Boundary Domain Integral Method Research Articles

Investigation of the heat transfer coefficient for a red clay brick

AbstractThere is a need for the development of materials for thermal isolation. Heat energy is used in different ways, for example, for house warming or production of electricity from solar power plants. However, to use the heat energy efficiently isolating materials are needed. There are different materials available to isolate houses and heat storage containers. However, those materials are expensive and some are over engineered. The best available material for thermal isolation would be air that has the thermal coefficient . Thus, materials that trap a large amount of air are good isolators. On the other hand, some isolating materials are harmful for the environment, for example, glass wool, asbestos. An alternative are red clay bricks because red clay is a natural product that is present everywhere on the planet. But the thermal coefficient of red clay bricks is high . To reduce the thermal coefficient cavities can be added in the brick. The cavities trap air and reduce the thermal coefficient of the brick.We simulate the heat transfer through a hollow brick. The holes in the brick are cavities. In order to determine the heat transfer coefficient we solve the heat flux through the hollow brick. For that a model of the brick is formulated as sequence of a solid and a fluid part. In the solid part only heat conduction is present and in the fluid heat transfer is present in form of conduction, convection and radiation. The flow field in the fluid part is modelled with the velocity‐vorticity formulation of the incompressible Navier‐Stokes. It is assumed that the Rayleigh number is below 106, hence, natural convective laminar fluid flow is present in the cavities. The cavities are also heated by heat radiation of the surrounding walls.The numerical realisation is done with the Boundary‐Domain Integral Method (BDIM). To handle suitable geometries the complexity of the BDIM is reduced from quadratic to logarithmic or almost linear by applying the ‐methodology. M is the number of unknown domain nodes. Numerical studies will be presented.

Numerical simulation of fluid flow with a fast Boundary‐Domain Integral Method

AbstractFor the simulation of fluid flow the velocity‐vorticity formulation of the Navier‐Stokes equations can be used. Approximating the first order time derivative in these equations by finite differences, essentially, the Yukawa equation for the velocity has to be solved with an inhomogeneity governed by the vorticity. Applying the Boundary Element Method to such problems is not straightforward. However, the Boundary‐Domain Integral Method (BDIM) can be used, which requires to solve beside the boundary integrals as well a domain integral due to the inhomogeneity. Classical BEM formulations have a quadratic complexity with respect to the unknowns on the boundary and, for the BDIM additionally, for the unknowns in the domain. This prevents the BDIM from being applied to real‐world problems.To reduce the quadratic complexity in BEM, approximation methods like the fast multipole method, ℋ‐matrices in combination with the adaptive cross approximation or the ℋ2 ‐matrix technique can be used. All these methods reduce the complexity to an almost linear order. Here, the ℋ2 ‐matrix technique is applied to reduce storage and computing time for the right hand side, i.e., for the matrix of the domain integrals. Based on interpolatory kernel expansions and an heuristic criterium for too small matrix entries an almost linear complexity can be achieved. The numerical studies focus on a lid‐driven cavity. The complexity reduction is demonstrated.

A point particle source model for conjugate heat and mass transfer in dispersed two-phase flows by BEM based methods

The paper reports on development of a Boundary Element Method (BEM) based two phase dispersed flow model, that allows an accurate and efficient heat and mass transfer coupling by using a moving point source model to account for particle–fluid interaction. The two-way coupling model highlights the efficient use of the elliptic fundamental solution and the Dirac delta distribution properties to accurately evaluate the heat and mass point particle source impact on the continuous phase, solved by the Boundary Domain Integral Method (BDIM). In addition to the BDIM model of the particle–fluid heat and mass transfer interaction, the two-phase flow case under consideration is extended to the case of porous spherical particle drying with internal moving drying front, which is solved by the Boundary Element Method. As the two-phase flow is considered to be dilute, the particle–fluid momentum exchange is covered by a one-way coupling algorithm, with Lagrangian particle tracking used for determination of particle positions and velocities in the flow. Two computational cases are presented, where 1000 and 10000 particles are dried in a stream of hot air. Comparison between the obtained drying times for the cases of the one-way and the two-way heat and mass transfer coupling results shows, that the developed two-way coupling model accurately captures the effect of moisture accumulation and temperature decrease in the fluid phase, leading to realistic computations of drying rates of particles in the flow.

