Solving general field equations in infinite domains with dual reciprocity boundary element method

The dual reciprocity boundary element method has been successfully employed to solve general field equations posed in a closed domain, i.e. interior problems. Up to now, however, little effort has been made to extend it to exterior problems (i.e. general field equations posed in an infinite region), which are commonly encountered in engineering practice. In this paper, the interpolation functions associated with exterior problems, which were proposed by Loeffler and Mansur (in Boundary Elements X, Vol. 2, Springer, 1988), are first examined. We have found that the choice of the arbitrary constant, the inclusion of which is necessary in those interpolation functions, has clear effects on the accuracy of the numerical results. A mapping transformation, through which any exterior problem can be solved by solving an equivalent interior problem, is then proposed. Although there are certain regularity conditions attached to such a mapping, they can be easily satisfied if the unknown function satisfies certain regularity conditions at infinity in the original exterior problem. A successful application of this mapping transformation to a transient heat transfer problem demonstrates the good performance of this approach.     

Arbitrary polygonal cloaks with multiple invisible regions

Arbitrary polygonal cloaks with multiple invisible regions are achieved by transformation optics. The original total space is divided into several layers. Choosing proper spatial mappings for these layers, a few empty layers in which objects can be hidden without bringing any scattering are available as extended invisible regions. Compared with the original polygonal cloaks the proposed one can provide multiple interior and/or exterior invisible regions simultaneously, and the objects can thus be hidden in the interior cavity and/or in the exterior space, or even be embedded into the gap between the cloaking shells.     

Nonradiating surface sources

The class of three-dimensional (3D) nonradiating sources to the inhomogeneous Helmholtz equation is generalized to include nonradiating surface sources that are localized on closed or infinite surfaces sigma and that generate no radiation throughout one of the two space regions whose common boundary is sigma. The class of non-radiating surface sources is shown to be identical to the class of so-called secondary surface sources that are generated from boundary values of a field radiated by a 3D source. In particular, it is proven that any secondary surface source radiating into the interior (exterior) of sigma is a nonradiating surface source relative to the exterior (interior) of sigma, and vice versa. Examples of nonradiating surface sources are included, and a discussion is given of possible applications of such sources.     

Line search methods for extended penalty function environments

Univariate minimization is considered for constrained optimization with the Kavlie-Moe extended interior penalty. An algorithm is derived using a linear-plus-hyperbolic fit suggested by R. L. Fox for interior penalties. A linear extension to the linear-plus-hyperbolic permits rapid convergence with extended interior alties. However, constraint deletion and approximation of remaining constraints before each minimization also characterize the environment. These and other complexities impede accurate fitting of larger intervals. Thus, following Murray, a mixed strategy is adopted with function comparison reductions of interval size to supplement curve fitting. Analysis shows how quadratic and cubic fit searches break down under certain limiting conditions of interior, extended and exterior penalties. A quadratic-extended linear curve derived for exterior penalties and the linear-plus-hyperbolic similarly fail on extended interior penalties. Examples of constrained optimization with extended interior penalties were run on the Stanford IBM 370/168 computer. A stiffened two-panel box provides eight optimization variables with many constraints on demensions and against failure. The extended linear-plus-hyperbolic surpasses cubic and linear-plus hyperbolic fitting, especially when the Fiacco-McCormick multiplier for the penalty becomes small. The value of feedback from curve fitting to decide function comparison use is also illustrated.     

Tackling Inhomogeneous Open Boundary Designs by the Finite-Element Method

In the course of designing and analyzing electromagnetic devices, open boundary field problems often need to be solved. The mapping of the infinite exterior of a finite circular region onto the interior of that closed circular region has emerged and established itself as one of the best methods. However, as a result of the work done in the past in solving homogenous exterior regions, this paper shows that this mapping method has been mistakenly taken to apply only when the exterior consists of air or, at best, some other homogeneous medium. This paper validates the method for problems with uniformly inhomogeneous exterior regions and makes possible the solution of a new class of problems by this powerful method.     

