Magnetostatic Systems Research Articles

Variational methods for solving numerically magnetostatic systems

Variational methods for solving numerically magnetostatic systems

Dot Sensitivity Analysis of the Current Region for Topology Optimization in Magnetostatic Systems

This paper proposes dot sensitivity analysis of a current region for topology optimization. A continuum shape sensitivity formula is employed to derive a dot sensitivity formula in a simple closed form. The derived dot sensitivity formula provides the information on where the current dots should be created for topology optimization. This is necessary to generate the current dots in a vacant region of the design domain. Because this dot sensitivity analysis can obtain various topologies of material distribution, the possibility of local optimum convergence is reduced. Furthermore, because this dot sensitivity analysis does not require a specific initial design, its numerical implementation is simple. This dot sensitivity method is combined with the level set method to perform topology optimization. To demonstrate the usefulness of the proposed method, it is applied to two optimization problems of magnetic resonance imaging.

Multiple Level-Set Methods for Optimal Design of Nonlinear Magnetostatic System

This paper proposes an optimal design method using continuum sensitivity analysis and the multiple level-set methods for multi-material shape optimization in nonlinear magnetostatic systems. The general shape sensitivity formula for the nonlinear magnetostatic system is derived based on the material derivative concept of continuum mechanics and the adjoint variable method. In the multiple level-set methods, the sign combination of level-set functions is used to identify the different material regions and their interfaces. The multi-material shapes are simultaneously deformed by solving the multiple level-set equations coupled with the velocity fields in the sensitivity formula. Two numerical examples are employed to show the usefulness of the proposed design method.

Design Sensitivity Analysis for Shape Optimization of Nonlinear Magnetostatic Systems

This paper discusses the sensitivity analysis for the shape optimization of a nonlinear magnetostatic system, evaluated both by direct and adjoint approaches. The calculations rely on the Lie derivative concept of differential geometry, where the flow is the velocity field associated with the modification of a geometrical parameter in the model. The resulting sensitivity formulas can be expressed naturally in a finite-element setting through a volume integral in the layer of elements connected to the surface undergoing shape modification. The accuracy of the methodology is analyzed on a 2-D model of an interior permanent magnet motor and on a 3-D model of a permanent magnet system.

Calculation of self-field coefficients for 2D magnetostatic systems

Physics and engineering students are introduced to the notion of a demagnetizing field in classical electromagnetism courses. This concept involves a formalism based on an integral formulation for calculating the coefficients of the demagnetizing tensor, i.e., a pure geometric quantity. For self-fields, the observation point is located inside the integration region which in turn leads to discontinuous integrands. Therefore, in order to avoid mathematical inconsistencies, special care must be taken when evaluating self-field coefficients, referred to here as self-terms. Given the complexity of this approach, in particular in 3D, it is certainly interesting from a pedagogical stand point to employ 2D systems as a first step for describing these kinds of coefficients. Thus, in this paper, the generalization of self-terms of the demagnetizing tensor is proven for 2D magnetostatic systems. Nonetheless, the structure of this proof pertains to many other situations given the fact that discontinuous integrands commonly arise in physics (e.g. integral solutions of PDEs which use a Green’s function).

A Magnetostatic MEMS Switch Designed for Portable Applications

A passive magnetostatic microelectromechanical systems (MEMS) switch using only one electroplated soft magnetic layer of nickel-iron (Ni <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">80</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> ) alloy was designed, fabricated, and characterized. The switch is composed of an electroplated Ni <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">80</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> plate supported by a pair of torsion bars from two sides. The Ni <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">80</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sub> plate is patterned into long and narrow strips to improve the sensitivity. The switch is actuated by bringing an external magnet closer to the switch. Therefore, no internal electrical power is consumed by the device for actuation. The magnetic field required to turn on the switch is 4.8 mT, and the initial contact resistance is 0.5 Ω with gold contacts. The switch has been tested to pass more than 34 million hot-switching cycles at 2-mA current at room temperature when packaged at the wafer level with SU-8 sealing. The die size is 2.1 × 1.94 × 1.1 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> . The magnetic switch of this paper has the potential to replace the conventional reed switch in portable electronics such as laptops, cellular phones, personal data assistants, pacemakers, and hearing aids.

Complementarity bilateral bounds on forces in magnet systems

PurposeIt is well known that complementarity can provide bilateral bounds on energy, in numerical approximations of nonlinear magnetostatics. Questions whether like force is, up to sign, the derivative of energy, can such bounds also apply to forces, torques, and other concepts?Design/methodology/approachThis is a discussion paper exploring the issues conceptually.FindingsOn the face of it, no, since inequalities are not preserved by differentiation. Yet, useful bounds can be given in some cases, as shown in the discussion, following an idea by Matagne.Originality/valueThe paper aids the understanding of nonlinear magnetostatic systems.

