Derivative Boundary Integral Equations Research Articles

Frequency domain Green’s function and boundary integral equations for multifield materials and quasicrystals

Present paper considers anisotropic elastic, magnetoelectroelastic and quasicrystal solids. It is shown that the equations of time-harmonic motion and constitutive relations for considered materials can be presented in the compact and unified form. The matrix approach is proposed for derivation of 3D time-harmonic Green’s functions for considered materials. The cases of each material type are studied separately. Two models of phason field dynamics in quasicrystals are considered in details. It is shown that in the case of phason field models based on the use of hydrodynamic formulations, phonon oscillations have an elastic nature, while the phason field leads to their decay due to phonon-phason interaction. The strict proof is given to the statement that the eigenvalues of the time-harmonic magnetoelectroelaticity problem are all positive. The paper also shows the application of the obtained time-harmonic Green’s functions in obtaining the boundary integral equations for the considered classes of problems. For this purpose, the novel approach is proposed, which utilizes only the symmetry of the unified material property tensor and the inertia matrix. Derived boundary integral equations can be used in the solution of boundary value problems for considered classes of materials.

A high-order BEM for acoustic problems in a subsonic uniform flow

The present work is to develop a boundary element method (BEM) with curved surface elements for 3D acoustic problems in a subsonic uniform flow. For exterior sound propagation problems in moving flows, the convected conventional boundary integral equation (CBIE) suffers from the irregular frequency issue which becomes more severe in the presence of moving flows. In this paper, a hybrid convected boundary integral formula based on the Burton-Miller method is implemented to avoid the fictitious eigenfrequency problem. However, the hypersingular integral involved in the convected normal derivative boundary integral equation (NDBIE) is one of the crucial troublesomeness in the implementation of BEM with curved continuous elements for 3D acoustic problems. Moreover, the presence of the Mach number further complicates the evaluation of the singular integrals. To resolve this dilemma, an effective method based on polar coordinate transformation (PCT) is developed to compute the singular integrals. By mapping physical coordinates to the intrinsic local coordinates, this method can effectually remove various kinds of singularities appearing in the CBIE and NDBIE. Additionally, a complete scheme on the high-order convected BEM including the scheme of collocation points is also constructed for acoustic problems in a subsonic uniform flow. Several numerical examples are elaborately designed to verify the accuracy of the developed method.

Heat Conduction in Anisotropic Medium with Perfectly Conductive Thread-Like Inclusions

Abstract The paper presents a novel approach for the analysis of steady-state heat conduction of solids containing perfectly conductive thread-like inhomogeneities. Modelling of a thread-like heat conductive inhomogeneity is reduced to determination of density of heat distributed along a spatial curve, which replaces the inclusion. Corresponding boundary integral equations are obtained for anisotropic solids with thread-like inclusions. Non-integral terms are computed in a closed form. It is shown that, nevertheless the singularity of the equation is 1/r, it is hypersingular, since the kernel is symmetric. Boundary element approach is adopted for solution of the obtained equations. Numerical example is presented for a rectilinear conductive thread, which verifies derived boundary integral equations.

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.

The Calculation of Potential Derivatives by Using NBIE for Anisotropic Potential Problems

AbstractThe natural boundary integral equation (NBIE) is developed to calculate potential derivatives for potential problems with anisotropic media. Firstly, the governing equation of the two-dimensional anisotropic potential problem is transformed into standard Laplace equation by a coordinate transformation method. Then a potential derivative boundary integral equation named as NBIE is extended to solve the anisotropic potential problem. The most important virtue of the NBIE is that the singularity of the integral kernel function is reduced by one order in comparison with the conventional potential derivative boundary integral equation(CDBIE). Therefore the new potential derivative boundary integral equation only contains strongly singular integrals rather than hyper-singular integrals. Thus the NBIE can calculate more accurate potential derivative results for both boundary nodes and interior points. Moreover, in combination with the analytical integral regularization algorithm of nearly singular integrals, the NBIE can obtain more accurate potential derivatives of interior points very close to the boundary than the CDBIE. Numerical examples on heat conduction in anisotropic media demonstrate the accuracy and efficiency of the NBIE.

