On a Galerkin boundary node method for potential problems

A Galerkin boundary node method (GBNM), for boundary only analysis of partial differential equations, is discussed in this paper. The GBNM combines an equivalent variational form of a boundary integral equation with the moving least-squares (MLS) approximations for generating the trial and test functions of the variational formulation. In this approach, only a nodal data structure on the boundary of a domain is required, and boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Formulations of the GBNM using boundary singular integral equations of the second kind for potential problems are developed. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method.     

Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces

We propose a fast, explicit numerical method for computing approximations for the immersed boundary problem in which the boundaries that separate the fluid into two regions are stiff. In the numerical computations of such problems, one frequently has to contend with numerical instability, as the stiff immersed boundaries exert large forces on the local fluid. When the boundary forces are treated explicitly, prohibitively small time-steps may be required to maintain numerical stability. On the other hand, when the boundary forces are treated implicitly, the restriction on the time-step size is reduced, but the solution of a large system of coupled non-linear equations may be required. In this work, we develop an efficient method that combines an integral equation approach with the immersed interface method. The present method treats the boundary forces explicitly. To reduce computational costs, the method uses an operator-splitting approach: large time-steps are used to update the non-stiff advection terms, and smaller substeps are used to advance the stiff boundary. At each substep, an integral equation is computed to yield fluid velocity local to the boundary; those velocity values are then used to update the boundary configuration. Fluid variables are computed over the entire domain, using the immersed interface method, only at the end of the large advection time-steps. Numerical results suggest that the present method compares favorably with an implementation of the immersed interface method that employs an explicit time-stepping and no fractional stepping.     

A meshless Galerkin method for Stokes problems using boundary integral equations

A meshless Galerkin scheme for the simulation of two-dimensional incompressible viscous fluid flows in primitive variables is described in this paper. This method combines a boundary integral formulation for the Stokes equation with the moving least-squares (MLS) approximations for construction of trial and test functions for Galerkin approximations. Unlike the domain-type method, this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns, thus it is especially suitable for the exterior problems. Compared to other meshless methods such as the boundary node method and the element free Galerkin method, in which the MLS is also introduced, boundary conditions do not present any difficulty in using this meshless method. The convergence and error estimates of this approach are presented. Numerical examples are also given to show the efficiency of the method.     

A combined boundary integral and vortex method for the numerical study of three-dimensional fluid flow systems

Vortex methods using vorticity–velocity formulations have become an increasingly powerful and popular means of studying complex fluid flow systems. The problem of combining an integral equation method and grid-free discrete vortex method (DVM) when studying three-dimensional wall-bounded flows is considered. While the normal boundary condition is satisfied by means of a boundary integral equation (BIE), we also consider the problem of recovering pressure from given vorticity and velocity fields when using Lagrangian DVMs in terms of a BIE. For validation purposes, vortical flow past a sphere and past a flat plate are considered, for which the commonly used method of images is available. Results of near-wall boundary-layer flow simulations are then presented as an illustration of the numerical scheme. The importance of hairpin vortices is highlighted. Finally, results on wall compliance fluid flow are displayed emphasizing the versatility of the numerical method.     

Global and local Trefftz boundary integral formulations for sound vibration

The sound-pressure field harmonically varying in time is governed by the Helmholtz equation. The Trefftz boundary integral equation method is presented to solve two-dimensional boundary value problems. Both direct and indirect BIE formulations are given. Non-singular Trefftz formulations lead to regular integrals counterpart to the conventional BIE with the singular fundamental solution. The paper presents also the local boundary integral equations with Trefftz functions as a test function. Physical fields are approximated by the moving least-square in the meshless implementation. Numerical results are given for a square patch test and a circular disc.     

An algorithm for velocity calculation in close proximity to body surfaces in panel methods

To model incompressible flow over a body of arbitrary geometry when using vortex methods, it is necessary to construct an irrotational field to impose the impermeability condition at the surface of the object. In order to achieve this impermeability, this paper uses a boundary integral equation based on the single-layer representation for the velocity potential. Specifically, we formulate this exterior Neumann problem in terms of a source/sink boundary integral equation. The solution to this integral equation is then coupled with an interpolation procedure which smoothes the transition between near-wall and interior regimes. We describe the numerical scheme embedding this strategy and discuss its accuracy and efficiency. For validation purposes, we consider the potential and vortical flow over a circular cylinder, for which an analytical solution and the commonly used method of images are available.     

Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions

In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate three-dimensional wave equation subject to given appropriate initial and Neumann's boundary conditions. The main drawback of methods in fully 3-D problems is the large computational costs. In the MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as a cube or a sphere. The point interpolation method with the help of radial basis functions is proposed to form shape functions in the frame of MLRPI. The local weak formulation using Heaviside step function converts the set of governing equations into local integral equations on local subdomains where Neumann's boundary condition is imposed naturally. A two-step time discretization technique with the help of the Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that the MLRPI method possesses reliable rates of convergence.     

A boundary integral method applied to plates of arbitrary plan form

An integral equation method for the solution of thin elastic plates of arbitrary plan form has been presented. The method involves embedding the real plate in a fictitious plate for which the Green's function is known. An unknown load vector is then introduced on the boundary of the real plate (line load and line normal moment). The deflection field due to both known transverse and unknown boundary loads can then be found everywhere by superposition. Satisfaction of the boundary conditions on the real plate results in a vector integral equation in the unknown boundary vector.In concept, any consistent set of boundary conditions will yield a solution. Practically, boundary conditions requiring higher derivatives of the deflection are both very cumbersome and yield singularities in the integral equations which cause numerical difficulties. For these reasons only clamped boundary conditions are treated numerically in the present paper.For interior bending moments and deflections (greater than distances of the order of one boundary subdivision from the boundary) the method is both highly accurate and inexpensive. Errors right on the boundary, e.g. the clamping moment in the clamped boundary condition case, can be appreciable, however. While this can be improved by a more sophisticated treatment of the unknown boundary vector in the numerical solution (increased expense) it is shown in the paper that a simple boundary extrapolation procedure gives excellent accuracy there.     

Design multiple-layer gradient coils using least-squares finite element method

The design of gradient coils for magnetic resonance imaging is an optimization task in which a specified distribution of the magnetic field inside a region of interest is generated by choosing an optimal distribution of a current density geometrically restricted to specified non-intersecting design surfaces, thereby defining the preferred coil conductor shapes. Instead of boundary integral type methods, which are widely used to design coils, this paper proposes an optimization method for designing multiple layer gradient coils based on a finite element discretization. The topology of the gradient coil is expressed by a scalar stream function. The distribution of the magnetic field inside the computational domain is calculated using the least-squares finite element method. The first-order sensitivity of the objective function is calculated using an adjoint equation method. The numerical operations needed, in order to obtain an effective optimization procedure, are discussed in detail. In order to illustrate the benefit of the proposed optimization method, example gradient coils located on multiple surfaces are computed and characterised.     

Smoothed surface gradients for panel methods

A weak form to compute the dipolar and monopolar surface gradients, related to a low-order panel method, is shown. The flow problem is formulated by means of a three-dimensional potential model and the discretization is based on Morino's formulation for the perturbation velocity potential. On the body surface, this representation reduces to a boundary integral equation with the source (or monopolar) and the doublet (or dipolar) densities. The first of the two is found by application of the boundary flow condition, and the second one is the unknown over the body surface. A lower panel method is used for the analytic integrations of both the monopolar and dipolar influence coefficients. The surface velocity field is computed after solving the linear system, with a strong and a weak form of the Stokes theorem, which is oriented to fairly non-structured panel meshes. The proposed method is validated by comparing the numerical results with analytical ones for an isolated sphere and includes a prediction over a car-like configuration.     

A frequency-domain BIEM combining DBIEs and TBIEs for 3-D crack-inclusion interaction analysis

The influence of a spherical elastic inclusion on a penny-shaped crack embedded in an infinite elastic matrix subjected to a time-harmonic crack-face or incident wave loading is investigated. A boundary integral equation method (BIEM) combining displacement boundary integral equations (DBIEs) on the matrix-inclusion interface and traction boundary integral equations (TBIEs) on the crack-surface is developed and applied for the numerical solution of the corresponding 3-D elastodynamic problem in the frequency domain. The singularity subtraction and mapping techniques in conjunction with a collocation scheme are implemented for the regularization and the discretization of the BIEs by taking into account the local structure of the solution at the crack-front. As numerical examples, the interaction of an elastic inclusion and a neighboring penny-shaped crack subjected to a tensile crack-surface loading or an incident plane longitudinal wave loading is investigated. The effects of the inclusion are assessed by the analysis of mixed-mode dynamic stress intensity factors (DSIFs) in dependence on the wave number, the material combination of the matrix and the inclusion, and the crack-inclusion orientation, size and distance.     

