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.     

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.     

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.     

Numerical evaluation of singular and finite-part integrals on curved surfaces using symbolic manipulation

The numerical integration of all singular surface integrals arising in 3-d boundary element methods is analyzed theoretically and computationally. For all weakly singular integrals arising in BEM, Duffy's triangular or local polar coordinates in conjunction with tensor product Gaussian quadrature are efficient and reliable for bothh-andp-boundary elements. Cauchy- and hypersingular surface integrals are reduced to weakly singular ones by analytic regularization which is done automatically by symbolic manipulation.     

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.     

Fracture analysis of plane piezoelectric/piezomagnetic multiphase composites under transient loading

The transient response of cracked composite materials made of piezoelectric and piezomagnetic phases, when subjected to in-plane magneto-electro-mechanical dynamic loads, is addressed in this paper by means of a mixed boundary element method (BEM) approach. Both the displacement and traction boundary integral equations (BIEs) are used to develop a single-domain formulation. The convolution integrals arising in the time-domain BEM are numerically computed by Lubich’s quadrature, which determines the integration weights from the Laplace transformed fundamental solution and a linear multistep method. The required Laplace-domain fundamental solution is derived by means of the Radon transform in the form of line integrals over a unit circumference. The singular and hypersingular BIEs are numerically evaluated in a precise and efficient manner by a regularization procedure based on a simple change of variable, as previously proposed by the authors for statics. Discontinuous quarter-point elements are used to properly capture the behavior of the extended crack opening displacements (ECOD) around the crack-tip and directly evaluate the field intensity factors (stress, electric displacement and magnetic induction intensity factors) from the computed nodal data. Numerical results are obtained to validate the formulation and illustrate its capabilities. The effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors is analyzed in detail for several crack configurations under impact loading.     

The Boundary Element Method with a Fast Multipole Accelerated Integration Technique for 3D Elastostatic Problems with Arbitrary Body Forces

A line integration boundary element method (LIBEM) is proposed for three-dimensional elastostatic problems with body forces. The method is a boundary-only discretization method like the traditional boundary element method (BEM), and the boundary elements created in BEM can be used directly in the proposed method for constructing the integral lines. Finally, the body forces are computed by summing one-dimensional integrals on straight lines. Background cells can be used to cut the lines into sub-lines to compute the integrals more easily and efficiently. To further reduce the computational time of LIBEM, the fast multipole method is applied to accelerate the method for large-scale computations and the details of the fast multipole line integration method for 3D elastostatic problems are given. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

A natural stress boundary integral equation for calculating the near boundary stress field

The stress computational accuracy of internal points by conventional boundary element method becomes more and more deteriorate as the points approach to the boundary due to the nearly singular integrals including nearly strong singularity and hyper-singularity. For calculating the boundary stress, a natural boundary integral equation in which the boundary variables are the displacements, tractions and natural boundary variables was established in the authors’ previous work. Herein, a natural stress boundary integral equation (NSBIE) is further proposed by introducing the natural variables to analyze the stress field of interior points. There are only nearly strong singular integrals in the NSBIE, i.e., the singularity is reduced by one order. The regularization algorithm proposed by the authors is taken over to deal with these singular integrals. Consequently, the NSBIE can analyze the stress field closer to the boundary. Numerical examples demonstrated that two orders of magnitude improvement in reducing the approaching degree can be achieved by NSBIE compared to the conventional one when the near boundary stress field is evaluated. Furthermore, this new way is extended to the multi-domain elasticity problem to calculate the stress field near the boundary and interface.     

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.     

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.     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

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 new BEM approach for reissner's plate bending

A new boundary element formulation for Reissner's plate bending is presented. This form of BEM has an advantage in that the bending stresses on the boundary can be calculated directly from the numerical solution, avoiding the use of tangential derivatives of displacement for finding plate bending stresses on the boundary. The effectiveness of the approach is also discussed through some test examples. In the present BEM formulation, the singular orders of the two kernels are the same as those in the standard BEM formulation of a Reissner's type plate—one of which is logarithmic singular and the other is 1/r singular.     

三维非定常扩散方程的虚边界元法

根据位势问题虚边界元法的基本思想,结合扩散方程与时间有关的基本解,提出了针对单层热势的三维非定常扩散方程虚边界元-配点法的一个具体实施方案.该方法既保留了边界元法的优点,也避免了传统边界元法中时间和空间上的奇异积分计算,采用较少的边界单元即可达到较高的精度.算例表明此方法的有效性和可行性,不过虚实边界比例选取范围比虚边界元方法应用于椭圆型问题时狭窄很多,对此本文进行了探讨,但还应继续从理论上加以论证.     

