Werner Renhart

Pareto optimality and particle swarm optimization

Real-world optimization problems often require the minimization/maximization of more than one objective, which, in general, conflict with each other. These problems (multiobjective optimization problems, vector optimization problems) are usually treated by using weighted sums or other decision-making schemes. An alternative way is to look for the pareto-optimal front. In this paper, the particle swarm algorithm is modified to detect the pareto-optimal front.

Numerical analysis of 3D magnetostatic fields

Formulations of three-dimensional magnetostatic fields are reviewed for their finite-element analysis. Partial differential equations and boundary conditions are set up for various kinds of potentials. Besides the method using two scalar potentials, several vector potential formulations are also discussed. Galerkin techniques combined with the finite element method are applied for the numerical solution of the boundary value problems. The effect of gauging the vector potential upon the numerical performance is investigated. Solutions by different formulations to a simple test problem and a benchmark problem involving relatively thin saturated iron plates are presented. The latter is compared to measured results. >

Performance of different vector potential formulations in solving multiply connected 3-D eddy current problems

The authors describe their numerical experiences in applying FEM (finite-element method) solution techniques to a 3-D (three-dimensional) eddy-current problem with a coil-driven multiply connected conductor, the benchmark problem No.7 of the International TEAM Workshops. Several formulations have been tried using a magnetic vector and electric scalar potential or an electric vector and a magnetic scalar in the conductor and a magnetic vector or scalar potential outside. The problem has been solved at two frequencies. The authors briefly describe the formulations used and compare the performance. Magnetic field and current density plots are also compared. The advantages and disadvantages of the various versions are pointed out. The use of a magnetic scalar potential H rather than a magnetic vector potential A outside the conductor and the hole substantially reduces the number of degrees of freedom and thus the computational effort. The versions using it in the conductor yield relatively ill-conditioned systems. Also, at the higher frequency, the conditioning deteriorates considerably. >

Computation of 3-D current driven skin effect problems using a current vector potential

A finite element formulation of current-driven eddy current problems in terms of a current vector potential and a magnetic scalar potential is developed. Since the traditional T- Omega method enforces zero net current in conductors, an impressed current vector potential T/sub 0/ is introduced in both conducting and nonconducting regions, describing an arbitrary current distribution with the prescribed net current in each conductor. The function T/sub 0/ is represented by edge elements, while nodal elements are used to approximate the current vector potential and the magnetic scalar potential. The tangential component of T is set to zero on the conductor-nonconductor interfaces. The method is validated by computing the solution to an axisymmetric problem. Problems involving a coil with several turns wound around an iron core are solved. >

SMES Optimization Benchmark Extended: Introducing Pareto Optimal Solutions Into TEAM22

In 1996, a superconducting magnetic energy storage arrangement was selected to become a benchmark problem for testing different optimization algorithms, both deterministic and stochastic ones. Since the forward problem can be solved semianalytically by Biot-Savarts law, this benchmark became quite popular. Nevertheless, the demands on optimization software have increased dramatically since then. To give an example, methods looking for Pareto-optimal points rather than for a single solution only have been introduced by several groups. In this paper, a proposal for an extended version of the benchmark problem will be made and some results will be presented.

Higher order evolution strategies for the global optimization of electromagnetic devices

Basic evolution strategies (ESs) utilizing simplified features of biological evolution like mutation and selection are applied to solve problems of parameter identification for the optimal design of electromagnetic devices. Their main advantage lies in their stable convergence behavior and their self-adapting stepwidth sigma . Higher order ( mu / rho , lambda ) evolution strategies can be successfully applied to multimodel problems. To ensure at least a result very near the global optimum, a theory of disaster can be introduced. After a certain number of standard generations, the stepwidth sigma is increased greatly for a specified number of generations. Descendants arrived at during these generations are more likely to escape a local minimum than ordinary descendants. ( mu / rho , lambda ) evolution strategies can be easily implemented on parallel computers, helping to reduce the real-time requirements for one generation. >

A finite element formulation for eddy current carrying ferromagnetic thin sheets

A finite element formulation for ferromagnetic thin sheets carrying time-harmonic eddy currents is presented. The static field outside the sheets is described by a magnetic scalar potential on one side of the sheet and by a magnetic vector potential on the other side. Special interface conditions resulting in a symmetric finite element system matrix are developed to take account of the sheet. A simple example is presented to demonstrate the efficiency of the method.

Application of eddy current formulations to magnetic resonance imaging

The calculation of 3-D radio-frequency (RF) fields with the aid of an eddy current formulation based on finite elements is described. It is shown that the displacement current density in the first Maxwell equation can be taken into account by making the conductivity a complex quantity. The validity of this formulation has been verified by calculating a conducting sphere influenced by a homogeneous time-harmonic RF field. The numerical results are compared with an analytical solution. The formulation described has been applied to calculate a 3-D phantom with regions of different conducting materials. For this phantom, measured results obtained by magnetic resonance imaging are compared. >

Robust target functions in electromagnetic design

Uncertainties in the design variables of non‐linear engineering optimization problems are often neglected. That could result in considerable deterioration of the target function value of an implemented design compared with the computed optimal solution. This effect can be reduced with robust optimization, where it is tried to achieve robust designs by actively embedding the uncertainties and robustness measures in the optimization process. A methodology for robust optimization of non‐linear problems is presented, including practical methods for the solution of such programs. The benefits of the approach are discussed in a numerical field calculation example.

Optimization of SMES solenoids with regard to their stray fields

Solenoidal SMES (Superconducting Magnetic Energy Storage) usually suffer form their remarkable stray field. A certain arrangement was devised that should reduce the extension of the far ranging stray field. Hence, a second solenoid was placed outside the SMES device with a current flowing in the opposite direction compared to the inner coil. Configurations with both coils in a single plane were regarded only. As an important factor the minimum energy to be stored was given. In a first step calculation, methods have been found that reveal data for the minimum geometrical dimensions of the SMES device in an analytical way. It was shown that the stray field, compared to a single coil arrangement could be reduced remarkably. Evolution strategies and a conjugate gradient method were used to improve the quality of the above solution. >