Finite elements for piezoelectric vibrations with open electricboundaries

The problem of a piezoelectric body vibrating in the free space of infinite extent has been posed as a problem with open electric boundaries. A three-dimensional finite element analysis of piezoelectric vibrations has been complemented by the modeling of the external electrostatic field. The infinite exterior region that adheres to Laplace's equation is represented as a single “superelement” obtained by the ballooning of the outer boundary. The spectral transformation Lanczos method is used to find approximations to some solutions of the generalized eigenvalue problem arising from the finite element discretization of interior and exterior regions. The proposed approach was tested on simple vibrators with different mesh discretization and element order. The method produces numerically stable results, and was used to estimate the influence of the exterior leakage field on electromechanical properties of free piezoelectric vibrations. Numerical tests show that for some materials, vibrator geometries and modes, favorable to the coupling with the exterior field, the resonant frequency and electrical parameter shifts due to open electric boundaries may be as great as several percent     

Finite element analysis of leakage magnetic flux from an induction heating system

The finite element method is applied to calculate the leakage magnetic field from an induction heating system consisting of an exciting coil and a conducting circular plate. The basic technique of the method is to draw a mathematical sphere to enclose the system. The total energy functional is assumed to be given by the interior and exterior functionals. The former is represented by using the usual axisymmetric triangular elements. On the other hand, the latter is given in a simple form by expanding exterior fields in terms of the solutions of a differential equation governing exterior empty spaces. The interior and exterior potentials are then matched on the spherical interface. After the validity of the method is verified by comparing results with analytic solutions for a single coil, the leakage magnetic flux from an induction heating system is computed at large distances.     

A comparison between a symmetric and a non-symmetric Galerkin finite element—boundary integral equation coupling for the two-dimensional exterior Stokes problem

Symmetric and non-symmetric Galerkin formulations are presented for the coupling of a finite element modelled interior region to a boundary integral supported exterior region for the two-dimensional steady state exterior Stokes problem. Both single and double-layer hydrodynamic potentials are used allowing a well conditioned symmetric matrix structure for the entire interior–exterior, velocity–pressure system when the exterior velocity boundary integral equation (VBIE) is augmented by a traction boundary integral equation (TBIE) with the pressure determined everywhere purely through the imposition of the divergence-free velocity condition. Corresponding non-symmetric formulations are obtained by additionally discretizing an associated pressure boundary integral equation (PBIE), where the associated kernel functions have singularities an order higher than in the VBIE, with a simple regularization of the new hyper-singular pressure kernel. Comparable solution convergence with mesh refinement for the symmetric and non-symmetric schemes is shown for stabilized and mixed velocity–pressure conforming finite element pairs using Lagrangian shape functions.     

Dual boundary integral equation formulation in antiplane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in antiplane elasticity using complex variable. Four kinds of boundary integral equation (BIE) are studied, and they are the first complex variable BIE for the interior region, the second complex variable BIE for the interior region, the first complex variable BIE for the exterior region, and the second complex variable BIE for the exterior region. The first BIE for the interior region is derived from the Somigliana identity, or the Betti’s reciprocal theorem in elasticity. A displacement versus traction operator is suggested. After using this operator, the second BIE for the interior region is derived. Similar derivations are performed for the first and second BIEs for the exterior region. In the case of the exterior boundary, two degenerate boundary cases are studied. One is the curved crack case, and other is the case of a deformable line. All kernels in the suggested BIEs are expressed in terms of complex variable.     

Flux and traction boundary elements without hypersingular or strongly singular integrals

The present paper deals with a boundary element formulation based on the traction elasticity boundary integral equation (potential derivative for Laplace's problem). The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require any change of co‐ordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. The formulation presented is completely general and valid for arbitrary shaped open or closed boundaries. Analytical expressions for all the required hypersingular or strongly singular integrals are given in the paper. To fulfil the continuity requirement over the primary density a simple BE discretization strategy is adopted. Continuous elements are used whereas the collocation points are shifted towards the interior of the elements. This paper pretends to contribute to the transformation of hypersingular boundary element formulations as something clear, general and easy to handle similar to in the classical formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

A Model for the Forward Problem in Magnetic Induction Tomography Using Boundary Integral Equations

We propose a new formulation for the forward problem in magnetic induction tomography (MIT). We formulate the problem in terms of interior and exterior boundary integral equations (BIEs), subject to appropriate boundary conditions. We then transform a standard exterior BIE involving the magnetic vector potential to a BIE involving the electric fields. This transformation eliminates two boundary conditions involving the magnetic vector potential and its normal derivative. This greatly reduces the computational complexity of the model. Here, we compare numerical solutions of the model to analytical solutions.      

