NURBS-enhanced line integration boundary element method for 2D elasticity problems with body forces

A NURBS-enhanced boundary element method for 2D elasticity problems with body forces is proposed in this paper. The non-uniform rational B-spline (NURBS) basis functions are applied to construct the geometry and the model can be reproduced exactly at all stages since the refinement will not change the shape of the boundary. Both open curves and closed curves are considered. The fields are approximated by the traditional Lagrangian basis functions in parameter space, rather than by the same NURBS basis functions for geometry approximation. The parametric boundary elements and collocation nodes are defined from the knot vector of the curve and the refinement of the NURBS curve is easy. Boundary conditions can be imposed directly since the Lagrangian basis functions have the property of delta function. In addition, most methods for the treatment of singular integrals in traditional boundary element method can be applied in the proposed method. To overcome the difficulty for evaluation of the domain integrals in problems with body forces, a line integration method is further applied in this paper to compute the domain integrals without additional volume discretizations. Numerical examples have shown the accuracy of the proposed method.     

A boundary element analysis of symmetric laminated composite shallow shells

This paper presents a boundary element formulation for the analysis of symmetric laminated composite shallow shells where only the boundary is discretized. Classical plate bending and plane elasticity formulations are coupled and effects of curvature are treated as body forces. Fundamental solutions for elastostatic formulations are used and body forces are written as a sum of approximation functions multiplied by unknown coefficients. Two approximation functions are used. Domain integrals which arise in the formulation are transformed into boundary integrals by the radial integration method. Results for the approximation functions are compared and the accuracy of the proposed formulation is assessed by results from literature. It was shown that results obtained with the approximation function called augmented thin plate spline present very good agreement with literature even for shells that are not so shallow.     

A new fast multipole boundary element method for two dimensional acoustic problems

A new fast multipole boundary element method (BEM) is presented in this paper for solving large-scale two dimensional (2D) acoustic problems based on the improved Burton–Miller formulation. This algorithm has several important improvements. The fast multipole BEM employs the improved Burton–Miller formulation, and successfully overcomes the non-uniqueness difficulty associated with the conventional BEM for exterior acoustic problems. The improved Burton–Miller formulation contains only weakly singular integrals, and avoids the numerical difficulties associated to the evaluation of the hypersingular integral, it leads to the numerical implementations more efficient and straightforward. Furthermore, the fast multipole method (FMM) and the approximate inverse preconditioned generalized minimum residual method (GMRES) iterative solver are adopted to greatly improve the overall computational efficiency. The numerical examples with Neumann boundary conditions are presented that clearly demonstrate the accuracy and efficiency of the developed fast multipole BEM for solving large-scale 2D acoustic problems in a wide range of frequencies.     

Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems

A new completely analytical integral algorithm is proposed and applied to the evaluation of nearly singular integrals in boundary element method (BEM) for two-dimensional anisotropic potential problems. The boundary layer effect and thin body effect are dealt with. The completely analytical integral formulas are suitable for the linear and non-isoparametric quadratic elements. The present algorithm applies the analytical formulas to treat nearly singular integrals. The potentials and fluxes at the interior points very close to boundary are evaluated. The unknown potentials and fluxes at boundary nodes for thin body problems with the thickness-to-length ratios from 1E−1 to 1E−8 are accurately calculated by the present algorithm. Numerical examples on heat conduction demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body effect in BEM. Furthermore, the present linear BEM is especially accurate and efficient for the numerical analysis of thin body problems.     

Graphics processing unit (GPU) accelerated fast multipole BEM with level-skip M2L for 3D elasticity problems

In order to accelerate fast multipole boundary element method (FMBEM), in terms of the intrinsic parallelism of boundary elements and the FMBEM tree structure, a series of CUDA based GPU parallel algorithms for different parts of FMBEM with level-skip M2L for 3D elasticity are presented. A rigid body motion method (RBMM) for the FMBEM is proposed based on special displacement boundary conditions to deal with strongly singular integration and free term coefficients. The numerical example results show that our parallel algorithms obviously accelerates the FMBEM and can be used in large scale engineering problems with wide applications in the future.     

A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods

The nearly singular integrals occur in the boundary integral equations when the source point is close to an integration element (as compared to its size) but not on the element. In this paper, the concept of a relative distance from a source point to the boundary element is introduced to describe possible influence of the singularity of the integrals. Then a semi-analytical algorithm is proposed for evaluating the nearly strongly singular and hypersingular integrals in the three-dimensional BEM. By using integration by parts, the nearly singular surface integrals on the elements are transformed to a series of line integrals along the contour of the element. The singular behavior, which appears as factor, is separated from remaining regular integrals. Consequently standard numerical quadrature can provide very accurate evaluation of the resulting line integrals. The semi-analytical algorithm is applied to analyzing the three-dimensional elasticity problems, such as very thin-walled structures. Meanwhile, the displacements and stresses at the interior points very close to its bounding surface are also determined efficiently. The results of the numerical investigation demonstrate the accuracy and effectiveness of the algorithm.     

