Boundary Element Method Formulation Research Articles

A coupled formulation of finite and boundary element methods for flexoelectric solids

A numerical method for analyzing the mechanical deformation and electric polarization of flexoelectric solids is proposed that considers the electric interactions among flexoelectric solids, electric charges, and their surrounding medium. The proposed method uses finite and boundary elements simultaneously. Finite elements are used to model flexoelectric solids, and boundary elements are applied along the boundary of the solids and an arbitrary surface encompassing the solids and electric charges. To account for the couple stress and flexoelectric terms, a C1 continuous finite element formulation is introduced. Next, a coupled formation between the finite elements and boundary elements is developed. The coupled method provides a converged solution for any system composed of flexoelectric solids and electric charges without prescribing electric boundary conditions on the boundary of solids. Several numerical examples prove that the method is convergent and effective for systems containing multiple solids and electric charges.

A New BEM Modeling Algorithm for Size-Dependent Thermopiezoelectric Problems in Smart Nanostructures

The main objective of this paper is to introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures. The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to describe the deformation and mechanical behaviors of smart nanostructures subjected to thermal, and piezoelectric loadings. Because of difficulty of experimental research problems associated with the proposed theory. Therefore, we propose a new boundary element method (BEM) formulation and algorithm for the solution of such problems, which involve temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation. The computational performance of the proposed methodology has been demonstrated by using the generalized modified shift-splitting (GMSS) iteration method to solve the linear systems resulting from the BEM discretization. GMSS advantages are investigated and compared with other iterative methods. The numerical results are depicted graphically to show the size-dependent effects of thermopiezoelectricity, thermoelasticity, piezoelectricity, and elasticity theories of nanostructures. The numerical results also show the effects of the size-dependent and piezoelectric on the displacement components. The validity, efficiency and accuracy of the proposed BEM formulation and algorithm have been demonstrated. The findings of the current study contribute to the further development of technological and industrial applications of smart nanostructures.

A Single-Layer Dual-Mesh Boundary Element Method for Multiscale Electromagnetic Modeling of Penetrable Objects in Layered Media

A surface integral representation of Maxwell's equations allows the efficient electromagnetic (EM) modeling of three-dimensional structures with a two-dimensional discretization, via the boundary element method (BEM). However, existing BEM formulations either lead to a poorly conditioned system matrix for multiscale problems, or are computationally expensive for objects embedded in layered substrates. This article presents a new BEM formulation which leverages the surface equivalence principle and Buffa-Christiansen basis functions defined on a dual mesh, to obtain a well-conditioned system matrix suitable for multiscale EM modeling. Unlike existing methods involving dual meshes, the proposed formulation avoids the double-layer potential operator for the surrounding medium, which may be a stratified substrate requiring the use of an advanced Green's function. This feature greatly alleviates the computational expense associated with the use of Buffa-Christiansen functions. Numerical examples drawn from several applications, including remote sensing, chip-level EM analysis, and metasurface modeling, demonstrate speed-ups ranging from 3x to 7x compared to state-of-the-art formulations.

Fundamental solutions and integral equations of first-order laminated composite beams

Laminated composite materials consist of oriented plies usually bonded together to provide desirable mechanical properties such as high stiffness and strength to weight ratio. The result is a feasible solution for many technological problems, principally in the naval, aerospace, and automotive industries. Laminated composite beams have been represented in many mathematical models since 3D elasticity to laminated beam theories such as discrete-layer approaches and equivalent single-layer models (ESL). In this article, the mathematical steps for deriving the fundamental solutions and integral equations of an ESL model, called first-order laminated composite beam theory (FOBT), under static loading are presented. The direct boundary element method (BEM) solution is properly addressed, incorporating topics such as: (a) explicit forms of influence matrices, (b) load vectors for concentrated loading and (trapezoidal, sinusoidal) distributed loading, and (c) reduction of BEM solution to the classical laminated beam theory. Numerical results are presented for beams under different static load cases, boundary conditions, and domain discontinuity problems associated with stepped beams and continuous beams. The results suggest the correctness and effectiveness of this direct BEM formulation.