Lamb Wave Interaction with Delaminations in CFRP Laminates

In this paper, we investigate Lamb wave interaction with delamination in an infinite carbon fiber reinforced plastics (CFRP) laminate by a hybrid method. The infinite CFRP laminate is divided into an exterior zone and an interior zone. In the exterior zone, the wave fields are expressed by wave mode expansion. In the interior zone, the wave fields are modeled by the finite element method (FEM). Considering the continuity condition at the boundary between the exterior and interior zones, the global wave fields can be calculated. Lastly, numerical examples show how a delamination in the laminate influences the mode conversion of different incident wave modes.     

Dual boundary integral equations for exterior problems

A dual integral formulation for the interior problem of the Laplace equation with a smooth boundary is extended to the exterior problem. Two regularized versions are proposed and compared with the interior problem. It is found that an additional free term is present in the second regularized version of the exterior problem. An analytical solution for a benchmark example in ISBE is derived by two methods, conformal mapping and the Poisson integral formula using symbolic software. The potential gradient on the boundary is calculated by using the hypersingular integral equation except on the two singular points where the potential is discontinuous instead of failure in ISBE benchmarks. Based on the matrix relations between the interior and exterior problems, the BEPO2D program for the interior problem can be easily reintegrated. This benchmark example was used to check the validity of the dual integral formulation, and the numerical results match the exact solution well.     

固体火箭发动机复合材料壳体承载力分析

为了提高固体火箭发动机(SRM)的外载荷承载能力,研究了其复合材料壳体的失效机制,提出了复合材料壳体的增强改进结构形式。通过提高复合材料外缠绕层的轴向刚度和横向弯曲刚度,使得连接区域内的内、外缠绕层的轴向变形相协调,改善了内、外缠绕层的轴向承载分配,使增强改进后的复合材料壳体结构的承载能力提高了124%,而结构质量增加低于10%。研究结果表明: SRM复合材料壳体承载能力的关键因素是连接区域内复合材料内、外缠绕层的刚度匹配设计,只有保证连接区域内的刚度匹配和位移变形相协调,才能充分发挥复合材料壳体的承载能力。     

存在内外质量交换的声空化泡动力学模型

对于空化微气泡的动力学研究多基于气泡内外无质量交换的Rayleigh模型。考虑了气泡内外的物质交换,泡内气体分子数量,即气泡平衡半径成为变量,进而将Rayleigh模型推广到有质量交换的情形。通过数值计算,获得稳定平衡半径,以及它与驱动声压的相关性。结果表明:在低声压驱动下,气泡稳定平衡半径随驱动声压增大而减小;而在高声压下,气泡稳定平衡半径随驱动声压增大而增大,稳定平衡半径-声压曲线存在一个转折点。     

模拟P波向无穷远域传播的一致边界

将人工边界设置在半无穷层单元和内部有限元区域的交界面上,建立了半无穷层单元的刚度矩阵后,得到了边界节点的动力平衡方程。任意给定激励圆频率,将边界节点系统的动力平衡方程转化为特征值方程。求解特征值方程得出边界节点系统的特征值和特征模态,利用模态叠加原理得到体现左半无穷层单元和右半无穷层单元对内部有限元区域作用的边界矩阵,这就是该文的一致边界。将其与内部有限元区域的刚度矩阵进行组装来模拟无穷远域介质对波的传播作用。最后用数值算例来说明一致边界的精确性和可行性。     

Magnetic resonance contrast agents from viral capsid shells: a comparison of exterior and interior cargo strategies

