Dual Reciprocity Boundary Element Method Research Articles

Identification of thermal conductivity for orthotropic FGMs by DT-DRBEM and L-M algorithm

ABSTRACTThe differential transformation dual reciprocity boundary element method (DT-DRBEM) combined with the Levenberg–Marquardt (L-M) algorithm is proposed to identify the thermal conductivity for orthotropic functionally gradient materials (FGMs). The original governing equation of two-dimensional transient heat conduction problem is transformed into a time-independent and recursive form equation by the differential transformation method (DTM). The analog equation method is used to convert the partial differential equation to a standard thermal diffusion equation. Then, the DRBEM is applied to solve the direct problem and the measured temperature is obtained. After that, the L-M algorithm is used to minimize the objective function and recover the desired thermal conductivity. The DT-DRBEM is compared with the finite difference DRBEM (FD-DRBEM). The L-M algorithm is also compared with the conjugate gradient method (CGM). The influences of different initial guesses, number of measurement points and random errors on the inverse results are also investigated. The initial guesses have little effect on the number of iteration. With the increase of measurement points and with the decrease of measurement errors, the results are more accurate.

DUAL RECIPROCITY BOUNDARY ELEMENT METHOD (DRBEM) DENGAN PREDIKTOR-KOREKTOR UNTUK MASALAH INFILTRASI STASIONER PADA SALURAN DENGAN LAPISAN IMPERMEABEL DENGAN ROOT WATER UPTAKE

This study aims to discussed about Dual Reciprocity Boundary Element Method (DRBEM) with a predictor-corrector for steady infiltration problems with impermeabel layer with root water uptake in homogeneous soils. Steady infiltration in homogeneous soils is governed by Richard equation. This equation is transformed using a set of transformation including Khirchoff, dimensionless variabels and dimensionless parameters into a type of modified Helmholtz equation. Furthermore with DRBEM, numerical solution of modified Helmholtz equation obtained. The proposed method is tested on problem involved infiltration from rectangular and trapezoidal channels with impermeabel layer with root water uptake. Keyword : infiltration, root water uptake, modified Helmholtz equation, Richard equation, DRBEM

A Computerized Boundary Element Algorithm for Modeling and Optimization of Complex Magneto-Thermoelastic Problems in MFGA Structures

Aims: The aim of this article is to propose a boundary integral equation algorithm for modeling and optimization of magneto-thermoelastic problems in multilayered functionally graded anisotropic (MFGA) structures.
 Study Design: Original research paper.
 Place and Duration of Study: Jamoum laboratory, January 2018.
 Methodology: a new dual reciprocity boundary element algorithm was implemented for solving the governing equations of magneto-thermoelastic problems in MFGA structures.
 Results: A numerical results demonstrate validity, accuracy, and efficiency of the presented technique.
 Conclusion: Our results thus confirm the validity, accuracy, and efficiency of the proposed technique. It is noted that the obtained dual reciprocity boundary element method (DRBEM) results are more accurate than the FEM results, the DRBEM is more efficient and easy to use than FEM because it only needs the boundary of the domain needs to be discretized.

Identification of boundary conditions for non-Fourier heat conduction problems by differential transformation DRBEM and improved cuckoo search algorithm

The differential transformation method is combined with the dual reciprocity boundary element method to solve the non-Fourier heat conduction problems in functionally gradient materials. The cuckoo search algorithm is improved by the Broyden–Fletcher–Goldfarb–Shanno algorithm to identify the boundary conditions for the heat conduction problems. The polynomial function related to coordinate and time is proposed to approximate the unknown boundary conditions. Numerical examples discuss the influences of measurement point numbers and measurement errors on inverse solutions. Numerical results demonstrate the effectiveness and accuracy of the proposed method.

Parameter Identification in MHD Duct Flow Cauchy Problem using the DRBEM

In this study, the Hartmann number (Ha) is identified through the inverse formulation of Cauchy problem for magnetohydrodynamic (MHD) flow in a duct with insulated but no-slip or variably slipping walls. The solutions for direct and inverse problems are obtained using the dual reciprocity boundary element method (DRBEM), and Tikhonov regularization with L-curve method is used for solving ill-conditioned linear system of equations resulted from the inverse problem. The velocity and the induced magnetic field profiles are simulated from the direct solution depicting the well-known MHD characteristics. The Hartmann number is reconstructed from the inverse problem with an accuracy of 10−4 to 10−6. The DRBEM provides overdetermined information needed in the inverse MHD flow problem since it gives both the solution and its normal derivatives on the boundary points.

