# Lazy Cohomology Generators: A Breakthrough in (Co)homology Computations for CEM

@article{Dlotko2014LazyCG, title={Lazy Cohomology Generators: A Breakthrough in (Co)homology Computations for CEM}, author={Pawel Dlotko and Ruben Specogna}, journal={IEEE Transactions on Magnetics}, year={2014}, volume={50}, pages={577-580} }

Computing the first cohomology group generators received great attention in computational electromagnetics as a theoretically sound and safe method to produce cuts required when eddy-current problems are solved with the magnetic scalar potential formulations. This paper exploits the novel concept of lazy cohomology generators and a fast and general algorithm to compute them. This graph-theoretic algorithm is much faster than all competing ones being the typical computational time in the order…

## 23 Citations

Fast Computation of Cuts With Reduced Support by Solving Maximum Circulation Problems

- Computer ScienceIEEE Transactions on Magnetics
- 2015

A technique to efficiently compute optimal cuts required to solve 3-D eddy current problems by magnetic scalar potential formulations is presented, based on a novel graph-theoretic algorithm to solve a maximum circulation network flow problem in unweighted graphs that typically runs in linear time.

Lean Cohomology Computation for Electromagnetic Modeling

- MathematicsIEEE Transactions on Magnetics
- 2018

Modifications to the Dłotko–Specogna algorithm are introduced to compute a regular basis of the first cohomology group in practically the same time as the generation of a lazy Cohomology basis.

Topoprocessor: An efficient computational topology toolbox for h-oriented eddy current formulations

- Mathematics2016 IEEE Conference on Electromagnetic Field Computation (CEFC)
- 2016

When solving eddy-current problems containing topologically non-trivial conductors with formulations using the magnetic scalar potential in the insulators, cohomology is recognized to be the only…

Topoprocessor: An Efficient Computational Topology Toolbox for h-Oriented Eddy Current Formulations

- MathematicsIEEE Transactions on Magnetics
- 2017

This paper introduces an upgrade in the Dłotko–Specogna (DS) algorithm that speeds up the execution for very complicated geometries, and provides a detailed comparison of computational resources needed for the topological pre-processing by the toolbox and the tool to compute a standard cohomology basis available in the mesh generator GMSH.

Lazy Cohomology Generators Enable the Use of Complementarity for Computing Halo Current Resistive Distribution in Fusion Reactors

- Computer ScienceIEEE Transactions on Magnetics
- 2014

This paper presents a novel algorithm to generate the absolute second cohomology group generators exploiting the idea of lazy Cohomology generators stored as sparse vectors that allows a saving of between four and five orders of magnitude computational time.

Fake Conductivity or Cohomology: Which to Use When Solving Eddy Current Problems With $h$ -Formulations?

- PhysicsIEEE Transactions on Magnetics
- 2019

There are two ways of solving eddy current problems with an edge element <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula>-conforming formulation in the case of…

- Formulation with Higher-Order Hierarchical Basis Functions for Nonsimply Connected Conductors

- Mathematics
- 2018

This paper presents in detail the extension of the - formulation for eddy currents based on higher-order hierarchical basis functions so that it can automatically deal with conductors of arbitrary…

Optimized cycle basis in volume integral formulations for large scale eddy-current problems

- Computer ScienceComput. Phys. Commun.
- 2021

Fast halo currents computation in fusion reactors by electrokinetic complementary formulations

- Computer Science
- 2013

A novel algorithm is presented to generate the absolute second Cohomology group generators exploiting the idea of lazy cohomology generators stored as sparse vectors to save orders of magnitude computational time.

Computation of stationary 3D halo currents in fusion devices with accuracy control

- PhysicsJ. Comput. Phys.
- 2014

## References

SHOWING 1-10 OF 19 REFERENCES

Physics inspired algorithms for (co)homology computation

- Computer ScienceArXiv
- 2012

This paper presents a physics inspired algorithm for first cohomology group computations on three-dimensional complexes that solves one of the most long-lasting problems in low-frequency computational electromagnetics.

Physics inspired algorithms for (co)homology computations of three-dimensional combinatorial manifolds with boundary

- MathematicsComput. Phys. Commun.
- 2013

A novel technique for cohomology computations in engineering practice

- Mathematics, Computer Science
- 2013

Efficient Cohomology Computation for Electromagnetic Modeling

- Computer Science
- 2010

An automatic, computationally efficient and provably general algorithm based on a rigorous algorithm to compute a cohomology basis of the insulating region with state-of-art reductions techniquesexpressly designed for cohomological computations over simplicial complexes is presented.

Optimal Cohomology Generators for 2-D Eddy-Current Problems in Linear Time

- MathematicsIEEE Transactions on Magnetics
- 2013

An automatic and efficient algorithm to find cohomology generators suitable for 2-D eddy-current problems formulated by means of complementary formulations that produces optimal representatives of generators.

A fast algorithm to compute cohomology group generators of orientable 2-manifolds

- MathematicsPattern Recognit. Lett.
- 2012

Cohomology in 3D Magneto-Quasistatics Modeling

- Mathematics
- 2013

Several definitions of cuts are surveyed, defined as generators of the first cohomology group over integers of a finite CW-complex, which has the virtue of providing an automatic, general and efficient algorithm for the computation of cuts.

Automatic generation of cuts on large-sized meshes for the T–Ω geometric eddy-current formulation

- Computer Science
- 2009

Greedy optimal homotopy and homology generators

- Mathematics, Computer ScienceSODA '05
- 2005

It is shown that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2-manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm.

Efficient generalized source field computation for h-oriented magnetostatic formulations

- Computer Science
- 2011

The aim of this paper is to present a generalization of STT called Extended Spanning Tree Technique (ESTT), which is provably general and it retains the STT computational efficiency.