Solution of two-dimensional diffusion–advection problems for non-isotropic media with spatially variable velocity field by the boundary element method

This work presents a boundary element method formulation for the solution of the diffusion–advection problem. The formulation, developed for two-dimensional problems, for non-isotropic media, considers a spatially variable velocity field. The only way to deal with such a kind of problem is employing a steady-state fundamental solution. Consequently, the basic BEM equation presents one domain integral related to the velocity components, and another one related to the time derivative of the variable of interest, which is approximated using a backward finite difference scheme. BEM results are compared with an available analytical solution in the first part of the first example, and with the results provided by a finite element method formulation, taken as reference ones, in the second part of the first example and in the second example. From the comparisons, one can observe a good agreement between the results furnished by the proposed formulation and the analytical and reference solutions.

Development of TD-BEM Formulation for Dynamic Analysis for Twin-Parallel Circular Tunnels in an Elastic Semi-Innite Medium

In order to simulate the propagation process of subway vibration of parallel tunnels in semi-infinite rocks or soils, time domain boundary element method (TD-BEM) formulation for analyzing the dynamic response of twin-parallel circular tunnels in an elastic semi-infinite medium is developed in this paper. The time domain boundary integral equations of displacement and stress for the elastodynamic problem are presented based on Betti’s reciprocal work theorem, ignoring contributions from initial conditions and body forces. In the process of establishing time domain boundary integral equations, some virtual boundaries are constructed between finite boundaries and the free boundary to form a boundary to refer to the time domain boundary integral equations for a single circular tunnel under dynamic loads. The numerical treatment and solving process of time domain boundary integral equations are given in detail, including temporal discretization, spatial discretization and the assembly of the influencing coefficients. In the process of the numerical implementation, infinite boundary elements are incorporated in time domain boundary element method formulation to satisfy stress free conditions on the ground surface, in addition, to reduce the discretization of the boundary of the ground surface. The applicability and efficiency of the presented time domain boundary element formulation are verified by a deliberately designed example.

A Novel BEM for Modeling and Simulation of 3T Nonlinear Generalized Anisotropic Micropolar-Thermoelasticity Theory withMemory Dependent Derivative

The main aim of this paper is to propose a new memory dependent derivative (MDD) theory which called threetemperature nonlinear generalized anisotropic micropolar-thermoelasticity. The system of governing equations of the problems associated with the proposed theory is extremely difficult or impossible to solve analytically due to nonlinearity, MDD diffusion, multi-variable nature, multi-stage processing and anisotropic properties of the considered material. Therefore, we propose a novel boundary element method (BEM) formulation for modeling and simulation of such system. The computational performance of the proposed technique has been investigated. The numerical results illustrate the effects of time delays and kernel functions on the nonlinear three-temperature and nonlinear displacement components. The numerical results also demonstrate the validity, efficiency and accuracy of the proposed methodology. The findings and solutions of this study contribute to the further development of industrial applications and devices typically include micropolar-thermoelastic materials.

A modal boundary element method for axisymmetric acoustic problems in an arbitrary uniform mean flow

This paper presents a new numerical analysis approach based on an improved Modal Boundary Element Method (MBEM) formulation for axisymmetric acoustic radiation and propagation problems in a uniform mean flow of arbitrary direction. It is based on the homogeneous Modal Convected Helmholtz Equation (MCHE) and its convected Green’s kernel using a Fourier transform method. In order to simplify the flow terms, a general modal boundary integral solution is formulated explicitly according to two new operators such as the particular and convected kernels. Through the use of modified operators, the improved MBEM approach with flow takes a convective form of the general MBEM approach and has a similar form of the nonflow MBEM formulation. The reference and reduced Helmholtz Integral Equations (HIEs) are implicitly taken into account a new nonreflecting Sommerfeld condition to solve far field axisymmetric regions in a uniform mean flow. For isolating the singular integrations, the modal convected Green’s kernel and its modified normal derivative are performed partly analytically in terms of Laplace coefficients and partly numerically in terms of Fourier coefficients. These coefficients are computed by recursion schemes and Gauss-Legendre quadrature standard formulae. Specifically, standard forms of the free term and its convected angle resulting from the singular integrals can be expressed only in terms of real angles in meridian plane. To demonstrate the application of the improved MBEM formulation, three exterior acoustic case studies are considered. These verification cases are based on new analytic formulations for axisymmetric acoustic sources, such as axisymmetric monopole, axial and radial dipole sources in the presence of an arbitrary uniform mean flow. Directivity plots obtained using the proposed technique are compared with the analytical results.

