Dual Reciprocity Boundary Element Method Research Articles

BEM performance with the addition of polynomials solving stationary thermal problems with functional gradation

This work presents the formulation and numerical results of the Dual Reciprocity Boundary Element Method (DRBEM), adapted to include the procedure known as the addition of polynomial functions. This technique is used to improve the interpolation scheme’s accuracy using radial basis functions. This idea works quite effectively in simple interpolation procedures, especially when the field profile to be approximated has similarities with added polynomials. Although applied as a supplementary resource in some works, the efficiency and restrictions of the procedure were not discussed in detail as necessary. In this article, the addition of functions is applied to solve non-homogeneous stationary heat transfer problems. The main objective is to analyze its capability, especially in reducing the demand for internal interpolation points. The three simulations show that although introducing polynomials can improve the results, other characteristics are mandatory for achieving suitable accuracy. Three examples of numerical simulation are solved and show that its effectivity is relative, mainly limited by the type of application.

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

A dual-reciprocity boundary element method based transient seepage model with nonlinear distributed point source dual-porosity gas reservoir

As a result of advancements in gas reservoir development technologies and a deeper understanding of seepage theory, numerous nonlinear seepage problems have emerged. Consequently, conventional analytical solution models are no longer applicable. Moreover, existing semi-analytical and numerical solution models suffer from issues like high computational complexity and limited flexibility, significantly constraining their practical utility. For this purpose, a gas reservoir transient seepage model is established based on the dual-reciprocity boundary element method (DRBEM). This model couples the wellbore storage coefficients with skin factor and introduces a novel dual-porosity transient interporosity seepage algorithm, enabling the resolution of transient seepage problems in dual-porosity gas reservoirs with nonlinear point source distributions. Additionally, it investigates the dynamic behavior of bottomhole pressure during the early production stage of gas wells, while capturing the dynamic pressure distribution within the seepage field. This approach offers the advantages of semi-analyticity, meshless, and eliminates the need for Laplace transforms, resulting in a significant reduction in computational complexity. Ultimately, the application of our model to well-pattern production with the production system transition issues, in conjunction with the log-log type curves of bottomhole pressure, and the dynamics of gas reservoir pressure, enables the analysis of interference characteristics between injection-production wells. This showcases the practical advantages of the model's application. The findings of this study provide valuable assistance for modeling and solving transient seepage problems in dual-porosity oil and gas reservoirs production. The application of DRBEM in unsteady-state diffusion, heat conduction, stress fields, and other potential field problems is relevant for reference.

Band structure calculations of three-dimensional solid-fluid coupling phononic crystals using dual reciprocity boundary element method and wavelet compression method

The algorithm for calculating the band structures of two-dimensional phononic crystals (PCs) using the boundary element method (BEM) has been proposed for many years. However, it has not yet been extended to three-dimensional (3D) PCs because the fundamental solutions of 3D dynamics are complex and are related to angular frequency. In this study, the BEM is applied to calculate the band structures of 3D solid-fluid coupling PCs by combining the dual reciprocity method. The use of the dual reciprocity method can help avoid nonlinear eigenvalue problems. To address the limitation of fully populated matrices in BEM, the wavelet compression method based on B-spline wavelet on the interval is adopted. Some small matrix entries, which are generated by the vanishing moment and local support characteristics, are set to zero using the provided truncation technique. This process results in the production of sparse matrices. The constructed generalized linear eigenvalue equations are modified to tackle the ill-conditioned matrix issues arising from the distinct fundamental solutions of solids and fluids. The results show that this method is superior to the finite element method in terms of calculation efficiency. Compared to conventional BEM, the current wavelet BEM can not only generate sparse matrices but also reduce the integration calculation time when handling large-scale problems.

Matric Flux Potential in Time Independent Infiltration Problems from a Single Triangular and a Trapezoidal Irrigation Channel

In this paper, steady infiltration problems into a homogeneous soil from a single triangular and trapezoidal irrigation channel are considered. The governing equation is Richard's equation that represents the movement of water in unsaturated soil. It is a non-linear equation and can be solved by linearizing to become a modified Helmholtz equation. Dual Reciprocity Boundary Element Methods (DRBEM) are used in this study to numerically solve the modified Helmholtz equation. Therefore, by using a provided solution, the numerical Matric Flux Potential (MFP) is calculated. This method was applied to the homogeneous soil problem of stationer infiltration from triangular and trapezoidal single irrigation. Both numerical solutions were compared. The result show that the MFP value from the triangular irrigation is higher than the trapezoidal irrigation. This indicates that content water from the triangular irrigation channel is higher than the trapezoidal irrigation channel.

