In this article the constant and the continuous linear boundary elements methods (BEMs) are given to obtain the numerical solution of the coupled equations in velocity and induced magnetic field for ...the steady magneto-hydrodynamic (MHD) flow through a pipe of rectangular and circular sections having arbitrary conducting walls. In recent decades, the MHD problem has been solved using some variations of BEM for some special boundary conditions at moderate Hartmann numbers up to 300. In this paper we develop this technique for a general boundary condition (arbitrary wall conductivity) at Hartmann numbers up to 105 by applying some new ideas. Numerical examples show the behavior of velocity and induced magnetic field across the sections. Results are also compared with the exact values and the results of some other numerical methods.
•Novel review of DRBEM (dual reciprocity boundary element method) and its applications in structural engineering.•Theroritcal background of the DRBEM.•Collection of more than 90 papers about the ...topic.•Illustration with examples from the literature of using different radial basis functions in the solution.•Highlighting the DRBEM cons and prons and recommending better future research extensions in this unique and practical subject.
The Dual Reciprocity Boundary Element Method (DRBEM) is a numerical technique that has been widely employed in structural engineering for solving boundary value problems. This method is based on the principle of reciprocity, which allows the application of boundary conditions at fictitious points on the actual domain of interest. In recent years, there has been a growing interest in the development and application of DRBEM formulations for different areas of engineering problems, such as heat transfer in solids, convection and diffusion problems, acoustics, and fluid flow in porous media.
This review paper provides a comprehensive overview of DRBEM formulations and focuses on their applications in structural engineering. The major advantages and limitations of DRBEM are discussed, along with the various approaches that employ different radial basis functions in its implementation are presented. The review also highlights several case studies that demonstrate the effectiveness of DRBEM in solving complex structural engineering problems, such as stress analysis, crack propagation in materials, and structural dynamics. Furthermore, the paper identifies several areas where DRBEM can be further improved.
Overall, this review paper provides valuable insights into the DRBEM method and its potential for solving complex structural engineering problems. The summary presented in this paper can help researchers and practitioners in the field of structural engineering to better understand the capabilities of DRBEM and its potential applications in their work.
In this paper, we present a scheme for cracks identification in three-dimensional linear elastic mechanical components. The scheme uses a boundary element method for solving the forward problem and ...the Nelder-Mead simplex numerical optimization algorithm coupled with a low discrepancy sequence in order to identify an embedded crack. The crack detection process is achieved through minimizing an objective function defined as the difference between measured strains and computed ones, at some specific sensors on the domain boundaries. Through the optimization procedure, the crack surface is modelled by geometrical parameters, which serve as identity variables. Numerical simulations are conducted to determine the identity parameters of an embedded elliptical crack, with measures randomly perturbed and the residual norm regularized in order to provide an efficient and numerically stable solution to measurement noise. The accuracy of this method is investigated in the identification of cracks over two examples. Through the treated examples, we showed that the method exhibits good stability with respect to measurement noise and convergent results could be achieved without restrictions on the selected initial values of the crack parameters.
This paper proposes a numerical method based on the dual reciprocity boundary elements method (DRBEM) to solve the stochastic partial differential equations (SPDEs). The concept of dual reciprocity ...method is used to convert the domain integral to the boundary. The conventional DRBEM starts with approximation of the source term of the original PDEs with radial basis functions (RBFs). Due to the fact that the nonhomogeneous term of SPDEs considered in this paper involves Wiener process, the traditional DRBEM cannot be applied. So a modification of it is suggested that has some advantages in comparison with the traditional DRBEM and can be developed for solving the SPDEs.
The time evolution is discretized by using the finite difference method, while the modified DRBEM is proposed for spatial variations of field variables. The noise term is approximated at the collocation points at each time step. We employ the generalized inverse multiquadrics (GIMQ) RBFs to approximate functions in the presented technique. To confirm the accuracy of the new approach, several examples are employed and simulation results are reported. Also the convergence of the new technique is studied numerically.
•A general framework is developed in regional heterogeneous reservoirs.•Arbitrary boundary shape is considered in the model.•New solutions for fractured horizontal well with finite conductivity are ...obtained.
