Triangular Boundary Research Articles

Generalized coordinate Bethe ansatz for non-diagonal boundaries

We compute the spectrum and the eigenstates of the open XXX model with non-diagonal (triangular) boundary matrices. Since the boundary matrices are not diagonal, the usual coordinate Bethe ansatz does not work anymore, and we use a generalization of it to solve the problem.

Horizontal decomposition of triangulated solids for the simulation of dip-coating processes

In dip-coating processes a three-dimensional object, e.g. an entire car body, is dipped into a liquid bath. In order to simulate such processes, the space surrounding the object is decomposed into the so-called flow volumes, for which each intersection with a horizontal plane is connected. At any time the liquid’s surface then has a unique level within such a flow volume, which greatly simplifies the simulation of the liquid. The decomposition into flow volumes corresponds to the Reeb graph of the object’s exterior (considered as 3-manifold with boundary) with respect to the height function. This article presents an algorithm which computes this decomposition for an object represented as oriented triangular boundary mesh. First critical vertices of the surface are identified, which include the upper and lower ends of flow volumes. Using local information about horizontal intersection planes near the critical points, a sweep plane algorithm then constructs the volume decomposition in a second step. It is shown that the method can deal with realistic data.

Numerical study of dynamic processes in a continuous medium with a crack initiated by a near-surface disturbance by means of the grid-characteristic method

The purpose of this work is to study the problem of the near-surface disturbance propagation in a massive rock containing various heterogeneities, i.e., empty or filled cracks. Numerical solutions have been obtained for problems of wave propagation in such highly heterogeneous media, including those taking into account the plastic properties of the rock that can be manifested in the vicinity of a seismic gap or a well bore. All kinds of elastic and elastoplastic waves are analyzed resulting from the propagation of the initial disturbance and the waves arising from the reflection from the cracks and from the boundaries of the integration domain. An investigation was carried out of wave identification by means of seismograms obtained at the receiver located near the ground surface. In this study, the grid-characteristic method is employed using computational grids with triangular meshes and boundary conditions formulated at the interface between the rock and the crack, and on free surfaces in an explicit form. The proposed numerical method is extremely general and is suitable for investigations of the processes of seismic waves’ interaction with heterogeneous inclusions because it ensures the construction of the most correct computational algorithms at the boundaries of the integration domain and at the medium’s interface.

Boundary element analysis of liquid sloshing in baffled tanks

The paper concerns the natural frequencies and mode shapes of liquid sloshing in three dimensional baffled tanks with arbitrary geometries. The liquid is considered to be inviscid and incompressible and amplitudes of oscillations are assumed to be small. The tank bottom and the baffle are treated as rigid. The boundary element method is used to solve the considered problem. Triangular curvilinear 6-node boundary elements are applied. In the present formulation is not necessary to introduce the zoning method, because the baffles are treated as double layers immersed in liquid. After discretization the problem is formulated as the standard eigenvalue problem, which is limited to free surface degrees of freedom only. Several examples are given to verify the proposed method.

Casting defects induced fatigue damage in aircraft frames of ZL205A aluminum alloy – A failure analysis

Two different types of aircraft frame components, which had collapsed respectively in their former vibration-fatigue performance tests, were submitted for failure analysis. The two failed frames were both made of aerial material ZL205A, a high-strength cast Al–Cu–Mn–Ti alloy. According to a series of experimental procedures including visual observations, X-ray detections, fractography inspections, microstructure examinations, mechanical tensile tests, hardness measurements and fluorescent penetrating inspections, it was indicated that the fracture was attributed to fatigue cracks which were induced by casting porosity defects at the external surfaces of frames. Numerous fine fatigue striations presented in the vicinity of casting porosities. Especially, it was observed of a special appearance of latitude–longitude crossed fatigue striations on the fracture surface due to the coupled stresses supplied by the former multi-directional vibration tests. The overload fast-rupture regions on fracture surfaces suggested the typical cleavage fracture mode, which was characterized by a number of river patterns and cleavage steps. The intergranular spatial dendrite-shaped casting porosities largely contributed to the local stress-concentrations in matrix materials. Triangular grain boundaries induced by the former casting burnt implied that the intergranular melting phenomenon had occurred. Furthermore, the effect of groove-shaped structure at roots of spatial convex-bodies on the edge of casting porosity was especially analyzed. And the influence of the casting porosity size on fatigue cracks was briefly discussed.

Boundary element approach for the slow viscous migration of spherical bubbles

This paper examines the slow viscous migration of a collection of $N\geq 1$ spherical bubbles immersed in a bounded Newtonian liquid under the action of prescribed uniform gravity field and/or arbitrary ambient Stokes flow. The liquid domain is either open or closed with fixed boundary(ies) where the ambient Stokes flow vanishes. The incurred translational velocity of each bubble is obtained by resorting to a well-posed boundary formulation which requires to invert $3N$ boundary-integral equations. Depending upon the selected Green tensor, these integral equations holds on the cluster's surface plus the boundaries or solely on the cluster's surface. The advocated numerical strategy resorts to quadratic triangular curvilinear boundary elements on each encoutered surface and enables one to accurately compute at a reasonable cpu time cost each bubble velocity. A special attention is paid, both theoretically and numerically, to the case of $N-$bubble cluster located near a solid and motionless plane wall with numerical results given and discussed for a few clusters subject to gravity effects and ambient linear or quadratic shear flows.