A hybrid full-Lagrangian technique for the static and dynamic analysis of magnetostatic MEMS

Magnetostatic microelectromechanical systems (MEMS) are based on the electromagnetic interactions between magnetic microstructures and active (coils) or passive (permanent magnets) magnetic field sources. They offer distinct advantages at the micrometer scale in strength, polarity and distance of actuation, when compared to other methods of actuation in MEMS. For proper understanding and detailed exploration of magnetostatic MEMS, it is important to have a reliable and efficient physical level simulation tool. In this paper, we propose an efficient technique, namely the hybrid full-Lagrangian technique for the static and dynamic analysis of magnetostatic MEMS. In this technique, the magnetostatic analysis needed to compute the magnetostatic force acting on the microstructure is performed using a hybrid BIE/Poisson approach. The Poisson equation is solved for the interior magnetostatic domains and the boundary integral equation (BIE) formulation of the potential equation is solved for the exterior magnetostatic domain and the different domains are coupled through interface conditions. A Lagrangian description of all the physical domains (magnetostatic, mechanical and fluidic) is used to eliminate geometry updates and rediscretization. The Lagrangian formulation along with the hybrid approach makes the proposed technique much more efficient than conventional tools for the analysis of magnetostatic MEMS. The new technique is used to simulate several magnetostatic MEMS switches and relays and validated by comparing numerical simulation results with experimental data. Dynamic analysis of a magnetostatic MEMS switch is performed using the new technique.

Fast Computation for Large Magnetostatic Systems Adapted for Micromagnetism

In this paper, an efficient method is developed for computing the magnetostatic field for ferromagnetic materials on large structured meshes. The problem is discretized using a finite volume approximation. The discrete operator is proved to preserve the main properties of the continuous model, and a lower estimate of its lower eigenvalue is given. Using the fact that the discrete operator has a block-Toeplitz structure for cubic meshes in parallelepipedic domains, a fast solving method is built. Based upon the use of fast Fourier transform, this method allows one to reduce the computational cost from n2 to O(n log(n)) but also to reduce the storage to O(n) instead of n2, where n is the number of cells in the mesh.

The solution of magnetostatic BEM systems of equations using iterative methods

A boundary element method for the solution of magnetostatics problems is developed. For large problems when using a direct linear equation solver such as LU decomposition, the decomposition phase of the algorithm comprises the predominant portion of the overall CPU time. A variety of indirect linear equation solvers are considered as a means of significantly reducing the CPU usage. Because of the excellent conditioning of the discretized linear systems, the number of iterations required of the indirect solvers to reach a prescribed residual is quite low and remains relatively constant with refined discretizations.

Derivation of a general sensitivity formula for shape optimization of 2-D magnetostatic systems by continuum approach

This paper presents a generalized sensitivity formula that is applicable to any objective function of the optimization problems in two-dimensional (2-D) linear magnetostatic systems. For the objective function defined on an interface boundary as well as in a region, the shape design sensitivity expression is derived in a closed form from the augmented Lagrangian method, the material derivative concept and the adjoint variable method. The resultant sensitivity formula is expressed as a line integral along the interface boundary undergoing shape modification. Three shape design problems, whose objective functions are defined on an interface boundary or in a region, are tested to validate the derived sensitivity formula.

Topology optimization of nonlinear magnetostatics

The topology optimization of nonlinear magnetostatic systems is studied using the finite-element method. The topology design sensitivity formulation of nonlinear magnetostatics is derived using the adjoint variable method. A computer program is developed using object-orient programming and applied to the topology optimization of a C-core actuator. A numerical study shows the effect of saturation by comparing linear and nonlinear topology optimization.

Upper and lower bounding-an equivalent circuit viewpoint

An alternative view of the behavior of magnetostatic and time harmonic systems when dual potential formulations are incorporated with a mesh refinement scheme is proposed. Its arguments are based on an equivalent circuit view of the field problem. The bounded nature of solutions is discussed in circuit terms. >

Analytic solutions for axisymmetric magnetostatic systems involving iron

The analytic solutions are given for three simple axisymmetric systems involving toroidal conductors and iron. The solutions can be used for testing and gauging the accuracy of three-dimensional numerical schemes. In each case it is demonstrated how the fields can be calculated from a spherical harmonic expansion. This provides a unified and efficient method for calculating the fields.

Unified approach to problems in electromagnetism

The development of a structure linking the various potentials and field vectors of the electric and magnetic fields is presented. The approach shows that it is usually possible to find alternative formulations, for a given problem, in a relatively simple manner. These alternative formulations, when correctly chosen, pose problems in dual or complementary form such that bounded solutions are possible. Such properties are particularly attractive when numerical solutions are being considered. The proposed structure is general, incorporating both electrostatic and magnetostatic systems as well as time varying field quantities.