Three dimensional nonlinear BEM formulations for the mechanical analysis of nonhomogeneous reinforced structural systems

This study presents a coupling formulation for the mechanical modelling of nonhomogeneous reinforced 3D structural systems. The coupling of dissimilar reinforced materials and components provides efficient designs. Particularly, it enables high stiffness and low weight mechanical components, which are largely desired in several engineering applications. In the present study, the Boundary Element Method (BEM) mechanically represents 3D solids through elastic and viscoelastic approaches. The 1D truss BEM elements represent the reinforcements, which allow for elastoplastic behaviour. The sub-region BEM technique models the nonhomogeneous systems whereas connection 1DBEM elements allow for crossing reinforcements along materials interfaces. Bond-slip behaviour has been accounted in the formulation by a nonlinear scheme without special link elements. Different bond-slip laws can be accounted, which enables the mechanical modelling of different adherence behaviours. The proposed formulations are applied in the mechanical analysis of four complex 3D applications. The results achieved by the proposed formulations are compared against the responses of equivalent numerical methods and experimental results available in the literature. The proposed formulations lead to accurate and stable results. Besides, the applications complexity demonstrates it.

Performance Comparison and Visualization with Different Computational Softwares for Predicting the Reservoir Pressure on Oil Production

This paper presents the performance comparison of some computation software for solving the boundary element method (BEM). BEM formulation is the numerical technique and high potential for solving the advance mathematical modeling to predict the production of oil well in arbitrarily shaped based on multiple leases reservoir. The limitation of data validation for ensuring that a program meets the accuracy of the mathematical modeling is considered as the research motivation of this paper. Thus, based on this limitation, there are three steps involved to validate the accuracy of the oil production simulation process. In the first step, identify the mathematical modeling based on partial differential equation (PDE) with Poisson-elliptic type to perform the BEM discretization. In the second step, implement the simulation of the 2D BEM discretization using COMSOL Multiphysic and MATLAB programming languages. In the last step, analyze the numerical performance indicators for both programming languages by using the validation of Fortran programming. The performance comparisons of numerical analysis are investigated in terms of percentage error, comparison graph and 2D visualization of pressure on oil production of multiple leases reservoir. According to the performance comparison, the structured programming in Fortran programming is the alternative software for implementing the accurate numerical simulation of BEM. As a conclusion, high-level language for numerical computation and numerical performance evaluation are satisfied to prove that Fortran is well suited for capturing the visualization of the production of oil well in arbitrarily shaped.

A boundary element method formulation based on the Caputo derivative for the solution of the anomalous diffusion problem

This work presents a boundary element method formulation for the solution of the anomalous diffusion problem. By keeping the fractional time derivative as it appears in the governing differential equation of the problem, and by employing a Weighted Residuals Method approach with the steady state fundamental solution for anisotropic media playing the role of the weighting function, one obtains the boundary integral equation of the proposed formulation. The presence of a domain integral with the fractional time derivative as part of its integrand, and the evaluation of this fractional time derivative as a Caputo derivative, constitute the main feature of the formulation. The analyses of some examples, in which the numerical results are always compared with the corresponding analytical solutions, show the robustness of the formulation, as accurate results are obtained even for small values of the order of the time derivative.

An enriched dual boundary element method formulation for linear elastic crack propagation

This study presents an eXtended Boundary Element Method (XBEM) formulation for simulating the linear elastic crack growth in two-dimensional domains. A displacement approximation enrichment based on the Williams first-order expansions is used to represent the near-tip square root behavior predicted by the linear elastic fracture mechanics. This strategy also enables the direct evaluation of the Stress Intensity Factors (SIFs) after a crack tip tying constraint is enforced at the crack tip to accommodate the additional enrichment parameters. Additionally, discontinuous functions are embedded into the displacement approximation of elements intercepted by cracks to avoid remeshing of these elements. For the discontinuous enrichment, new equations are provided by imposing displacement continuity conditions at the crack mouth. Shifted enrichment functions are used to preserve the physical meaning of the nodal parameters and to reduce the singularity order of the integral kernels containing the enrichment terms in the XBEM formulation. Furthermore, an enriched traction approximation is proposed to apply concentrated forces and support points along the boundary. When considering the XBEM formulation, changes in the conventional BEM are minimal, which facilitates the implementation of the extended approach into existing codes. Three numerical examples demonstrate the application of the proposed XBEM formulation in fracture modeling.

