Добірка наукової літератури з теми "Numerical methods, FEM, MOM, singular functions"

This paper presents comparative studies on different numerical methods like method of moments (MOM), Boundary Element Method (BEM), Finite element method (FEM), Finite difference method (FDM), Charge Simulation method (CSM) and Surface charge method. The evaluation of the capacitance of various structures having different geometrical shapes is importance to study the behavior of electrostatic charge analysis. The MOM is based upon the transformation of an integral equation, into a matrix equation by employing expansion of the unknown in terms of known basis functions with unknown coefficients such as charge distribution and hence the capacitance is to be determined. To illustrate the usefulness of this technique, apply these methods to the computation of capacitance of different conducting shapes. This paper reviews the results of computing the capacitance-per-unit length with the other methods. The capacitance of charged conducting plates is reviewed by different methods.

This paper presents a cell-based smoothed extended finite element method (CS-XFEM) to analyze fractures in piezoelectric materials. The method, which combines the cell-based smoothed finite element method (CS-FEM) and the extended finite element method (XFEM), shows advantages of both methods. The crack tip enrichment functions are specially derived to represent the characteristics of the displacement field and electric field around the crack tip in piezoelectric materials. With the help of the smoothing technique, integrating the singular derivatives of the crack tip enrichment functions is avoided by transforming interior integration into boundary integration. This is a significant advantage over XFEM. Numerical examples are presented to highlight the accuracy of the proposed CS-XFEM with the analytical solutions and the XFEM results.

This paper presents an efficient numerical technique for stress analysis of three-dimensional infinite media containing cracks and localized complex regions. To enhance the computational efficiency of the boundary element methods generally found inefficient to treat nonlinearities and non-homogeneous data present within a domain and the finite element method (FEM) potentially demanding substantial computational cost in the modeling of an unbounded medium containing cracks, a coupling procedure exploiting positive features of both the FEM and a symmetric Galerkin boundary element method (SGBEM) is proposed. The former is utilized to model a finite, small part of the domain containing a complex region whereas the latter is employed to treat the remaining unbounded part possibly containing cracks. Use of boundary integral equations to form the key governing equation for the unbounded region offers essential benefits including the reduction of the spatial dimension and the corresponding discretization effort without the domain truncation. In addition, all involved boundary integral equations contain only weakly singular kernels thus allowing continuous interpolation functions to be utilized in the approximation and also easing the numerical integration. Nonlinearities and other complex behaviors within the localized regions are efficiently modeled by utilizing vast features of the FEM. A selected set of results is then reported to demonstrate the accuracy and capability of the technique.

The Finite Element Method (FEM) and Method of Moments (MoM) are popular numerical methods for solving complex problems encountered in almost any field of engineering. The structure to be studied is divided into small cells (2D or 3D) and the relevant equations can be numerically solved. The modes of the structure are typically obtained by solving a generalized eigenvalue equation (FEM) or performing a matrix inversion (MoM); scattering problems formulated in term of integral equations are typically solved with MoM. For problems with smooth surfaces or other regular features, high order finite-element techniques based on the use of (hierarchical) curl-conforming for FEM or divergence-conforming vector bases for MoM successfully improve accuracy and efficiency (greatly reducing the dimension of the matrices and minimizing CPU time). High degree polynomial expansion functions often do not improve the solution accuracy, or do not provide as rapid convergence as anticipated, when dealing with geometries containing sharp edges or corners. The slow convergence observed in these cases is a consequence of the non-analytic nature of the solution in the vicinity of the singular point. To improve the accuracy of these problems, special basis functions are being developed that incorporate the singular field behavior.Conversely, by using a $5^{th}$ or $6^{th} $ order base with only four triangles one obtains far better results than with 2748 triangles and order 0 (“classic FEM” implementation). In the following the reader will be presented with the development of high order polynomial basis for 2D structures (typically waveguide cross sections) in terms of scalar, vector and full-wave analysis; then with especially developed functions which greatly enhance the accuracy of modes that are affected by corner singularities. Chapters 2 and 3 will show the development of high order polynomial bases for 3D FEM (used to study cavities) founded on tetrahedral or triangular prism based cells. Chapter 4 will provide the results from the development of divergence-conforming high order hierarchical singular bases for quadrangular cells.