Unstructured partial- and net-current based coarse mesh finite difference acceleration applied to the extended step characteristics method in NEWT

The NEWT (NEW Transport algorithm) code is a multi-group discrete ordinates neutral-particle transport code with flexible meshing capabilities. This code employs the Extended Step Characteristic spatial discretization approach using arbitrary polygonal mesh cells. Until recently, the coarse mesh finite difference acceleration scheme in NEWT for fission source iteration has been available only for rectangular domain boundaries because of the limitation to rectangular coarse meshes. Therefore no acceleration scheme has been available for triangular or hexagonal problem boundaries. A conventional and a new partial-current based coarse mesh finite difference acceleration schemes with unstructured coarse meshes have been implemented within NEWT to support any form of domain boundaries. The computational results show that the new acceleration schemes works well, with performance often improved over the earlier two-level rectangular approach.

Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions

Asymmetric discrete triangular distributions are introduced in order to extend the symmetric ones serving for discrete associated kernels in the nonparametric estimation for discrete functions. The extension from one to two orders around the mode provides a large family of discrete distributions having a finite support. Establishing a bridge between Dirac and discrete uniform distributions, some different shapes are also obtained and their properties are investigated. In particular, the mean and variance are pointed out. Applications to discrete kernel estimators are given with a solution to a boundary bias problem.

ON FIELD REPRESENTATION FOR A TRIANGULAR BUILDING BLOCK WHEN MODELING H-PLANE WAVEGUIDE STRUCTURES

A homogeneously filled triangular region with waveguide arms is considered as a possible building block when analyzing complicated H-plane structures by means of generalized scattering matrices. For triangular boundaries, a specific field representation is proposed. It is based on the domain-product technique and ensures a good convergence of the used series.Numerical examples demonstrating reliability and efficiency of the approach are presented.

Analyzing static three-dimensional elastic domains with a new infinite boundary element formulation

A new two-dimensionally mapped infinite boundary element (IBE) is presented. The formulation is based on a triangular boundary element (BE) with linear shape functions instead of the quadrilateral IBEs usually found in the literature. The infinite solids analyzed are assumed to be three-dimensional, linear-elastic and isotropic, and Kelvin fundamental solutions are employed. One advantage of the proposed formulation over quadratic or higher order elements is that no additional degrees of freedom are added to the original BE mesh by the presence of the IBEs. Thus, the IBEs allow the mesh to be reduced without compromising the accuracy of the result. Two examples are presented, in which the numerical results show good agreement with authors using quadrilateral IBEs and analytical solutions.

Integral and geometrical means in the analytical evaluation of the BEM integrals for a 3D Laplace equation

By using integral theorems and geometrical interpretations, the analytical formulas for the coefficients occurring in boundary element method (BEM) equations for a 3D Laplace equation were found for arbitrary planar polygonal boundary elements with constant approximation. Closed forms for the gradients of the coefficients were also obtained. In addition, an analytical formula for planar triangular boundary elements with a linear approximation of potential was given. The formulas obtained are appropriate especially in singular and nearly singular cases.

Regularization of hypersingular integrals in 3-D fracture mechanics: Triangular BE, and piecewise-constant and piecewise-linear approximations

In this article the hypersingular integrals that arise when boundary integral equation (BIE) methods are used to solve fracture mechanics problems are considered. An approach for hypersingular integral regularization is based on the theory of distribution and Green's theorems. This approach is applied for regularization of the hypersingular integrals over triangular boundary elements (BEs) for the case of piecewise-constant and piecewise-linear approximations. The hypersingular integrals are transformed into regular contour integrals that can be easily calculated analytically.

3D BEM for the general piezoelectric solids

In this paper we present a new scheme of three-dimensional boundary element method for the general anisotropic piezoelectric solids. We use the Radon transform representation of the three-dimensional fundamental solutions of piezoelectricity and integrate them analytically over the triangular boundary element with the linear interpolation. This reduces the computation for the system matrices G and H from the standard singular surface integrations to the simple regular line integrations and enables a drastic reduction of the computation time. The integrand of the line integral consists of the product of a function dependent and another function independent on the location vectors representing the source and observation points. The latter function depends only on the material and element properties and thus calculated only once for each element and saved for a repeated use in the calculation of G and H matrices and in the post-processing. Exploitation of this favorable structure results in the further reduction of the computation time for very large systems. The implementation of the proposed method with numerical examples will be presented.

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.

Crystal Structure of In2O3(ZnO)m Superlattice Wires and Their Photoluminescence Properties

One-dimensional In2O3(ZnO)m superlattice wires were first synthesized on silicon substrates by evaporating a mixture of In and ZnO powders. The high-resolution transmission electron microscopy image indicates that the wires have a superlattice structure along their length, which consists of alternating stacks of an InO2− octahedral layer, as an inversion boundary, and an InO+(ZnO)m slab. Each slab is further separated by many triangular (zigzag) shape-contrast boundaries. They are secondary polarity inversion boundaries formed by 5-fold coordinated In and Zn atoms. The formation of such superlattice nanostructures is explained by the zigzag model. The PL properties of the superlattice nanostructures are discussed with regard to its temperature dependence for the first time.