A fast full-wave solver for calculating ultrasound propagation in the body

Therapeutic ultrasound is a promising non-invasive method for inducing various beneficial biological effects in the human body. In cancer treatment applications, high-power ultrasound is focused at a target tissue volume to ablate the malignant tumour. The success of the procedure depends on the ability to accurately focus ultrasound and destroy the target tissue volume through coagulative necrosis whilst preserving the surrounding healthy tissue. Patient-specific treatment planning strategies are therefore being developed to increase the efficacy of such therapies, while reducing any damage to healthy tissue. These strategies require to use high-performance computing methods to solve ultrasound wave propagation in the body quickly and accurately. For realistic clinical scenarios, all numerical methods which employ volumetric meshes require several hours or days to solve the full-wave propagation on a computer cluster. The boundary element method (BEM) is an efficient approach for modelling the wave field because only the boundaries of the hard and soft tissue regions require discretisation. This paper presents a multiple-domain BEM formulation with a novel preconditioner for solving the Helmholtz transmission problem (HTP). This new formulation is efficient at high-frequencies and where high-contrast materials are present. Numerical experiments are performed to solve the HTP in multiple domains comprising: (i) human ribs, an idealised abdominal fat layer and liver tissue, (ii) a human kidney with a perinephric fat layer, exposed to the acoustic field generated by a high-intensity focused ultrasound (HIFU) array transducer. The time required to solve the equations associated with these problems on a single workstation is of the order of minutes. These results demonstrate the great potential of this new BEM formulation for accurately and quickly solving ultrasound wave propagation problems in large anatomical domains which is essential for developing treatment planning strategies.

Cohesive crack propagation analysis using a dipole BEM formulation with tangent operator

This work deals with the use of Boundary Element Method (BEM) formulations in fracture problems in a 2D approach. BEM is a suitable method for solving this type of problem, since the absence of a domain mesh translates into a more efficient modeling of regions with high stress concentration. Besides, reducing the dimensionality of the mesh results in a convenient remeshing process. Regarding the nonlinear BEM formulation, an alternative to the classic dual formulation is used, with the introduction of an initial stress field to represent the cohesive zone, based on the concept of dipoles. This formulation is particularly interesting because it is able to represent mathematically the presence of the fracture process zone (FPZ) with only three algebraic equations (related to stress correction) per source point located in the crack path. In contrast, dual BEM formulation requires four algebraic equations (displacements and forces) per source point. As for the effects relevant to the nonlinear system, two distinct iterative solving algorithms are used, Constant Operator (CO) and Tangent Operator (TO). Only the elastic stiffness of the structure in the search of equilibrium point is considered in the first one, while the latter takes into account the mechanical degradation properties. Some examples are presented to validate the use of the Dipole BEM/TO formulation in crack propagation analysis and also in multiple crack analysis. Finally, this formulation proves to be a powerful alternative to other classic methods, considering the results presented.

SLIM: A Well-Conditioned Single-Source Boundary Element Method for Modeling Lossy Conductors in Layered Media

The boundary element method (BEM) enables the efficient electromagnetic modelling of lossy conductors with a surface-based discretization. Existing BEM techniques for conductor modelling require either expensive dual basis functions or the use of both single- and double-layer potential operators to obtain a well-conditioned system matrix. The associated computational cost is particularly significant when conductors are embedded in stratified media, and the expensive multilayer Green's function (MGF) must be invoked. In this work, a novel single-source BEM formulation is proposed, which leads to a well-conditioned system matrix without the need for dual basis functions. The proposed single-layer impedance matrix (SLIM) formulation does not require the double-layer potential to model the background medium, which reduces the cost associated with the MGF. The accuracy and efficiency of the proposed method is demonstrated through realistic examples drawn from different applications.

Subtraction singularity technique applied to the regularization of singular and hypersingular integrals in high-order curved boundary elements in plane anisotropic elasticity

The numerical solutions of boundary integral equations by the Boundary Element Method (BEM) have been applied in several areas of computational engineering and science such as elasticity and fracture mechanics. The BEM formulations often require the evaluation of complex singular and hypersingular integrals. Therefore, BEM requires special integration schemes for singular elements. The Subtraction Singularity Technique (SST) is a general procedure for evaluating such integrals, which allows for the integral over high-order curved boundary elements. The SST regularises these kernels through the Taylor expansion of the integral kernels around the source point. This study presents the expressions from Taylor expansions, which are required for the regularization of singular and hypersingular boundary integral equations of plane linear anisotropic elasticity. These expressions have been implemented into an academic BEM code, which enables the integration over high-order straight and curved boundary elements. The proposed scheme leads to excellent performance. The results obtained by the proposed scheme are in excellent agreement with reference responses available in the literature.

