Boundary integral equation for tangential derivative of flux in Laplace and Helmholtz equations

In this paper, the boundary integral equations (BIEs) for the tangential derivative of flux in Laplace and Helmholtz equations are presented. These integral representations can be used in order to solve several problems in the boundary element method (BEM): cubic solutions including degrees of freedom in flux's tangential derivative value (Hermitian interpolation), nodal sensitivity, analytic gradients in optimization problems, or tangential derivative evaluation in problems that require the computation of such variable (elasticity problems in BEM). The analysis has been developed for 2D formulation. Kernels for tangential derivative of flux lead to high‐order singularities (O(1/r³)). The limit to the boundary analysis has been carried out. Based on this analysis, regularization formulae have been obtained in order to use such BIE in numerical codes. A set of numerical benchmarks have been carried out in order to validate theoretical and practical aspects, by considering known analytic solutions for the test problems. The results show that the tangential BIEs have been properly developed and implemented. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite parts of singular and hypersingular integrals with irregular boundary source points

The subject of this paper is the evaluation of finite parts (FPs) of certain singular and hypersingular integrals, that appear in boundary integral equations (BIEs), when the source point is an irregular boundary point (situated at a corner on a one-dimensional plane curve or at a corner or edge on a two-dimensional surface). Two issues addressed in this paper are: an unified, consistent and practical definition of a FP with an irregular boundary source point, and numerical evaluation of such integrals together with solution strategies for hypersingular BIEs (HBIEs). The proposed formulation is compared with others that are available in the literature and interesting connections are made between this formulation and those of other researchers.     

Analytical integrations in 3D BEM for elliptic problems: Evaluation and implementation

The present publication deals with 3D elliptic boundary value problems (potential, Stokes, elasticity) in the framework of linear, isotropic, and homogeneous materials. Numerical approximation of the unique solution is achieved by 3D boundary element methods (BEMs). Adopting polynomial test and shape functions of arbitrary degree on flat triangular discretizations, the closed form of integrals that are involved in the 3D BEMs is proposed and discussed. Analyses are performed for all operators (single layer, double layer, hypersingular). The Lebesgue integrals are solved working in a local coordinate system. For singular integrals, both a limit to the boundary as well as the finite part of Hadamard (Lectures on Cauchy's Problem in Linear Partial Differential Equations. Yale University Press: New Haven, CT, U.S.A., 1923) approach have been considered. Copyright © 2010 John Wiley & Sons, Ltd.     

A direct approach for boundary integral equations with high‐order singularities

Boundary integral equations with extremely singular (i.e., more than hypersingular) kernels would be useful in several fields of applied mechanics, particularly when second‐ and third‐order derivatives of the primary variable are required. However, their definition and numerical treatment pose several problems. In this paper, it is shown how to obtain these boundary integral equations with still unnamed singularities and, moreover, how to efficiently and reliably compute all the singular integrals. This is done by extending in full generality the so‐called direct approach. Only for definiteness, the method is presented for the analysis of the deflection of thin elastic plates. Numerical results concerning integrals with singularities up to order r⁻⁴ are presented to validate the proposed algorithm. Copyright © 2000 John Wiley & Sons, Ltd.     

A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non‐smooth boundaries

The aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1D and 2D weakly singular and singular integral equations. The introduction of a smoothing transformation, besides smoothing the solution, allows also the use of a very simple and efficient collocation method based on Chebyshev polynomials of the first kind and their zeros. Further, we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements, which are given by weakly singular, singular or nearly singular integrals. Several numerical tests are given to point out the efficiency of the numerical approach we propose. Copyright © 2003 John Wiley & Sons, Ltd.     

