Complex Variable Boundary Integral Equation Research Articles

Quaternion Boundary Element Method for Coupled Exterior and Interior Magnetostatic Fields

In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic problem. The present quaternion-valued BEM is developed by discretizing the quaternion-valued boundary integral equation (BIE). The quaternion-valued BIE can be seen as an extension of the generalized complex variable BIE in 3-D space. In other words, quaternion algebra is an extension of the complex variable in 3-D space. To derive quaternion-valued BIEs, the quaternion-valued Stokes’ theorem is utilized. The quaternion-valued BIEs are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). An analytical scheme is developed to evaluate the CPV by introducing a simple quaternion-valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called “numerical boundary layer” phenomena appear and are remedied by using a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. It is noted that all derivations are done in quaternion algebra. In addition, quaternion algebra is more flexible than vector algebra in solving some 3-D problems from the point view of algebraic space. In deriving BIEs for exterior fields, the conditions at infinity for the quaternion-valued functions are carefully examined. Later, a magnetic sphere in a uniform magnetic field is considered. This problem is a magnetostatic problem of coupled exterior and interior magnetostatic fields. Finally, we apply the present approach to solve the magnetostatic problem. By comparing with exact solutions, the validity of the present approach is checked.

Amended influence matrix method for removal of rigid motion in the interior BVP for plane elasticity

A conventional complex variable boundary integral equation (CVBIE) in plane elasticity is provided. After using the Somigliana identity between a particular fundamental stress field and a physical stress field, an additional integral equality is obtained. By adding both sides of this integral equality to both sides of the conventional CVBIE, the amended boundary integral equation (BIE) is obtained. The method based on the discretization of the amended BIE is called the amended influence matrix method. With this method, for the Neumann boundary value problem (BVP) of an interior region, a unique solution for the displacement can be obtained. Several numerical examples are provided to prove the efficiency of the suggested method.

Numerical solution of Eshelby's elastic inclusion problem in plane elasticity by using boundary integral equation

This paper provides a numerical solution for Eshelby's elastic inclusions in an plane elasticity based on the complex variable boundary integral equation (CVBIE) method. An inclusion with arbitrary shape is embedded in the infinite matrix. In the inclusion, the constant eigenstains are assumed. No remote loading is applied to the matrix. The displacements from the assumed eigenstrains are evaluated exactly, which in turn are the generalized loading in the problem. The proposed problem is reduced to solve interior and exterior boundary value problems simultaneously. For the elliptical inclusion case, the computed stresses along the interface of the inclusion side are nearly uniform. One more numerical example is devoted to a square-type inclusion with round corner.

Numerical solution for the degenerate scale problem in plane elasticity using null field CVBIE

This paper provides a numerical solution for the degenerate scale problem in plane elasticity using the null field complex variable boundary integral equation (CVBIE). After performing the coordinate transformation, the CVBIE can be formulated in the normal scale. After making discretization, a linear algebraic equation is obtained. The influence matrix in the normal scale is invertible. By introducing two basic solutions, the degenerate scale problem is finally solved. Several numerical examples are given.

Generalized complex variable boundary integral equation for stress fields and torsional rigidity in torsion problems

Theory of complex variables is a very powerful mathematical technique for solving two-dimensional problems satisfying the Laplace equation. On the basis of the conventional Cauchy integral formula, the conventional complex variable boundary integral equation (CVBIE) can be constructed. The limitation is that the conventional CVBIE is only suitable for holomorphic (analytic) functions, however. To solve for a complex-valued harmonic-function pair without satisfying the Cauchy–Riemann equations, we propose a new boundary element method (BEM) based on the general Cauchy integral formula. The general Cauchy integral formula is derived by using the Borel–Pompeiu formula. The difference between the present CVBIE and the conventional CVBIE is that the former one has two boundary integrals instead of only one boundary integral in the latter one. When the unknown field is a holomorphic function, the present CVBIE can be reduced to the conventional CVBIE. Therefore, the conventional Cauchy integral formula can be viewed as a special case applicable to a holomorphic function. To examine the present CVBIE, we consider several torsion problems in this paper since the two shear stress fields satisfy the Laplace equation but do not satisfy the Cauchy–Riemann equations. Using the present CVBIE, we can directly solve the stress fields and the torsional rigidity simultaneously. Finally, several examples, including a circular bar containing an eccentric inclusion (with dissimilar materials) or hole, a circular bar, elliptical bar, equilateral triangular bar, rectangular bar, asteroid bar and circular bar with keyway, were demonstrated to check the validity of the present method.

