Some geometrical and evolutionary procedures for optimizing the calculation times of 3D current fields by the finite element method

As well known, in the finite element method, the calculation and the location of the elements of the matrix C of the coefficients requires a lot of calculation times and memory employment especially for 3D problems. Besides, once the matrix C is properly filled, the solution of the system of linear equations is computationally expensive.

The paper consists of two parts. In the first part, to quickly calculate and store only the non‐null terms of the matrix of the system, a geometrical analysis on three‐dimensional domains has been carried out. The second part of the paper deals with the solution of the system of linear equations and proposes a procedure for increasing the solution speed: the traditional method of the conjugate gradient is hybridized with an adequate genetic algorithm (Genetic Conjugate Gradient).

The proposed geometrical procedure allows us to calculate the non‐null terms and their location within the matrix C by simple recursive formulas. The results concerning the genetic conjugate gradient show that the convergence to the solution of the linear system is obtained in a much smaller number iterations and the calculation time is also significantly decreased.

The approach proposed to analyze the geometrical space has been turned out to be very useful in terms of memory saving and computational cost. The genetic conjugate gradient is an original hybrid method to solve large scale problems quicker than the traditional conjugate gradient. An application of the method has been shown for current fields generated by grounding electrodes.