Hybrid finite element-boundary integral algorithm to solve the problem of scattering from a finite array of cavities with multilayer stratified dielectric coating

This work presents a hybrid finite element-boundary integral algorithm to solve the problem of scattering from a finite array of two-dimensional cavities engraved in a perfectly electric conducting screen covered with multilayer stratified dielectric coating. The solution region is divided into interior regions containing the cavities and the region exterior to the cavities. The finite element formulation is applied only inside the interior regions to derive a linear system of equations associated with unknown field values. Using a two-boundary formulation, the surface integral equation employing a closed-form multilayer Green's function in the spatial domain is applied at the opening of the cavities as a boundary constraint to truncate the solution region. The closed-form Green's function in the spatial domain for multilayer planar coating is expressed in terms of complex images using the generalized pencil-of-function method in conjunction with a two-level sampling approach. Placing the truncation boundary at the opening of the cavities and inside the dielectric coating results in a highly efficient solution in terms of computational resources, which makes the algorithm well suited for optimization problems involving scattering from grating surfaces. The near fields are generated for array of cavities with different dimensions and inhomogeneous fillings covered with dielectric layers.     

Symmetric Elasticity Problem for a Layer with Tunnel Cavities Covered with a Diaphragm

A new procedure is proposed for the solution of mixed three-dimensional symmetric problems of the theory of elasticity for layers weakened by through tunnel cavities. The boundary-value problem is reduced to a system of 3k (k = 1, 2, ...) one-dimensional singular integral equations. The results of numerical evaluation of the characteristic stresses are presented.     

Hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of cavities with stratified dielectric coating

This work presents a hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of two-dimensional cavities engraved in a perfectly electric conducting screen covered with a stratified dielectric layer. The solution region is divided into interior regions containing the cavities and the region exterior to the cavities. The finite-element formulation is applied only inside the interior regions to derive a linear system of equations associated with unknown field values. Using a two-boundary formulation, the surface integral equation employing the grounded dielectric slab Green's function in the spatial domain is applied at the opening of the cavities as a boundary constraint to truncate the solution region. Placing the truncation boundary at the opening of the cavities and inside the dielectric layer results in a highly efficient solution in terms of computational resources, which makes the algorithm well suited for the optimization problems involving scattering from grating surfaces. The near fields are generated for an array of cavities with different dimensions and inhomogeneous fillings covered with dielectric layers.     

Sounding of finite solid bodies by way of topological derivative

This paper is concerned with an application of the concept of topological derivative to elastic‐wave imaging of finite solid bodies containing cavities. Building on the approach originally proposed in the (elastostatic) theory of shape optimization, the topological derivative, which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity‐free (reference) body, is used as a void indicator through an assembly of sampling points where it attains negative values. The computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary‐value problems for the reference body, which are here formulated as boundary integral equations. For a comprehensive treatment of the subject, formulas for topological sensitivity are obtained using three alternative methodologies, namely (i) direct differentiation approach, (ii) adjoint field method, and (iii) limiting form of the shape sensitivity analysis. The competing techniques are further shown to lead to distinct computational procedures. Methodologies (i) and (ii) are implemented within a BEM‐based platform and validated against an analytical solution. A set of numerical results is included to illustrate the utility of topological derivative for 3D elastic‐wave sounding of solid bodies; an approach that may perform best when used as a pre‐conditioning tool for more accurate, gradient‐based imaging algorithms. Despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution, the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

Master equation for misaligned cavities

Misaligned cavities can be described by a 3 × 3 extension of the well-known ABCD matrix approach to aligned cavities. There is a corresponding Huygens integral (HI) formulation for the evolution of the field within such a cavity. We derive a differential ‘master’ equation (ME) formulation which yields identical eigenmodes and eigenvalues to this HI. This ME is fully equivalent to the HI for linear problems, while providing a more convenient route to the analysis of nonlinear behaviour in such cavities.     

Solving kinematics problems by efficient interval partitioning

In this study, a generic solution methodology is presented for solving constraint satisfaction problems with smooth continuous constraint functions. We apply the proposed method in solving small scale kinematics problems with the goal of identifying all real solutions to a given problem. The developed approach assumes a collaborative methodology that integrates interval partitioning, a new interval inference method, and local search. The resulting methodology aims at reducing the search space and discarding infeasible sub-spaces effectively and reliably during the early stages of the search so that exact solutions can be identified faster by local methods.     

