Modified Boundary Element Method Research Articles

An improved tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors

The contact quality between geometric features significantly affects the assembly accuracy and dynamic performance of mechanical products. However, accurate and efficient contact problem solving is currently a common challenge in the fields of tolerance analysis and tooth contact analysis (TCA). In this paper, a tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors is proposed. Specifically, tolerance modeling of systematic and random deviations on tooth surfaces is represented by skin model shapes (SMS). Elastic-plastic contact behaviors including contact deformation, status, and stress distribution are modeled and calculated by a modified boundary element method (BEM). The accuracy and efficiency of this BEM algorithm are verified by comparing with the finite element method (FEM), indicating that the BEM can greatly improve the computational efficiency while maintaining high solution accuracy. A case study for a pair of spur gears demonstrates the significant effects of contact behaviors on transmission performance. This research provides an effective method to introduce mechanical contact behaviors in tolerance analysis of gear mechanisms. In addition, the contact simulation results can also be adopted in the analysis of vibration, reliability, and other aspects.

Modified BEM for scattering analysis by a flaw at interface in an anisotropic multi-layered plate

In this paper, a modified boundary element method (BEM) for time-harmonic scattering computation analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic plates is proposed. The aim of modification is to correct the spurious scattering introduced by inevitable model truncation at far-field in the traditional BEM model. The far-field wave displacement fields beyond the truncation points of BEM model are assumed to be the superposition of orthogonal propagating guided wave patterns, and are finally incorporated into BEM equation systems as the modified items to account for the contribution of infinite boundaries traditionally omitted. This method is simple and elegant, which has advantages on dimension reduction and needs no absorption medium or perfectly matched layer to suppress the reflected waves compared to FEM. Also, we can obtain the reflection and transmission coefficients of each mode directly without post-processing. This modified BEM is implemented to solve wave scattering problems due to a cavity-type defect at the interface in an infinite anisotropic multi-layered plate. The formulation is finally validated for several benchmark problems.

Intelligent Structural Defect Reconstruction Using the Fusion of Multi-Frequency and Multi-Mode Acoustic Data

Quantitative detection of defects in structures is always a hot research topic in the field of guided wave inverse scattering. Research studies on how to effectively extract the defect-related information encompassed in the multi-frequency and multi-modes scattered wave signals for reconstructions of defects have been paid attention in recent decades. In this paper, a novel deep learning-based quantitative guided wave inverse scattering technique has been proposed to intelligently realize the end-to-end mapping of the multi-frequency, multi-modes scattered signals to defect profiles with high levels of accuracy and efficiency. Based on the manifold distribution principle, the data patterns of scattered SH-wave signals have been investigated, owing to leveraging the capability of the intelligent encoder-projection-decoder neural network. Following that, the manifold-learning oriented network has been trained using the data generated by the modified boundary element method. Several numerical examples have been examined to demonstrate the correctness and efficiency of the proposed reconstruction approach. It has been concluded that this novel data-driven technique intelligently enables the high-quality solution to inverse scattering problems and provides a valuable insight into the development of practical approaches to quantitative detection using multi-frequency and multi-modal acoustic data from scattered ultrasonic guided waves.

Three dimensional modified BEM analysis of forward scattering problems in elastic solids

In this paper, a three-dimensional modified boundary element method (BEM) is proposed to solve the guided wave scattering problem by cavity-type flaws in an infinite plate. The aim of modification is to correct the artificial scattering introduced by inevitable modal truncation at far-field in the traditional BEM model. The far-field wave displacement fields beyond the BEM model are assumed to be the summation of orthogonal propagating guided wave patterns, and are finally incorporated into BEM equation sets as the modified items to account for the contribution of infinite boundaries traditionally omitted. With this improvement, the spurious reflected waves are eliminated. The method is formulated for a three-dimensional homogeneous, isotropic, elastic plate and its implementation is discussed in details. The formulation is finally validated for several benchmark problems.

Mixed Boundary Value Problem for an Anisotropic Thermoelastic Half-Space Containing Thin Inhomogeneities

Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and the extended Stroh formalism which allows writing the general solution of thermoelastic problems in terms of certain analytic functions. In addition, with the help of it, it is possible to convert the volume integrals included in the equation into contour integrals, which, in turn, will allow the use of the method of boundary elements. For modelling of solids with thin inhomogeneities, a coupling principle for continua of different dimensions is used. Applying the theory of complex variable functions, in particular, Cauchy integral formula and Sokhotski–Plemelj formula, the Somigliana type boundary integral equations are constructed for thermoelastic anisotropic half-space. The obtained integral equations are introduced into the modified boundary element method. A numerical analysis of the influence of boundary conditions on the half-space boundary and relative rigidity of the thin inhomogeneity on the intensity of stresses at the inclusions is carried out.

