Preconditioned nonlinear conjugate gradient method for micromagnetic energy minimization

Abstract

Fast computation of demagnetization curves is essential for the computational design of soft magnetic sensors or permanent magnet materials. We show that a sparse preconditioner for a nonlinear conjugate gradient energy minimizer can lead to a speed up by a factor of 3 and 7 for computing hysteresis in soft magnetic and hard magnetic materials, respectively. As a preconditioner an approximation of the Hessian of the Lagrangian is used, which only takes local field terms into account. Preconditioning requires a few additional sparse matrix vector multiplications per iteration of the nonlinear conjugate gradient method, which is used for minimizing the energy for a given external field. The time to solution for computing the demagnetization curve scales almost linearly with problem size.

A comparative study of gradient-based and meta-heuristic optimization methods using Griewank benchmark function

“…However, if the function to be minimized, F(x), is not quadratic, the nonlinear conjugate gradient method is applied to update iteratively the solution vector x, until the convergence is reached as shown in Algorithm 2. [7] Algorithm 2: Nonlinear conjugate gradient method Task: Minimize F(x). Initialize:…”

“…According to Exl et al [7] the nonlinear conjugate gradient method that use (3) instead of ( 2) are believed to have more efficient convergence characteristics due the self-correcting behavior of term (g j+1 − g j ).…”

A comparative study of gradient-based and meta-heuristic optimization methods using Griewank benchmark function/ Um estudo comparativo de métodos de otimização baseados em gradientes e meta-heurísticos usando a função de benchmark do Griewank

“…However, if the function to be minimized, F(x), is not quadratic, the nonlinear conjugate gradient method is applied to update iteratively the solution vector x, until the convergence is reached as shown in Algorithm 2. [7] The first steps of nonlinear conjugate gradient method are similar to steepest descent method. However, the search direction is updated by assuming the orthogonality between next and previous gradient present in b proposed originally by Fletcher and…”

Computational Design of Rare-Earth Reduced Permanent Magnets

“…A Python script controlling the open-source CAD software Salome [12] introduces the grain boundary phase with a specific thickness and produces the finite element mesh. For these synthetic microstructures the demagnetization curve is computed through minimization of the micromagnetic energy with a preconditioned nonlinear conjugate gradient method [13]. The search for higher coercive fields, µ0Hc, and energy density products, (BH)max, is managed via the open-source optimization framework Dakota [14].…”