Properties of integral operators and solutions for complex variable boundary integral equation in plane elasticity for multiply connected regions

This paper studies properties of integral operators and solutions for CVBIE (complex variable boundary integral equation) in plane elasticity for multiply connected regions. Four cases for considered regions are studied. For the individual case, we study (a) the domain field equality, (b) the null field BIE and (c) the usual CVBIE. Properties of integral operators or the kernels are studied in detail, which is based on the properties of Cauchy type integral. The Neumann problem is considered. It is shown that for finite region cases (Sections 2 and 3) the CVBIE allow three modes of rigid motion along contours under the traction free condition. In addition, for infinite region cases (Sections 4 and 5) the CVBIE does not allow three modes of rigid motion.

Numerical solution of the t-version complex variable boundary integral equation for the interior region in plane elasticity

This paper suggests the t-version complex variable boundary integral equation (CVBIE) for the interior region in plane elasticity. All the kernels in the t-version CVBIE are expressed explicitly. The property for the operator acting upon the displacement is studied. It is proved that there are three types of rigid mode movement of displacement for the t-version CVBIE, if the boundary is assumed under a traction free condition. Discretization of the t-version CVBIE is suggested. For the hypersingular integral, the integration is carried out exactly in the concept of Hadamard׳s finite part integral. Two particular examples which have known solution beforehand are used to examine the accuracy in computation. The Neumann and the Dirichlet boundary value problems are examined numerically. It is proved that the computation error is acceptable in the examples.

Solution for Eshelby's elastic inclusions in a finite plate using boundary integral equation method

Abstract This paper provides a solution for Eshelby's elastic inclusions in a finite plate based on the complex variable boundary integral equation (CVBIE) method. In the formulation, an inclusion with Eshelby's eigenstrains is embedded in an elliptic plate, and the exterior boundary is applied by some static loading. Two BIEs are suggested in the present study. One of BIEs is used for the finite inclusion region, and the other is used for region bounded by interface and the exterior boundary. After the discretization of BIEs, a numerical solution is suggested. In the solution, an inverse matrix technique is suggested which can eliminate one unknown vector in advance. Three numerical examples under different generalized loadings are provided. Interaction between the prescribed eigenstrains and the static loading along the exterior boundary is studied in detail.

Properties of integral operators in complex variable boundary integral equation in plane elasticity

This paper investigates properties of integral operators in complex variable boundary integral equation in plane elasticity, which is derived from the Somigliana identity in the complex variable form. The generalized Sokhotski-Plemelj`s formulae are used to obtain the BIE in complex variable. The properties of some integral operators in the interior problem are studied in detail. The Neumann and Dirichlet problems are analyzed. The prior condition for solution is studied. The solvability of the formulated problems is addressed. Similar analysis is carried out for the exterior problem. It is found that the properties of some integral operators in the exterior boundary value problem (BVP) are quite different from their counterparts in the interior BVP.

Solution for dissimilar elastic inclusions in a finite plate using boundary integral equation method

This paper studies the boundary value problem for a finite plate containing two dissimilar inclusions. The matrix and the two inclusions have different elastic properties. The loadings applied along the outer boundary are in equilibrium. The mentioned problem is decomposed into three boundary value problems (BVPs). Two of them are interior BVP for the elastic inclusions, while the other is a BVP for the triply-connected region. Three problems are connected together through the common displacements and tractions along the interface boundaries. Explicit form for the complex variable boundary integral equation (CVBIE) is derived. After discretization of relevant BIEs, the solutions are evaluated numerically. Three numerical examples for different elastic constant combinations are provided.

An iteration approach for multiple notch problem based on complex variable boundary integral equation

This paper provides an iteration approach for the solution of multiple notch problem, which is based on the complex variable boundary integral equation (CVBIE). The contours of notches are applied by some loadings. The source points are assumed on the boundary of individual notch and the displacements along the boundaries become unknowns to be investigated. After discretization of the BIE, many influence matrices are obtained. One does not need to assemble many influence matrices into a larger matrix. This will considerably reduce the work in the program. The displacements along the many boundaries can be obtained from an iteration. There is no limitation for the configuration of notches. Several numerical examples are provided to prove the efficiency of the suggested approach.

Boundary integral equation method for periodic dissimilar elastic inclusions in an infinite plate

This paper provides a theoretical derivation and a numerical solution for an infinite plate with periodic dissimilar inclusions. The matrix and periodic inclusions have different elastic properties, and a remote loading is applied for the infinite matrix. The original problem is decomposed into two problems, one is the interior boundary value problem (BVP) and the other is the exterior BVP. Both BVPs are formulated in the form of complex variable boundary integral equation (CVBIE). In addition, both BIEs are solved numerically after discretization of the integral equations. For the exterior BVP with periodic notches, the remainder estimation technique (RET) is suggested to evaluate the influence to the central notch from many (from Nth to infinity) neighboring notches. This will significantly improve the efficiency of solution. Two numerical examples are provided. In the examples, the ratio of the two shear moduli of elasticity changes from near 0 (1.d–6), 0.1, 0.5, 1, 2 to 10.

