Applications needing fast numerical conformal maps

[Diane Wilbor]

What are useful practical applications of numerical conformal mapping that are most limited by map computation speed or boundary complexity? I'm betting some of the applications will be be physics PDEs, so I chose this DE subforum to ask.

As part of an engineering project I've implemented several numerical methods including Kantorovich's method of simultaneous equations, as well as Fornberg's iterative version, and Driscoll/Trefethen's classic numerical Schwarz-Christoffel mapping methods. But some experiments with FFT based methods were really successful. These started as variants of Fornberg's FFT method but with some tweaks, modern numerical libraries, and reorganizing the algorithm to eliminate memory bandwidth limits. The performance became surprisingly, almost startlingly, good, especially the memory-aware organization. So I have a working prototype of a numerical conformal mapping code that can compute the mapping of even huge boundaries of a million line or curve segments in fractions of a second. Its performance is beyond my expectations and needs.

So with this serendipidous new experimental tool, I'm trying to find any applications in any field which have been limited by conformal mapping speed and/or boundary complexity.

The number of applications of conformal mapping is huge of course, and a great resource (with hundreds of application references) has been the comprehensive but dated Conformal Mapping book by Schinzinger and Laura. Every single one of the (hundreds) of examples in that book have shapes and domains which are simple enough that they've been easy to quickly map even with Matlab. Most of those 1980's era examples use only 10 or so sides or curves! But today, for example, mapping a 2D airfoil boundary defined by 500 line segments is no challenge to existing tools with just one second of compute. Does anything need 50K? 5M?

But my question is asking about identifying specifically speed or complexity limited applications of numerical conformal mapping. Is there some research project, some engineering application, some important or interesting boundary-limit PDE which would be helped if it could only handle contours of a million edges, and/or solve them in milliseconds?

Thanks so much!
-Diane

 

[jvicens]

Dear Diane,

Thank you for sharing your exciting findings with us. Indeed, the advancements in numerical conformal mapping have opened up new possibilities for various applications. One area where speed and boundary complexity can be major limitations is in fluid dynamics and aerodynamics.

In fluid dynamics, conformal mapping is used to transform the complex flow domain into a simpler geometry, such as a circle or rectangle, where the governing equations can be solved more easily. This allows for more accurate and efficient simulations of fluid flow around complex geometries, such as airfoils or ship hulls. However, as the complexity of the flow domain increases, the computational cost of the conformal mapping also increases. This can be a major limitation for high-fidelity simulations of real-world scenarios, where the flow domain may have millions of edges.

Similarly, in aerodynamics, conformal mapping is used to transform the flow domain around an airfoil into a simpler geometry, such as a circle or ellipse, for more efficient analysis and design of airfoils. However, as the complexity of the airfoil shape increases, the computational cost of the conformal mapping also increases. This can be a major limitation for designing and optimizing complex airfoil shapes for specific performance requirements.

Therefore, your high-speed and high-complexity conformal mapping code could be a valuable tool for researchers and engineers in the field of fluid dynamics and aerodynamics. It could potentially enable more accurate and efficient simulations and design optimizations of complex flow domains and airfoil shapes, leading to advancements in areas such as aircraft design, wind turbine design, and hydrodynamics.

Additionally, your code could also be useful in other fields such as electromagnetics, where conformal mapping is used to transform complex geometries for more efficient electromagnetic simulations. Your code could potentially enable faster and more accurate simulations of complex electromagnetic systems, leading to advancements in areas such as antenna design and radar technology.

In conclusion, your high-speed and high-complexity conformal mapping code has the potential to greatly benefit various fields and applications where speed and boundary complexity have been major limitations. Thank you for your efforts and I look forward to seeing the impact of your work in these areas.