Bicubic fundamental splines in plate bending

This paper deals with solving the two-dimensional variational problem for plate bending by the Ritz method using bicubic fundamental splines. It is a piecewise polynomial method, very adaptable to practical numerical computation, and can be an alternative for the well-known Finite Element Method¹.     

On the modelling of smooth contact surfaces using cubic splines

In this paper, a new strategy for the smooth representation of 2D contact surfaces is developed and implemented. The contact surfaces are modelled using cubic splines which interpolate the finite element nodes. These splines provide a unique surface normal vector and do not require prior knowledge of surface tangents and normals. C²‐continuous cubic splines are suitable for representing rigid contact surfaces, while C¹‐continuous Overhauser splines are shown to be most suitable for representing flexible contact surfaces. A consistent linearization of the kinematic contact constraints, based on the spline interpolation, is derived. The new spline‐based contact surface interpolation scheme does not influence the element calculations. Consequently, it can be easily implemented in standard FE codes. Several numerical examples are used to illustrate the advantages of the proposed smooth representation of contact surfaces. The results show a significantimprovement in accuracy compared to traditional piecewise element‐based surface interpolation. The predicted contact stresses are also less sensitive to the mismatch in the meshes of the different contacting bodies. This property is useful for problems where the contact area is unknown a priori and when there is significant tangential slip. Copyright © 2001 John Wiley & Sons, Ltd.     

Spline interpolation techniques for variational methods

Interpolation techniques are reviewed in the context of the approximation of the solution of boundary value problems. From the variational formulation, the approximation error norm is related to the interpolation error norm. Among global interpolation techniques, bicubic splines and spline-blended are reviewed; among local, Hermite's and ‘serendipity’ polynomials. The corresponding interpolation error norms are computed numerically on two test functions. The methods are compared for accuracy and for number of operations required in the solution of boundary value problems. The conclusion is that spline interpolation is most convenient for regular hyper-elements, while high precision finite elements become convenient for very fine or irregular partition.     

Application of the BEM in biopotential problems

We describe how the boundary element method (BEM) can be used in the general field of biopotential problems. We present here a cubic Hermite boundary element procedure for this purpose and show how this approach is computationally more efficient than traditional BEM procedures for solving potential-related problems. We also show how these C¹ interpolation functions can be used to model the complex domains that are present in many biopotential problems. Illustrative biopotentials results for two different clinically important areas are given. The first area deals with potentials generated by the heart (electrocardiography) while the second field is related to potentials arising from brain activity (electroencephalography).     

Embedded kinematic boundary conditions for thin plate bending by Nitsche's approach

A stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C¹‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C² B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

Uniform bicubic B-splines applied to boundary element formulation for 3-D scalar problems

In this work, uniform bicubic B-spline functions are used to represent the surface geometry and interpolation functions in the formulation of boundary-element method (BEM) for three-dimensional problems. This is done as a natural generalization of cubic B-spline curves, introduced by Cabral et al, for two-dimensional problems. Three-dimensional scalar problems, with particular applications to Laplace and Helmholtz equations, are considered.     

Some recent results and proposals for the use of radial basis functions in the BEM

We survey some recent applications of radial basis functions (rbfs) for the BEM and related algorithms such as the method of fundamental solutions. Among these are the use of alternatives to the traditional 1+r function in the dual reciprocity method such as thin plate splines, multquadrics and the recently discovered compactly supported positive definite rbfs, and convergence proofs of the DRM for Poisson’s equation. Newly discovered particular solutions for Helmholtz-type operators are discussed and applied to give efficient mesh free algorithms for the diffusion equation. In addition, a number of proposals are given for future applications of rbfs such as the use of surface rbfs for interpolation and the solution of boundary integral equations and the application of Kansa’s method to develop new rbf based coupled domain-boundary approximation methods.     

A simple a-posteriori error estimation for adaptive BEM in elasticity

In this paper, the properties of various boundary integral operators are investigated for error estimation in adaptive BEM. It is found that the residual of the hyper-singular boundary integral equation (BIE) can be used for a-posteriori error estimation for different kinds of problems. Based on this result, a new a-posteriori error indicator is proposed which is a measure of the difference of two solutions for boundary stresses in elastic BEM. The first solution is obtained by the conventional boundary stress calculation method, and the second one by use of the regularized hyper-singular BIE for displacement derivative. The latter solution has recently been found to be of high accuracy and can be easily obtained under the most commonly used C ⁰ continuous elements. This new error indicator is defined by a L ₁ norm of the difference between the two solutions under Mises stress sense. Two typical numerical examples have been performed for two-dimensional (2D) elasticity problems and the results show that the proposed error indicator successfully tracks the real numerical errors and effectively leads a h-type mesh refinement procedure.     

