Three-dimensional Boundary Element Analysis Research Articles

Hybrid nearly singular integration for three-dimensional isogeometric boundary element analysis of coatings and other thin structures

The isogeometric boundary element method (IGABEM) has great potential for the simulation of elasticity problems because of its exact geometric representation and good approximation properties. These advantages can be extended to thin structures, including coatings, but the development of an accurate and efficient method to deal with the large number of nearly singular integrals existing in the IGABEM presents a great challenge for very thin sections. In this paper, we propose a new sinh+ scheme for weakly, strongly and hyper near-singular integrals arising in 3D IGABEM for thermoelastic problems, based on the sinh transformation method and adaptive integral method. The presented scheme is efficient, since it combines the advantages of both methods: (1) when the thickness δ of coatings/thin structures is moderately small, an accurate and efficient integral result will be obtained by the adaptive integral method; (2) when δ is very small, the nearly singular integrals are computed by the sinh+ scheme efficiently. With the introduction of NURBS in IGABEM, truncation errors arising in the Taylor expansion cannot be ignored. Based on the values of these errors, the computed knot spans are further divided into several sub-knot spans and different methods will be used to evaluate the integral over each sub-knot span in the new scheme. In addition, based on the analytical extension of the NURBS surface, an adaptation of the sinh transformation method is proposed which can evaluate the near-singular integrals accurately for cases in which the projection point lies outside of the considered knot span. Several numerical examples are presented to validate the accuracy and efficiency of the 3D IGABEM based on the sinh+ scheme in the analysis of thermoelastic problems.

Prediction of fatigue crack propagation and fractography of rail steel

The aim of the present research is to determine the initiation and propagation of cracks and its geometric specifications. Metallurgical factors, especially the non-metallic inclusions have a significant impact on the fatigue life of railways. Since all non-metallic inclusions in a measure of steel alloy is not dangerous, determining how and which one of them affects the fatigue process of the rails is of the present studies' concern. Afterwards, the results of the research on fracture surface of fatigue specimens will be presented. Numerical and experimental methods are used to investigate the behavior of fatigue crack growth in the specimen. For this purpose, the fatigue crack growth and hardness tests were carried out and have been checked by fractography studies on fractured specimens. The hardness of the specimens was measured in different points by using Rockwell micro-hardness experiment. Afterwards, a three-dimensional boundary element method is used for fatigue crack growth under stress field. The modified Paris model is used to estimate fatigue crack growth rates. Finally, three-dimensional boundary element analysis results obtained show good agreement with those achieved in experimental tests.

Three-dimensional boundary element analysis for anisotropic elastic solids and its extension to piezoelectric and magnetoelectroelastic solids

This article presents three-dimensional (3D) analysis of generally anisotropic elastic, piezoelectric, piezomagnetic, and magneto-electro-elastic solids by boundary element method (BEM). The associated Green's functions of displacements and tractions in both complex and real forms together with their first derivatives are derived completely in the first time by using Radon–Stroh formalism. Like the Stroh formalism for two-dimensional problems, the solutions for the solids with these different material types all bear exactly the same mathematical forms distinguished by the contents and dimensions of the related matrices and vectors. This feature provides a big advantage for computer programming. To emphasize the development of 3D-BEM, the system of algebraic equations to calculate the nodal displacements and nodal tractions along the boundary of 3D solids is formulated. The relations for calculating strains and stresses at the boundary nodes as well as at the interior points are also derived. After the employment of Green's functions to the newly formulated 3D-BEM, we found that the 3D-BEM with complex form solution is computationally more efficient than that with real form solution, although the opposite performance was observed for a single point calculation of Green's function. To verify the correctness of our solutions, numerical examples of a cube with or without through hole are presented as illustrations of our successful implementation in the BEM.

A general algorithm for evaluating nearly strong-singular (and beyond) integrals in three-dimensional boundary element analysis

One of the typical and most significant issues of almost all boundary element analyses is the accurate evaluation of nearly singular boundary element integrals. In this study, we review some numerical techniques used currently to calculate nearly singular integrals and propose an improved algorithm to calculate integrals with nearly strong (and beyond) singularities. This new method has full generally and can be easily included in any existing computer code. The method is tested in general three-dimensional boundary element analysis. Comparison of this method with some of the existing methods is also presented. It is shown that several orders of magnitude improvement in relative errors can be obtained using the proposed method when compared to a straightforward implementation of Gaussian quadrature.

