Boundary Integral Equation Analysis Research Articles

Vibrations of a strip with delamination in the framework of the one-parameter Aifantis model of gradient elasticity theory

The problems on in-plane and anti-planar steady-state vibrations of an isotropic elastic strip with delamination at the lower boundary has been investigated. The goal of the study is to analyze the stress-strain state in the crack tips areas and to construct a crack opening function being the main mechanical characteristics in the crack theory problems. The problems under study have been solved in the framework of the nonclassical gradient elasticity theory (GET) on the basis of the one-parameter model proposed by Aifantis. The boundary integral equations (BIE) are obtained with respect to crack opening functions or their derivatives. The analysis of BIEs is carried out, regular and irregular parts are distinguished, the obtained BIEs with singular (e.g., with hypersingular, with cubic singularity) integrals are solved via collocation methods, approximating Chebyshev polynomials, quadrature formulas for singular integrals. For the in-plane problem solution, the simplified Ru-Aifantis method has been applied. The Ru-Aifantis method allows to divide the initial boundary value problem into two sub-problems - the classical linear elasticity theory (LTE) problem and the simplified boundary value problem for finding the gradient solution which includes the solution found via the classical theory. For each of the problems, semi-analytical expressions for the functions of crack opening have been constructed, and the analysis of the stress-strain state in the area of crack tips has been carried out. The problems have also been solved in the case of a crack with small relative length, the analysis of BIE depending on small parameters ratio has been carried out, and explicit expressions for the crack opening functions have been obtained. Numerical calculations have been performed; the applicability conditions for the asymptotic method are determined, and a comparative analysis of the results obtained on the basis of GET and LTE models, depending on the values of the gradient parameter and the delamination length, is realized.

Shape optimization of heterogeneous materials based on isogeometric boundary element method

In this paper, the isogeometric boundary element method (IGABEM) is used to optimize the shape of heterogeneous materials. In contrast to the isogeometric finite element method (IGAFEM), the iterative optimization algorithm based on IGABEM can be implemented directly from Computer-Aided Design (CAD) without returning the optimization results to CAD designers. The discontinuous element method is extended to IGABEM which deals with corner point problems of inhomogeneous materials. After the singularity of sensitivity analysis of boundary integral equations is demonstrated, the power series expansion method (PSEM) is applied to IGABEM to evaluate various degrees of singularity about sensitivity analysis, which shows more accuracy and efficiency than the element sub-division method (ESDM). A set of control points on the geometric boundary are chosen as design variables, which can be passed from the design model to the analysis model, and the objective function is the elastic energy increment. Finally, several numerical examples in 2D and 3D problems are presented to demonstrate the validity and robustness of the present method.

First-Kind Boundary Integral Equations for the Hodge--Helmholtz Operator

We adapt the variational approach to the analysis of first-kind boundary integral equations associated with strongly elliptic partial differential operators from [M. Costabel, SIAM J. Math. Anal., ...

Runge–Kutta convolution coercivity and its use for time-dependent boundary integral equations

Abstract A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretizations. It is known that this coercivity property is inherited by convolution quadrature time discretization based on A-stable multistep methods, which are of order at most 2. Here we study the question as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge–Kutta methods and hence for methods of arbitrary order. As an illustration the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretization of a nonlinear boundary integral equation that originates from a nonlinear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretization.

Boundary integral equation analysis for suspension of spheres in Stokes flow

We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.

Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

On fractional Langevin equation involving two fractional orders

In numerical analysis, it is frequently needed to examine how far a numerical solution is from the exact one. To investigate this issue quantitatively, we need a tool to measure the difference between them and obviously this task is accomplished by the aid of an appropriate norm on a certain space of functions. For example, Sobolev spaces are indispensable part of theoretical analysis of partial differential equations and boundary integral equations, as well as are necessary for the analysis of some numerical methods for the solving of such equations. But most of articles that appear in this field usually use ‖.‖∞ in the space of C[a, b] which is very restrictive. In this paper, we introduce a new norm that is convenient for the fractional and singular differential equations. Using this norm, the existence and uniqueness of initial value problems for nonlinear Langevin equation with two different fractional orders are studied. In fact, the obtained results could be used for the classical cases. Finally, by two examples we show that we cannot always speak about the existence and uniqueness of solutions just by using the previous methods.

