Seepage analysis in multi-domain general anisotropic media by three-dimensional boundary elements

A three-dimensional boundary element solution for the seepage analysis in multi-domain general anisotropic media has been developed based on the transformation approach. Using analytical eigenvalues and eigenvectors of the hydraulic conductivity tensor, a closed-form coordinate transformation matrix has been provided to transform the quadratic form of governing equation of seepage for the general anisotropic media to the Laplace equation. This transformation allows the analysis to be carried out using any standard BEM codes for the potential theory on the transformed space by adding small pre- and post-processing routines. With this transformation, any physical quantity like the total head remains unchanged at corresponding nodes on the physical and transformed space, and the normal gradient across the domain boundaries should also be transformed. In multi-domain problems, compatibility equations (equality of the potential on corresponding nodes on the interface) and equilibrium equations (conservation of the flux across the interface boundaries of adjacent domains) on the corresponding nodes of interface between two neighbor domains are needed for boundary element method. In the transformed space, the compatibility equation remains unchanged. However, due to the distortion of boundaries in the mapped space and therefore misalignment of the unit outward normal vectors along the inter-domain boundaries, the equilibrium of the normal fluxes have to be transformed accordingly. Based on the proposed transformation, the normal to boundary flux boundary conditions in the mapped space and the transformed equilibrium equation for interface of adjacent zones have been given in this paper. Examples have been solved with the proposed scheme and the results were verified with the finite element method. Excellent agreement of the results shows the veracity of the proposed transformation and the formulas given for transformation of equilibrium equation for multi-domain general anisotropic media.     

Frictional contact analysis of functionally graded materials with Lagrange finite block method

Based on the one‐dimensional differential matrix derived from the Lagrange series expansion, the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials. In this paper, the formulation of the Lagrange finite block method with boundary type in the strong form is presented and applied to non‐conforming contact problems for the functionally graded materials subjected to either static or dynamic loads. The first order partial differential matrices are only needed both in the governing equations and in the Neumann boundary condition. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate of global system to the normalized coordinate with eight seeds. Time dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin's inversion method is applied to determine all the physical values in the time domain. Conforming and non‐conforming contacts are investigated by using the iterative algorithm with full load technique. Illustrative numerical examples are given and comparisons have been made with analytical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

An interface integral equation method for solving general multi‐medium mechanics problems

In this paper, based on the general stress–strain relationship, displacement and stress boundary‐domain integral equations are established for single medium with varying material properties. From the established integral equations, single interface integral equations are derived for solving general multi‐medium mechanics problems by making use of the variation feature of the material properties. The displacement and stress interface integral equations derived in this paper can be applied to solve non‐homogeneous, anisotropic, and non‐linear multi‐medium problems in a unified way. By imposing some assumptions on the derived integral equations, detailed expressions for some specific mechanics problems are deduced, and a few numerical examples are given to demonstrate the correctness and robustness of the derived displacement and stress interface integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic analysis of interfacial crack problems in anisotropic bi-materials by a time-domain BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study dynamic interfacial crack problems in two-dimensional (2D), piecewise homogeneous, anisotropic and linear elastic bi-materials. The bi-material system is divided into two homogeneous sub-domains along the interface and the traditional displacement boundary integral equations (BIEs) are applied on the boundary of each sub-domain. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. A displacement extrapolation technique is used to determine the complex dynamic stress intensity factors (SIFs). Numerical examples for computing the complex dynamic SIFs are presented and discussed to demonstrate the accuracy and the efficiency of the present time-domain BEM.     

Equilibrium on line method (ELM) for imposition of Neumann boundary conditions in the finite point method (FPM)

Equilibrium on line method (ELM) for imposition of Neumann boundary conditions in the finite point method (FPM) is presented. In contrary to weak‐form‐based methods, strong‐form‐based methods such as the FPM are often unstable and less accurate, especially for problems governed by partial differential equations with Neumann (derivative) boundary conditions. In this paper, a truly meshless approach for imposition of Neumann boundary conditions in the FPM is proposed and adopted for 2D elasticity analyses. In the proposed method, equilibrium on lines on the Neumann boundary conditions is satisfied as Neumann boundary condition equations. Numerical studies show that this method for imposition of Neumann boundary is simple to implement and computationally efficient and also leads to more stable and accurate results. Copyright © 2006 John Wiley & Sons, Ltd.     

Analysis of anisotropic Kirchhoff plates using a novel hypersingular BEM

In this article a hypersingular boundary element method (BEM) for bending of thin anisotropic plates is presented. A new complex variable fundamental solution is implemented in the algorithm. For spatial discretization a collocation method with discontinuous quadratic elements is adopted. The domain integrals arising from the transversely applied load are transformed analytically into boundary integrals by means of the radial integration technique. The considered numerical examples prove that the novel BEM formulation presented in this study is much more efficient than previous formulations developed for the analysis of this kind of problems.     