A ONE-WAY COUPLED APPROACH FOR MULTISCALE CHARACTERIZATION OF FILLING OF DUAL-SCALE FIBROUS REINFORCEMENTS CONSIDERING AIR COMPRESSIBILITY AND DISSOLUTION IN LUMPED FASHION

The filling characterization of dual-scale fibrous reinforcements is challenging due to the presence of subdomains with dissimilar permeabilities, existence of wicking effects, and combination of air compressibility and dissolution phenomena. These factors cause flow imbalances inside the representative unitary cell (RUC), which lead to void formation and influence the behavior of macroscopic field variables, affecting the parts manufacturing by liquid composite molding (LCM). Here, the filling characterization of woven fabrics used in LCM is done using one-way coupled simulations. Once RUC geometry is characterized by scanning-electron microscopy (SEM), and stereomicroscopy, standard thickness test, and resin viscosity are measured, the multiphase finite volume method-volume of fluid (FVM-VOF) model of ANSYS Fluent is used for the three-dimensional filling of the RUC, incorporating an experimentally calibrated air entrapment parameter (λ) to consider air compressibility and dissolution; then, a lumped function for the coupling term with macroscopic equations is obtained in terms of volume-averaged variables. This function is used in the equivalent Darcy macroscopic model, which is solved using a dual-reciprocity boundary element method (DR-BEM) algorithm. By considering a single value of λ during the simulation, neglecting wicking effects, and normalizing physical variables, unified injection pressure-independent results for the local tows saturation and normalized pressure fields at mesoscopic scale were obtained, as well as for global tows saturation and normalized pressure and fluid front profiles at macroscopic scale, thus simplifying the filling characterization of reinforcements. Numerical results are coherent with unidirectional injection experiments at both scales.

Non-iterative reconstruction of time-domain sound pressure and rapid prediction of large-scale sound field based on IG-DRBEM and POD-RBF

In this paper, a novel non-iterative reconstruction method of time-domain sound pressure and rapid prediction scheme of large-scale sound field are proposed respectively based on the isogeometric dual reciprocity boundary element method (IG-DRBEM) and proper orthogonal decomposition - radial basis functions (POD-RBF). By establishing the non-iterative inversion theory in the meaning of least squares, infinite domain acoustic holography is achieved with a small number of measured sound pressure information. Furthermore, the POD-RBF network is adopted to predict large-scale sound pressure. The presented work further expands the scope of application of IG-DRBEM, and gives full play to its advantage of not having to calculate the coefficient matrix repeatedly when analyzing the sound field at different moments. Moreover, in order to satisfy the integral regularization conditions, a suitable convergence condition about variation coefficients is investigated. Numerical results show that the proposed method has high accuracy and good stability even when the sound pressure of complex airship model is predicted with large-scale freedom.

Comparisons between direct interpolation and reciprocity techniques of the boundary element method for solving two-dimensional Helmholtz problems

PurposeThis work compares the performance of the three boundary element techniques for solving Helmholtz problems: dual reciprocity, multiple reciprocity and direct interpolation. All techniques transform domain integrals into boundary integrals, despite using different principles to reach this purpose.Design/methodology/approachComparisons here performed include the solution of eigenvalue and response by frequency scanning, analyzing many features that are not comprehensively discussed in the literature, as follows: the type of boundary conditions, suitable number of degrees of freedom, modal content, number of primitives in the multiple reciprocity method (MRM) and the requirement of internal interpolation points in techniques that use radial basis functions as dual reciprocity and direct interpolation.FindingsAmong the other aspects, this work can conclude that the solution of the eigenvalue and response problems confirmed the reasonable accuracy of the dual reciprocity boundary element method (DRBEM) only for the calculation of the first natural frequencies. Concerning the direct interpolation boundary element method (DIBEM), its interpolation characteristic allows more accessibility for solving more elaborate problems. Despite requiring a greater number of interpolating internal points, the DIBEM has presented higher-quality results for the eigenvalue and response problems. The MRM results were satisfactory in terms of accuracy just for the low range of frequencies; however, the neglected higher-order primitives impact the accuracy of the dynamic response as a whole.Originality/valueThere are safe alternatives for solving engineering stationary dynamic problems using the boundary element method (BEM), but there are no suitable comparisons between these different techniques. This paper presents the particularities and detailed comparisons approaching the accuracy of the three important BEM techniques, aiming at response and frequency evaluation, which are not found in the specialized literature.

