Boundary Collocation Method Research Articles

Stress Intensity Factors for Cracked Finite Plates with Mixed Boundary Condition

The mixed boundary problems for finite plates with one crack or two collinear cracks are studied. Complex stress functions that satisfy the equilibrium equations and compatibility conditions in the cracked plate as well as the stress condition on crack surfaces are presented. Four models, that is, a square plate with one crack or with two collinear cracks and an airfoil-shaped plate with one crack or with two collinear cracks, are established. The unknown coefficients of the complex stress functions are determined by using boundary collocation method (BCM). The effects of crack orientation, crack distance, and boundary condition on SIFs are investigated by combining with BCM, and the corresponding photoelastic experiments are conducted. The test results generally agree with the BCM calculation results.

Singular boundary method for inverse heat conduction problems in general anisotropic media

This study documents the first attempt to apply the singular boundary method, a recently developed meshless boundary collocation method, to the solution of inverse anisotropic heat conduction problems. The method cures the perplexing fictitious boundary issue associated with the method of fundamental solutions while inheriting the merits of the latter of being truly meshless, integration-free and easy to programme. Thanks to its boundary-only discretization and semi-analytical nature, the method can be viewed as an ideal candidate for the solution of inverse problems. Four benchmark examples indicate that the proposed method, in connection with proper regularization techniques, is accurate, computationally efficient and numerically stable for the solution of inverse problems with various levels of noisy Cauchy data.

A semi-analytical analysis of the elastic buckling of cracked thin plates under axial compression using actual non-uniform stress distribution

An improved hybrid semi-analytical method for calculating elastic buckling load of a thin plate with a central straight through-thickness crack subject to axial compression is proposed. In the study, the actual non-uniform in-plane stress distribution is firstly conducted by using Muskhelishvili's complex variable formulation in conjunction with boundary collocation method. A deflection shape function, satisfying not only the outer boundary conditions but also the inner boundary conditions of the crack edges, is obtained by using domain decomposition method. Finally the buckling load of a cracked plate using Raleigh–Ritz energy method is calculated based on the actual in-plane stress distribution and the reasonable deflection shape function obtained. The effects of crack length, plate's aspect ratio are studied for thin plates with different boundary conditions. Results obtained from the proposed method are in good agreement with the existing numerical results and experimental ones. It is finally shown that the proposed method, based on a correct non-uniform in-plane stress distribution, is more accurate than the few existing analytical methods based on a uniform in-plane stress distribution.

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1] , [2] . The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points. The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.

Comparison of different methods for choosing the collocation points in the boundary collocation method for 2D-harmonic problems with special purpose Trefftz functions

Abstract Based on special purpose Trefftz functions, this paper presents a comparison of different methods for choosing collocation points when a boundary collocation method is used to fulfill the boundary conditions. First, it is shown that for some geometries when applying a boundary collocation method with special purpose Trefftz functions, equidistant collocation points give unacceptable results. Next, for four 2-D harmonic boundary value problems, four cases of different placements of the collocation points are described and compared. Based on the results of the comparison of the test methods for the location of the collocation points it is shown that one of the proposed methods for placing the collocation points is simple to implement and accurate. The essence of this method is an adaptive determination of the consequent collocation points.

Approximate solution to a singular, plane mixed boundary-value problem for Laplace’s equation in a curved rectangle

In this paper, we use a hybrid method based on a variant of Trefftz’s method (TM), in combination with the usual Boundary Collocation Method (BCM) to find the approximate solution to a singular, two-dimensional mixed boundary-value problem for Laplace’s equation in a rectangular sheet with one curved side. After expressing the solution as a finite linear combination of harmonic trial functions, the usual BCM is used to enforce the boundary condition on the curved side, while a variant of TM is applied to the three remaining sides. The singularity at one corner of the rectangle is treated via the enrichment of the expansion with a specially built harmonic function which has a singularity at one corner. The procedure ultimately produces a rectangular set of linear algebraic equations, which is solved by QR factorization method. Numerical results are presented and discussed, in order to assess the efficiency of the proposed method.

Heat Conduction Through Eccentric Annuli: An Appraisal of Analytical, Semi-Analytical, and Approximate Techniques

The problem of conduction heat transfer through an eccentric annulus with the inner surface kept at a constant temperature and the outer surface subjected to convective condition is solved by three different techniques. A perturbation analysis yields an analytical expression for temperature profile for small values of eccentricity. The two dimensional conduction problem has also been solved by a two dimensional semi-analytical technique in which the condition at the outer periphery is matched by a collocation technique. Finally, a one dimensional approximate technique namely sector method has been used to solve the same problem. The sector method does not require any numerical technique yet yields remarkable accuracy. Next, the heat flow through an eccentric insulation surrounding a circular cylinder was considered. It has been demonstrated that the sector method is effective also in determining the geometry of the critical insulation in this case over a wide range of radius ratio and Biot number. Finally, both all the three methods, i.e., the perturbation technique, the boundary collocation method and the sector method have been applied to determine the geometry of the critical as well as crossover perimeter of insulation around a circular cylinder when the insulation is provided eccentrically.