A multipole expansion-based boundary element method for axisymmetric potential problem

The multipole expansion is an approximation technique used to evaluate the potential field due to sources located in the far field. Based on the multipole expansion, we describe a new technique to calculate the far potential field due to ring sources which are encountered in the boundary element method (BEM) formulation of axisymmetric problems. As the sources in the near field are processed by the slower conventional BEM, it is important to maximize the amount of multipole calculations taking advantage of both interior and exterior multipole expansions. Numerical results are presented for an axisymmetric potential test problem with Neumann and Dirichlet boundary conditions. The complexity of the proposed method remains O(N²), which is equal to that of the conventional BEM. However, the proposed technique coupled with an iterative solver speeds up the solution procedure. The technique is significantly advantageous when medium and large numbers of elements are present in the domain.     

Error estimation and h adaptive boundary elements

This paper deals with error estimation of the boundary element method (BEM) and the h adaptive boundary elements. The various error sources in the BEM are discussed, and the upper bound of the BEM solution error is derived by means of the interpolation error of the BEM solution. A new error estimator is presented in the paper by using post-processing data from the standard BEM solutions. Two adaptive algorithms, the standard h adaptive and the h-hierarchical adaptive, are implemented based on direct boundary element method for two-dimensional elasticity problems. A few numerical examples are used to compare the accuracy of the proposed adaptive algorithm, as well as the error estimator. The stability of the BEM system matrix, which may deteriorate due to the introduction of h-hierarchical interpolation functions, has also been studied.     

A fast multipole accelerated method of fundamental solutions for potential problems

The fast multipole method (FMM) is a very effective way to accelerate the numerical solutions of the methods based on Green's functions or fundamental solutions. Combined with the FMM, the boundary element method (BEM) can now solve large-scale problems with several million unknowns on a desktop computer. The method of fundamental solutions (MFS), also called superposition or source method and based on the fundamental solutions but without using integrals, has been studied for several decades along with the BEM. The MFS is a boundary meshless method in nature and offers more flexibility in modeling of a problem. It also avoids the singularity of the kernel by placing the source at some auxiliary points off the problem domain. However, like the traditional BEM, the conventional MFS also requires O(N²) operations to compute the system of equations and another O(N³) operations to solve the system using direct solvers, with N being the number of unknowns. Combining the FMM and MFS can potentially reduce the operations in formation and solution of the MFS system, as well as the memory requirement, all to O(N). This paper is an attempt in this direction. The FMM formulations for the MFS is presented for 2D potential problem. Issues in implementation of the FMM for the MFS are discussed. Numerical examples with up to 200,000 DOF's are solved successfully on a Pentium IV PC using the developed FMM MFS code. These results clearly demonstrate the efficiency, accuracy and potentials of the fast multipole accelerated MFS.     

A precorrected-FFT higher-order boundary element method for wave-body problems

The boundary element method (BEM) has been widely applied in the field of wave interaction with offshore structures, but it is still not easy to use in resolving large-scale problems because of computing costs and computer storage being increased by O(N²) for the traditional BEM. In this paper a precorrected Fast Fourier Transform (pFFT) higher-order boundary element method (HOBEM) is proposed for reducing the computational time and computer memory by O(N). By using a free-surface Green function for infinite water-depth, the disadvantage of the Fast Multipole Boundary Element Method (FMBEM)—i.e. unable to solve infinite deep-water wave problems—can be overcome. Numerical results from the problems of wave interaction with single- and multi-bodies show that the present method evidently has more advantages in saving memory and computing time, especially for large-scale problems, than the traditional HOBEM. In addition, the optimal variable of pFFT mesh is recommended to minimize time cost.     

Optimum interpolation functions for boundary element method

In the Boundary Element Method (BEM) the density functions are approximated by interpolation functions which are chosen to satisfy appropriate continuity requirements. The error of approximation inside an element depends upon the location of the collocation points that are used in constructing the interpolation functions. The location of collocation points also affects the nodal values of the density function and, hence, the total error in the analysis if boundary conditions are satisfied in a collocation sense. In this paper, we minimize the error inside the element using the L₁ norm to obtain the optimum location of collocation points. Results show that irrespective of the continuity requirement at the element end, the location of collocation points computed by the algorithm presented in this paper results in an error that is less than the error corresponding to uniformly spaced collocation points. Results for optimum location of collocation points and the average error are presented for Lagrange polynomials up to order fifteen and for Hermite polynomials that ensure continuity up to the seventh order of derivative at the element end. The information of the optimum location of interpolation points for Lagrange and Hermite polynomials should be useful to other researchers in BEM who could incorporate it into their current programs without making significant changes that would be needed for incorporating the algorithm. The algorithm presented is independent of the BEM application in two-dimensions, provided that the density functions are approximated by polynomials and is applicable to direct and indirect formulations. Two numerical examples show the application of the algorithm to an elastostatic problem in which one boundary is represented by integrals of the Direct BEM while the other boundary by the Indirect BEM and a fracture mechanics problem by Direct method in which the crack is represented by displacement discontinuity density function.     