A general algorithm for evaluating nearly singular integrals in anisotropic three-dimensional boundary element analysis

This paper presents a new general method for the evaluation of nearly singular boundary element integrals arising in anisotropic three-dimensional (3D) boundary element analysis. It is shown that the original nearly singular integrals can be transformed into, using a sinh function, an element-by-element sum of regular integrals, each one expressed in terms of intrinsic (local) coordinates. As a consequence, the actual computation can be performed by using standard n×n Gaussian quadrature and the procedure can be easily included in any existing computer code. This new method has full generality and, therefore, can be applied to a wide class of integrals. The numerical results demonstrate the accuracy and efficiency of the method, along with its insensitivity to the location of the nearly singular points. It is shown that several orders of magnitude improvement in relative errors can be obtained using this transformation when compared to a straightforward implementation of Gaussian quadrature.

Study on Force-Transmissibility of a Magnetic Gear by Using 3-D Boundary Element Analysis

This paper presents three-dimensional boundary element analysis for coupled magnetic field and moving problem. The method is applied to improve transmission characteristics of a magnetic gear and the usefulness is demonstrated. The gear has two movers consisting of permanent magnets and they rotate with different velocity. The double gap moving problem is very time consuming, because the mesh has to be regenerated at each time step or prepared many patterns previously. The use of the boundary element method is suitable for this problem, because meshes are required around only surfaces of considering objects and the objects can be rotated any angle according to the moving simulation. In this paper, the technique to reduce unknown variables in the boundary element equations utilizing symmetrical construction is proposed.

The effect of three-dimensional crack growth on the force distribution and meshing stiffness of a spur gear: Ideal and misaligned contacts

This article presents a routine for prediction of the crack growth path in gears. Crack simulation helps in determining failure modes and crack detecting by vibration monitoring. To simulate crack propagation an effective method is developed by which the meshing force is calculated based on the compliance of meshing teeth in every stage of crack growth. In this method the compliance matrix of the tooth is calculated using three-dimensional boundary element analysis without the need for simultaneously modelling two meshing gears. The force distribution on the contact area is then calculated using Hertz theory and compliance matrices of meshing teeth. The method is used in prediction of the crack propagation path, meshing force, and meshing stiffness for two cases of ideal and misaligned meshing of a spur gear set. The crack propagation path is very different in the two cases of meshing and affects force distribution on a contact zone. Meshing stiffness reduces as crack propagates but the amount of reduction depends on the position of the contact line and crack propagation path.

Stray currents of ESP well casings

Comprehensive field measurements and simulation using three-dimensional (3D) boundary element analysis system (BEASY) software package of electrical submersible pump (ESP) well casings have been conducted to (a) investigate the effectiveness and applicability of pulse cathodic protection (PCP) system, (b) find out the sources of stray currents, (c) suggest a practical solution to reduce the galvanic cell current between the steel well casings and the four copper earthing rods used in the ESP earthing system, and (d) check the possibility of utilizing unused well casings as anodebed energized by the conventional cathodic protection (CP) to protect many nearby well casings. Field measurements and theoretical simulation reveal that the PCP current is DC with superimposed ripples having the same frequency of the voltage pulses (∼3 kHz). Hence there is a doubt about the effectiveness of the PCP systems for protecting deep well casings. The naturally created galvanic cells generate a pure DC current up to 1000 mA, and it can be reversed by using aluminum earthing rods instead of copper ones. Simulation has proven the superiority of utilizing unused well casings as anodes and it represents a promising means for having quasi-uniform current-density profiles along the axial length of the protected well casings.

Failure of curved brittle layer systems from radial cracking in concentrated surface loading

A study was made of radial crack evolution in curved brittle layers on compliant support substrates. Three-dimensional boundary element analysis was used to compute the stepwise growth of radial cracks that initiate at the bottom surfaces of glass on polymeric support layers, from initiation to final failure. The algorithm calculates reconstituted displacement fields in the near-tip region of the extending cracks, enabling direct evaluation of stress-intensity factors. Available experimental data on the same material systems with prescribed surface curvatures were used to validate the essential features of the predicted crack evolution, particularly the stability conditions prior to ultimate failure. It was shown that the critical loads to failure diminish with increasing surface curvature. Generalization of the ensuing fracture mechanics to include alternative brittle-layer/polymer-substrate systems enabled an explicit expression for the critical load to failure in terms of material properties and layer thicknesses. Implications concerning practical layer systems, particularly dental crowns, are briefly discussed.