Effect of surface/interface stress on nanoscale phononic crystals using complete Gurtin-Murdoch model

The gradual miniaturization of acoustic devices requires us to study phononic crystals (PCs) at nanoscale, which should consider the effect of surface/interface stress. The first key point in this study is the complete use of the Gurtin-Murdoch model, and thereby the treatment of the surface displacement gradient term, which was avoided by all previous work on the nanoscale PCs. The second key point is that the dual reciprocity boundary element method is first applied to analyze the band structures of two-dimensional nanoscale PCs. The effect of surface/interface stress on band structures is discussed by investigating four typical nanoscale systems: nanoholes in a soft matrix, nanoholes in a stiff matrix, soft nanoscatterers in a stiff matrix and stiff nanoscatterers in a soft matrix. The numerical results demonstrate that the band structures obtained applying complete Gurtin-Murdoch model ascend comparing with those using simplified model, which enhances theoretical basis for designing nanoscale PCs. Furthermore, the effect of surface displacement gradient term is not obvious for the last three systems, and the displacement gradient term and surface tension should be ignored simultaneously when simplifying calculations. This study can provide good support for the control of hypersound at nanoscale.

Geometric inverse estimation for the inner wall of a furnace under transient heat conduction based on dual reciprocity boundary element method

A boundary shape estimation problem for the furnace’s inner surface is solved using the dual reciprocity boundary element method (DRBEM) and the conjugate gradient method (CGM) under transient heat conduction. The DRBEM is utilized to eliminate the drawbacks of traditional numerical methods and classical boundary element method of discretizing the entire computational domain, with only boundary discretization. The inversion results are obtained by applying the CGM to minimize the objective function, in which the sensitivity coefficients are calculated with the complex variable derivation method (CVDM), making the calculation precise and independent of step size. To verify the accuracy of the DRBEM in solving the transient heat conduction problem, the influencing factors including radial-basis function, the number of internal collocation points, and time step size are investigated. The influences of measurement time interval, future time step, initial guess, measurement error, and the number and position of measurement points on the inversion results are analyzed. Meanwhile, the effectiveness of the proposed approach is tested by numerical examples, and the inversion results show that it is stable, accurate, and efficient for identifying different and complicated unknown boundary shapes of the furnace.

Implementation of DRBEM for coupled sine-Gordon equations

In this study, two-dimensional, time dependent, nonlinear, coupled sine-Gordon equations (CSGE) is solved using the dual reciprocity boundary element method (DRBEM). The DRBEM provides to transform the domain integral, caused by the inhomogeneous terms, into the boundary integral by approximating with thin plate spline. DRBEM formulation is derived using the fundamental solution of the modified Helmholtz equations (MHE)’s. In order to write the original equations in the form of MHE, time dervatives are expanded using finite difference. The principle purpose of this study is to show that DRBEM is a successful alternative for the solution of CSGE. Method is tested with the examples which have the analitical solution. The study also includes the solutions of other examples in the literature. Numerical experiments show that obtained results are in good agreement with other studies where the problem has been solved before and thus DRBEM is effective and accurate for the solution of CSGE.

The Problem Parameters Effects on Transient Behavior of MHD Duct Flow

The present study focuses the effects of Reynolds number Re and magnetic Reynolds number Rm on the transient behavior of the MHD flow. The incompressible, electrically conducting and viscous fluid flows through a long pipe subjected to magnetic field B0(t)=B0f(t). B0 is the intensity and f(t) is the time varying function of the magnetic field which is chosen as polynomial, trigonometric, exponential and logarithmic function to illustrate the problem parameters effects. The Re and Rm effects on the behavior of the flow at transient levels are studied with these functions by taking Hartmann number Ha value as 20. The unsteady MHD equations in coupled form are treated by using the dual reciprocity boundary element method (DRBEM). The study reveals that, when Re or Rm increases the time level where the flow elongates is postponed to a further time level. Moreover, the increase in Re flattens the flow as in the increase of Hartmann number. However, the increase in Rm increases the flow magnitude. The transient flow and induced current contours are demonstrated for several Re and Rm values. After the flow elongates, the flow and induced current lines preserve the behavior for polynomial, exponential and logarithmic type f(t) while trigonometric type f(t) causes the flow to show periodic behavior.