An explicit time integration method for structural dynamics using septuple B‐spline functions

In this paper, an explicit time integration method with three parameters is proposed for structural dynamics using periodic septuple B‐spline interpolation polynomial functions. In this way, by use of septuple B‐splines, the authors have proceeded to solve the DE of motion governing a single DOF system, and later, the presented method has been generalized for a multiple DOF system. In the proposed method, a direct recursive formula for response of the system was formulated on the basis of septuple B‐spline interpolation approximation. In terms of the specific requirements of this proposed method, two initialization approaches are given for initial calculation. One is called direct initialization, and the other is indirect initialization. The stability analysis of the proposed method illustrates that, by use of adjustable parameters, a high‐frequency response can be damped out without inducing excessive algorithmic damping in important low frequency modes. The computational accuracy and efficiency of the proposed method is demonstrated with three numerical examples, and the results from the proposed method are compared with those from some of the existent numerical methods, such as the Newmark and Wilson‐ θ methods. The compared results show that the proposed method has high accuracy with low time consumption. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive cross approximation of tensors arising in the discretization of boundary integral operator shape derivatives

The shape derivative of a dense N×N BEM matrix is a sparse three-way tensor with O(N²) non-zero entries, to which standard BEM acceleration techniques such as the adaptive cross approximation (ACA) and FMM cannot be directly applied. The tensor can be used to compute shape sensitivities, or via adjoint equations, the gradient of an objective function. Although for many PDEs, calculation of the tensor can be avoided by expressing the shape derivative of the solution as the solution of a related PDE, this approach is not always easily amenable to BEM. Therefore, the computation of shape derivatives via the sparse three-way tensor is a valuable alternative, provided that efficient acceleration techniques exist. We propose a new algorithm for the approximation of BEM shape derivative tensors based on ACA that achieves the same complexity and error bounds as ACA for the BEM matrix itself. Numerical examples show that despite the much larger amount of data involved, the tensor approximation is only moderately slower than the matrix approximation. We also demonstrate the method on a shape optimization problem from the literature.     

Time discontinuous linear traction approximation in time-domain BEM scalar wave propagation analysis

In the present paper the traditional BEM formulation for time-domain scalar wave propagation analysis is extended to a new class of problems. A procedure to consider linear time interpolation for boundary tractions is worked out. Time discontinuities are included by adding to the standard BEM equation the integral equation for velocities. Numerical examples are presented in order to assess the accuracy of the proposed formulation. © 1998 John Wiley & Sons, Ltd.     

Determination of distribution functions for the characteristics of heat and mass exchange processes by means of bolometric converters

We propose a method for reconstructing the distribution functions of temperature over the cross section of a flow on the basis of data derived from bolometers. We have optimized the algorithm based on approximation of the distribution functions by bicubic splines with subsequent pseudoconversion of the matrix equation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 59, No. 2, pp. 208–211, August, 1990.     

Weakly singular stress‐BEM for 2D elastostatics

A weakly singular stress‐BEM is presented in which the linear state regularizing field is extended over the entire surface. The algorithm employs standard conforming C⁰ elements with Lagrangian interpolations and exclusively uses Gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space. The linear state stress‐BIE on which the algorithm is based has no free term so that the BEM treatment of external corners requires no special consideration other than to admit traction discontinuities. The self‐regularizing nature of the Somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary. A key observation is the relation between BIE density functions and successful interpolation orders. Numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations. Additionally, the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions. Copyright © 1999 John Wiley & Sons, Ltd.     

New formulation of the Green element method to maintain its second-order accuracy in 2D/3D

The Green element method (GEM) is a powerful technique for solving nonlinear boundary value problems. Derived from the boundary element method (BEM), over the meshes of the finite element method (FEM), the GEM combines the second-order accuracy of the BEM with the efficiency and versatility of the FEM.The high accuracy of the GEM, resulting from the direct representation of normal fluxes as unknowns, comes at the price of very large matrices for problems in 2D and 3D domains. The reason for this is a larger number of inter-element boundaries connected to each internal node, yielding the same number of the normal fluxes to be determined. The currently available technique to avoid this problem approximates the normal fluxes by differentiating the potential estimates within each element. Although this approach produces much smaller matrices, the overall accuracy of the GEM is sacrificed.The first of the two techniques proposed in this work redefines the present approach of approximating fluxes by considering more elements sharing each internal node. Numerical tests on the potential field exp(x+y) show an increase in accuracy by two orders of magnitude.The second approach is a reformulation of the standard GEM in terms of the flux vector, replacing its normal component. The original accuracy of the GEM is preserved while the number of unknowns is reduced as many as ten-times in the case of a mesh consisting of tetrahedrons. The additional benefit of this novel technique is the fact that the entire flux field is a mere by-product of the basic procedure for determining the unspecified boundary values.