MHD flow in a circular pipe with arbitrarily conducting slipping walls

In this study, the magnetohydrodynamic (MHD) flow is simulated in a circular pipe with slipping and arbitrarily conducting boundary. The 2D governing coupled equations in terms of the velocity and the induced magnetic field are solved by the Dual Reciprocity Boundary Element Method (DRBEM). The discretized system of equations is solved in one stroke without introducing an iteration which reduces the computational cost. It is shown that, the flow decelerates, Hartmann layers enlarge through the top and the bottom of the pipe and induced current lines align as the wall conductivity or Hartmann number increases. An increase in the slip length accelerates the flow, shrinks the stagnant region, diminishes the boundary layers and retards the effect of the wall conductivity increase. The DRBEM is an advantageous method in solving MHD flow especially with slipping and arbitrarily conducting boundary conditions, since it enables to insert both the unknowns and their normal derivatives in slip and conductivity wall conditions.

MHD Natural Convection Ferrofluid Flow in Semi-Annulus Enclosures

This study examines the natural convection flow and the heat transfer of a water-based ferrofluid consisting Fe3O4 nanoparticles in a semi-annulus cavity with a circular outer and sinusoidal inner walls under the effect of a magnetic field produced by multiple nodal magnetic sources. The equations governing the ferrofluid flow are numerically solved using the dual reciprocity boundary element method. That is the equations are transformed into integral equations only on the boundary by using the fundamental solution of the Laplace equation and treating all other terms as nonhomegeneity through radial basis function approximation, which results in a discretized system of small size. The numerical computations are performed for several physical parameters namely, Hartmann, Rayleigh and magnetic numbers, solid volume fraction of nanoparticles, the number of undulation and the amplitude of sinusoidal wall. The obtained results show that increasing Rayleigh number, solid volume fraction and amplitude of sine waves enhance the average Nusselt number whereas it decreases with an increase in Hartmann number.

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation ε is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.

Unsteady MHD mixed convection flow in a lid-driven cavity with a heated wavy wall

The unsteady mixed convection flow under the effect of an externally applied uniform magnetic field is investigated numerically in a lid-driven cavity with a sinusoidal wavy wall. The governing equations given in terms of stream function, vorticity and temperature are discretized by using the dual reciprocity boundary element method (DRBEM) in space and a two-level time integration scheme is employed in time discretization. The simulations focus on the effects of several physical parameters, namely Hartmann and Rayleigh numbers, Joule heating parameter, the inclination angle of magnetic field and number of undulation of wavy wall on the flow and temperature distribution. Moreover, the stability analysis of the numerical method is performed in terms of time increment and time relaxation parameters for various combinations of the problem controlling parameters.

Complex Fourier element shape functions for analysis of 2D static and transient dynamic problems using dual reciprocity boundary element method

In this paper, the boundary element method is reformulated using new complex Fourier shape functions for solving two-dimensional (2D) elastostatic and dynamic problems. For approximating the geometry of boundaries and the state variables (displacements and tractions) of Navier's differential equation, the dual reciprocity (DR) boundary element method (BEM) is reconsidered by employing complex Fourier shape functions. After enriching a class of radial basis functions (RBFs), called complex Fourier RBFs, the interpolation functions of a complex Fourier boundary element framework are derived. To do so, polynomial terms are added to the functional expansion that only employs complex Fourier RBF in the approximation. In addition to polynomial function fields, the participation of exponential and trigonometric ones has also increased robustness and efficiency in the interpolation. Another interesting feature is that no Runge phenomenon happens in equispaced complex Fourier macroelements, unlike equispaced classic Lagrange ones. In the end, several numerical examples are solved to illustrate the efficiency and accuracy of the suggested complex Fourier shape functions and in comparison with the classic Lagrange ones, the proposed shape functions result in much more accurate and stable outcomes.

Identification of transient boundary conditions with improved cuckoo search algorithm and polynomial approximation

The cuckoo search (CS) algorithm combined with Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm (CS-BFGS) is proposed to identify time-dependent boundary conditions for 2-D transient heat conduction problems in functionally gradient materials. Firstly the dual reciprocity boundary element method (DRBEM) is used to solve the direct problem. Then taking the unknown boundary conditions as a polynomial function of coordinates with time-dependent coefficients, the CS-BFGS is applied to obtain the unknown coefficients of the polynomial. As a result, the transient boundary conditions are evaluated. The convergence speed of the CS-BFGS algorithm is faster than the CS algorithm. What's more, the effect of the polynomial degree is discussed. As the polynomial degree increases, the inverse results are more accurate but the iterative number and computation time also increase. Finally, the influences of the position and number of measurement points, and random errors on the inverse results are investigated. With the measurement points closer to the boundary, with the increase of measurement point number and with the decrease of measurement errors, the results are more accurate.