Three-dimensional singular boundary elements for corner and edge singularities in potential problems

It is well known that the spatial derivative of the potential field governed by the Laplace and Poisson equations can become infinite at corners (in two and three dimensions) and edges (in three dimensions). Conventional elements in the finite element and boundary element methods do not give accurate results at these singular locations. This paper describes the formulation and implementation of new singular elements for three-dimensional boundary element analysis of corner and edge singularities in potential problems. Unlike the standard element, the singular element shape functions incorporate the correct singular behavior at the edges and corners, specifically the eigenvalues, in the formulation. The singular elements are used to solve some numerical examples in electrostatics, and it is shown that they can improve the accuracy of the results for capacitance and electrostatic forces quite significantly. The effects of the size of the singular elements are also investigated.

Mapped infinite elements for three‐dimensional multi‐region boundary element analysis

AbstractA formulation for an infinite boundary element (BE) is presented, which allows the modelling of infinite surfaces. The concept is that the finite surface is mapped to an infinite surface using special mapping functions. Using such mapping functions together with linear and quadratic interpolation for the displacements and the tractions, respectively, the desired decay behaviour can be modelled. The implementation of the proposed infinite elements becomes straightforward, since the Cauchy principal value, as well as the free term, are evaluated for the finite and infinite BEs with exactly the same techniques. The element developed can be used in a multi‐region BE analysis of piecewise homogeneous domains, or for domains with joints and faults. The accuracy of the element is tested on some benchmark problems. Finally, a practical application in tunnelling is shown. Copyright © 2004 John Wiley & Sons, Ltd.

Three-dimensional piezoelectric boundary element analysis of transversely isotropic half-space

In this paper, we present a boundary element method (BEM) solution technique for studying the three-dimensional transversely-isotropic piezoelectric half-space problems. The use of mixed alternative point force solutions for half and full-space problems presented are necessary to overcome the computation difficulties especially in the calculation of the derivatives with respect to z. Infinite boundary elements are introduced to model the surface of the half-space only when stresses at the internal points are required to be evaluated. The integration over the infinite boundary elements is bounded and some limitations of the infinite element construction are relaxed. Closed-form solutions for uniformly distributed mechanical and electrical loads acting on a circular area on the surface of half-space are derived. This theoretical work serves as a good verification tool for numerical computation. In this paper, the numerical and theoretical results show good agreement. Numerical analysis via the finite element method (FEM) is also carried out using the commercial solver ANSYS. These FEM results are used to verify against the accuracy of the BEM solution. Finally, numerical results for the case of Hertzian pressure applied to an imperfect half-space are presented. The effects of the coupled mechanical–electrical influences as well as the presence of voids are examined.

Two-dimensional and three-dimensional magnification factors, Mk, for non-load-carrying fillet welds cruciform joints

Two-dimensional M k factors at the weld toe of non-load-carrying transverse welded attachments are derived indirectly using the principle of superposition and stresses obtained from boundary element analysis. The results are in good agreement with that obtained by Maddox and Andrews [1] for different ratios of l/ T, where l is the weld toe to weld toe distance, and T is the main plate thickness. Further three-dimensional boundary element analysis using realistic crack shapes, confirm that l/ T is the most important parameter. Two-dimensional edge cracks analyses overestimate and underestimate the deepest and surface points M k values by as much as 15% and 65% respectively. Parametric equations for three-dimensional M k are proposed for design purposes.

Three-dimensional boundary element analysis of drop deformation in confined flow for Newtonian and viscoelastic systems

An adaptive (Lagrangian) boundary element approach is proposed for the general three-dimensional drop deformation in confined flow. The adaptive method is stable as it includes remeshing capabilities of the deforming interface between drop and suspending fluid, and thus can handle large deformations. Both drop and surrounding fluid are viscous incompressible and can be Newtonian or viscoelastic. A boundary-only formulation is implemented for fluids obeying the linear Jeffrey's constitutive equation. Similarly to the formulation for two-dimensional Newtonian fluids (Khayat RE, Luciani A, Utracki LA. Boundary element analysis of planar drop deformation in confined flow. Part I. Newtonian fluids. Engineering Analysis of Boundary Elements 1997; 19: 279), the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a cylindrical tube. The accuracy of the method is assessed by comparison with the analytical solution for two-phase radial spherical flow, leading to good agreement. The influence of mesh refinement is examined for a drop in simple shear flow. Copyright © 2000 John Wiley & Sons, Ltd.