Interaction of Two Parallel Cracks in REBCO Bulk Superconductors under Applied Magnetic Field**Supported by the National Natural Science Foundation of China under Grant Nos 11372096 and 11402073, the National Research Foundation for the Doctoral Program of Higher Education of China, and the Natural Science Foundation of Jiangsu Province under Grant No BK20130824.

Interactions of two collinear and parallel a/b-plane cracks in REBCO (where RE is a rare-earth element and usually Y is adopted) bulk superconductors under the Lorentz force resulted from the applied magnetic field are studied. By using the derived boundary integral equation for the crack problem of a cylindrical bulk superconductor under the applied magnetic field, we comprehensively investigate the stress intensity factor (SIF) of modes I and II at the crack tips of the two collinear and parallel cracks with their sizes, relative positions and the applied magnetic field. The calculated results show that in most cases, the SIF of mode I is found to be about tens of times of the one of mode II, and all the SIFs are always proportional to increase in the applied magnetic field, and the cracks near the center are more dangerous due to the larger Lorentz force.

The natural boundary integral equation of the orthotropic potential problem

The governing equation of two-dimensional orthotropic potential problem is transformed into standard Laplace equation by the coordinate transformation method. Then a novel potential derivative boundary integral equation termed natural boundary integral equation (NBIE) is established for the two-dimensional orthotropic potential problem. The NBIE reduces the singularity by one order when compared with the conventional potential derivative boundary integral equation (CDBIE). Thus the potential derivative of the two-dimensional orthotropic potential problem can be computed more accurately by using the NBIE at the same boundary mesh. Furthermore, after using the analytical integral regularization algorithm of nearly singular integrals, the NBIE can obtain more accurate potential derivatives of interior points which are very close to the boundary than the CDBIE. Numerical examples verify the accuracy and efficiency of the present method.

Sustaining ‘super’ midwives: Building resilience in midwifery students

The governing equation of two-dimensional orthotropic potential problem is transformed into standard Laplace equation by the coordinate transformation method. Then a novel potential derivative boundary integral equation termed natural boundary integral equation (NBIE) is established for the two-dimensional orthotropic potential problem. The NBIE reduces the singularity by one order when compared with the conventional potential derivative boundary integral equation (CDBIE). Thus the potential derivative of the two-dimensional orthotropic potential problem can be computed more accurately by using the NBIE at the same boundary mesh. Furthermore, after using the analytical integral regularization algorithm of nearly singular integrals, the NBIE can obtain more accurate potential derivatives of interior points which are very close to the boundary than the CDBIE. Numerical examples verify the accuracy and efficiency of the present method.

Analysis of 3D Planar Crack Problems by Body Force Method

Derivation of the integral equation for general 3D crack problems was examined based on the theory of body force method. In the present analysis, stress intensity factors (SIFs) along a front of arbitrary shaped 3D planar crack are obtained directly only by solving simultaneous equations expressing a boundary condition. The crack surface is discretized using number of triangular elements and the variation of the force doublet embedded in each triangle is assumed at constant. The derived boundary integral equation was transformed into a set of simultaneous equations and was solved computationally. In order to improve the accuracy of the numerically examined boundary integral, a polar transformation scheme combined with Tayler expansion of the fundamental solutions is introduced. Not only a single crack problem but also an interference among coplanar cracks can be calculated using the unique program developed in this research. It was verified that as the number of triangular elements increases, the evaluated SIF converges to the reference solution.