Analysis of dynamic crack propagation in two-dimensional elastic bodies by coupling the boundary element method and the bond-based peridynamics

A novel method for predicting dynamic crack propagation based on coupling the boundary element method (BEM) and bond-based peridynamics (BBPD) is developed in this work. The special feature of this method is that it can take full advantages of both the BEM and PD to achieve a higher level of accuracy and efficiency. Based on the scale of the structure and the location of cracks, the considered domain can be divided into a non-cracked region and a cracked region. For the non-cracked region, a meshfree boundary-domain integral equation method (meshfree-BDIEM) is employed in the analysis to reduce the dimension by one and to increase the computational efficiency. The bond-based PD is applied to simulate the cracked region, which can model the initiation and propagation of the cracks automatically. The boundary nodes from the BEM on the interfaces can interact with the material points from the PD directly. By using the displacement continuity and force equilibrium conditions on the interfaces, a combined model is obtained by merging the mass, stiffness, and force matrices from each region of the domain. Both the implicit and explicit BEM-PD coupled solution methods can deliver accurate results without inducing the ghost forces. Several benchmark problems of dynamic crack propagation have been modeled by using the method, which demonstrate that the developed BEM-PD coupled approach can be an efficient numerical tool to model the dynamic crack propagation problems.

A new periodic cell model of aerosol diffusion deposition in a fibrous filter

A new periodic cell model for the convective-diffusive transport of small aerosol particles in a fibrous filter is developed. The proposed periodic model takes into account the influence of the fiber diffusion wake on the particle deposition. This allows us to predict the efficiency of a filter more accurately compared to the single-fiber approximations. The fluid flow is described in the framework of Stokes flow in a rectangular periodic cell. New form of periodic boundary conditions for the particle transport equation in the cell is suggested. The equations of fluid flow and particle transport are solved using the Boundary Element Method (BEM) and Boundary Domain Integral Method (BDIM).The diffusive deposition of nanoaerosols in the filter, modeled as a row of the large number of circular cylinders located perpendicular to the flow direction, is studied numerically using the many fibers model. It is shown, that predictions of the new periodic model are in the better agreement with the many fibers model for the filters of greater length. It is also shown, that the widely used approximation for single fiber deposition efficiency (Stechkina, 1967) overestimates the efficiency of multi-fiber filter compared to periodic model. The overestimation increases with increase of Peclet number.The approximate formula for the single fiber particle deposition efficiency in a periodic cell for a square and staggered cylinder arrays is derived. The formula accounts for the effect of the Peclet number and the hydrodynamic factor. The predictions of the derived formula are compared to the numerical data from the work of Hosseini and Tafreshi (2010) for the fibrous filter comprised of many randomly arranged cylinders. The predictions of the diffusive deposition efficiency obtained using the derived formula for the staggered fibers array are shown to be in a good agreement with the numerical model predictions.

A stable computation on local boundary-domain integral method for elliptic PDEs

Many local integral methods are based on an integral formulation over small and heavily overlapping stencils with local Radial Basis Functions (RBFs) interpolations. These functions have become an extremely effective tool for interpolation on scattered node sets, however the ill-conditioning of the interpolation matrix – when the RBF shape parameter tends to zero corresponding to best accuracy – is a major drawback. Several stabilizing methods have been developed to deal with this near flat RBFs in global approaches but there are not many applications to local integral methods. In this paper we present a new method called Stabilized Local Boundary Domain Integral Method (LBDIM-St) with a stable calculation of the local RBF approximation for small shape parameter that stabilizes the numerical error. We present accuracy results for some Partial Differential Equations (PDEs) such as Poisson, convection–diffusion, thermal boundary layer and an elliptic equation with variable coefficients.