The dual reciprocity boundary element formulation for convection-diffusion-reaction problems with variable velocity field using different radial basis functions

This paper presents a dual reciprocity boundary element method (DRBEM) formulation for the solution of steady-state convection-diffusion-reaction problems with variable velocity field at moderately high Péclet number. This scheme is based on utilising the fundamental solution of the convection-diffusion-reaction equation with constant coefficients. In this case, we decompose the velocity field into an average and a perturbation, with the latter being treated using a dual reciprocity approximation to convert the domain integrals arising in the boundary element formulation into equivalent boundary integrals. A proposed approach is implemented to treat the convective terms with variable velocity, for which the concentration is expanded as a series of functions. Four numerical experiments are included with available analytical solutions, to establish the validity of the approach and to demonstrate the efficiency of the proposed method.

Convective Flow of Blood in Square and Circular Cavities

Abstract In this study, the fully developed, steady, laminar ow of blood is studied in a long pipe with square and circular cross-sections subjected to a magnetic field generated by an electric wire. Temperature difference between the walls causes heat transfer within the fluid by the displacement of the magnetizable fluid particles in the cavity. The governing equations are the coupled Navier-Stokes and energy equations including magnetization terms. The axial velocity is also computed with the obtained plane velocity. The Dual Reciprocity Boundary Element Method (DRBEM) is used by taking all the terms other than Laplacian as in- homogeneity which transforms the partial differential equations into the boundary integral equations. Numerical results are given for increasing values of Magnetic (Mn) and Rayleigh (Ra) numbers. The numerical results reveal that an increase in Mn accelerates the plane velocity in the cavity but decelerates the axial velocity around the magnetic source. Pressure increases through the channel starting from the magnetic source. Isotherms show the cooling of the channel with high Mn and Ra only leaving a thin hot layer near the top heated wall. As Ra increases viscous effect is reduced leaving its place to convection in the channel. The use of DRBEM has considerably small computational expense compared to domain type methods.

A novel dual reciprocity boundary element formulation for two-dimensional transient convection–diffusion–reaction problems with variable velocity

This paper describes a new formulation of the dual reciprocity boundary element method (DRBEM) for two-dimensional transient convection–diffusion–reaction problems with variable velocity. The formulation decomposes the velocity field into an average and a perturbation part, with the latter being treated using a dual reciprocity approximation to convert the domain integrals arising in the boundary element formulation into equivalent boundary integrals. The integral representation formula for the convection–diffusion–reaction problem with variable velocity is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. A finite difference method (FDM) is used to simulate the time evolution procedure for solving the resulting system of equations. Numerical applications are included for three different benchmark examples for which analytical solutions are available, to establish the validity of the proposed approach and to demonstrate its efficiency. Finally, results obtained show that the DRBEM results are in excellent agreement with the analytical solutions and do not present oscillations or damping of the wave front, as it appears in other numerical techniques.

DBEM and DRBEM solutions to 2D transient convection-diffusion-reaction type equations

The present study focuses on the numerical solution of the transient convection-diffusion-reaction equation by transforming it into modified Helmholtz equation through an exponential type transformation. In the spatial discretization of the problem domain two different boundary element methods (BEM), namely the domain BEM (DBEM) and the dual reciprocity BEM (DRBEM), are employed which are combined with an implicit backward finite difference time integration. The BEM techniques differ in the sense of treating the leftover domain integral. That is, in DBEM the domain integral is kept and calculated by quadrature while it is reduced to an equivalent boundary interal by means of radial basis functions in DRBEM. The numerical simulations are first carried out for several values of diffusion coefficient in the transient convection-diffusion-reaction equation. The results reveal that DBEM gives more accurate results for smaller values of diffusion coefficient compared to DRBEM. Thus, DBEM is further used for the solution of the coupled transient convection-diffusion type magnetohydrodynamic flow equations. The results are presented by equivelocity and current lines for various values of problem parameters, which show the well-known characteristics of the MHD flow.