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.     

An infinite boundary element formulation for three-dimensional potential problems

An infinite boundary element (IBE) is presented for the analysis of three-dimensional potential problems in an unbounded medium. The IBE formulations are done to allow their coupling with the finite element (FE) matrices for finite domains and to obtain the overall matrices without destroying the banded structure of the FE matrices. The infinite body is divided into a number of zones whose contributions are expressed in terms of the nodal quantities at FE nodes by employing suitable decay functions and performing mainly analytical integrations of the boundary element kernels. The continuity and compatibility conditions for the potential and the flux at the FE-IBE interface are developed. The relationships for the contributions of the IBE flux vectors to the FE load vectors are given. The final equations for the IBE are obtained in the usual FE stiffness-load vector form and are easily assembled with the FE matrices for the finite object. A series of numerical examples in heat transfer and electromagnetics were solved and compared with alternative solutions to demonstrate the validity of the present formulations.     

An anisotropic elastic formulation for configurational forces in stress space

A new variational principle for an anisotropic elastic formulation in stress space is constructed, the Euler–Lagrange equations of which are the equations of compatibility (in terms of stress), the equilibrium equations and the traction boundary condition. Such a principle can be used to extend recently obtained configurational balance laws in stress space to the case of anisotropy.     

A parallel domain decomposition BEM for diffusion problems with surface reactions

A parallel domain decomposition boundary element method (BEM) is developed for the solution of three-dimensional multispecies diffusion problems. The chemical species are uncoupled in the interior of the domain but couple at the boundary through a nonlinear surface reaction equation. The method of lines is used whereby time is discretized using the finite difference method and space is discretized using the boundary element method. The original problem is transformed into a sequence of nonhomogeneous modified Helmholtz equations. A Schwarz Neumann–Neumann iteration scheme is used to satisfy interfacial boundary conditions between subdomains. A segregated solver based on a quasi-predictor–corrector time integrator is used to satisfy the nonlinear boundary conditions on the reactive surfaces. The accuracy and parallel efficiency of the method is demonstrated through a benchmark problem.     

Subdomain radial basis collocation method for heterogeneous media

Strong form collocation in conjunction with radial basis approximation functions offer implementation simplicity and exponential convergence in solving partial differential equations. However, the smoothness and nonlocality of radial basis functions pose considerable difficulties in solving problems with local features and heterogeneity. In this work, we propose a simple subdomain strong form collocation method, in which the approximation in each subdomain is constructed separately. Proper interface conditions are then imposed on the interface. Under the subdomain strong form collocation construction, it is shown that both Neumann and Dirichlet boundary conditions should be imposed on the interface to achieve the optimum convergence. Error analysis and numerical tests consistently confirm the need to impose the optimal interface conditions. The performance of the proposed methods in dealing with heterogeneous media is also validated. Copyright © 2009 John Wiley & Sons, Ltd.     

Meshless local Petrov–Galerkin (MLPG) method in combination with finite element and boundary element approaches

The meshless local Petrov–Galerkin (MLPG) method is an effective truly meshless method for solving partial differential equations using moving least squares (MLS) interpolants. It is, however, computationally expensive for some problems. A coupled MLPG/finite element (FE) method and a coupled MLPG/boundary element (BE) method are proposed in this paper to improve the solution efficiency. A procedure is developed for the coupled MLPG/FE method and the coupled MLPG/BE method so that the continuity and compatibility are preserved on the interface of the two domains where the MLPG and FE or BE methods are applied. The validity and efficiency of the MLPG/FE and MLPG/BE methods are demonstrated through a number of examples. Received 6 June 2000     

Cracks in anisotropic elastic media

A numerical method for 3D problems of cracks in anisotropic media is developed, and this is based on the variational approach to the crack opening problem. Properties of the pseudodifferential operator of the crack equilibrium problem are considered. Numerical examples are also presented.     

Scattering of a plane wave by an anisotropic ferrite-coated conducting sphere

On the basis of the three-dimensional spherical vector wave functions in ferrite anisotropic media, and the fact that the first and second spherical vector wave functions in ferrite anisotropic media satisfy the same differential equations, the electromagnetic fields in homogeneous ferrite anisotropic media can be expressed as the addition of the first and second spherical vector wave functions in ferrite anisotropic media. Applying the continue boundary condition of the tangential component of electromagnetic fields in the interface between the ferrite anisotropic medium and free space, and the tangential electric field vanishing in the interface of the conducting sphere, the expansion coefficients of electromagnetic fields in terms of spherical vector wave function in ferrite medium and the scattering fields in free space can be derived. The theoretical analysis and numerical result show that when the radius of a conducting sphere approaches zero, the present method can be reduced to that of the homogeneous ferrite anisotropic sphere. The present method can be applied to the analyses of related microwave devices, antennas and the character of radar targets.     