Use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicators

In this work, a new global reanalysis technique for the efficient computation of stresses and error indicators in two‐dimensional elastostatic problems is presented. In the context of the boundary element method, the global reanalysis technique can be viewed as a post‐processing activity that is carried out once an analysis using Lagrangian elements has been performed. To do the reanalysis, the functional representation for the displacements is changed from Lagrangian to Hermite, introducing the nodal values of the tangential derivatives of those quantities as additional degrees of freedom. Next, assuming that the nodal values of the displacements and the tractions remain practically unchanged from the ones obtained in the analysis using Lagrangian elements, the tangent derivative boundary integral equations are collocated at each functional node in order to determine the additional degrees of freedom that were introduced. Under this scheme, a second system of equations is generated and, once it is solved, the nodal values of the tangential derivatives of the displacements are obtained. This approach gives more accurate results for the stresses at the nodes since it avoids the need to differentiate the shape functions in order to obtain the normal strain in the tangential direction. When compared with the use of Hermite elements, the global reanalysis technique has the attraction that the user does not have to give as input data the additional information required by this type of elements. Another important feature of the proposed approach is that an efficient error indicator for the values of the stresses can also be obtained comparing the values for the stresses obtained through the use of Lagrangian elements and the global reanalysis technique. Copyright © 2001 John Wiley & Sons, Ltd.     

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

This paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier–Stokes equations. The method computes high‐accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution ( u , p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two‐dimensional problems possessing a closed‐form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Stable and time-dissipative finite element methods for the incompressible Navier–Stokes equations in advection dominated flows

This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1-D scalar advection–diffusion and that leads to non-oscillatory solutions as the streamline upwinding algorithms for 2-D scalar advection–diffusion and incompressible Navier–Stokes. This method retains the satisfaction of the Babuska–Brezzi condition and, thus, leads to optimal performance in the incompressible limit. This method, when, combined with the recently proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier–Stokes equations ideal for either diffusive or advection-dominated flows. Examples from scalar advection–diffusion and the solution of the incompressible Navier–Stokes equations are presented.     

A finite element formulation of the outlet gradient boundary condition for convective–diffusive transport problems

A simple finite element formulation of the outlet gradient boundary condition is presented in the general context of convective–diffusive transport processes. Basically, the method is based on an upstream evaluation of the dependent variable gradient along open boundaries. Boundary normal unit vectors and gradient operators are evaluated using covariant bases and metric tensors, which allow handling finite elements of mixed dimensions. Even though the presented method has implications for many fields where diffusion processes are involved, discussion and illustrative examples address more particularly the framework of contaminant transport in porous media, in which the outlet gradient concentration is classically, but wrongly assumed to be zero. Copyright © 2004 John Wiley & Sons, Ltd.     

A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations

We present a scheme for solving two‐dimensional semilinear reaction–diffusion equations using an expanded mixed finite element method. To linearize the mixed‐method equations, we use a two‐grid algorithm based on the Newton iteration method. The solution of a non‐linear system on the fine space is reduced to the solution of two small (one linear and one non‐linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Modification of the boundary integral equation for the Navier–Lamé equations in 2D elastoplastic problems and its numerical solving

The paper presents the generalization of the modification of classical boundary integral equation and obtaining parametric integral equation system for 2D elastoplastic problems. The modification was made to obtain such equations for which numerical solving does not require application of finite or boundary elements. This was achieved through the use of curves and surfaces for modeling introduced at the stage of analytical modification of the classic boundary integral equation. For approximation of plastic strains the Lagrange polynomials with various number and arrangement of interpolation nodes were used. Reliability of the modification was verified on examples with analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

An energy and momentum conserving method for rigid–flexible body dynamics

In this work, an energy and momentum conserving method is developed for doing coupled flexible and rigid body dynamics. The main focus is on the bilateral connection of flexible finite elements to rigid bodies. The coupling of rigid bodies at joints is also introduced. Existing conserving algorithms for individual (un‐coupled) rigid and flexible bodies are exploited and modified for the coupled system. By using the appropriate rigid body rotational update and generalized force definitions, the resulting rigid–flexible and rigid–rigid systems are unconditionally stable and conserve linear and angular momentum. The conservation and stability properties are demonstrated in numerical simulation. Published in 2001 by John Wiley & Sons, Ltd.     

Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

FE/BE coupling for an acoustic fluid–structure interaction problem. Residual a posteriori error estimates

In this paper, we developed an a posteriori error analysis of a coupling of finite elements and boundary elements for a fluid–structure interaction problem in two and three dimensions. This problem is governed by the acoustic and the elastodynamic equations in time‐harmonic vibration. Our methods combined integral equations for the exterior fluid and FEMs for the elastic structure. It is well‐known that because of the reduction of the boundary value problem to boundary integral equations, the solution is not unique in general. However, because of superposition of various potentials, we consider a boundary integral equation that is uniquely solvable and avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures were considered; one is based on the nonsymmetric formulation and the other is based on a symmetric formulation. For both formulations, we derived reliable residual a posteriori error estimates. From the estimators we computed local error indicators that allowed us to develop an adaptive mesh refinement strategy. For the two‐dimensional case we performed an adaptive algorithm on triangles, and for the three‐dimensional case we used hanging nodes on hexahedrons. Numerical experiments underline our theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Acoustic modelling by BEM–FEM coupling procedures taking into account explicit and implicit multi‐domain decomposition techniques

The numerical modelling of interacting acoustic media by boundary element method–finite element method (BEM–FEM) coupling procedures is discussed here, taking into account time‐domain approaches. In this study, the global model is divided into different sub‐domains and each sub‐domain is analysed independently (considering BEM or FEM discretizations): the interaction between the different sub‐domains of the global model is accomplished by interface procedures. Numerical formulations based on FEM explicit and implicit time‐marching schemes are discussed, resulting in direct and optimized iterative BEM–FEM coupling techniques. A multi‐level time‐step algorithm is considered in order to improve the flexibility, accuracy and stability (especially when conditionally stable time‐marching procedures are employed) of the coupled analysis. At the end of the paper, numerical examples are presented, illustrating the potentialities and robustness of the proposed methodologies. Copyright © 2008 John Wiley & Sons, Ltd.     

Dispersion analysis of a non‐conforming finite element method for the Helmholtz and elastodynamic equations

We investigate the dispersive properties of a non‐conforming finite element method to solve the two‐dimensional Helmholtz and elastodynamics equations. The study is performed by deriving and analysing the dispersion relations and by evaluating the derived quantities, such as the dimensionless phase and group velocities. Also the phase difference between exact and numerical solutions is investigated. The studied method, which yields a linear spatial approximation, is shown to be less dispersive than a conforming bilinear finite element method in the two cases shown herein. Moreover, it almost halves the number of points per wavelength necessary to reach a given accuracy when calculating the mentioned velocities in both cases here presented. Copyright © 2003 John Wiley & Sons, Ltd.     

A novel method for integrating first‐ and second‐order differential equations in elastohydrodynamic lubrication for the solution of smooth isothermal,line contact problems

This paper contains details of recent developments in the analysis of elastohydrodynamic lubrication problems using the finite element method. A steady state isothermal finite element formulation of the smooth line contact problem with Newtonian lubricant behaviour is presented containing both first‐ and second‐order formulations of the hydrodynamic equation. Previous problems with the limited range of applicability of both first‐ and second‐order finite difference solutions have been overcome by summing both the first‐ and second‐order equations' weighted contributions. Application of the method to a range of problems unattainable by either single first‐ or second‐order formulations is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Uniformly convergent non‐standard finite difference methods for singularly perturbed differential‐difference equations with delay and advance

A new class of fitted operator finite difference methods are constructed via non‐standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second‐order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be ε‐uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.     

A high‐order immersed boundary discontinuous‐Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

We adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low‐order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses post‐processing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous‐Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous‐Galerkin‐based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427‐454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher‐order geometric primitives. Secondly, we employ higher‐order shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re‐entrant corner. Results are compared to a low‐order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.