The Dual-Reciprocity Boundary Element Method Solution for Gas Recovery from Unconventional Reservoirs with Discrete Fracture Networks

SummaryIn this paper, we present a novel application of the dual-reciprocity boundary-element formulation (DRBEM) to model compressible (gas) fluid flow in tight and shale-gas reservoirs containing arbitrary distributed finite- or infinite-conductivity discrete fractures. Compared with the standard boundary-element method (BEM), the DRBEM transforms the nonlinear domain integrals at the righthand side (RHS) of BEM formulations for nonlinear partial differential equations into equivalent boundary integrals. This transformation allows retention of the domain-integral-free, boundary-integral-only character of standard BEM approaches. The proposed approach is based on coupling DRBEM with the finite-volume method (FVM) in which a multidimensional system is solved by integrating over a line with random fractures. The resulting system of equations is solved simultaneously for fracture and matrix boundary conditions by combining DRBEM and FVM without invoking any approximation for pressure-dependent nonlinear terms such as gas viscosity and compressibility. Numerical examples and field cases are presented to test the validity and showcase the capabilities of the proposed approach. The proposed model provides a general framework that can be applied to a variety of well and fracture geometries and operating schedules, and it is used to analyze production behavior for these complex systems. To the best of the authors’ knowledge, this is the first successful application of the dual-reciprocity principle to the BEM analysis of massively fractured horizontal wells (MFHWs) performance in natural-gas formations in which nonlinear, pressure-dependent gas properties are captured without approximation.

BEM-FORM Model for the Probabilistic Response of Circular Tunnels in Elastic Media

Problems involving cavities or excavations are widely addressed in geomechanics, in both analytical and numerical approaches. The boundary element method (BEM) is well-known as an interesting choice for half plane problems, providing accurate results at a low computational cost. This work deals with the probabilistic analysis of circular tunnels embedded in elastic media, coupling a BEM formulation to a structural reliability model. The gravitational loading and material parameters are treated as random variables, whose statistical description is taken from the literature. The loadings considered include the vertical overburden stress and the lateral earth pressure. Regarding the reliability evaluation, first order reliability method (FORM) and Monte Carlo simulation technique are employed, being compared in terms of accuracy. Regarding the BEM model, the Multiple Reciprocity Method (MRM) is used in the evaluation of domain integrals, and the subregion technique is employed for the analysis of the tunnel lining. Some analyses are presented, in order to validate the coupled BEM-FORM model and apply it to the estimation of failure probability, evaluating the influence of the random variables taken into account in the probabilistic response.

Dual boundary element method for elastoplastic fracture mechanics of shear deformable plate

A dual boundary element method formulation for elastoplastic fracture mechanics of shear deformable plate is reported in this work. The formulation can be viewed as an extension to the two-dimensional problem. The boundaries of crack surfaces are modeled using discontinuous elements of quadratic elements to fulfill the continuity requirements of displacements and tractions at collocation points for displacement and traction integral equations. Rectangular element of 9-nodes quadrilateral is used to discretize the domain. Semi-discontinuous rectangular elements are placed along the boundary including the crack surfaces in order to force the cell nodes to be internal points. Two examples are presented. The first example isa centre-cracked plate subjected to tensile, bending and combined of the two loads. The second example is rectangular plate with double edge symmetric cracks subjected to combined tensile and bending load. J-type integrals are calculated.

The Boundary Element Method applied to the solution of the diffusion-wave problem

A Boundary Element Method formulation is developed for the solution of the two-dimensional diffusion-wave problem, which is governed by a partial differential equation presenting a time fractional derivative of order α, with 1.0 < α < 2.0. In the proposed formulation, the fractional derivative is transferred to the Laplacian through the Riemann–Liouville integro-differential operator; then, the basic integral equation of the method is obtained through the Weighted Residual Method, with the fundamental solution of the Laplace equation as the weighting function. In the final expression, the presence of additional terms containing the history contribution of the boundary variables constitutes the main difference between the proposed formulation and the standard one. The proposed formulation, however, works well for 1.5 ≤ α < 2.0, producing results with good agreement with the analytical solutions and with the Finite Difference ones.