This dissertation presents new singular curl- and divergence- conforming vector bases that incorporate the edge conditions. Singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester–Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. The curl (divergence) conforming singular bases guarantee tangential (normal) continuity along the edges of the elements allowing for the discontinuity of normal (tangential) components, adequate modeling of the curl (divergence), and removal of spurious modes (solutions). These singular high-order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems. Sample numerical results confirm the faster convergence of these bases on wedge problems.

The paper presents an overview of the smoothed finite element methods (S-FEM) which are formulated by combining the existing standard FEM with the strain smoothing techniques used in the meshfree methods. The S-FEM family includes five models: CS-FEM, NS-FEM, ES-FEM, FS-FEM and α-FEM (a combination of NS-FEM and FEM). It was originally formulated for problems of linear elastic solid mechanics and found to have five major properties: (1) S-FEM models are always “softer” than the standard FEM, offering possibilities to overcome the so-called overly-stiff phenomenon encountered in the standard the FEM models; (2) S-FEM models give more freedom and convenience in constructing shape functions for special purposes or enrichments (e.g, various degree of singular field near the crack-tip, highly oscillating fields, etc.); (3) S-FEM models allow the use of distorted elements and general n-sided polygonal elements; (4) NS-FEM offers a simpler tool to estimate the bounds of solutions for many types of problems; (5) the αFEM can offer solutions of very high accuracy. With these properties, the S-FEM has rapidly attracted interests of many. Studies have been published on theoretical aspects of S-FEMs or modified S-FEMs or the related numerical methods. In addition, the applications of the S-FEM have been also extended to many different areas such as analyses of plate and shell structures, analyses of structures using new materials (piezo, composite, FGM), limit and shakedown analyses, geometrical nonlinear and material nonlinear analyses, acoustic analyses, analyses of singular problems (crack, fracture), and analyses of fluid-structure interaction problems.

The general approach of numerical treatment of integro-differential equation of the flat crack problem is considered. It consists in presenting the crack surface loading as the set of the polynomial functions of two Cartesian coordinates while the corresponding crack surface displacements are chosen as the similar polynomials multiplied by the function of form (FoF) which reflects the required singularity of their behavior. To find the relations matrixes between these two sets a new effective numerical procedure for the integration over the area of arbitrary shape crack is developed. In based on the classical hyper-singular method, i.e. Laplace operator is initially analytically applied to the integral part of equation and the resulting hyper singular equation is subsequently considered. The presented approach can be implemented with any variant of FoF, but Oore-Burns FoF, which was earlier suggested in their famous 3D weight function method, is supposed to be the most accurate and universal. It takes into account all points of crack contour, which provides perfect physical conditionality of the solution, but such FoF is relatively heavy in implementation and of low computational speed. The special procedure is developed for the approximation of the crack contour of arbitrary shape by the circular and straight segments. It allows to easily obtain analytical expression for Oore-Burns FoF, which greatly increases the calculation speed and accuracy. The accuracy of the considered method is confirmed by the examples of the circular, elliptic, semicircular and square cracks at different polynomial laws of loading. The developed methods are used in the implemented procedure for crack growth simulation. It allows to model growth of crack of arbitrary shape at arbitrary polynomial loading, at that all contour points are taken into account and can expand with their own speeds each. Procedure has high accuracy and don’t need complex and high-cost re-meshing process between the iterations unlike FEM or other numerical methods. At that usage of Oore-Burns FoF provides high flexibility of the presented approach: unlike similar theoretical methods, where FoF calculation procedure is rigidly connected with the crack shape, which complicates the adequate crack growth modeling, the used FoF automatically takes into account all points of crack contour, even if its shape became complex during the growth. Presented crack growth procedure can be effectively used to test accuracy and correctness of correspondent numerical methods, including the newest XFEM approach.