Inhomogeneity reconstructions in tendon ducts via boundary integral equations

In this study, as an alternative to the formerly presented investigations, Newton-type numerical algorithms are proposed to find location and shape of an air void inside of a tendon duct and to identify gathered metallic bars in a concrete column. The simulated structures are illuminated by four acoustic sources at a fixed frequency such that the scattered field is measured in a near-field region at 128 points. According to the nature of physical problems, the Dirichlet boundary condition is employed to model air-filled cavities and transmission conditions are assumed for metallic objects. Additionally, conductive boundary conditions are suggested for a more realistic representation of the inhomogeneities for the rusty metallic skin of the duct. Potential approaches are used to derive boundary integral equations. The proper treatment of the ill-conditioned equations is established via Tikhonov regularization. Applicability of the proposed inversion algorithms is tested with realistic parameters for different scenarios using noisy scattered field data and accurate numerical results are presented at 10kHz for the unknown physical properties of the duct׳s skin.

Boundary integral equations and Green׳s functions for 2D thermoelectroelastic bimaterial

This paper presents a comprehensive study on the 2D boundary integral equations, Green׳s functions and boundary element method for thermoelectroelastic bimaterials containing cracks and thin inclusions. Based on the extended Stroh formalism, complex variable approach and the Cauchy integral formula, the paper derives integral formulae for the Stroh complex functions, and Somigliana type integral identities for 2D thermoelectroelastic bimaterial. The kernels arising in the integral formulae are obtained explicitly and in a closed-form. It is proved that these kernels are fundamental solutions for a line extended force and a line heat. The far-field mechanical, electric and thermal load and internal volume load are accounted for in the obtained integral formulae. The latter allow to derive boundary integral equations for a bimaterial containing holes, cracks and thin inclusions, and to develop the corresponding boundary element approach. Special tip boundary elements used in the analysis allow accurate determination of the stress and electric displacement intensity factors for cracks and thin deformable inclusions. Several numerical examples are considered that show the validity and efficiency of the developed boundary element approach in the analysis of defective thermoelectroelastic anisotropic bimaterials.

Boundary integral equations for 2D elasticity and its application in discrete dislocation dynamics simulation in finite body: 1. General theory

A unified approach, originating from Cauchy integral theorem, is presented to derive boundary integral equations for two dimensional elasticity problems. Several sets of boundary integral equations are derived and their relations are revealed. Explicit expressions for materials with different symmetry planes are listed. Special attention is given to the formulation that is based on the tractions and the tangential derivatives of displacements along solid boundary, since its integral kernels have the weakest singularities. The formulation is further extended to include singular points, such as dislocations and line forces, in a finite body, so that the singular stress field can be directly obtained from solving the integral equations on the external boundary, without involving the linear superposition technique that was often used in the literature. Its application in simulating discrete dislocation motion in a finite solid body is discussed.

Boundary integral equations for 2D thermoelectroelasticity of a half-space with cracks and thin inclusions

Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelectroelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and Stroh orthogonality relations to obtain the integral formulae for the Stroh complex functions, which are piecewise-analytic in the complex half-plane with holes and opened mathematical cuts. Further application of the Stroh formalism allows derivation of the Somigliana type integral formulae and boundary integral equations for a thermoelectroelastic half-space. The kernels of these equations correspond to the fundamental solutions of heat transfer, electroelasticity and thermoelectroelasticity for a half-space. It is shown that the difference between the obtained fundamental solution of thermoelectroelasticity and those presented in literature is due to the fact, that present solution additionally accounts for extended displacement and stress continuity conditions, thus, it is physically correct. Obtained integral equations are introduced into the boundary element approach. Numerical examples validate derived boundary integral equations, show their efficiency and accuracy.

A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities

Abstract This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.

Analyzing interaction between coplanar square cracks using an efficient boundary element-free method

This paper examines the interaction between coplanar square cracks by combining the moving least-squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element-free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.

Square Cracks in Three-Dimensional Transversely Isotropic Solids

This paper presents a study for the square crack in a three-dimensional infinite transversely isotropic medium, which can model the fracture damage of rock that displays transversely isotropic behavior. The study is based on a newly derived boundary integral equation. To carry out the numerical simulation, the crack opening displacement is first expressed as the product of the weight functions and the characteristic terms, and the unknown weight is approximated with the moving least-square approximation. A boundary type numerical scheme is established, and the effect of the orientation of the principle axis on the stress intensity factor is studied. The interaction between two coplanar square cracks are also modeled and discussed.