Integral Equations for Electromagnetic Scattering at Multi-Screens

In (X. Claeys and R. Hiptmair, Integral equations on multi-screens. Integral Equ Oper Theory, 77(2):167–197, 2013) we developed a framework for the analysis of boundary integral equations for acoustic scattering at so-called multi-screens, which are arbitrary arrangements of thin panels made of impenetrable material. In this article we extend these considerations to boundary integral equations for electromagnetic scattering. We view tangential multi-traces of vector fields from the perspective of quotient spaces and introduce the notion of single-traces and spaces of jumps. We also derive representation formulas and establish key properties of the involved potentials and related boundary operators. Their coercivity will be proved using a splitting of jump fields. Another new aspect emerges in the form of surface differential operators linking various trace spaces.

A numerical boundary integral equation analysis for standard linear viscoelastic media made of functionally graded materials

Although structures made of functionally graded materials have been studied by many researchers, no research may be found in literature on boundary element analysis of the functionally graded viscoelastic structures. In the present paper, a 2D boundary element formulation capable of modeling time-dependent functionally graded materials (FGM) is presented. A numerical implementation of the Somigliana identity in terms of the displacements is developed to solve 2D problems of the exponentially graded viscoelasticity. The FGM concept can be applied to various materials, for structural and functional purposes. In this model, only Green functions of the nonhomogeneous elastostatic problems are needed with material properties that vary continuously along a given dimension. Results reveal that the boundary element approach can successfully be employed for the present complicated problem for arbitrary time histories of the applied loads and arbitrary boundary conditions, without the need to use relaxation functions or mathematical transformations.

Boundary integral equation analysis on the sphere

We present a systematic analysis of the integral operators of potential theory that arise when solving the Helmholtz or Maxwell equations in the exterior (or interior) of a sphere in the frequency domain. After obtaining expressions for the signatures of layer potentials in the spherical harmonic or vector spherical harmonic basis, we turn to a consideration of various integral equations that have been proposed in the literature for problems of acoustic and electromagnetic scattering. The selection of certain parameters in "combined field" and "Calderon-preconditioned" formulations is shown to have a significant impact on condition number, extending earlier work by Kress and others.

Boundary Integral Equation Analysis of Spoof Localized Surface Plasmons Excited in a Perfectly Conducting Cylinder with Longitudinal Corrugations

The main purpose of this paper is to apply the boundary integral equation (BIE) method to the analysis of spoof localized surface plasmons (spoof LSPs) excited in a perfectly conducting cylinder with longitudinal corrugations. Frequency domain BIE schemes based on electric field integral equation (EFIE), magnetic field integral equation (MFIE) and combined field integral equation (CFIE) formulations are used to solve two-dimensional electromagnetic (EM) problems of scattering from the cylinder illuminated by a transverse electric plane wave. In this approach effects of spoof LSPs are included in the secondary surface current and charge densities resulting from the interaction between the plane wave and the cylinder. Numerical results obtained with the BIE schemes are validated by comparison with that of a recently proposed modal solution based on the metamaterial approximation.

Boundary Integral Equation Analysis of an Inhomogeneous Medium Made of Functionally Graded Materials

The present work develops direct graded boundary integral equation formulation for behavior investigation of the inhomogeneous media made of functionally graded materials. The isoparametric boundary elements, the elastostatic governing equations and a weighted residual technique are implemented with the material characteristics that vary continuously along a given dimension. The resulting algorithm is capable of solving the quasistatic problems for elastic functionally graded media with a variety of the boundary conditions and loadings. The inhomogeneous media is made of a ceramic–metal mixture, in which the material properties vary continuously according to a power law graded distribution in a given direction. Avoiding the use of internal elements in the graded boundary element formulation is one of the main objectives of this paper, which results only in numerical discretization of the boundaries of the considered media. Some examples with continuously inhomogeneous isotropic materials were provided under different boundary conditions to evaluate the proposed numerical formulation for the FGMs.

A mathematical boundary integral equation analysis of standard viscoelastic solid polymers

Structural analysis of viscoelastic solid polymers is one of the most important subjects in engineering structures. Several attempts have been so far made for the integral equation approach to viscoelastic problems. From the basic assumptions of viscoelastic constitutive equations and weighted residual techniques, a simple but effective boundary element formulation (BEF) is implemented for the standard linear solid (SLS) viscoelastic models. The SLS model provides an approximate representation of the observed behavior of a real polymer in its viscoelastic range. This formulation needs only Kelvin’s fundamental solution of isotropic elastostatics with material constants prescribed as explicit functions of time. This approach avoids the use of relaxation functions and mathematical transformations, and it is able to solve the quasistatic viscoelastic problems with any load time-dependence and boundary conditions. As an application, a numerical example is provided to validate the proposed formulation. The problem of the pressurization of thick-walled cylindrical viscoelastic tanks made of PMMA polymer is completely analyzed by this approach.

