Computation of the magnetostatic interaction between linearly magnetized polyhedrons

In this paper we present a method to accurately compute the energy of the magnetostatic interaction between linearly (or uniformly, as a special case) magnetized polyhedrons. The method has applications in finite element micromagnetics, or more generally in computing the magnetostatic interaction when the magnetization is represented using the finite element method (FEM). The magnetostatic energy is described by a six-fold integral that is singular when the interaction regions overlap, making direct numerical evaluation problematic. To resolve the singularity, we evaluate four of the six iterated integrals analytically resulting in a 2d integral over the surface of a polyhedron, which is nonsingular and can be integrated numerically. This provides a more accurate and efficient way of computing the magnetostatic energy integral compared to existing approaches. The method was developed to facilitate the evaluation of the demagnetizing interaction between neighouring elements in finite-element micromagnetics and provides a possibility to compute the demagnetizing field using efficient fast multipole or tree code algorithms

Computing the demagnetizing tensor for finite difference micromagnetic simulations via numerical integration

In the finite difference method which is commonly used in computational micromagnetics, the demagnetizing field is usually computed as a convolution of the magnetization vector field with the demagnetizing tensor that describes the magnetostatic field of a cuboidal cell with constant magnetization. An analytical expression for the demagnetizing tensor is available, however at distances far from the cuboidal cell, the numerical evaluation of the analytical expression can be very inaccurate. Due to this large-distance inaccuracy numerical packages such as OOMMF compute the demagnetizing tensor using the explicit formula at distances close to the originating cell, but at distances far from the originating cell a formula based on an asymptotic expansion has to be used. In this work, we describe a method to calculate the demagnetizing field by numerical evaluation of the multidimensional integral in the demagnetization tensor terms using a sparse grid integration scheme. This method improves the accuracy of computation at intermediate distances from the origin. We compute and report the accuracy of (i) the numerical evaluation of the exact tensor expression which is best for short distances, (ii) the asymptotic expansion best suited for large distances, and (iii) the new method based on numerical integration, which is superior to methods (i) and (ii) for intermediate distances. For all three methods, we show the measurements of accuracy and execution time as a function of distance, for calculations using single precision (4-byte) and double precision (8-byte) floating point arithmetic. We make recommendations for the choice of scheme order and integrating coefficients for the numerical integration method (iii)

Nmag micromagnetic simulation tool - software engineering lessons learned

We review design and development decisions and their impact for the open source code Nmag from a software engineering in computational science point of view. We summarise lessons learned and recommendations for future computational science projects. Key lessons include that encapsulating the simulation functionality in a library of a general purpose language, here Python, provides great flexibility in using the software. The choice of Python for the top-level user interface was very well received by users from the science and engineering community. The from-source installation in which required external libraries and dependencies are compiled from a tarball was remarkably robust. In places, the code is a lot more ambitious than necessary, which introduces unnecessary complexity and reduces main- tainability. Tests distributed with the package are useful, although more unit tests and continuous integration would have been desirable. The detailed documentation, together with a tutorial for the usage of the system, was perceived as one of its main strengths by the community.Comment: 7 pages, 5 figures, Software Engineering for Science, ICSE201

Normal modes of carbon nanotubes: similarities and differences with their continuum counterpart

Carbon nanotubes (CNTs) possess a range of unusually interesting and useful physicochemical properties. In this paper, the mechanical properties of single wall CNTs are investigated via free vibration normal modes using molecular mechanics models. The forcefield used is empirical and the usual assumptions of potential energy contributions coming from bondstretching, bond angle bending, and bond twisting for two, three, and four atom interactions respectively, are made. The validity of continuum behaviour is examined by comparing the modal spacing obtained from the molecular mechanics models and that obtained from classical continuum elastodynamics. The breakdown of continuum behaviour is systematically characterised for various combinations of length to diameter ratio as well as for the number of atoms per circumference

Finite element optimizations for efficient non-linear electrical tomography reconstruction

Electrical Tomography can produce accurate results only if the underlying 2D or 3D volume discretization is chosen suitably for the applied numerical algorithm. We give general indications where and how to optimize a finite element discretization of a volume under investigation to enable efficient computation of potential distributions and the reconstruction of materials. For this, we present an error estimator and material-gradient indicator as a driver for adaptive mesh refinement and show how finite element mesh properties affect the efficiency and accuracy of the solutions

A systematic approach to multiphysics extensions of finite-element-based micromagnetic simulations: Nmag

Extensions of the basic micromagnetic model that include effects such as spin-current interaction, diffusion of thermal energy or anisotropic magnetoresistance are often studied by performing simulations that use case-specific ad-hoc extensions of widely used software packages such as OOMMF or Magpar. We present the novel software framework 'Nmag' that handles specifications of micromagnetic systems at a sufficiently abstract level to enable users with little programming experience to automatically translate a description of a large class of dynamical multifield equations plus a description of the system's geometry into a working simulation. Conceptually, this is a step towards a higher-level abstract notation for classical multifield mutliphysics simulations, similar to the change from assembly language to a higher level human-and-machine-readable formula notation for mathematical terms (FORTRAN) half a century ago. We demonstrate the capability of this approach through two examples, showing 1) a reduced dimensionality model coupling two copies of the micromagnetic sector and 2) the computation of a spatial current density distribution for anisotropic magnetoresistance (AMR). For cross-wise validation purposes, we also show how Nmag compares to the OOMMF and Magpar packages on a selected micromagnetic toy system. We furthermore, briefly discuss the limiations of our framework and related conceptual questions