Analytic and numerical solutions for linear and nonlinear multidimensional wave equations

We develop three reliable iterative methods for solving the nonlinear 1D, 2D and 3D second-order wave equation and compare the results with a discretization-based solver. The iterative Tamimi–Ansari method (TAM), Daftardar–Jafari method (DJM) and the Banach contraction method (BCM) are used to obtain the exact solution for linear equations. For nonlinear equations and practical problems, however, one obtains the approximate solutions that converge to the exact solution, if one exists. The convergence analysis of the three methods is shown using the fixed-point theorem. The methods prove to be quite efficient and well suited to solve this kind of problems. We present several examples that demonstrate the accuracy and efficiency of the methods. We also compare the methods with a method based on discretization (Boundary Domain Integral Method (BDIM)). The BDIM uses a standard domain grid and discretizes the integral form of the governing equations. The iterative methods were developed with Mathematica® 10, while BDIM is a proprietary development.

Boundary-domain integral method and homotopy analysis method for systems of nonlinear boundary value problems in environmental engineering

In this paper we present the development of numerical methods for two problems in chemical engineering: adsorption of carbon dioxide into phenyl glycidyl ether and diffusion and reaction processes in a porous catalyst. Both problems are modelled by systems of two coupled nonlinear ordinary differential equations. Dirichlet and Neumann type boundary conditions are considered. We develop the standard homotopy analysis method and the boundary-domain element method to solve for the unknown steady-state concentrations of reactants. The validity of the proposed methods is checked and demonstrated by numerical examples. Convergence rates are compared and the advantages and drawbacks of the proposed methods are discussed and compared to the Mathematica NDSOLVE solver. We show, that the proposed homotopy analysis method features exponential convergence rate and is highly accurate and efficient. As low as 6 terms are needed to reach error norms of Furthermore, the boundary-domain integral approach is also capable of solving the same problem very efficiently, using a very coarse computational mesh (21 nodes).

An accelerated Boundary‐Domain Integral Method for three‐dimensional fluid flow analysis

AbstractThe rapid growth of computer power enables performing of more and more complex numerical simulations. However, the need to develop efficient numerical algorithms is always present. In this study, we propose an acceleration of an algorithm to solve the nonlinear system of Navier‐Stokes equations. We present an accelerated Boundary‐Domain Integral method (BDIM). The BDIM is a boundary element method (BEM) based method, thus its computational demands scale as 𝒪(N2), where N is the number of grid nodes. This is due to the fact that the discretization procedure leads to fully populated matrices. However, the computational demands can be reduced to a linear‐logarithmic 𝒪(N log N) complexity, with the implementation of an approximation procedure. In this study, we solve the velocity‐vorticity formulation of the Navier‐Stokes equations and develop an accelerated algorithm for determination of the boundary vorticity in 3D based on ℋ‐matrix and the adaptive cross approximation (ACA) method. We test the algorithm using coupled nanofluid flow and heat transfer. As expected results show that, the acceleration (matrix compression rate) has a negative influence on the solution accuracy.

Stochastic modelling of nanofluids using the fast Boundary-Domain Integral Method

In this paper, we couple a numerical method aimed at simulation of flow and heat transfer of nanofluids with stochastic modelling of input and material parameters. In order to simulate nanofluids, an in-house numerical method was developed, based on the solution of 3D velocity–vorticity formulation of Navier–Stokes equations. A fast Boundary-Domain Integral Method has been employed to solve the governing equations and set up the deterministic flow and heat transfer solver. The developed algorithm is used to simulate natural convection of a nanofluid in a closed cavity. The uncertainty present in the input parameters is propagated to the output of interest via the Stochastic Collocation Method. The stochastic mean, variance, and higher-order moments of the output values are presented. The non-intrusive nature of the Stochastic Collocation Method facilitates the previously validated deterministic code to remain unchanged. The stochastic analysis reveals that the uncertainty of input parameters influences the output results most in the areas where high flow field gradients appear.

