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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 Motz problem can be considered as a benchmark problem for testing the performance of numerical methods in the solution of elliptic problems with boundary singularities. The purpose of this paper is to address the solution of the Motz problem using the radial basis function (RBF) method, which is a truly meshfree scheme.

Both the global RBF collocation method (also known as Kansa's method) and the recently proposed local RBF‐based differential quadrature (LRBFDQ) method are considered. In both cases, it is shown that the accuracy of the solution can be significantly increased by using special functions which capture the behavior of the singularity. In the case of global collocation, the functional space spanned by the RBF is enlarged by adding singular functions which capture the behavior of the local singular solution. In the case of local collocation, the problem is modified appropriately in order to eliminate the singularities from the formulation.

The paper shows that the exponential convergence both with increasing resolution and increasing shape parameter, which is typical of the RBF method, is lost in problems containing singularities. The accuracy of the solution can be increased by collocation of the partial differential equation (PDE) at boundary nodes. However, in order to restore the exponential convergence of the RBF method, it is necessary to use special functions which capture the behavior of the solution near the discontinuity.

The paper uses Motz's problem as a prototype for problems described by elliptic partial differential equations with boundary singularities. However, the results obtained in the paper are applicable to a wide range of problems containing boundaries with conditions which change from Dirichlet to Neumann, thus leading to singularities in the first derivatives.

The paper shows that both the global RBF collocation method and the LRBFDQ method, are truly meshless methods which can be very useful for the solution of elliptic problems with boundary singularities. In particular, when complemented with special functions that capture the behavior of the solution near the discontinuity, the method exhibits exponential convergence both with resolution and with shape parameter.

We compare a recently proposed generalized eigensystem approach and a new modified generalized eigensystem approach to more widely used truncated singular value decomposition and zero‐order Tikhonov regularization for solving multidimensional elliptic inverse problems. As a test case, we use a finite element representation of a homogeneous eccentric spheres model of the inverse problem of electrocardiography. Special attention is paid to numerical issues of accuracy, convergence, and robustness. While the new generalized eigensystem methods are substantially more demanding computationally, they exhibit improved accuracy and convergence compared with widely used methods and offer substantially better robustness.

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.

The real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem.

To generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh.

Adaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS.

The proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes.

Although many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh.

In spite of the vehicle, magnetic field interference can be reduced by some measures and techniques in ammunition design and manufacturing stage, the corruption of the vehicle magnetic field can still reach hundreds to thousands of nanoteslas. Besides, the magnetic field that the ferromagnetic materials generate in response to the strong magnetic field in the vicinity of the body. So, a real-time and accurate vehicle magnetic field calibration method is needed to improve the real-time measurement accuracy of the geomagnetic field for spinning projectiles.

Unlike the past two-step calibration method, the algorithm uses a linear model to calibrate the magnetic measurement error in real-time. In the method, the elliptical model of magnetometer measurement is established to convert the coefficients of hard and soft iron errors into the parameters of the elliptic equation. Then, the parameters are estimated by recursive least square estimator in real-time. Finally, the initial conditions for the estimator are established using prior knowledge method or static calibration method.

Studies show the proposed algorithm has remarkable estimation accuracy and robustness and it realizes calibration the magnetic measurement error in real-time. A turntable experiments indicate that the post-calibration residuals approximate the measurement noise of the magnetometer and the roll accuracy is better than 1°. The algorithm is restricted to biaxial magnetometers’ calibration in real-time as expressed in this paper. It, however, should be possible to broaden this method’s applicability to triaxial magnetometers' calibration in real-time.

Unlike the past two-step calibration method, the algorithm uses a linear model to calibrate the magnetic measurement error in real-time and the calculation is small. Besides, it does not take up storage space. The proposed algorithm has remarkable estimation accuracy and robustness and it realizes calibration the magnetic measurement error in real time. The algorithm is restricted to biaxial magnetometers’ calibration in real-time as expressed in this paper. It, however, should be possible to broaden this method’s applicability to triaxial magnetometers’ calibration in real-time.

It has been well recognized that interface problems often contain strong singularities which make conventional numerical approaches such as uniform h‐ or p‐version of finite element methods (FEMs) inefficient. In this paper, the partition‐of‐unity finite element method (PUFEM) is applied to obtain solution for interface problems with severe singularities. In the present approach, asymptotical expansions of the analytical solutions near the interface singularities are employed to enhance the accuracy of the solution. Three different enrichment schemes for interface problems are presented, and their performances are studied. Compared to other numerical approaches such as h‐p version of FEM, the main advantages of the present method include: easy and simple formulation; highly flexible enrichment configurations; no special treatment needed for numerical integration and boundary conditions; and highly effective in terms of computational efficiency. Numerical examples are included to illustrate the robustness and performance of the three schemes in conjunction with uniform h‐ or p‐refinements. It shows that the present PUFEM formulations can significantly improve the accuracy of solution. Very often, improved convergence rate is obtained through enrichment in conjunction with p‐refinement.

The purpose of this paper is to develop pseudospectral meshless radial point Hermit interpolation (PSMRPHI) for applying to the Motz problem.

The author aims to propose a kind of PSMRPHI method.

Based on the Motz problem, the author aims also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods.

Although the PSMRPHI method has been infrequently used in applications, the author proves it is more accurate and trustworthy than the PSMRPI method.

A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique.

By linearizing the free vibration problem of elastic membranes into a series of linear equivalent problems, reliable a posteriori point-wise error estimator is constructed via EEP super-convergent technique. Hierarchical local mesh refinement is incorporated to better deal with tough problems.

Several classical examples were analyzed, confirming the effectiveness of the EEP-based error estimation and overall adaptive procedure equipped with a local mesh refinement scheme. The computational results show that the adaptively-generated meshes reasonably catch the difficulties inherent in the problems and the procedure yields both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm.

By reasonable linearization, the linear-problem-based EEP technique is successfully transferred to two-dimensional eigenproblems with local mesh refinement incorporated to effectively and flexibly deal with singularity problems. The corresponding adaptive strategy can produce both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm and thus can be expected to apply to other types of eigenproblems.

The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry).

This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes.

The significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry).

It has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.