DRBEM solution of the Cauchy MHD duct flow with a slipping perturbed boundary

In this study, the MHD flow direct and Cauchy problems are solved in a rectangular duct with a perturbed, curved, and slip upper boundary. The aim is to recompute the slipping velocity and the slip length by using the asymptotic analysis with respect to the perturbation parameter ϵ and solving MHD flow equations for the first order and the corrector solutions in the rectangular duct. Hence, without discretizing the curved boundary, we are able to obtain the solution of MHD flow in the duct with curved perturbed boundary. The dual reciprocity boundary element method (DRBEM) is utilized for solving the coupled MHD equations as a whole. The discretization of Cauchy problems leads ill-conditioned systems of linear algebraic equations, hence a regularization technique is necessary. In this study, the Tikhonov regularization is used to obtain the solution of the Cauchy MHD problems. The main advantage of the DRBEM is discretizing only the boundary and providing both the velocity and its normal derivative values on the boundary. This enables us to recover the slip length on the perturbed boundary through the slip boundary condition.

DRBEM-FDM solution of a chemotaxis–haptotaxis model for cancer invasion

In this paper, the invasion model focusing on the effect of chemotaxis and haptotaxis is considered which involves the interactions between the cancer cells, the normal cells and the matrix degrading enzyme (MDE) that is secreted by the cancer cells to invade the surrounding tissue. The corresponding model consists of a system of nonlinear reaction–diffusion-transport equations and a combined application of dual reciprocity boundary element method (DRBEM) and finite difference method (FDM) is used to solve this system numerically. For the space discretization of the equations for cancer cells, MDE, plasminogen activator inhibitor and plasmin DRBEM is made use of. To this end, the fundamental solution of Laplace equation is used and the rest of the terms are considered as the nonhomogenity. Due to chemotaxis and haptotaxis terms, the nonhomogenity for the cancer cell equation contains the space derivatives of other unknowns and FDM is used in combination with DRBEM for their discretization. The time dependent ordinary differential equations (ODEs) of the DRBEM and DRBEM-FDM discretized equations for the cancer cell density and MDE concentration, respectively, and the ODE for the normal cell density are solved by using a combination of forward and backward Euler methods. The difficulties of the numerical solution in two-space dimensions, due to the nonlinear terms in the modeling system, are overcome using the boundary-only nature of DRBEM which gives the solution with a lower computational cost. Effects of haptotaxis, chemotaxis, proliferation of cells and plasminogen activation system on the invasion are analyzed and the numerical solutions are in good agreement with the expected behavior of the modeling phenomena.

A New Hybrid Method for Numerical Simulation of Combined Conductive-Radiative Heat Transfer in Participating Medium

This paper presents a new hybrid method for combined conductive-radiative heat transfer in two-dimensional enclosures. The participating medium is radiatively absorbing, emitting, scattering and gray. The radiative transport equation is solved by the discrete ordinate method, while the energy equation is solved by the dual reciprocity boundary element method. The various influencing factors i.e., radiation-conduction parameter, surface emissivity and single scattering albedo are investigated to demonstrate the feasibility and efficiency of the present method.

Dynamic analysis of three-dimensional polycrystalline materials using the boundary element method

A new computational framework to analyse the microscale dynamic behaviour of three-dimensional polycrystalline materials with different lattice structures is presented. The absence of analytical solutions for these stochastic materials has been a challenge in validating the numerical results. In macroscale analysis, when the number of crystal aggregates in the microscale is large, polycrystalline aggregates exhibit an effective isotropic nature. To model the elastodynamic effects, random crystalline orientations and morphology configurations are used for each polycrystalline aggregate. The recently proposed fundamental solution based on double Fourier series for general anisotropy coupled to the dual-reciprocity boundary element method is used. A drastic reduction in the degrees of freedom is achieved owing to the nature of the boundary mesh. The stochastic time-dependent displacement wave under various boundary conditions is evaluated, and the validation is carried out using homogenisation over the grain surfaces. An assessment of the effective macroscopic properties of the available analytical isotropic models is proposed, wherein the convergence is evaluated using statistical samples. Numerical results are presented using a large number of simulations to obtain a good confidence interval.

Numerical simulation of unsteady mixed convection of nanofluid in a lid-driven square cavity

The behavior of unsteady mixed convection flow of $Cu-$water based nanofluids is investigated numerically inside a square lid-driven partially heated flow below. Dual Reciprocity Boundary Element Method is used to solve stream function-vorticity form of the governing equations of the problem. The need of time integration scheme is eliminated by transforming the vorticity transport and energy equations to modified Helmholtz equations. This procedure also diminishes the stability problems. The numerical results are given for several values of Reynolds number $(Re)$, Rayleigh number $(Ra)$, heat source location $(D)$, heat source length $(B)$ and solid volume fraction $(\phi)$. The steady-state results are in good agreement with the results available in the literature.