In-Plane Free Vibration of Circular and Annular FG Disks

The analysis of the in-plane free vibration of the circular and annular FG disks by a meshfree boundary-domain integral equation method is presented in this paper. The material properties of the disks are assumed to vary in the radial direction obeying an exponential law. Based on the two-dimensional linear elastic theory, the motion equations of the FG disks are derived by using the static fundamental solutions. Radial integration method as an efficient tool is adopted to treat the domain integrals which are raised due to the material inhomogeneous and inertial effects. The natural frequencies and associate mode shapes are calculated for the FG disks with combinations of free and clamped boundary conditions. Parametric studies are also conducted to study the effects of the material gradients, radius ratios and boundary conditions on the frequency of the FG disks.

Dynamic behaviors of tapered bi-directional functionally graded beams with various boundary conditions under action of a moving harmonic load

In this paper, the dynamic response of beams made of bi-directional functionally graded materials (FGMs) with varied cross-sections subjected to a moving harmonic load has been studied by a developed boundary-domain integral equation method based on the two-dimensional linear elastic theory. It is assumed that the elastic modulus and mass density of the FG beams vary separately following each exponential distribution parameters through longitudinal or transvers direction. Then, the elastostatic homogeneous fundamental solution was substituted into the dynamic governing equation for deriving the vibration boundary-domain integral equation of the tapered FG beams. A meshfree scheme can be achieved by applying the radial integration technique to transform the domain integrals into boundary integrals. Houbolt's algorithm is employed to obtain the dynamic response of the tapered FG beams. The effectiveness of the present method applied to the non-uniformed FG beams is verified by comparing numerical results with those available for free vibration of tapered beams of linearly variable width or depth. A detailed parametric study is carried out to investigate the influences of the taper ratios, material gradients, gradation directions and boundary conditions as well as the moving load frequencies and velocities on the dynamic behaviors of tapered FG beams.

Solving the fluid flow problems with boundary domain integral method

The boundary-domain integral method with the analytical integrals over the individual boundaries and internal cells for the solving the boundary value problems of fluid flow problems is developed. The axisymmetric potential flow of an incompressible fluid around a sphere, the fluid flow in a porous medium with spatially inhomogeneous permeability and the problem of convective-diffusive transport of a gas suspension in a periodic cell are calculated The comparison of the numerical results with analytical solutions show the efficiency and good accuracy of the developed boundary-domain integral method.

Fast boundary-domain integral method for heat transfer simulations

In this paper, we present a method to decrease the computational cost of the Boundary-Domain Integral Method. We focus on the solution of the velocity–vorticity formulation of the Navier–Stokes equation for incompressible 3D fluid flow. The objective is to accelerate the solution of the boundary vorticity values. In order to reduce the computational cost of the Boundary-Domain Integral Method, we employ the Adaptive Cross Approximation algorithm in combination with the hierarchical matrix structure. The hierarchical matrix structure enables higher compression rates when individual matrix parts are approximated by low-rank matrices. We use the fundamental solution of the modified Helmholtz equation to improve further the approximation accuracy when solving the boundary vorticity values.The developed algorithm is used to simulate natural convection in a closed cavity. By comparing the simulation results with a published benchmark, we are able to assess the influence of the approximation techniques and give a recommendation on parameter values that lead to optimal compression characteristics when simulating the heat and mass transfer processes.