Dynamic elastoplastic analysis of 3-D structures by the domain/boundary element method

The main objective of this paper is to present a general three-dimensional boundary element methodology for solving transient dynamic elastoplastic problems. The elastostatic fundamental solution is used in writing the integral representation and this creates in addition to the surface integrals, volume integrals due to inertia and inelasticity. Thus, an interior discretization in addition to the usual surface discretization is necessary. Isoparametric linear quadrilateral elements are used for the surface discretization and isoparametric linear hexahedra for the interior discretization. Advanced numerical integration techniques for singular and nearly singular integrals are employed. Houbolt's step-by-step numerical time integration algorithm is used to provide the dynamic response. Numerical examples are presented to illustrate the method and demonstrate its accuracy.     

Analysis of thin piezoelectric solids by the boundary element method

The piezoelectric boundary integral equation (BIE) formulation is applied to analyze thin piezoelectric solids, such as thin piezoelectric films and coatings, using the boundary element method (BEM). The nearly singular integrals existing in the piezoelectric BIE as applied to thin piezoelectric solids are addressed for the 2-D case. An efficient analytical method to deal with the nearly singular integrals in the piezoelectric BIE is developed to accurately compute these integrals in the piezoelectric BEM, no matter how close the source point is to the element of integration. Promising BEM results with only a small number of elements are obtained for thin films and coatings with the thickness-to-length ratio as small as 10⁻⁶, which is sufficient for modeling many thin piezoelectric films as used in smart materials and micro-electro-mechanical systems.     

Boundary element methods for transient convective diffusion. Part I: General formulation and 1D implementation

A general formulation of higher-order boundary element methods (BEM) is presented for time-dependent convective diffusion problems in one- and multi-dimensions. Free-space time-dependent convective diffusion fundamental solutions originally proposed by Carslaw and Jaeger are used to obtain the boundary integral formulation. Linear, quadratic and quartic time interpolation functions are introduced in this paper for approximate representation of time-dependent boundary temperatures and normal fluxes. Closed form time integration of the kernels is mandatory to attain both accuracy and efficiency of the numerical approach. A complete set of time integrals for the one-dimensional formulation is presented here for the first time in the literature.     

Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method

In this paper, an advanced boundary element method (BEM) is developed for solving three-dimensional (3D) anisotropic heat conduction problems in thin-walled structures. The troublesome nearly singular integrals, which are crucial in the applications of the BEM to thin structures, are calculated efficiently by using a nonlinear coordinate transformation method. For the test problems studied, promising BEM results with only a small number of boundary elements have been obtained when the thickness of the structure is in the orders of micro-scales (10^?6), which is sufficient for modeling most thin-walled structures as used in, for example, smart materials and thin layered coating systems. The advantages, disadvantages as well as potential applications of the proposed method, as compared with the finite element method (FEM), are also discussed.     

Fast multipole method based solution of electrostatic and magnetostatic field problems

Abstact Applications of boundary element methods (BEM) to the solution of static field problems in electrical engineering are considered in this paper. The choice of a suitable BEM formulation for electrostatics, steady current flow fields or magnetostatics is discussed from user's point of view. The dense BEM matrix is compressed with an enhanced fast multipole method (FMM) which combines well-known BEM techniques with the FMM approach. An adaptive grouping scheme for problem oriented meshes is presented along with a discussion on the influence of the mesh to the efficiency of the FMM. The computational costs of the FMM algorithm are analyzed for typical problems in practice. Finally, some electrostatic and magnetostatic numerical examples demonstrate the simple usability and the efficiency of the FMM. Communicated by: U. Langer     

GPU-accelerated indirect boundary element method for voxel model analyses with fast multipole method

An indirect boundary element method (BEM) that uses the fast multipole method (FMM) was accelerated using graphics processing units (GPUs) to reduce the time required to calculate a three-dimensional electrostatic field. The BEM is designed to handle cubic voxel models and is specialized to consider square voxel walls as boundary surface elements. The FMM handles the interactions among the surface charge elements and directly outputs surface integrals of the fields over each individual element. The CPU code was originally developed for field analysis in human voxel models derived from anatomical images. FMM processes are programmed using the NVIDIA Compute Unified Device Architecture (CUDA) with double-precision floating-point arithmetic on the basis of a shared pseudocode template. The electric field induced by DC-current application between two electrodes is calculated for two models with 499,629 (model 1) and 1,458,813 (model 2) surface elements. The calculation times were measured with a four-GPU configuration (two NVIDIA GTX295 cards) with four CPU cores (an Intel Core i7-975 processor). The times required by a linear system solver are 31 s and 186 s for models 1 and 2, respectively. The speed-up ratios of the FMM range from 5.9 to 8.2 for model 1 and from 5.0 to 5.6 for model 2. The calculation speed for element-interaction in this BEM analysis was comparable to that of particle-interaction using FMM on a GPU.     

Treatment of singular integral caused by employing linear boundary elements for two-dimensional elastostatic problems

The boundary element method employing linear elements has the advantage of high precision in calculations. However, there exists a problem of singularity when the diagonal terms of the boundary influence coefficients are calculated. A method by which the singular terms are cancelled is proposed in this paper. A computer program for the solution of two-dimensional elastostatic problems using linear boundary elements is developed and verified through two examples. The technique dealing with the nodes at corners is discussed. The results show that the present method of solving singular integrals is credible.     