A threedimensional boundary element analysis of elasto‐plastic cracked prolems

AbstractBased on the theoretical background, the formulation for the three‐dimensional elasto‐plastic analysis has been enhanced and implemented in the computer program. The algorithm utilities the initial stamen approach to satisfy the material non‐linearity. The discrimination inculpates the continues and the discontinuous boundary elements. Also included are the domain cells with the linear and quadratic approximation. The non‐linear boundary element results are presented to demonstrate the accuracy and usefulness of the method.

Three-dimensional boundary element analysis of fatigue crack growth in linear and non-linear fracture problems

This paper presents general boundary element procedures for the elastic and elastoplastic numerical modelling of three-dimensional fatigue crack growth. The elastic and elastoplastic boundary element formulations are briefly introduced, followed by the description of the crack propagation strategies. In both cases, fatigue crack growth is numerically modelled through an incremental crack extension procedure. For each crack extension the dual boundary element method is used to perform a single region analysis of the cracked component and the corresponding fracture parameter ( K or J depending on if the analysis is elastic or elastoplastic) is computed along the crack front. New positions of the crack front are determined using the Paris law. The capabilities of the proposed procedures are demonstrated by comparison of numerical predictions with experimental results.

Three-dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies

A frequency domain boundary element methodology of solving three dimensional electromagnetic wave scattering problems by dielectric particles is reported. The method utilizes a computationally attractive surface integral equation containing only weakly and strongly singular integrals in the contrast to most formulations involving not only strongly singular but hypersingular integrals as well. The main advantage of this integral equation is the fact that its strongly singular part is similar to the one appearing in the corresponding integral equation of dynamic elasticity. Thus, well known advanced integration techniques used successfully in elastic scattering problems can be directly applied to the present analysis. Both continuous and discontinuous quadratic elements are employed in order to accurately treat dielectric scatterers with smooth and piecewise smooth boundaries. Numerical examples dealing with three dimensional electromagnetic wave scattering problems demonstrate the accuracy and efficiency of the proposed boundary element formulation.

3-D infinite boundary elements for half-space problems

This paper presents a new infinite element approach for three-dimensional boundary element analysis of half-space problems. The strongly singular integrals over the infinite surface are evaluated analytically by transforming the surface integrals into line integrals. The illustrative numerical results demonstrate the potential of the formulation.

Theoretical Evaluation of Permeability of Porous Media Including Microscopic Internal Flow.

The behavior of flow in porous media has been investigated using a model of a homogeneous medium obeying Darcy's law. As the permeability reflects the microstructure of porous media, theoretical analysis is performed for evaluation of the permeability of the medium, including the microscopic internal flow. The fluid flow in the medium is assumed to be Stokes flow. Homogenization theory with multiscale asymptotic expansion is used to derive Darcy's law from governing equations for Stokes flow theoretically. The microstructure of the medium is modeled as a body-centered cubic array of sphere particles connected to each other periodically. A cubic unit cell is placed in the microscopic field of the medium to represent a boundary value problem of the internal flow. The internal flow is formulated using intergral equations. Permeability and microscopic fluid flow of the medium are shown by three-dimensional boundary element analysis on the cubic unit cell.

3-D seismic response analysis of long lined tunnels in half-space

The dynamic response of infinitely long lined tunnels with a uniform cross-section buried into an elastic or viscoelastic half-space to body and surface harmonic seismic waves is numerically determined by a special direct boundary element method in the frequency domain. The waves have an arbitrary direction of propagation with respect to the axis of the tunnel and this renders the problem three-dimensional. However, this problem is effectively reduced to a two-dimensional one by a coordinate transformation and appropriate integration of the full-space dynamic fundamental solution along the direction of the tunnel axis. Quadratic isoparametric boundary line elements and advanced numerical integration techniques for the treatment of singular integrals produce results of high accuracy. Numerical results are presented for the case of an infinitely long lined tunnel of circular cross-section and compared against those of a full three-dimensional boundary element analysis, as well as those of other methods. Thus the proposed method is illustrated and its merits demonstrated.