A New Boundary Element Analysis of 3-D Acoustic Fields Avoiding the Fictitious Eigenfrequency Problem

This paper is concerned with a new approach for avoiding the fictitious eigenfrequency problem in boundary element analysis of three-dimensional acoustic problems governed by Helmholtz equation. It is well known that, in solving without any care an external acoustic problem with internal sub-domains by means of the boundary integral equation, its numerical solution is violated at so-called fictitious eigenfrequencies corresponding to the internal sub-domains. The present paper proposes a new boundary element analysis to circumvent such a fictitious eigenfrequency problem by using dual boundary integral equation for nodal points on the boundary. One equation is the combined integral equation proposed by Burton-Miller. The other equation is the normal derivative boundary integral equation multiplied by the same coupling parameter as in the Burton-Miller expression. The quadrilateral element is employed in this study, and the Burton-Miller combined boundary integral equation is used at the middle nodes of element, while only the normal derivative boundary integral equation multiplied by the same coupling parameter is applied to the vertex nodes of element, and vice versa. The proposed approach is implemented, and its validity and effectiveness are demonstrated through numerical computation of the typical problems.

A New Design Sensitivity Analysis of Acoustic Problems Based on Bem Avoiding Fictitious Eigenfrequency Issue

This paper presents a new design sensitivity analysis based on the boundary element method avoiding the fictitious eigenfrequency issue in acoustic problems. The direct differentiation method is applied to the derivation of sensitivity formulas. In solving an external acoustic problem with internal sub-domains by means of the boundary integral equation without any care, is numerical solution is violated at the so-called fictitious eigenfrequencies corresponding to the internal subdomains. The present paper proposes a new boundary element sensitivity analysis avoiding such a fictitious eigenfrequency problem. This is based on the dual boundary integral equation, proposed previously by the authors, and the integral expressions are differentiated directly with respect to the design parameters. One equation is the combined boundary integral equation proposed by Burton-Miller and the other is the normal derivative boundary integral equation multiplied by the same coupling parameter as in the Burton-Miller expression. The quadrilateral element is employed in this study. The Burton-Miller integral expression is used only at the middle nodes of element, while the normal derivative boundary integral equation multiplied by the same coupling parameter is applied to the vertex nodes of element, and vice versa. The effectiveness of the proposed approach is illustrated through some numerical examples for three-dimensional problems.

Second-order shape optimization using wavelet BEM

This present paper is concerned with second-order methods for a class of shape optimization problems. We employ a complete boundary integral representation of the shape Hessian which involves first- and second-order derivatives of the state and the adjoint state function, as well as normal derivatives of its local shape derivatives. We introduce a boundary integral formulation to compute these quantities. The derived boundary integral equations are solved efficiently by a wavelet Galerkin scheme. A numerical example validates that, in spite of the higher effort of the Newton method compared to first-order algorithms, we obtain more accurate solutions in less computational time.

A Boundary Element Analysis for Avoiding the Fictitious Eigenfrequency Problem in Acoustic Field (2nd Report, Revised Version)

This paper is concerned with an approach for avoiding the fictitious eigenfrequency problem in boundary element analysis of two-dimensional acoustic problems governed by Helmholtz equation. It is well known that in solving without any care the external acoustic problem by means of the boundary integral equation, the solution is violated at the eigenfrequencies of the internal problem. The present paper shows the effectiveness to circumvent the fictitious eigenfrequency problem by using the combined integral equation, which is originally proposed by Burton-Miller for constant elements, when employing the quadratic boundary element. On the other hand, we also propose a new approach to avoid that problem, which is for the quadratic boundary element. In this new approach, the Burton-Miller integral equation is used at the end points of element, while the normal derivative boundary integral equation multiplied by some parameter is applied to the other node of element. The proposed approach is implemented, and its effectiveness is demonstrated through comparison of the numerical results obtained by the developed computer code with other solutions.