Numerical solution for an inverse MRI problem using a regularised boundary element method

We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils used in magnetic resonance imaging (MRI) and is formulated using its corresponding integral representation according to potential theory. A constant boundary element method (BEM) which satisfies the continuity equation for the current density, i.e. divergence-free BEM, and was originally proposed by Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005;26B:67–80], is presented based on geometrical arguments with respect to the linear (flat) triangular boundary elements employed in order to emphasise its possible extension to further higher-order divergence-free interpolations. Since the discretised BEM system is ill-posed and hence the associated least-squares solution may be inaccurate and/or physically meaningless, the Tikhonov regularisation method is employed in order to retrieve accurate and physically correct solutions. A rigorous numerical approximation for the calculation the magnetic energy, which reduces the errors induced by employing the approach of Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005;26B:67–80], is also proposed.

Analysis of the early stages of stress-induced notch growth

The early stages of preexisting small void growth into larger voids of characteristic shapes caused by stress-induced migration in aluminum lines were studied through biaxial thermal stress simulation. Preexisting small voids, referred to as notches, were assumed to be hemicylindrical, triangular, and grain boundary edge notches. The author showed that the notches have the potential to grow into voids of varying shape such as wedgelike, grainlike, and slitlike voids. The void shape depends strongly on the stress concentration and the free surface boundary conditions. The stress concentration suppresses void growth. The free surface influences not only the shape of the voids but also their size.

Intrinsic Viscosity of Proteins and Platonic Solids by Boundary Element Methods.

The boundary element (BE) method is used to implement a very precise computation of the intrinsic viscosity for rigid molecules of arbitrary shape. The formulation, included in our program BEST, is tested against the analytical Simha formula for ellipsoids of revolution, and the results are essentially numerically exact. Previously unavailable, very precise results for a series of Platonic solids are also presented. The formulation includes the optional determination of the center of viscosity; however, for globular proteins, the difference compared to the computation based on the centroid is insignificant. The main application is to a series of 30 proteins ranging in molecular weight from 12 to 465 kD. The computation starts from the crystal structure as obtained from the Protein Data Bank, and a hydration thickness of 1.1 Å obtained in previous work with BEST was used. The results (extrapolated to an infinite number of triangular boundary elements) for the proteins are separated into two groups: monomeric and multimeric proteins. The agreement with experimental measurements of the intrinsic viscosity in the case of monomeric proteins is excellent and within experimental error of 5%, demonstrating that the solution and crystal structure are hydrodynamically equivalent. However, for some multimeric proteins, we observe strong systematic deviations around -20%, which we interpret as a systematic deviation of the solution structure from the crystal structure. A possible description of the structural change is deduced by using simple ellipsoid model parameters. A method to obtain intrinsic viscosity values for proteins to 1-2% accuracy (better than experimental error) on the basis of a single BE computation (avoiding the need for an extrapolation on the number of surface triangles) is also presented.

A spectral collocation method with triangular boundary elements

A spectral collocation method is proposed for solving integral equations arising from boundary integral formulations over surfaces discretized into flat or curved triangular elements. In the numerical approximation, a function of interest defined over a triangular element is approximated with an arbitrary-degree complete polynomial of two local triangle barycentric coordinates. Collocation points are then deployed at the nodes of a triangular Lobatto grid constructed on the basis of the zeros of the Lobatto polynomials, so that the number of collocation points over each element is equal to the number of terms in a complete polynomial expansion. The node interpolation functions are computed from the Proriol polynomial base using the generalized Vandermonde matrix approach. The spectral element method is applied to solve integral equations of the second kind arising from the double-layer representation of a harmonic function in the interior or exterior of a sphere. The numerical results confirm a rapid convergence with respect to the order of the polynomial expansion.

Effect of Yttrium on Morphologies of Precipitation Phases along Grain Boundaries of As-Cast ZK60 Magnesium Alloy

Morphologies of precipitation phases along grain boundaries of the as-cast Mg-6.0Zn-0.45Zr, Mg-6.0Zn-1.2Y and Mg-6.0Zn-0.6Zr-1.0Y alloys were investigated. The experimental results indicated that the main precipitation phase along grain boundary of the as-cast Mg-6.0Zn-0.45Zr was a Mg-Zn binary compound. There were two kinds of precipitation phases along grain boundaries for Mg-6.0Zn-1.2Y and Mg-6.0Zn-0.6Zr-1.0Y alloys. One grew at triangular grain boundary, which was fish-bone -like in shape and a Mg-Zn binary compound. The other distributed mainly around the grain, which was net-like in shape and a Mg-Zn-Y ternary compound. In addition, there were lots of granular phases, extending to the interior of the grain, at the rim of grain boundary phases. Y was found to affect obviously morphology of precipitation phases along grain boundary of the as-cast Mg-Zn-Zr (ZK60) alloys.