MHD forced convection flow in an infinite channel with a rotating cylinder

The numerical investigation of forced convection flows under the influence of a uniform magnetic field is carried out in an infinite channel with an isothermal rotating circular cylinder placed in the center of the channel. The problem is discretized using the dual reciprocity boundary element method with constant elements. The analysis is done on how the controlling parameters affect both the fluid flow and heat transfer properties. It is observed that increasing the intensity of the magnetic field suppresses the impact of the rotational cylinder and decreases the flow velocity at the center while it enhances the heat transfer rate which causes a rise in the average Nusselt number along the heated walls. On the other hand, the heat transfer rate along the horizontal walls is an increasing or decreasing function of the rotational velocity of the cylinder, and the direction of cylinder rotation causes the boundary layers to become thicker or thinner close to the heated surfaces. Thus, the combined effects of the spinning cylinder and the magnetic field can be used to regulate the characteristic of fluid flow as well as the thermal topology of an infinite channel.

Influence of electric pulse characteristics on the cellular internalization of chemotherapeutic drugs and cell survival fraction in electroporated and vasoconstricted cancer tissues using boundary element techniques.

Electroporation has emerged as a suitable technique to induce the pore formation in the cell membrane of cancer tissues, facilitating the cellular internalization of chemotherapeutic drugs. An adequate selection of the electric pulse characteristics is crucial to guarantee the efficiency of this technique, minimizing the adverse effects. In the present work, the dual reciprocity boundary element method (DR-BEM) is applied for the simulation of drug transport in the extracellular and intracellular space of cancer tissues subjected to the application of controlled electric pulses, using a continuum tumour cord approach, and considering both the electro-permeabilization and vasoconstriction phenomena. The developed DR-BEM algorithm is validated with numerical and experimental results previously published, obtaining a satisfactory accuracy and convergence. Using the DR-BEM code, a study about the influence of the magnitude of electric field (E) and pulse spacing (dpulses) on the time behavior and spatial distribution of the internalized drug, as well as on the cell survival fraction, is carried out. In general, the change of drug concentration, drug exposure and cell survival fraction with the parameters E and dpulses is ruled by two important factors: the balance between the electro-permeabilization and vasoconstriction phenomena, and the relative importance of the sources of cell death (electric pulses and drug cytotoxicity); these two factors, in turn, significantly depend on the reversible and irreversible thresholds considered for the electric field.

DRBEM solution of singularly perturbed coupled MHD flow equations

In this study, the numerical solution of singularly perturbed magnetohydrodynamic (MHD) flow in a square duct with no-slip, and insulated or perfectly conducting walls is investigated by using the Dual Reciprocity Boundary Element Method (DRBEM). The steady, laminar and fully-developed MHD flow of an incompressible, viscous, and electrically conducting fluid in a long channel of square cross-section (duct) is driven by a pressure gradient. The governing MHD flow equations are convection–diffusion type and coupled in terms of the velocity V(x,y) and induced magnetic field B(x,y). When the intensity of horizontally applied external magnetic field is high, the Hartmann number (Ha) which is the coefficient of convection terms becomes large, that is, the coupled MHD flow equations become convection dominated. In other words, the coefficient of the diffusion terms is very small giving singularly perturbed MHD duct flow which exhibits thin boundary layers. The numerical scheme uses Shishkin mesh which depends on the number of points on one side and Ha. Numerical results obtained by DRBEM reveal that the well-known characteristics of the MHD flow problem are found as Ha increases and it is possible to obtain V(x,y) and B(x,y) for large values of Hartmann number up to Ha=1000.

A suggested dynamic soil – structure interaction analysis

In this paper, a new methodology for time domain analysis of buildings on raft foundations considering soil – structure interaction is proposed. Sub-structuring technique is used to separate the building as super-structure and the underneath soil as sub-structure. The super-structure can be modeled using any numerical method. However, in this paper the super-structure is modeled via the BEM to consider the real interaction area between column and slabs. The dynamic load is considered as an earthquake acceleration record that can be transformed to equivalent dynamic loads acting on the super-structure floors. The sub-structure is analyzed using the dual reciprocity boundary element method as closed domain. New iterative coupling technique is proposed between the super and sub-structures to reduce the computational effort and required storage. An example is presented to demonstrate the strength and the practicality of the proposed methodology.