Green's function method for axisymmetric flows: analysis of the Taylor-Couette flow

A numerical method for the solution of the Navier-Stokes equations in rotationally symmetric flow problems is presented. The numerical procedure is based on a boundary integral equation formulation with the fundamental solution of the Stokes' equation accounting for the rotational symmetry. The proposed methodology has been applied to the study of the Taylor-Couette flow between two concentric rotating cylinders of infinite axial length. A comparison with the available theoretical, experimental or numerical findings is performed to evaluate the accuracy of the present results. As predicted by the analytical theory and confirmed by the experiments, multiple solutions that are found for Reynolds numbers higher than the critical value, indicate the proposed methodology as a useful tool to get physical insight on the instabilities occurring in the solution of the Navier-Stokes equations.     

Null-field Integral Equations for Stress Field around Circular Holes under Antiplane Shear

In this paper, we derive the null-field integral equation for a medium containing circular cavities with arbitrary radii and positions under uniformly remote shear. To fully capture the circular geometries, separate expressions of fundamental solutions in the polar coordinate and Fourier series for boundary densities are adopted. By moving the null-field point to the boundary, singular integrals are transformed to series sums after introducing the concept of degenerate kernels. The solution is formulated in a manner of a semi-analytical form since error purely attributes to the truncation of Fourier series. The two-hole problems are revisited to demonstrate the validity of our method. The bounded-domain approaches using either displacement or stress approaches are also employed. The proposed formulation has been generalized to multiple cavities in a straightforward way without any difficulty.     

Transient heat conduction analysis of solids with small open-ended tubular cavities by boundary face method

This paper applies the boundary face method (BFM) to solve transient heat conduction problems for the first time. Rather than using a transformation scheme, a direct solution of the boundary integral equation (BIE) with time domain fundamental solution is performed in this application. To avoid the domain integrals, the boundary integral equation is solved by the time stepping convolution method. For problems on structures that contain a large number of open-ended tubular shaped cavities in small diameters, a curvilinear tube element is employed to approximate the variables on the cavity surface. Furthermore, to perform integration and boundary variable approximation on the end faces that are intersected by the tubular cavity, a triangular element with negative part is adopted. With the two types of specified elements, the BFM is implemented to solve transient heat conduction problems on structures with open-ended tubular shaped cavities of small size which are usually inconvenience in finite element implementations. Three numerical examples on different structures are presented to illustrate the validity and efficiency of the method.     

Boundary integral Equations Method in two- and three-dimensional problems of elastodynamics

In this paper the boundary integral equations method (BIEM) are considered for elastodynamic initial boundary value problems. It's known two approaches are discerned for account time. First of one is a combination of BIEM with Laplace (Fourier) transformation. This approach was suggested and realized by Cruse T.E. and Rizzo F. J. By them BIE in Laplace transformation space were obtained, investigated and some concrete problems were solved. This method was developed also by Manolis G. D., Beskos D. and other scholars for some dynamic problems solving.The second approach using retarding potentials was considered by Brebbia C. A., Fujiki K., Fukui T., Kato S., Kishima T., Kobayashi S., Nishimura N., Niwa Y., Manolis G. D. Mansur W.J. (for 2D elastodynamics), Chutoryansky N.M. (for 3D elastodynamics). Detailed review of abroad scholars elaborating BIEM was made by Beskos D. [7].This paper discusses BIEM for 2 and 3D elastodynamics on the base of the second approach. The fundamental solutions, integral representations and boundary integral equations are constructed by means distributions theory for the general case of anisotropic elastic media. It's suggested some new results concerning special regularization of singularities on the wave fronts of the integral equations kernels. The illustrative numerical examples concern the scattering of elastic waves on cavities embedded in an infinite isotropic medium. So, it's shown the numerical results of waves diffraction on the one and two cavities of arched and rectangular forms in 2 and 3D cases. These results show quite stability of the elaborating algorithm.     

Method of effective stress field in three-dimensional problems of interaction of plane cracks

We propose a modification of the method of boundary integral equations capable of the efficient solution of the problems of interaction of large numbers of arbitrarily oriented plane cracks on the engineering level. The approach is based on the determination of the effective stress field formed in the vicinity of a fixed crack by neighboring cracks interacting with this crack. The reliability of the results obtained by the method of effective stress field is checked by comparing with the exact solution of the problem of interaction of two plane circular cracks for different mutual orientations of the cracks. The efficiency of the proposed approach is illustrated by an example of interaction of an aperiodic system of six cracks located in different planes.     

