Uwe Pahner

A parametric finite element environment tuned for numerical optimization

Nowadays, numerical optimization in combination with finite element (FE) analysis plays an important role in the design of electromagnetic devices. To apply any kind of optimization algorithm, a parametric description of the FE problem is required and the optimization task must be formulated. Most optimization tasks described in the literature, feature either specially developed algorithms for a specific optimization task, or extensions to standard finite element packages. Here, a 2D parametric FE environment is presented, which is designed to be best suited for numerical optimization while maintaining its general applicability. Particular attention is paid to the symbolic description of the model, minimized computation time and the user friendly definition of the optimization task.

Adaptive coupling of differential evolution and multiquadrics approximation for the tuning of the optimization process

Recently, the combination of global convergent stochastic search methods with approximation schemes based on radial basis functions has been introduced. This paper presents a new approach: instead of a procedural sequencing of the approximation algorithm and optimization algorithm, this optimization scheme is characterized by a direct and adaptive coupling of both algorithms. An approximation of the feasible space is constructed and updated during the progress of the evolutionary search. If the approximation fulfils particular accuracy criteria, the evolutionary search algorithm starts sampling the approximation (indirect search) instead of directly sampling the objective function. This can lead to a significant reduction of function calls, which is desirable if the function evaluation is computational expensive (e.g. involving finite element analysis steps).

Computation Of A Highly Saturated Permanent Magnet Synchronous Motor For A Hybrid Electric Vehicle

In the framework of the development of a drive system for the implementation in a hybrid electric vehicle, a 45 kW 6-pole permanent magnet synchronous motor (PMSM) is designed. Due to the rotor design with inset permanent magnets the machine parameters are dependant on the operating point due to saturation effects. Numerical computations using the finite element method (FEM) are performed to build up a lumped parameter model of the machine.

Optimizing ferromagnetic characteristics to reduce Newton iterations

The standard Newton iteration scheme to solve a non‐linear system of equations obtained from the finite element methods is based on the updating of the field dependent element reluctivity. Usually, the manufacturer of the ferromagnetic material provides a BH‐characteristic as diagram or in the form of a table of data samples. The influence of the material properties, in particular their accurate numerical representation, is significant for the rate of convergence during the Newton iterations. Here, a numerical optimization aiming at a technically smooth non‐linear characteristic is performed to obtain a higher rate of convergence of the Newton iteration scheme.

A parallel implementation of a parametric optimization environment-numerical optimization of an inductor for traction drive systems

Optimum design is defined as a design that is the best possible solution. All design variables are determined simultaneously to satisfy a set of constraints and optimize a set of objectives. Two parametric FE pre-processors and a general purpose optimization environment are presented. Due to its open architecture, finite element as well as analytical models can be implemented. Stochastic algorithms usually require substantially more function evaluations compared to gradient methods, which increases the elapsed computation time. However, the stochastic algorithms feature unmatched simplicity in the setup of an optimization model. A parallel implementation of the evolution strategy is presented, which offers one way to reduce the elapsed computation time. An optimization task is discussed to outline the general application range of the developed tools. The optimum design of an inductor used in a traction drive system is described in detail. Special attention is paid to the formulation of the quality function.

Different Formulations in Axisymmetric Magnetostatic Problems

In the paper a number of formulations for axisymmetric finite element analysis are compared. Different types of auxiliary potentials and the accuracy of the results with these potentials are discussed. Special attention will be paid to a new and very simple way of enhancing the local accuracy of the post-processing results near the axis. A model containing high permeability near the axis is taken as example for demonstrating the benefits of the new mixed potential as well as the new post-processing approach.

Selected projection methods to improve the convergence of non‐linear problems

Short computation times required for the design and numerical optimisation of electromagnetic devices with the finite element method are obtained using an adaptive mesh refinement algorithm. Less time is spent on the initial mesh generation, while the time needed for refinement is negligible when special data structures are used. But an even more significant reduction in total computation time is achieved with the initialisation of the solution on the generated mesh. Fewer Newton steps are required to solve non‐linear problems compared to a zero initial solution, while the time needed for the projection of the solution is far less than the time needed for a Newton step.

A Combined Finite Element — Circuit Model of a Squirrel-Cage Induction Motor

The paper describes the results of a squirrel-cage induction motor analysis taking saturation and motor end effects into account. The main part of the induction motor analysis is formed by an iterative method earlier developed by Belmans et al.1 allowing the inclusion of saturation in the solution. The method can be summarised as follows: The procedure starts (for current-driven mode of operation) by solving two static non-linear problems, with real and imaginary part of the stator currents as input. From these solutions an averaged reluctivity vector is generated. This vector is used with the time-harmonic solver to obtain a first estimate of the induced rotor currents. After the time-harmonic solving, two new static problems can be defined, but now with the real and imaginary part of both stator and rotor currents as input. A more accurate reluctivity vector can be extracted. This procedure is repeated until convergence is reached. For the voltage-driven mode, the procedure starts with a time-harmonic solution generating a first estimate of both stator and rotor currents. During the time-harmonic solution, end effects of the motor are taken into account. This is done by linking the finite element model with external sources and impedances 2. The time-harmonic solver solves both the finite element model and the circuit equations from the external circuit. Temperature effects are included by a variation of the conductivity of the stator winding and the rotor conductors. Since the end-effect parameters are introduced in the model as lumped parameters, values must be assigned to them. These values are calculated from analytical formulas or using a finite-element approach.

Highly accurate 3D field gradient computation using local post-solving

To enhance the accuracy of finite element based computations of field quantities and their derivatives, a post-solving technique with superconvergent properties is presented. This technique uses an analytical expression for the magnetic field potential inside a closed region. The coefficients of the analytical expression are evaluated using a FE solution as boundary condition. All second derivatives of the analytical expression are calculated, leading to values for the flux density B and its derivatives. These values are highly accurate, even when based upon a FE solution. As an example, the (edge) write gradient in a notch write head is evaluated.

The numerical optimization of an inductor for traction drive systems-a parametric optimization environment

Optimum design is defined as a design that is feasible and the best solution possible. All design variables are determined simultaneously so as to satisfy a set of constraints and optimize a set of objectives. Here, a parametric preprocessor and a general purpose optimization environment are presented. Both are implemented in MATLAB, providing the graphical interface and controlling external processes. Due to the open architecture of the package, finite element as well as analytical models can be implemented. An optimization task is discussed to outline the general application range of the tools. The optimum design of an inductor used in a traction drive system is described in detail. Special attention is paid to the formulation of the quality function.