S.R.H. Hoole

Inverse problem methodology and finite elements in the identification of cracks, sources, materials, and their geometry in inaccessible locations

Inverse problem methodology is extended, through the more difficult geometric differentiation of finite-element matrices, to identify the location, material, and value of unknown sources within an inaccessible region using exterior measurements. This is done through the definition of an object function that vanishes at its minimum when the externally measured electric field matches the electric field given by an assumed configuration that is optimized to match measurements. The method is demonstrated by identifying the shape, permittivity, charge, and location of an electrostatic source through exterior measurement. The procedure is then extended to eddy current problems for the identification of the location and shape of cracks in metallic structures. An example demonstrates that when dealing with eddy current problems the least squares object function used by others has multiple local minima so that gradient methods have to be combined with search methods to identify the one absolute minimum. Procedures are also given for handling situations with no cracks and overdescribed cracks. >

Artificial neural networks in the solution of inverse electromagnetic field problems

The use of artificial neural networks in the solution of inverse electromagnetic field problems is investigated. It is shown that artificial neural networks, while being no panacea, have a role to play in a limited domain of applications-that is, while it is ineffective to train networks to cover a broad class of devices, it is indeed possible to develop well-trained networks that function effectively over a narrow range of performance of a particular class of device. Particularly if one knows the desired geometry approximately and uses training sets around this geometry, simple neural networks with a few training sets can be used to do an effective job. However, neural networks cannot be used efficiently without such prior knowledge. >

Geometric parametrization and constrained optimization techniques in the design of salient pole synchronous machines

The shape optimization of magnetic devices is efficiently performed with field calculation and sensitivity analysis based on the finite element method. Several sequential unconstrained optimization techniques are discussed and evaluated with respect to their application in engineering design. The optimization of the geometry of a salient pole generator so as to achieve a desired field configuration in the airgap is used as an illustrative numerical example to demonstrate the geometric parametrization technique, emphasize the importance of constraints in engineering design, and highlight the advantageous features of the augmented Lagrangian multiplier method for nonlinear constrained optimization. For the required geometric parametrization a recent novel use of structural mapping is extended to incorporate constrained optimization. The associated equations of structural mapping are presented. >

The subregion method in magnetic field analysis and design optimization

A method that subdivides the finite element solution region into subregions is introduced for the efficient synthesis of magnetic devices. The coefficient matrices of the different subregions are assembled separately and reduced only to those degrees of freedom that are associated with the nodes at subregion interfaces. The reduced matrices of all subregions are used to assemble the final global matrix, which is solved for the reduced system. It is in this repeated analysis of similar field problems that the subregion method is applied with significant computational savings: in synthesis only those subregions that enclose changes in the design have to be assembled and reduced for the modified design. Thus, the computational effort for reanalysis is reduced to the area of design modifications. The subregion approach is successfully applied to the procedure of device synthesis, where a large number of field computations is required in the iterative search for the optimal design. The subregion method is extended to the calculation of the potential gradient directly from the finite element equations. >

Finite element electromagnetic field computation on the Sequent Symmetry 81 parallel computer

The fact that finite-element field analysis algorithms lend themselves ideally to parallelization is exploited to implement a finite-element analysis program for electromagnetic field computation on the Sequence Symmetry 81 parallel computer with three processors. Since in terms of waiting time, the maximum gains are to be made in matrix solution, special attention was given to parallelizing the solution part of the finite-element analysis. An outline of how parallelization could be exploited in most finite element operations is given, although the actual implementation of parallelism on the Sequent Symmetry 81 computer was in sparsity computation, matrix assembly, and the matrix solution areas. In all cases, the algorithms were modified to suit the parallel programming application rather than allowing the compiler to parallelize an existing algorithm. >

Higher finite element derivatives for the quick synthesis of electromagnetic devices

A fast, robust scheme for synthesizing electromagnetic devices from the higher derivatives of object functions is presented. A method of computing derivatives of any given order from a finite element solution employing trial functions of any order is presented. These derivatives are used in expanding the object function in device optimization as a Taylor series. The series truncates after the eighth term or before because of subsequent Taylor terms being vanishingly small in the vicinity of the minimum. This series is used to predict the optimum of the device by solving the single polynomial equation by Newtons scheme. The procedure is shown to result in much faster convergence towards the optimum. >

The impedance boundary condition in the boundary element-vector potential formulation

The impedance boundary condition has been used with great profit to eliminate large regions from the solution to solve eddy current problems. In the boundary element formulation, the reduction of the order of singularities is convenient, if not critical. A vector-potential boundary-element formulation with the impedance boundary condition has a lower order of singularity in relation to the magnetic field intensity and has computational simplicity compared to the finite-element implementation. It offers great advantages in open boundary problems. Some important lessons are offered for those accustomed to finite-element formulations. >

Experimental validation of the impedance boundary condition and a review of its limitations

The impedance boundary condition, which is an approximate boundary condition applicable at the surfaces of materials experiencing pronounced skin effect, is discussed. Its use allows the elimination of bodies made of such materials from the field solution region at great savings in cost. However, the boundary condition is an approximation that is applicable only under certain limited conditions and is justified from Maxwells equations. The author validates the impedance boundary condition using a laboratory experiment and reviews its limitations so that its proper use can be established. >

Reflections off aircraft and the shape optimization of a ridged waveguide

The analysis of TE and TM waves traveling down a guide is a well-known art. However, it is the inverse problem that is more relevant in industrial design. That is, for a given cut-off frequency or attenuation limit, to synthesize the guide. The methodology for solving this inverse problem is laid down and demonstrated using a ridged guide as example. The procedure relies on defining an object function and minimizing it using its gradients with respect to the parameters of design interest. >

Flux density and energy perturbations in adaptive finite element mesh generation

A simple, easy-to-use scheme for the adaptive refinement of finite element meshes is introduced. The changes in the flux densities and energy contributions of finite elements from iteration to iteration are monitored, and elements in which the changes are more than a preassigned percentage are selected for refinement. Working with first-order triangular elements, whenever a triangle has to be refined, each of the three edges of the triangle is cut into two by placing a node in the middle. This scheme presents a natural extension of the nodal perturbation scheme. It is shown that combining the three criteria results in more accuracy, and studiedly moving the location of a new node on an edge away from the middle reduces the computational cost. >