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Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.

The purpose of this paper is to present a computationally efficient approach for the stochastic modeling of an inhomogeneous reluctivity of magnetic materials. These materials can be part of electrical machines such as a single-phase transformer (a benchmark example that is considered in this paper). The approach is based on the Karhunen–Loève expansion (KLE). The stochastic model is further used to study the statistics of the self-inductance of the primary coil as a quantity of interest (QoI).

The computation of the KLE requires solving a generalized eigenvalue problem with dense matrices. The eigenvalues and the eigenfunction are computed by using the Lanczos method that needs only matrix vector multiplications. The complexity of performing matrix vector multiplications with dense matrices is reduced by using hierarchical matrices.

The suggested approach is used to study the impact of the spatial variability in the magnetic reluctivity on the QoI. The statistics of this parameter are influenced by the correlation lengths of the random reluctivity. Both, the mean value and the standard deviation increase as the correlation length of the random reluctivity increases.

The KLE, computed by using hierarchical matrices, is used for uncertainty quantification of low frequency electrical machines as a computationally efficient approach in terms of memory requirement, as well as computation time.

A new surface error calculation method for layered manufacturing processes is proposed in this paper. The developed method is used to generate the layers by adaptively varying the thickness of the layers based on the surface approximation errors. Traditionally, the surface errors are calculated using local approximation techniques. In this paper, the surface approximation errors are calculated more accurately by marching through the surface points and determining the distances between layers and the surface points. Using the calculated distances, the adaptive layers are generated for both traditional two‐dimensional layer and ruled‐layer approximation methods. It has been shown that layered manufacturing (rapid prototyping) processes can achieve better accuracy and efficiency using the proposed surface error calculation and the adaptive ruled layer approximation methods. Computer implementation and illustrative examples are also presented in this paper.

The purpose of this paper is to propose an adaptive neural-sliding mode-based observer for the estimation and reconstruction of unknown faults and disturbances for time-varying nonlinear systems such as aircraft, to ensure preciseness in the diagnosis of fault magnitude as well as the shape without enhancement of system complexity and cost. Fault-tolerant control (FTC) strategy based on adaptive neural-sliding mode is also proposed in the existence of faults for ensuring the stability of the faulty system.

In this paper, three strategies are presented: adaptive radial basis functions neural network (ARBFNN), conventional radial basis functions neural network (CRBFNN) and integral-chain differentiator. For the purpose of enhancement of fault diagnosis and isolation, a new sliding mode-based concept is introduced for the weight updating parameters of radial basis functions neural network (RBFNN).The main objective of updating the weight parameters adaptively is to enhance the effectiveness of fault diagnosis and isolation without increasing the computational complexities of the system. Results depict the effectiveness of the proposed ARBFNN approach in fault detection (FD) and approximation compared to CRBFNN, integral-chain differentiator and schemes existing in literature. In the second step, the FTC strategy is presented separately for each observer in the presence of unknown faults and failures for ensuring the stability of the system, which is validated on Boeing 747 100/200 aircraft.

The proposed adaptive neural-sliding mode approach is investigated, which depicts more effectiveness in numerous situations such as faults, disturbances and uncertainties compared to algorithms used in literature. In this paper, both the fault approximation and isolation and the fault tolerance approaches are studied.

For the enhancement of safety level as well as for avoiding any kind of damage, timely FD and fault tolerance have always had a significant role; therefore, the algorithms proposed in this research ensure the tolerance of faults and failures, which plays a vital role in practical life for avoiding any kind of damage.

In this study, a new neural-sliding mode concept is adopted for the adaptive faults approximation and reconstruction, and then the FTC algorithms are studied for each observer separately, whereas in previous studies, only the fault detection and isolation (FDI) or the fault tolerance problems were studied. Results demonstrate the effectiveness of the proposed strategy compared to the approaches given in the literature.

Current rapid prototyping processes are mainly based on layered manufacturing techniques using 2.5D slices. Defines manufacturing by means of 2.5D slices as a zero order approximation. A disadvantage of this approximation is the staircase effect, requiring thin layers to be used. If the outer surfaces of the slices can be inclined, speaks of a first order approximation. This approximation is achieved by linear interpolation between adjacent contours, resulting in ruled slices. Describes a method to approximate a given model geometry in a layered fashion not exceeding a user‐defined error δ using either a zero or a first order approximation and an adaptive layer thickness. Analyses the model geometry for curvature and inclination in order to determine the adaptive layer thickness. Provides a method for matching corresponding contours from adjacent slices. Several test objects have been processed using both zero and first order approximation. Shows that the first order approximation significantly reduces the number of required layers for a given δ when compared to the zero order approximation.