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

5th Biennial IEEE Conference on Electromagnetic Field Computation(CEFC)

The following topics were dealt with: formulations/methodology; biomedical applications; power devices; edge elements; boundary elements/integral methods; materials; electroheating; waveguides and cavities; scattering and microwave fields; strip lines; finite elements and electric circuits; time domain simulation; transients; inverse problems and optimisation; boundary conditions; lightning; nondestructive evaluation; eddy currents; meshes; software; neural networks; knowledge engineering; nonlinear and iterative problems; postprocessing; force computation; radiation; electric field problems; and magnetic fields     

Numerical analysis of the efficiency of multilayer-coated gratings using integral method

An analysis is made of a rigorous and an approximate approach to the solution of the diffraction problem for a multilayer-coated X-ray grating by the integral equation formalism. Whereas a rigorous analysis involving the integral method requires a lot of computer resources, even for gratings with a small number of layers, the approximate approach based on a modification of the solution of the integral equation at the lower boundary with a finite conductivity is practically independent of the number of layers and is readily tractable with the use of a standard PC. The efficiencies of multilayer gratings measured at grazing angles with synchrotron soft X-ray radiation are compared with the values calculated using the integral approaches for ideal groove profiles.     

A DECOMPOSITION APPROACH TO PROJECT COMPRESSION WITH CONCAVE ACTIVITY COST FUNCTIONS

This article develops a solution methodology for project time compression problems in CPM/PERT type networks with convex or concave activity cost-duration functions. The proposed procedure actually approximates these relationships by piece-wise linear time-cost curves. The solution procedure is based on the Benders decomposition approach and seeks to minimize die total direct cost of a project subject to activity precedence relationships, as well as upper/lower bounds on activity durations. The computational efficiency of the proposed decomposition methodology is also discussed.     

A variational formulation in fracture mechanics

A variational approach to linear elasticity problems is considered. The family of variational principles is proposed based on the linear theory of elasticity and the method of integrodifferential relations. The idea of this approach is that the constitutive relation is specified by an integral equality instead of the local Hooke’s law and the modified boundary value problem is reduced to the minimization of a nonnegative functional over all admissible displacements and equilibrium stresses. The conditions of decomposition on two separated problems with respect to displacements and stresses are found for the variational problems formulated and the relation between the approach under consideration and the minimum principles for potential and complementary energies is shown. The effective local and integral criteria of solution quality are proposed. A numerical algorithm based on the piecewise polynomial approximations of displacement and stress fields over an arbitrary domain triangulation are worked out to obtained numerical solutions and estimate their convergence rates. Numerical results for 2D linear elasticity problems with cracks are presented and discussed.     

A simple and efficient XFEM approach for 3-D cracks simulations

In this work, a simple and efficient XFEM approach has been presented to solve 3-D crack problems in linear elastic materials. In XFEM, displacement approximation is enriched by additional functions using the concept of partition of unity. In the proposed approach, a crack front is divided into a number of piecewise curve segments to avoid an iterative solution. A nearest point on the crack front from an arbitrary (Gauss) point is obtained for each crack segment. In crack front elements, the level set functions are approximated by higher order shape functions which assure the accurate modeling of the crack front. The values of stress intensity factors are obtained from XFEM solution by domain based interaction integral approach. Many benchmark crack problems are solved by the proposed XFEM approach. A convergence study has been conducted for few test problems. The results obtained by proposed XFEM approach are compared with the analytical/reference solutions.     

A VDQ-based multifield approach to the 2D compressible nonlinear elasticity

A numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu-Washizu principle, considering displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix-vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ-based approach. Some well-known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.     

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.     

Analysis of an orthotropic strip containing multiple defects bonded between two piezoelectric layers

In this paper, the problem of multiple defects in an orthotropic layer bonded between two piezoelectric layers is considered. The analysis is based on the stress fields caused by Volterra-type screw dislocation in the orthotropic strip. The solution for the dislocation is obtained by means of the complex Fourier transform. The dislocation solution is then employed as strain nuclei to derive singular integral equations for a medium weakened by multiple defects. These equations, as a class of Cauchy singular equations, are solved numerically for dislocation density functions. A number of examples is given for various crack orientations and material properties. At the end, it is shown that the effect of the properties and defect geometries on the stress intensity factors and hoop stress for cavities can be highly significant.