Exact integration for the hypersingular boundary integral equation of two-dimensional elastostatics

This paper presents an exact integration for the hypersingular boundary integral equation of two-dimensional elastostatics. The boundary is discretized by straight segments and the physical variables are approximated by discontinuous quadratic elements. The integral for the hypersingular boundary integral equation analysis is given in a closed form. It is proven that using the exact integration for discontinuous boundary element, the singular integral in the Cauchy Principal Value and the hypersingular integral in the Hadamard Finite Part can be obtained straightforward without special treatment. Two numerical examples are implemented to verify the correctness of the derived exact integration.

Elastic and piezoelectric fields in quantum wire semiconductor structures – A boundary integral equation analysis

AbstractThis paper presents a rigorous analysis on the elastic and piezoelectric fields in quantum wire (QWR) semiconductor structures using the accurate boundary integral equation method. The materials modelled include both zincblende and wurtzite III–V semiconductors, and both the well‐accepted inclusion and structural inhomogeneity models are discussed. While the difference in the strain field between the inclusion and inhomogeneity models is consistent with previously published results, within about 10%, the difference in the induced piezoelectric fields between these two models can be very large, sometimes even with different signs (for zincblende semiconductors). The electric field distribution in polygonal QWRs with different sides is also studied and it is found that the electric field in triangular and square QWRs can be very different from the fields in polygonal QWRs with more than four sides. These findings should provide important considerations and trend analyses for both growth and modelling associated with QWR semiconductor structures. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Q factor and emission pattern control of the WG modes in notched microdisk resonators

Two-dimensional (2-D) boundary integral equation analysis of a notched circular microdisk resonator is presented. Results obtained provide accurate description of optical modes, free from the staircasing and discretization errors of other numerical techniques. Splitting of the double degenerate whispering-gallery (WG) modes and directional light output is demonstrated. The effect of the notch depth and width on the resonance wavelengths, Q factors, and emission patterns is studied. Further improvement of the directionality is demonstrated in an elliptical notched microdisk. Applications of the notched resonators to the design of microdisk lasers, oscillators, and biosensors are discussed.

Boundary-integral equation analysis of twisted internally cracked axisymmetric bimaterial elastic solids

Green's function is obtained for the infinite bimaterial elastic solid, containing an internal circular interface crack, loaded by a unit tangential co-axial circular source. An axisymmetric direct boundary integral equation (BIE) is used for the analysis of a finite bimaterial axisymmetric body containing an internal circular interface crack and a finite homogeneous cracked cylinder, both under torsional loading. Using the proposed technique, no discretization of the crack surface is necessary. Numerical results for both examples as obtained by the proposed method are presented and discussed.

Mindlin's problem for an anisotropic piezoelectric half–space with general boundary conditions

This paper considers Mindlin's problem in an anisotropic and piezoelectric halfspace with general boundary conditions, including 16 different sets of surface conditions. The Green's function due to...

Time-domain BIE analysis of large three-dimensional electromagnetic scattering problems

A time-domain boundary integral equation (BIE) solution of the magnetic field integral equation (MFIE) for large electromagnetic scattering problems is presented. It employs isoparametric curvilinear quadratic elements to model fields, geometry, and time dependence, eliminating staircasing problems. The approach is implicit, which seems to provide both stability and permits arbitrary local mesh refinement to model geometrically difficult regions without the significant cost penalty explicit methods suffer. Error dependence on discretization is investigated; accurate results are obtained with as few as five nodes per wavelength. The performance both on large scatterers and on low-radar cross section (RCS) scatterers is demonstrated, including the six wavelength NASA almond, two spheres, a thirteen wavelength missile, and a high-Q cavity.

Boundary Integral Equation Method Applied to the Strength Design of Gear. 3rd Report. Corner Problems and Efficient Evaluation of Integrals.

In this paper, in order to develop a more efficient and more convenient Procedure for the strength design of a spur gear, substantial effort is devoted to improving the accuracy in the boundary integral equation analyses. First, corner problems associated with a gear tip are mathematically avoided by adopting Priwalow's formulae. Next, a systematic technique using higher-order shape functions is proposed for the numerical evaluation of singular integrals. Then, a Stieltjes integral is used to deal with a discontinuous change of the boundary direction a corners. The accuracy is investigated using some numerical examples and comparison is made with the results in previous reports.