Numerical solution of elastic inclusion problem using complex variable boundary integral equation

Based on a complex variable boundary integral equation (CVBIE) suggested previously, this paper provides a numerical solution for the elastic inclusion problem using CVBIE. A dissimilar elastic inclusion is embedded in the infinite matrix. The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for the elastic inclusion, while the other is an exterior BVP for the matrix with notch. Both problems are connected by conventional boundary integral equations (BIEs) in complex variables. After performing discretization for the coupled BIEs, the inverse matrix technique is suggested to solve the relevant algebraic equations. Based on the properties of some integral operators, three ways for the inverse matrix technique are suggested. Several numerical examples are carried out to prove the efficiency of the suggested method.

Boundary integral equation method for two dissimilar elastic inclusions in an infinite plate

This paper provides a numerical solution for an infinite plate containing two dissimilar elastic inclusions, which is based on complex variable boundary integral equation (CVBIE). The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for two elastic inclusions, while other is an exterior BVP for the matrix with notches. After performing discretization for the coupled boundary integral equations (BIEs), a system of algebraic equations is formulated. The inverse matrix technique is suggested to solve the relevant algebraic equations, which can avoid using the assembling of some matrices. Several numerical examples are carried out to prove the efficiency of suggested method and the hoop stress along the interface boundary is evaluated.

Solutions of periodic notch problems with arbitrary configuration by using boundary integral equation and superposition method

This paper studies the periodic notch problem with arbitrary configuration. A complex variable boundary integral equation (CVBIE) is suggested to solve the problem. The periodic notch problem is considered as a superposition of infinite single notch problems. The influence on the domain point from the assumed boundary traction on the notch contour is reduced to formulate a matrix. In this paper, this matrix is formulated completely after the relevant BIE is solved in a matrix representation. The remainder estimation technique is suggested to evaluate the influence to the central notch from many (form N-th to infinity) neighboring notches. Many computed results for the stress concentration factor for the elliptic and square notches are carried out. The stacking effect is also studied.

MULTIPLE AND PERIODIC NOTCH PROBLEMS OF ELASTIC HALF-PLANE BY USING BIE BASED ON GREEN'S FUNCTION METHOD

This paper provides a modified complex potential for the fundamental solution, which is composed of a principal part and a complementary part. The suggested complex potential satisfies the traction free condition along the boundary of half-plane in advance. After using the Somigliana identity or the Betti's reciprocal theorem between the physical field and the field of the fundamental solution, a complex variable boundary integral equation (BIE) for the notch problem in elastic half-plane is obtained. A compact derivation for the BIE is presented. By using the BIE, multiple notch problems of elastic half-plane can be solved numerically. In the method, there is no limitation for the configuration of notches. Many problems with the elliptic notch or the square notch are solved successfully. For the periodic notch problem, the remainder estimation technique is suggested. This technique provides an effective way for the solution of periodic notch problem.

Dual boundary integral equation formulation in antiplane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in antiplane elasticity using complex variable. Four kinds of boundary integral equation (BIE) are studied, and they are the first complex variable BIE for the interior region, the second complex variable BIE for the interior region, the first complex variable BIE for the exterior region, and the second complex variable BIE for the exterior region. The first BIE for the interior region is derived from the Somigliana identity, or the Betti’s reciprocal theorem in elasticity. A displacement versus traction operator is suggested. After using this operator, the second BIE for the interior region is derived. Similar derivations are performed for the first and second BIEs for the exterior region. In the case of the exterior boundary, two degenerate boundary cases are studied. One is the curved crack case, and other is the case of a deformable line. All kernels in the suggested BIEs are expressed in terms of complex variable.

Transient thermal stresses in a medium with a circular cavity with surface effects

A two-dimensional, transient, uncoupled thermoelastic problem of an infinite medium with a circular nano-scale cavity is considered. The analysis is based on the generalized Gurtin and Murdoch model [Murdoch, A.I., 2005. Some fundamental aspects of surface modelling. Journal of Elasticity 80, 33–52.] where the surface of the cavity possesses its own surface tension and thermomechanical properties. A semi-analytical solution for the problem is obtained using a complex variable boundary integral equation method and the Laplace transform. Several examples are presented to study the significance of surface thermomechanical properties and surface tension, and to compare the results obtained using the generalized Gurtin and Murdoch model and a thin interphase layer model.

Complex variable BEM for thermo- and poroelasticity

A stationary thermoelastic (poroelastic) boundary element method is suggested based on the Complex Variable Hypersingular Boundary Integral Equation. The method is developed for heterogeneous blocky media. Various conditions on the contacts between the blocks are considered, namely discontinuity of temperature (pore pressure) or discontinuity of heat (fluid) flux. The problem of the basic integrals calculation is discussed and numerical examples are presented to demonstrate the potential of the method.