The focus of this paper is to establish a new general model framework considering both hydraulic fractures with finite conductivity and heterogeneous characteristics of multiple connected regions by using boundary element method. With this framework, we can calculate the pressure performance of any heterogeneous reservoir. The innovation is that the complex fracture flow, boundary shape and heterogeneous characteristics can be flexibly considered in the model. Transient pressure of fractured horizontal wells in heterogeneous reservoir is obtained by using Stehfest numerical inversion method. Subsequently, this proposed regional heterogeneity model was validated by using numerical simulation methods. The results show that the proposed heterogeneous model can be divided into five flow segments, which are bilinear flow, linear flow, radial flow, first boundary dominated flow and second boundary dominated flow. However, the research results show that the presence or absence of some flow sections is closely related to the regional permeability ratio and the regional storage capacity ratio. As the two most important parameters of heterogeneous reservoir, the permeability ratio and regional storage capacity of heterogeneous reservoir will have a huge impact on the pressure response and pressure field. The sensitivity analysis shows that these two factors will lead to the deformation of the pressure field distribution and affect the flow characteristics, and cause some flow segments to fail to appear. The smaller the regional storage capacity ratio, the deeper the V-shape, and the more fluid supplied from the outer zone to the inner zone. This study also suggests that the fracture conductivity mainly affects the early flow characteristics, and the smaller the conductivity, the greater the pressure loss. In addition, irregular boundary will also cause important deformation of pressure field, and the area size will affect the late flow characteristics, and the smaller the area, the greater the pressure loss. The new semi-analytical solutions can form a series of typical curves and be applied to the well test analysis of multi-stage fracturing horizontal wells in heterogeneous and unconventional oil and gas reservoirs. The findings of this study can help for better understanding of the influence of reservoir heterogeneity on pressure response and find some applications in well testing for inverse problems in heterogeneous reservoirs.
Non-uniform rational B-splines (NURBS) are a convenient way to integrate CAD software and analysis codes, saving time from the operator and allowing efficient solution schemes that can be employed in ...the analysis of complex mechanical problems. In this paper, the Isogeometric Boundary Element Method coupled with Bézier extraction of NURBS and conventional BEM are used for analysis of 2D contact problems under cyclic loads. A node-pair approach is used for the analysis of the slip/stick state. Furthermore, the extent of the contact area is continuously updated to account for the nonlinear geometrical behaviour of the problem. The Newton–Raphson’s method is used for solving the non-linear system. A comparison to analytical results is presented to assess the performance and efficiency of the proposed formulation. Both BEM and IGABEM show good agreement with the exact solution when it is available. On most examples, they are equivalent with some advantage for IGABEM, though the former is slightly more accurate in some situations. This is probably due to the smoothness of NURBS not being able to describe sharp edges on tractions. As expected, IGABEM incurs in higher computational cost due to the basis being more complex than conventional Lagrangian polynomials.
The uniqueness of solution of boundary integral equations (BIEs) is studied here when geometry of boundary and unknown functions are assumed piecewise constant. In fact we will show BIEs with 3-times ...monotone radial kernels have unique piecewise constant solution. In this paper nonnegative radial function Fδ3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if δ3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than Fδ3/2. Note that there is a smart relation between δ3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln(c0r) as fundamental solution of Laplace equation. As a result, an upper bound for condition number of constant Galerkin BEMs system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Also non-singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non-singular BIEs are used. Finally some numerical examples are presented which confirm the analytical results.
•Barycentric mesh refinements used in dual basis functions worsen computational performance of Calderón preconditioners•Our new fast preconditioners significantly reduce assembly and computation ...times for 3D Helmholtz boundary integral operators•These efficient preconditioners inherit the good spectral properties of the Calderón preconditioners•Several numerical experiments in three dimensions validate our claims and point towards further enhancement
Calderón multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local low-order basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refined. This stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of L2-pairings. Based on coarser quadrature rules over dual cells and H-matrix compression, we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of Calderón preconditioning for the three-dimensional Helmholtz weakly and hyper-singular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements.
In this work, two new techniques of the Boundary Elements Method are coupled for solving eigenvalues in three-dimensional piecewise scalar problems. The Domain Superposition Technique (DST) is used ...to approach the sectorial homogeneities, modeling the domain as a sum of a homogeneous surrounding sector and other complementary ones with different constitutive properties. The Direct Interpolation Technique (DIBEM) with Radial Basis Functions transform domain integrals, in this case referring to the inertia, into boundary integrals. Thus, after discretization, a classical boundary element matrix system is formed in which the natural frequencies are achieved solving an eigenvalue problem. Considering the absence of analytical solutions for three-dimensional dynamic problems, benchmarks are taken by the Finite Element Results, using finer meshes.
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