Forward and Inverse Analysis of Love Wave Scattering by Interface Cavities

Love waves have great potential in geological inspection and ultrasonic nondestructive testing for near-surface underground characteristics. A thorough and effective utilization of the Love wave requires a better understanding of its scattering phenomenon. The paper studies the problem of Love wave scattering by cavity-like flaws on the interface between the upper layer and the lower half-plane. For the forward analysis, we suggest a modified boundary element method (BEM) incorporating the far-field displacement patterns, which can effectively eliminate fictitious reflections introduced by model truncation. For inverse analysis, we propose a quantitative reconstruction procedure for the flaw shape using reflection coefficients of the first-order Love wave. By theoretical deduction, it can be proved that the cavity’s geometric shape is approximately expressed as an inverse spatial Fourier transform of far-field reflection coefficients in the wavenumber domain. Numerical examples are given by substituting the reflection coefficients obtained from the forward analysis into the inversion algorithm, and high consistency is shown between the reconstructed flaw images and the geometric characteristics of the actual flaws.

A potential flow model with viscous dissipation based on a modified boundary element method

Conventional potential flow models are known to exaggerate predictions in water wave resonance problems, attributed to the inviscid fluid assumption in the potential theory. In this paper, we present a modified potential flow model incorporating the viscous dissipation effect. We introduce some ’dissipative surfaces’ inside the fluid domain which are defined such that the normal velocity through these surfaces remains continuous but a pressure drop occurs across them, representing physically the viscous dissipation. In formulating the boundary value problem using Green theorem by a boundary element method (BEM), modified boundary integral equations are deduced to include the integral over the dissipative surfaces. We apply this model to three cases where overestimation is reported using classical potential models: gap resonance, monocolumn moonpool resonance and tuned wave surge converter. Numerical results show that the dissipative surface is effective to dampen the responses at resonance. Validation is carried out by comparisons against either experimental data or analytical solutions. The spikes in the response amplitude operators (RAOs) produced from the non-dissipative model are removed with inclusion of dissipation. Importantly, the proposed model with dissipation effect favorably retains the same level of computational efficiency as with the classical potential flow model.

Forward and Inverse Studies on Scattering of Rayleigh Wave at Surface Flaws

The Rayleigh wave has been frequently applied in geological seismic inspection and ultrasonic non-destructive testing, due to its low attenuation and dispersion. A thorough and effective utilization of Rayleigh wave requires better understanding of its scattering phenomenon. The paper analyzes the scattering of Rayleigh wave at the canyon-shaped flaws on the surface, both in forward and inverse aspects. Firstly, we suggest a modified boundary element method (BEM) incorporating the far-field displacement patterns into the traditional BEM equation set. Results show that the modified BEM is an efficient and accurate approach for calculating far-field reflection coefficients. Secondly, we propose an inverse reconstruction procedure for the flaw shape using reflection coefficients of Rayleigh wave. By theoretical deduction, it can be proved that the objective function of flaw depth d(x1) is approximately expressed as an inverse Fourier transform of reflection coefficients in wavenumber domain. Numerical examples are given by substituting the reflection coefficients obtained from the forward analysis into the inversion algorithm, and good agreements are shown between the reconstructed flaw images and the geometric characteristics of the actual flaws.

Forward Analysis of Love-Wave Scattering due to a Cavity-Like Defect

This paper presents a modified boundary element method (BEM) to solve the scattering problem of Love surface wave from a two-dimensional cavity defect. Because of the truncation of BEM models at a far distance from the cavity, spurious reflected waves are generated. In order to eliminate the unwanted reflections, the guided Love-wave displacement patterns are assumed on the far-field infinite boundaries previously omitted by model truncation, and they are incorporated into the BEM equation set as modified items. The numerical results are verified by theoretical solutions of far-field Green’s functions. Additional parametric studies are performed to find out the influence of truncation distance and defects’ geometric characters on the accuracy of scattered wave solutions.

