had problems with coupling the zones together. I came up with a very simple way to avoid those problems.

The motiviation was to limit the size of the grid for the RANSE code. At the time we were working on this the Cray supercomputer we used had less computational power and less memory than PCs have today. Today concerns about grid sizes may be less, but such an approach could still reduce turn-around times. It might be possible to develop a zonal method for free-surface problems which would provide the outer boundary conditions including the radiation condition for the RANSE code.

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### DCockeySenior Member

What is non-linear in your method? Free surface boundary conditions?

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### BYDEJunior Member

I still don't see the practical advantages.
So basically you input nodes and respective normal vectors, rather than the corner points and then calculate centroid and normal vector from them (as in other panel methods). Is it right?
But you need to calculate the normals to input, so you still do that effort, just out of the code.
Second, the grid generation process takes almost zero time on an average PC.
The location of the coll.point on the actual surface and the actual normal vector (not the one of the panel) are used since a long time. You can check
Webster W.C. (1975), J. of Ship Research, vol.19, no. 14

In my bem code for example, the input is the IGES file with the NURBS surface(s) of the hull. Then using NURBS manipulation algorithms it creates the grid of quadrilateral or triangular panels, find control/collocation/corner points and normal vector (plus a bunch of other data). All that takes a fraction of a second...

That's one methodology. But there are several others, numerical techniques. Some of the best (imo), involve the addition-then-subtraction of a function that de-singularize the kernel of the integrals. So that a gaussian quadrature (or other quadratures) can be used.
Most of them originated from
Landweber and Macagno (1969), Iowa Institute of Hydraulic research, report No.123

This would be a very favourable feature, but should be elaborated in more details.

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### BYDEJunior Member

I still find difficult to understand this part of your concept.
Zonal methods are used (though not often) in ship hydrodynamics, but it's a different subdivision of zones, not 'inner' and 'outer'.

There are boundary layer calculations implemented in the panel methods, but I don't think you mean this.

In a RANS computation such as car/ship you mainly look at the results on the body itself, so the domain is always kept as small as possible. But big enough to obtain a good convergence.
The only way I can think to use a PM to help the RANS solver is in term of initialize the solution, but more the near-body solution I'd say and not the farfield where the flow is basically equal to the free-stream flow.

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### BYDEJunior Member

yes exactly.
Basically the solution is iterative and the free-surface is updated at each iteration to a more accurate prediction

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