A meshfree boundary-domain integral equation method for free vibration analysis of the functionally graded beams with open edged cracks

A dynamic analysis may be required either because a crack is excited by time dependent loads or because a crack under static loading conditions propagates so rapidly that the effects of the inertia forces are important and the inertia effects cannot be neglected. In this paper, free vibration of the functionally graded beams with open edged cracks are analyzed by a meshfree boundary-domain integral equation method. Elastostatic fundamental solutions are used as weight functions to generate the weighted residual statements of the equations of motion. Fundamental solutions are obtained by considering the inhomogeneous and inertia effects. Numerical results compared well with that of the analytical methods. Comprehensive parametric study investigates the effects of the material gradients and directions, crack length and depth ratios, as well as boundary conditions on the free vibration responses on the cracked FG beams, which demonstrate the present method states high efficiency and accuracy. Besides, the developed method can be used to identify the crack size and location of the FG beams.

Solution of Nonhomogeneous Helmholtz Equation with Variable Coefficient Using Boundary Domain Integral Method

The Boundary Domain Integral Method (BDIM) is applied to the solution of the nonhomogeneous Helmholtz equation with variable coefficient. The analytical formulas for the integrals over the individual boundaries and domain integrals are used to increase the accuracy of the numerical approach. Comparisons of the developed BDIM with the analytical solutions for the homogeneous Helmholtz equation with constant coefficient and the nonhomogeneous Helmholtz equation with variable coefficient are given.

Boundary-domain integral method for vorticity transport equation with variable viscosity

Boundary-domain integral method for vorticity transport equation with variable viscosity

Forced vibration analysis of functionally graded beams by the meshfree boundary-domain integral equation method

Forced vibration of two-dimensional functionally graded beams is studied in this paper by a developed meshfree boundary-domain integral equation method. Material properties of the functionally graded beams are assumed varying continuously either in longitudinal or transvers direction following the exponential function. The boundary-domain integral equations are derived by using the elastostatic fundamental solutions based on the two-dimensional elastic theory. Radial integral method (RIM) is employed to transform the domain integrals into boundary integrals. A meshfree scheme is achieved through assuming the displacements and accelerations in the domain integrals by a combination of the radial basis function and polynomials with time dependent coefficients. Wilson-θ, Houbolt as well as two kinds of damped Newmark's algorithms are applied to accomplish the time integration. The forced vibration of the functionally graded beam subjected by the harmonic loading and transient loading are investigated in detail. Numerical examples demonstrate that the four mentioned time integral schemes are all adapted well to the developed meshfree boundary-domain integral equation method for analyzing the forced vibration of homogeneous structures. For the analysis of FG structures, it is shown that the damped Newmark's algorithm can achieve more stable and accurate results.

Free vibration analysis of the functionally graded sandwich beams by a meshfree boundary-domain integral equation method

Free vibration of the functionally graded (FG) sandwich beams are studied by a meshfree boundary-domain integral equation method. Two sandwich beams, namely, FG core with homogeneous face sheets and homogeneous core with FG face sheets are considered in the study. Based on the two-dimensional elasticity theory, the boundary-domain integral equations are derived by applying the elastostatic fundamental solutions. Radial integration method is employed to transform the domain integrals related to material nonhomogeneity and inertia effect into boundary integrals, hence a meshfree scheme involving boundary integrals only is achieved. Each layer in the sandwich beam is modeled separately and the equilibrium and compatibility conditions are enforced at the interfaces to derive the equations for the whole sandwich beam. Extensive parametric study is presented to examine the influence of material composition, material gradient, layer thickness proportion, thickness to length ratio and boundary conditions on the free vibration of FG sandwich beams.

Velocity–vorticity RANS turbulence modeling by boundary element method

Turbulent flow over various geometries is studied numerically. Incompressible set of Navier–Stokes equations is considered and solved by boundary domain integral method (BDIM). Governing equations are written in velocity–vorticity form. Turbulence models used are based on eddy-viscosity hypothesis. Integral form of equations, discretization and the solution algorithm are presented. The algorithm is tested with two separate test cases. The first is the turbulent channel flow for two different Reynolds numbers: Re τ = 180 and Re τ = 395 . Results show very good agreement with corresponding DNS data. The second test case is the flow over backward facing step for Reynolds number Re h = 5000 , which shows good agreement with literature data on mean reattachment length.