Revisit of the Indirect Boundary Element Method: Necessary and Sufficient Formulation

Although the boundary element method (BEM) has been developed over forty years, the single-layer potential approach is incomplete for solving not only the interior 2D problem in case of a degenerate scale but also the exterior problem with bounded potential at infinity for any scale. The indirect boundary element method (IBEM) is revisited to examine the uniqueness of the solution by using the necessary and sufficient boundary integral equation (BIE). For the necessary and sufficient BIE, a free constant and an extra constraint are simultaneously introduced into the conventional IBEM. The reason why a free constant and an extra constraint are both required is clearly explained by using the degenerate kernel. In order to complete the range of the IBEM lacking a constant term in the case of a degenerate scale, we provide a complete base with a constant. On the other hand, the formulation of the IBEM does not contain a constant field in the degenerate kernel expansion for the exterior problem. To satisfy the bounded potential at infinity, the integration of boundary density is enforced to be zero. Besides, sources can be distributed on either the real boundary or the auxiliary (artificial) boundary in this IBEM. The enriched IBEM is not only free of the degenerate-scale problem for the interior problem but also satisfies the bounded potential at infinity for the exterior problem. Finally, three examples, a circular domain, an infinite domain with two circular holes and an eccentric annulus were demonstrated to illustrate the validity and the effectiveness of the necessary and sufficient BIE.     

Dynamic crack analysis in piezoelectric solids with non-linear electrical and mechanical boundary conditions by a time-domain BEM

This paper presents advanced transient dynamic crack analysis in two-dimensional (2D), homogeneous and linear piezoelectric solids using non-linear mechanical and electrical crack-face boundary conditions. Stationary cracks in infinite and finite piezoelectric solids subjected to impact loadings are considered. For this purpose a time-domain boundary element method (TDBEM) is developed. A Galerkin-method is implemented for the spatial discretization, while a collocation method is applied for the temporal discretization. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. An iterative solution algorithm is developed to consider the non-linear semi-permeable electrical crack-face boundary conditions. Furthermore, an additional iteration scheme for crack-face contact analysis is implemented at time-steps when a physically meaningless crack-face intersection occurs. Several numerical examples are presented and discussed to show the effects of the electrical crack-face boundary conditions on the dynamic intensity factors.     

Development of novel PZT thin films for active sliders based on head load/unload on demand systems

 Micromachined active sliders based on head load/unload on demand systems is an interesting concept technology for ultra-high magnetic recording density of more than 100 Gb/in². The active sliders that we proposed use PZT thin films as a microactuator and control the slider flying height of less than 10 nm. It is necessary to develop high performance microactuators in order to achieve active sliders operating at very low applied voltage. This paper describes the development of novel PZT thin films for active sliders. The sol–gel fabrication process for PZT thin films is developed and the fundamental characteristics for the PZT thin films are investigated. It is confirmed that the PZT thin films have good ferroelectric properties. Furthermore, novel thin film microactuators are proposed. The feature is that the sol–gel PZT thin films (thickness 540 nm) are deposited on the sputtered PZT thin films (thickness 300 nm) fabricated on bottom Pt/Ti electrodes. Therefore, the novel thin films consist of a thermal SiO₂ layer and the sputtered and sol–gel PZT thin films layers sandwiched with upper Pt and bottom Pt/Ti electrodes on a Si slider material. Fabricating the diaphragm microactuator, the piezoelectric properties for the novel composite PZT thin films are studied. As a result, the piezoelectric strain constant d ₃₁ for the novel PZT thin films is identified to be 130 × 10⁻¹² m/V. This value is higher than conventional monolithic PZT thin films and it is found that the novel composite PZT thin films have the good piezoelectric properties. This suggests the feasibility of realizing active sliders operating at lower voltage under about 10 V. Received: 22 June 2001/Accepted: 17 October 2001     

A 2.5D coupled FE–BE methodology for the dynamic interaction between longitudinally invariant structures and a layered halfspace

This paper presents a general 2.5D coupled finite element–boundary element methodology for the computation of the dynamic interaction between a layered soil and structures with a longitudinally invariant geometry, such as railway tracks, roads, tunnels, dams, and pipelines. The classical 2.5D finite element method is combined with a novel 2.5D boundary element method. A regularized 2.5D boundary integral equation is derived that avoids the evaluation of singular traction integrals. The 2.5D Green’s functions of a layered halfspace, computed with the direct stiffness method, are used in a boundary element method formulation. This avoids meshing of the free surface and the layer interfaces with boundary elements and effectively reduces the computational efforts and storage requirements. The proposed technique is applied to four examples: a road on the surface of a halfspace, a tunnel embedded in a layered halfspace, a dike on a halfspace and a vibration isolating screen in the soil.     

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.