РОЗРОБКА АЛГОРИТМІВ ТА ПРОГРАМНИХ КОМПОНЕНТІВ МОДЕЛЮВАННЯ КВАЗІСТАЦІОНАРНИХ МАГНІТНИХ ПОЛІВ

The magnetometric section of the marine areas geophysics is characterized by the transition to a qualitatively new level of research, pipeline tracing, search for sunken ships, aerial bombs and unexploded ordnance, study of sunken objects of cultural heritage, identification of such sunken objects (inverse magnetometry problem), etc. The identification of such objects is called the inverse problem of marine magnetometry, and it belongs to the class of incorrectly posed task. The first step in solving this task is to find solution of the direct problem. A direct problem is to determine the anomalous field created by objects of a certain shape, density under certain conditions of immersion and accumulation of information about these objects. Problems of determining the anomalous field of a thin-walled objects are of particular interest. The solution to this problem is largely determined by the existence of an effective system of programs for solving three-dimensional direct magnetometry problems – finding the fields of typical objects (by using sets of such objects you can approximate real objects). The article describes the operation of the main algorithm and subroutines that are part of the main algorithm of the developed MBEM software for modeling the magnetic fields of thin-walled structures. MBEM is based on the developed modified boundary element method, based on the classical boundary element method and the secondary source method. Numerical verification has been carried out and the efficiency of the developed algorithms and software components has been established on the problems of calculating the magnetic field of thin-walled structures for which solutions are known. The example of the software complex work is given – the calculation of the resulting field of a thin-walled pipe, namely the simulation of the tension vector in the plane that is under the object at a distance of one half meter.

Frustration of the total reflection by a hidden scatterer.

The scattering of a wave by a cylindric object hidden within the dielectric is studied. The total reflection is frustrated because of the scattering. Magnetic and electric fields at the dielectric surface and the energy flux are calculated by the modified boundary element method at near-field and far-field distances. The total reflection is shown to rearrange the shape of a scattering indicatrix.

Boundary Integral Equations for an Anisotropic Bimaterial with Thermally Imperfect Interface and Internal Inhomogeneities

Abstract This paper studies a thermoelastic anisotropic bimaterial with thermally imperfect interface and internal inhomogeneities. Based on the complex variable calculus and the extended Stroh formalism a new approach is proposed for obtaining the Somigliana type integral formulae and corresponding boundary integral equations for a thermoelastic bimaterial consisting of two half-spaces with different thermal and mechanical properties. The half-spaces are bonded together with mechanically perfect and thermally imperfect interface, which model interfacial adhesive layers present in bimaterial solids. Obtained integral equations are introduced into the modified boundary element method that allows solving arbitrary 2D thermoelacticity problems for anisotropic bimaterial solids with imperfect thin thermo-resistant inter-facial layer, which half-spaces contain cracks and thin inclusions. Presented numerical examples show the effect of thermal resistance of the bimaterial interface on the stress intensity factors at thin inhomogeneities.

Application of a Modified Boundary-Element Method to Problems of the Elasticity Theory in Piecewise Homogeneous Anisotropic Media

UDC 539.3 We study the problem of determination of the stress-strain state of a thin elastic orthotropic plate with orthotropic inclusions of any shape under loads applied to its edges. To solve the problem, we propose a modified direct boundary-element method capable of the simultaneous determination of all necessary quantities on the contact surfaces of each inclusion with the matrix by using a two-dimensional approximation of the components of the vector of displacements and stress tensor in every element of the discretization. The governing input relations are obtained for all components of the vector of displacements and stress tensor on the boundary and at inner points of the body under consideration. Recent years are marked by the extensive application of materials and structural elements with anisotropic properties. The use of these materials makes it possible to get the required operating characteristics of strength and stiffness. For their efficient and safe application, we need the exact and reliable numerical methods aimed at the determination of their stress-strain state. The customary numerical methods based on the discretization of the domain itself (finite-element method) or its boundary (boundary-element method) exhibit high efficiency in the case where the corresponding fundamental solutions of the problem under study are available. The presence of anisotropy increases the number of elastic constants in Hooke’s law and, hence, complicates the procedure of construction of the fundamental solutions. However, the fundamental solutions of the two-dimensional problems of elasticity theory are, as a rule, available [5, 6]. The analysis of the stress-strain state for these class of problems was carried out, e.g., in [4, 7, 8]. If we use the ordinary boundary-element method, then the solution of boundary integral equations enables one to find only a part of the components of the stress tensor, namely, the components appearing in the contact conditions. To find the other components of the stress tensor, it is necessary to perform the limit transition both in the integral representations of the vector of displacements and in their differentiated representations. The two-dimensional approximation of displacements on the boundaries of inclusions [1] removes the necessity of differentiation of the representations and makes it possible to determine all components of the vector of displacements and stress tensor on the boundaries of the analyzed bodies already directly from the boundary integral equations. In what follows, we apply this approach to the construction of constitutive relations for the plane problem of elasticity for orthotropic bodies. Consider a thin elastic orthotropic plate with orthotropic inclusions of any shape. The plate suffers the action of muss forces Xi and distributed forces pi applied to its edges. Thus, it can be regarded as an elastic orthotropic matrix with elastic orthotropic inclusions.