Two high-relaxivity nanoscale magnetic resonance contrast agents have been built using bacteriophage MS2 as a biomolecular scaffold. Protein capsid shells were functionalized on either the exterior or interior surface to display multiple copies of an aldehyde functional group. Subsequently, approximately 90 heteropodal bis(hydroxypyridonate)terephthalamide ligands were attached to these sites through oxime condensation reactions. Upon complexation with Gd3+, contrast agents with ionic relaxivities of up to 41.6 mM-1 s-1 (30 MHz, 25 degrees C) and total molecular relaxivities of up to 3900 mM-1 s-1 (30 MHz, 25 degrees C) were produced. Capsids sequestering the Gd-chelates on the interior surface (attached through tyrosine residues) not only provided higher relaxivities than their exterior functionalized counterparts (which relied on lysine modification) but also exhibited improved water solubility and capsid stability. The attachment functional cargo to the interior surface is envisioned to minimize its influences on biodistribution, yielding significant advantages for tissue targeting by additional groups attached to the capsid exterior.     

Exterior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder: semiclassical scattering theory analysis

We use the semiclassical limit of electromagnetic wave scattering theory to determine the properties of the exterior caustics of a diagonally incident plane wave scattered by an infinitely long homogeneous dielectric circular cylinder in both the near zone and the far zone. The transmission caustic has an exterior/interior cusp transition as the tilt angle of the incident beam is increased, and each of the rainbow caustics has a farzone rainbow/exterior cusp transition and an exterior/interior cusp transition as the incident beam tilt angle is increased. We experimentally observe and analyze both transitions of the first-order rainbow. We also compare the predictions of the semiclassical approximation with those of ray theory and exact electromagnetic wave scattering theory.     

体外预应力混凝土连续梁的弯矩重分布试验研究

进行了3根体外预应力混凝土两跨连续梁受力全过程试验。试验表明,自加载至受拉区混凝土开裂前,连续梁处于弹性阶段,边支座、中支座反力、跨中截面和中支座截面弯矩的实测值与采用弹性理论计算值接近。受拉区混凝土开裂后至非预应力受拉钢筋屈服,边支座反力及跨中截面弯矩实测值开始向大于弹性理论计算值的方向偏离;而中支座反力及中支座截面弯矩实测值则向小于弹性理论计算值的方向偏离。当梁内受拉非预应力筋屈服后,边支座、中支座反力的实测值以及跨中截面弯矩和中支座截面弯矩实测值与弹性理论计算值的偏差进一步增大,这种偏差在试验梁破坏时达到最大。3根试验梁中支座截面弯矩重分布值分别为12.8%、16.9%及14.6%。试验实测值还与4个不同设计规范的弯矩重分布计算值进行了比较。结果表明:采用美国ACI 318-95规范及中国GB 50010-2010规范计算的中支座截面弯矩重分布值均小于试验实测值;除一根编号为B5的梁外,加拿大A23.3-M84规范的预测值与试验值最为接近;而英国BS8110规范则偏于不安全。实际设计中,可按中国规范公式来计算体外预应力混凝土连续梁的弯矩重分布,但必须合理确定体外预应力筋的极限应力。     

Dual boundary integral equation formulation in plane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in plane elasticity using a complex variable. Four kinds of BIE are studied, and they are: (1) the first complex variable BIE for the interior region, (2) the second complex variable BIE for the interior region, (3) the first complex variable BIE for the exterior region, and (4) the second complex variable BIE for the exterior region. Using the Somigliana identity and letting the domain point approach a boundary point, the first complex variable BIE is obtained. Displacement versus traction operator is suggested. Using this operator and letting the domain point approach a boundary point, the second complex variable BIE is obtained. When the domain point approaches a boundary point, all limit processes are performed exactly through the generalized Sokhotski–Plemelj’s formulae. For the exterior problems, two degenerate boundary cases, the curved crack and the deformable curved line, are studied. Particularly, for the degenerate boundary case, or the shrinking curved crack case, four kinds of BIE are obtained.