Accuracy estimation of the 3-D BEM analysis of elastostatic problems by the zooming method

A p-version zooming method for BEM is developed for 3-D elastostatic problems. The objective of this report is to develop an accuracy guarantee method for 3-D BEM analysis.First, the relations between the element degree and the analysis error of the displacement and of the traction are investigated by the numerical experiments of a thick-walled spheroid subjected to the internal or external pressure. Next, on the basis of these relations an accuracy guarantee method for the 3-D zooming method is proposed. In this method, the existence range of the analysis error of the displacement or of the traction at the point of particular interest is given by the sum of two errors. One is the analysis error of the displacement on the zooming boundary and the other is the analysis error of the traction at the point of particular interest in the zooming region. Finally, an adaptive analysis program using this method is developed and applied to the analysis of various 3-D elastostatic problems. The usefulness of this method is illustrated by these application results.     

Transient analysis of the dynamic stress intensity factors using SGBEM for frequency-domain elastodynamics

In this paper, a two-dimensional symmetric-Galerkin boundary integral formulation for elastodynamic fracture analysis in the frequency domain is described. The numerical implementation is carried out with quadratic elements, allowing the use of an improved quarter-point element for accurately determining frequency responses of the dynamic stress intensity factors (DSIFs). To deal with singular and hypersingular integrals, the formulation is decomposed into two parts: the first part is identical to that for elastostatics while the second part contains at most logarithmic singularities. The treatment of the elastostatic singular and hypersingular singular integrals employs an exterior limit to the boundary, while the weakly singular integrals in the second part are handled by Gauss quadrature. Time histories (transient responses) of the DSIFs can be obtained in a post-processing step by applying the standard fast Fourier transform (FFT) and algorithm to the frequency responses of these DSIFs. Several test examples are presented for the calculation of the DSIFs due to two types of impact loading: Heaviside step loading and blast loading. The results suggest that the combination of the symmetric-Galerkin boundary element method and standard FFT algorithms in determining transient responses of the DSIFs is a robust and effective technique.     

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.     

The application of boundary element method to the resistance calculation problem in designing flat panel displays

Several techniques are presented for the efficient resistance calculation of wiring structures in flat panel display (FPD). The techniques are based on two‐dimensional boundary element method (BEM), suitable for the geometry characteristics of the FPD structures. With an automatic strategy for boundary element partition and the analytical BEM‐coupled approach, the proposed resistance solver shows good accuracy and fast computational speed. Numerical experiments demonstrate that the solver can be more than 10,000 times faster than the finite difference solver raphael while preserving good accuracy. And the proposed techniques accelerate the original BEM remarkably. Structures from real FPD designs have validated the efficiency of the proposed techniques.     

CUDA架构下的三维弹性静力学边界元并行计算

针对传统边界元法计算量大、计算效率低的问题,以三维弹性静力学的边界元法为对象,将基于CUDA的GPU并行计算应用到其边界元计算中,提出了基于CUDA架构的GPU并行算法.该算法首先对不同类型的边界元系数积分进行并行性分析,描述了相关的GPU并行算法,然后阐述了边界元方程组的求解方法及其并行策略.实验结果表明,文中算法较传统算法具有显著的加速效果.     

Structural shape optimisation using boundary elements and the biological growth method

A numerical evolutionary procedure for the structural optimisation for stress reduction of two-dimensional structures is presented in this paper. The proposed procedure couples the biological growth method (BGM) with the boundary element method (BEM). The boundary-only intrinsic characteristic of BEM together with its accuracy in the boundary displacement and stress solutions make BEM especially attractive for solving shape-optimisation problems. Two formulations of BEM are used in this work: the standard for two-dimensional elastostatics for the stress analysis and the dual reciprocity method (DRM), which is used to model the swelling or shrinking of the material. Two examples are analysed to illustrate the proposed methodology and to demonstrate its versatility and robustness.     

An Internet-based computing platform for the boundary element method

Computers combined with the Internet are dramatically changing the engineering practices in design and analysis. More and more engineering software applications are moving to the Internet, prompted by the promising advantages, such as easy access for users everywhere with an Internet connection, easy upgrade of the software, easy control of the software. All these advantages will contribute to faster and cost-effective engineering software development and applications. Successful applications of the Internet computing depend largely on the speed of data transfer on the network. The boundary element method (BEM) has the inherent advantages over other domain-based numerical methods, because the size of the BEM model files are always considerably smaller, leading to faster data transfer on the current network. In this paper, a successful investigation in implementing the BEM on the Internet is presented. A BEM code for 2D elastostatic problems is used as the first solver in this work. A graphical-user interface (GUI) for the pre- and post-processing using Java, which provides platform-independent applications, is developed and implemented on the Internet. Demonstration problems using the developed platform clearly show the feasibility of the Internet-based computing and the potentials of this platform for future BEM development with further research.