Comparison study on the numerical stability of dual reciprocity boundary element method for the MHD slip flow problem

In this paper, unsteady Magnetohydrodynamics (MHD) flow in a cavity with insulated and slipping walls is studied. The nonlinear coupled MHD equations containing the velocity and the induced magnetic field are solved by the dual reciprocity boundary element method (DRBEM). The time derivative is discretized by the explicit Euler or the central difference methods. The grid independence is tested. The numerical stability is analyzed through the spectral radius of the coefficient matrix and the performances of the two time discretization methods are compared. In the studied ranges, the proper choices of relaxation parameter and time increment are found. The influences of Hartmann number (Ha), Reynolds number (Re), magnetic Reynolds number (Rm), and the slip length on the numerical stability are investigated. The flow behaviors in the steady-state and transient time levels are presented. It is found that relaxation parameter and time step close to one guarantee the numerical stability in the studied range. Increasing Re alters the stability condition more than Rm increment. DRBEM with the central difference method is the appropriate numerical technique considering the computational cost and the stability for solving MHD slip flow problems. The flow elongation time is postponed for large Rm and Re.

An integral framework for computational thermo-elastic homogenization of polycrystalline materials

A grain scale framework for thermo-elastic analysis and computational homogenization of polycrystalline materials is proposed. The morphology of crystal aggregates is represented employing Voronoi tessellations, which retain the main statistical features of polycrystalline materials. The behaviour of the individual grains is modelled starting from an integral representation for anisotropic thermo-elasticity, which is numerically addressed through a dual reciprocity boundary element method. The integrity of the aggregate is enforced through suitable intergranular thermo-elastic continuity conditions. By virtue of the features of the underlying formulation, the polycrystalline thermo-elastic problem is expressed in terms of grain boundary variables only, thus simplifying the subsequent task of meshing and reducing the overall computational cost of the analysis, ultimately providing an appealing tool for multiscale applications. The framework has been tailored for computational thermo-elastic homogenization of polycrystalline materials and it has been applied to the statistical computational homogenization of SiC and Al2O3 polycrystals, with accurate results confirming its robustness and effectiveness. The extension of the proposed framework to multiscale modelling of materials failure in thermally active environments is eventually discussed.

Isogeometric dual reciprocity BEM for solving non-Fourier transient heat transfer problems in FGMs with uncertainty analysis

In this paper, the precise integration multi-patch isogeometric dual reciprocity boundary element method (IG-DRBEM) of the non-Fourier transient heat transfer problem in functionally graded materials (FGMs) is proposed, and the stochastic analysis of multidimensional material uncertainty by hyperbolic truncated adaptive sparse polynomial chaos expansion (PCE) is established based on the determine theory. Up to now, most research results of the isogeometric analysis boundary element method (IGABEM) in transient heat transfer problems are based on Fourier's law. The main reason is the higher-order derivative term with respect to time in the non-Fourier heat transfer control equation leads to additional domain integrals in the boundary-domain integral equation. The dual reciprocity method (DRM) can transform the domain integral to the boundary conveniently and maintain the dimension reduction advantage of the BEM. The introduction of the precise integration method (PIM) can improve the stability and accuracy of the time-domain solution. In the random analysis based on IG-DRBEM, a sparse PCE strategy with hyperbolic truncation is constructed to overcome the dimensional disaster difficulty of traditional PCE. Through the proposed adaptive iterative process, a relatively small number of PCE terms are retained compared with the full PCE on the premise of meeting the error estimation requirements. The presented method is verified to have good accuracy and high efficiency by the complex model similar to actual engineering, which further expands the application extent of IG-DRBEM.

On numerical study of constrained coupled shape optimization problems based on a new shape derivative method

AbstractIn this article, we deal with a numerical method for the approximation of a class of coupled shape optimization problems, which consist in minimizing an appropriate general volume cost functional subjected to coupled boundary value problems by means of a Neumann boundary transmission condition. We show the existence of the shape derivative of the cost functional and express it by means of support functions, using a new formula of shape derivative on a family of convex domains. This allows us to avoid the disadvantages related to the classical shape derivative method using vectors field. Then the numerical discretization is performed using the dual reciprocity boundary element method in order to avert the remeshing task required for the finite element method. Finally, we give some numerical results, based on the gradient method, showing the efficiency of the proposed approach.