Radial Slit Maps of Bounded Multiply Connected Regions

In this paper we present a boundary integral equation method for the numerical conformal mapping of a bounded multiply connected region onto a radial slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Application of the method of moments to acoustic scattering from fluid-filled cylinders using three different formulations

In this work, we present three different formulations, namely the pressure field integral equation formulation (PFIE), the velocity field integral equation formulation (VFIE), and the combined field integral equation formulation (CFIE) for solving acoustic scattering problems associated with two-dimensional fluid-filled bodies of arbitrary cross-section. In particular using the boundary conditions on the surface of the body, two equivalent problems, each valid for the outside and inside regions of the scatterer, are derived. By properly selecting the associated equations for these equivalent problems, the three different formulations are derived. The PFIE, VFIE, and CFIE are then solved by approximating the cylindrical cross-section by linear segments and employing the method of moments. Further, it is shown that the moment matrices generated by the PFIE and VFIE are ill-conditioned at resonant frequencies of the cylinder, whereas the CFIE generates a well-conditioned matrix at all frequencies. The solution techniques presented in this work are simple, efficient and applicable to truly arbitrary geometries. Numerical results are presented for certain canonical shapes and compared with other available data.     

Determination of a control parameter in a one-dimensional parabolic equation using the moving least-square approximation

In this paper the approximation of moving least-square (MLS) is used for finding the solution of a one-dimensional parabolic inverse problem with source control parameter. Comparing with other numerical methods based on meshes such as finite difference method, finite element method and boundary element method, etc. the MLS approximation has merits of simpler numerical procedures, lower computation cost and arbitrary nodes. The result of a numerical example is presented.     

A modified boundary integral evolution formulation for the wave equation

We apply a modified boundary integral formulation otherwise known as the Green element method (GEM) to the solution of the two-dimensional scalar wave equation.GEM essentially combines three techniques namely: (a) finite difference approximation of the time term (b) finite element discretization of the problem domain and (c) boundary integral replication of the governing equation. These unique and advantageous characteristics of GEM facilitates a direct numerical approximation of the governing equation and obviate the need for converting the governing partial differential equation to a Helmholtz-type Laplace operator equation for an easier boundary element manipulation. C₁ continuity of the computed solutions is established by using Overhauser elements. Numerical tests show a reasonably close agreement with analytical results. Though in the case of the Overhauser GEM solutions, the level of accuracy obtained does not in all cases justify the extra numerical rigor.     

Effects of slip on sheet-driven flow and heat transfer of a non-Newtonian fluid past a stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHFcase) in presence of a uniform heat source or sink. Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field, heat source (sink) parameter and the third grade fluid parameters on the velocity, skin friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.     

A MLPG(LBIE) approach in combination with BEM

The local boundary integral equation (LBIE) approach is a promising meshless method, recently proposed as an effective alternative to the boundary element method (BEM), for solving non-homogeneous, anisotropic and non-linear problems. Since the LBIE method utilizes in its weak form fundamental solutions as test functions, it can be considered as one of the six meshless local Petrov–Galerkin (MLPG) methods proposed by Atluri and coworkers. This explains the use of the initials MLPG(LBIE) in the title of the present paper. This work addresses a coupling of a new MLPG(LBIE) method, recently proposed by the authors for elastodynamic problems, and the BEM. Because both methods conclude to a final system of linear equations expressed in terms of nodal displacement and tractions, their combination is accomplished directly with no further transformations as it happens in other MLPG/BEM formulations as well as in typical hybrid finite element method/BEM schemes. The coupling approach is demonstrated for static and frequency domain elastodynamic problems. Three representative examples are provided in order to illustrate the achieved accuracy of the proposed here MLPG(LBIE)/BE methodology.     

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 $\epsilon$ 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.     

Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.