Antiplane Deformation of Anisotropic Bodies with Periodic Systems of Thin Inhomogeneities

The integral equations of antiplane shear deformation of anisotropic solids with periodic sets of thin ribbonlike inclusions are constructed. A modified boundary-element method is used to obtain the numerical solutions of specific problems. We compute the stress intensity factors for anisotropic bodies with one, two, or three columns of parallel defects. The comparison with the analytic solutions confirms the validity of the obtained results for cracks and absolutely rigid inclusions. We also study the effect of changes in the degree of anisotropy (orthotropy) of the material on the stress-strain state of the body and on the stress intensity factors in the vicinity of the tips of thin inhomogeneities.

Plasmons excited by an evanescent wave in a periodic array of nanowires

Unusual plasmons, excited by an evanescent wave in narrow slits in a periodic array of nanowires overlying a dielectric substrate, are found from Maxwell’s equations by a modified boundary element method. The evanescent wave appears when the incident plane wave runs onto the boundary from the substrate under the angle of total internal reflection. On the basis of Floquet’s theorem, the effective Green’s function is derived by reducing the infinite structure to one elementary cell. The method is tested by simulation of the known Rayleigh–Wood’s anomalies. The plasmon resonance is found to have a sharp dependence on the angle of incidence.

Модифицированный метод граничных элементов для решения связных задач математической физики

The modified boundary element method for physico-mathematical modeling of the multifactorial processes is offered for discussion. Physical modeling based on the Onsager's theorem about the relationship between generalized forces and fluxes where we assume the coefficients of reciprocity are nonlinear. The approach is illustrated by the example of the strain diagram. Mathematical modeling is based on a modification of the BEM where all incorrect procedures of the numerical differentiation and integration were replaced by preliminary analytical calculations.

Modified boundary element method for problems on oscillations of flat membranes

A modified boundary element method (MBEM) for hyperbolic problems is exemplified by solving the problem of oscillations of a flat membrane. The modification of the method consists in the analytical computation of the components of the influence vector; the integration is carried out not over all the components of the boundary but only once over a specially chosen base element.

An improved boundary element formulation for the motion of spherical bubbles

We suggest a modified boundary element method for modeling the potential flow caused by the motion of many spherical bubbles. The problem for the fluid velocity potential is reduced to a Fredholm integral equation of the second kind. The kernel of that integral equation has no singularity; as a result, its numerical solution does not encounter any difficulties. The matrix representation of the integral equation is diagonally dominant and is well suited to handle multiple bubble system.

The stress-strain state of a plate with a multilayer inclusion

A modified boundary-element method is proposed to analyze the stress-strain state of a plate with a multilayer inclusion of arbitrary shape. The two-dimensional approximation of displacements on each boundary element makes it possible to determine all the stress/strain components at the interface between the inclusion and each layer. The stress state of a plate with an elliptical and a rectangular multilayer inclusion under unidirectional tension (a load uniformly distributed along the edges) is analyzed as numerical examples. The numerical examples demonstrate high accuracy and computational efficiency of the method

Singularities in anisotropic steady‐state heat conduction using a boundary element method

AbstractIn many heat conduction problems, boundaries with sharp corners or abrupt changes in the boundary conditions give rise to singularities of various types which tend to slow down the rate of convergence with decreasing mesh size of any standard numerical method used for obtaining the solution. In this paper, it is shown how this difficulty may be overcome in the case of an anisotropic medium by a modified boundary element method. The standard boundary element method is modified to take account of the form of the singularity, without appreciably increasing the amount of computation involved. Two test examples, the first with a singularity caused by an abrupt change in a boundary condition and the second with a singularity caused by a sharp re‐entrant corner, are investigated and numerical results are presented. Copyright © 2002 John Wiley & Sons, Ltd.