A Monte Carlo solution of transport equations

A local method is developed for solving partial differential transport equations. The method is local in the sense that the value of the unknown solution of these equations can be calculated at arbitrary space and time coordinates directly rather than extracting its value from the field solution as done when using current numerical methods for solution. The proposed method is based on an analogy between the partial differential operator of transport equations and the infinitesimal generator of Itô processes, the Itô formula, the Dynkin formula, and Monte Carlo simulation. The method can be applied to solve transport problems with Dirichlet and Neumann boundary conditions. The solution of transport problems with Neumann boundary conditions is less simple because it requires the use of reflected Brownian motion and Itô processes.     

Crack tip stress intensities in viscoelastic anisotropic bimaterials and the use of the M-integral

The time evolution of the stress intensity factors (S.I.F's) at both tips of a finite crack lying near the interface of a viscoelastic anisotropic bimaterial, is studied. The simultaneous integral equations for the dislocation density of the crack developed in [2], are now used in the Laplace transformed domain. Their numerical solution and the solution via Neumann series are used for the determination of the Laplace trasformed S.I.F's. In the case of rectilinear anisotropies the latter are extracted from the M(p) integral which has been evaluated along a circle at infinity and along the interface. Numerical results for the real time dependence of the S.I.F's for two different anisotropies and geometries are also discussed.     

Three-step multi-domain BEM solver for nonhomogeneous material problems

A three-step solution technique is presented for solving two-dimensional (2D) and three-dimensional (3D) nonhomogeneous material problems using the multi-domain boundary element method. The discretized boundary element formulation expressed in terms of normalized displacements and tractions is written for each sub-domain. The first step is to eliminate internal variables at the individual domain level. The second step is to eliminate boundary unknowns defined over nodes used only by the domain itself. And the third step is to establish the system of equations according to the compatibility of displacements and equilibrium of tractions at common interface nodes. Discontinuous elements are utilized to model the traction discontinuity across corner nodes. The distinct feature of the three-step solver is that only interface displacements are unknowns in the final system of equations and the coefficient matrix is blocked sparse. As a result, large-scale 3D problems can be solved efficiently. Three numerical examples for 2D and 3D problems are given to demonstrate the effectiveness of the presented technique.     

SGBEM with Lagrange multipliers applied to elastic domain decomposition problems with curved interfaces using non‐matching meshes

An original approach to the solution of linear elastic domain decomposition problems by the symmetric Galerkin boundary element method is developed. The approach is based on searching for the saddle‐point of a new potential energy functional with Lagrange multipliers. The interfaces can be either straight or curved, open or closed. The two coupling conditions, equilibrium and compatibility, along an interface are fulfilled in a weak sense by means of Lagrange multipliers (interface displacements and tractions), which enables non‐matching meshes to be used at both sides of interfaces between subdomains. The accuracy and robustness of the method is tested by several numerical examples, where the numerical results are compared with the analytical solution of the solved problems, and the convergence rates of two error norms are evaluated for h‐refinements of matching and non‐matching boundary element meshes. Copyright © 2010 John Wiley & Sons, Ltd.     

Formulation for an FE and BE coupled problem and its application to the earmuff-earcanal system

A direct coupling procedure of the finite element method (FEM) and the boundary element method (BEM) for structure-acoustic-cavity problems has been introduced. The Laplace transformed matrix equations for the structure and acoustic cavity are coupled directly satisfying the necessary equilibrium and compatibility conditions. The coupled FEM-BEM technique is verified using a box-type cavity for the steady-state and the transient predictions compared with analytical results. The verified coupled FEM-BEM code is utilized in the analysis of the earmuff-earcanal system.     

A mixed DOF collocation method for elastic problems in heterogeneous structure

In this paper, a collocation method with mixed degrees of freedom (DOFs) is proposed for heterogeneous structures. Local tractions of the outer and interface boundaries are introduced as DOFs in the mixed collocation scheme. Then, the equilibrium equations of all the nodes and the outer boundary conditions are discretized and assembled into the global stiffness matrix. A local force equilibrium equation for modeling the stress discontinuity through the interface is developed and added into the global stiffness matrix as well. With those contributions, a statically determined stiffness matrix is obtained. Numerical examples show that the present method is superior to the classical mixed collocation method in the heterogeneous structure because it improves the accuracy and the convergence and remains the efficiency. Besides, almost constant convergence rates of displacements and stresses are found in all the examples, even for three-dimensional problems.