Presents an efficiency study of different refinement procedures for the p‐version of the adaptive finite element method in two‐dimensional elasticity problems. The refinement strategy, based on the estimated error in energy norm, attempts an optimal distribution of the nodeless degrees of freedom associated with the basic approximation parameter of the order p of the hierarchical shape functions. This procedure is combined with appropriate matrix‐handling techniques and equation solvers in order to achieve a solution of a given accuracy with the minimum computational resources in terms of computing time and storage. To this extent, convergence studies are performed with constant and variable adaptivity indices, with error estimators based on global and elemental approaches and with domain decomposition matrix‐handling techniques and the preconditioned conjugate gradient solver.

To reduce the computational complexity per step from O(n²) to O(n) for optimization based on quadratic surrogates, where n is the number of design variables.

Applying nonlinear optimization strategies directly to complex multidisciplinary systems can be prohibitively expensive when the complexity of the simulation codes is large. Increasingly, response surface approximations (RSAs), and specifically quadratic approximations, are being integrated with nonlinear optimizers in order to reduce the CPU time required for the optimization of complex multidisciplinary systems. For evaluation by the optimizer, RSAs provide a computationally inexpensive lower fidelity representation of the system performance. The curse of dimensionality is a major drawback in the implementation of these approximations as the amount of required data grows quadratically with the number n of design variables in the problem. In this paper a novel technique to reduce the magnitude of the sampling from O(n²) to O(n) is presented.

The technique uses prior information to approximate the eigenvectors of the Hessian matrix of the RSA and only requires the eigenvalues to be computed by response surface techniques. The technique is implemented in a sequential approximate optimization algorithm and applied to engineering problems of variable size and characteristics. Results demonstrate that a reduction in the data required per step from O(n²) to O(n) points can be accomplished without significantly compromising the performance of the optimization algorithm.

A reduction in the time (number of system analyses) required per step from O(n²) to O(n) is significant, even more so as n increases. The novelty lies in how only O(n) system analyses can be used to approximate a Hessian matrix whose estimation normally requires O(n²) system analyses.

Current rapid prototyping processes are mainly based on layered manufacturing techniques using 2.5D slices. Defines manufacturing by means of 2.5D slices as a zero order approximation. A disadvantage of this approximation is the staircase effect, requiring thin layers to be used. If the outer surfaces of the slices can be inclined, speaks of a first order approximation. This approximation is achieved by linear interpolation between adjacent contours, resulting in ruled slices. Describes a method to approximate a given model geometry in a layered fashion not exceeding a user defined error δ using either a zero or a first order approximation and an adaptive layer thickness. Analyses the model geometry for curvature and inclination in order to determine the adaptive layer thickness. Provides a method for matching corresponding contours from adjacent slices. Several test objects have been processed using both zero and first order approximation. Shows that the first order approximation significantly reduces the number of required layers for a given δ when compared to the zero order approximation.

The purpose of this paper is to extend the generalized finite‐difference calculus of flexible local approximation methods (FLAME) to problems where local analytical solutions are unavailable.

FLAME uses accurate local approximations of the solution to generate difference schemes with small consistency errors. When local analytical approximations are too complicated, semi‐analytical or numerical ones can be used instead. In the paper, this strategy is applied to electrostatic multi‐particle simulations and to electromagnetic wave propagation and scattering. The FLAME basis is constructed by solving small local finite‐element problems or, alternatively, by a local multipole‐multicenter expansion. As yet another alternative, adaptive FLAME is applied to problems of wave propagation in electromagnetic (photonic) crystals.

Numerical examples demonstrate the high rate of convergence of new five‐ and nine‐point schemes in 2D and seven‐ and 19‐point schemes in 3D. The accuracy of FLAME is much higher than that of the standard FD scheme. This paves the way for solving problems with a large number of particles on relatively coarse grids. FLAME with numerical bases has particular advantages for the multi‐particle model of a random or quasi‐random medium.

Irregular stencils produced by local refinement may adversely affect the accuracy. This drawback could be rectified by least squares FLAME, where the number of stencil nodes can be much greater than the number of basis functions, making the method more robust and less sensitive to the irregularities of the stencils.

Previous applications of FLAME were limited to purely analytical basis functions. The present paper shows that numerical bases can be successfully used in FLAME when analytical ones are not available.