An analytical model for the longitudinal permeability of aligned fibrous media

Abstract This paper presents an analytical series solution of the longitudinal fluid flow in porous media consisting of aligned rigid fibers, which, in turn, is used to establish a relationship for the longitudinal permeability as a function of the fiber packing geometry, fiber volume fraction and the fiber radius. The analytical series solution is developed for rectangular and staggered packing arrangements of the fibers using the boundary collocation method where the constants in the series solution are solved for numerically. As the number of boundary collocation points is increased, the analytical solution is shown to converge with a finite element solution of the identical flow situation for all fiber volume fractions and packing arrangements. In addition, the permeability results are presented in a dimensionless form as a function of the fiber volume fraction and fiber packing arrangement for a general applicability and easy use of the results for predicting the longitudinal permeability of fiber tows consisting of aligned rigid fibers.

A general method to determine the stress intensity factor of multiple collinear cracks

In this study, the issue of multiple collinear cracks in a finite plate is studied, and the overall stress field as well as stress intensity factors (SIFs) near crack tips are determined. Based on Muskelishvili’s results, the problem is formulated using complex functions in terms of unknown coefficients, which are obtained by enforcing the external boundaries using a least-square boundary collocation method. These functions satisfy all the required conditions, and can be used to calculate the overall stress field, displacement field and the SIFs at the crack tips. Examples of a rectangular plate with one, two and three cracks, respectively, have been calculated, and a set of photoelastic experiments were performed and the results are supportive. At the end, the effect of confining stress, friction coefficient, crack orientation and the distance between crack tips on SIF are investigated.

Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media

The singular boundary method (SBM) is a recent strong-form meshless boundary collocation method. Like the method of fundamental solutions (MFS), the SBM uses the fundamental solution of the governing differential equation of interest as the basis function and is mathematically simple, truly meshless, accurate, and easy-to-program. Unlike the MFS, the SBM, however, uses the concept of the origin intensity factor to isolate the singularity of the fundamental solutions and overcomes the fictitious boundary issue which has long perplexed the MFS. This study makes the first attempt to apply the SBM to steady-state heat conduction in three-dimensional (3D) anisotropic materials. Five benchmark numerical examples demonstrate that the SBM is accurate, convergent, stable, and computationally efficient in solving this kind of problems.

An analytical method for the determination of stress and strain concentrations in textile-reinforced GF/PP composites with elliptical cutout and a finite outer boundary and its numerical verification

Stress concentrations in the vicinity of cutouts can often be regarded to be the limiting factor for a whole structure. As a further development of prior research at the Institute of Lightweight Engineering and Polymer Technology, an analytical method for the determination of the whole stress-strain fields in the vicinity of holes in multilayered textile-reinforced composites has been developed, which takes into consideration the influences of a finite outer boundary of the specimen. The analytical method is based on the classical laminate theory and the use of complex-valued potential functions. To account for the shape of the specimen, the method of conformal mappings is applied for the inner boundary, while a combination of boundary collocation and least squares method is used for the outer boundary. The method allows a layer-by-layer analysis of stress concentrations. For the verification of the developed calculation model, extensive experimental and numerical finite-element (FE) studies have been carried out on multilayered GF/PP plates with different laminate layups, notches, and specimen dimensions. The comparison of the experimentally or numerically determined results with the analytically calculated ones shows a very good correlation, of which the numerical studies are presented here for the first time. In a second step, the applicable boundary conditions on the outer boundary have been extended in such a way that varying stress and moment resultants can be applied, so that the calculation method can be used as an analytical sub-model in combination with FE techniques.

Stochastic boundary collocation and spectral methods for solving PDEs

We develop a stochastic boundary method (SBM) which can be considered as a randomized version of the method of fundamental solutions (MFS). We suggest solving the large system of linear equations for the weights in the expansion over the fundamental solutions by a randomized SVD method introduced by Sabelfeld and Mozartova (2011). In addition, we also deal with solving inhomogeneous problems where we use the integral representation through the Green integral formula. The relevant volume integrals are calculated by a Monte Carlo integration technique which uses the symmetry of the Green function. We construct also a stochastic boundary method based on the spectral inversion of the Poisson formula representing the solution in a disc. This is done for the Laplace equation, and the system of elasticity equations. We stress that the stochastic boundary method proposed is of high generality; it can be applied to any bounded and unbounded domain with any boundary condition provided the existence and uniqueness of the solution are proven. We present a series of numerical results which illustrate the performance of the suggested methods.

Crane Mechanics Crack Growth Life Analysis Method Research Based SIF Amplitude Fitting

At present, the crane metal structure cracks are ignored based on the traditional strength theory in the crane engineering design. As enlarging of crane mechanics, although their maximum stress don’t exceed allowed stress, recently considerable number of serious crane accidents happened, that are caused by the fracture because of crack faults. It is essential to analyze crack growth based on fracture mechanics in design or diagnose. The crack analysis consists of determination of the crack growth trajectory and calculation of crack growth rate. In generally, the crack of crane is supposed as I-mode in the engineer application. So the crack growth trajectory can be determinated without calculation, which simply cracks trajectory analysis. Then the crack growth rate will be calculated according to the Paris formula. The Paris’s formula is put forward as the form of the relation between stress value and crack length often. To improve the precision, a new method is put forward in the paper. According to the method, the form of Paris’s formula is function of the stress intensity factor (SIF). The new method is valid and high precision through the comparing with the boundary collocation method and the traditional method. And then an example of crack growth analysis on crane is given; the analysis procedure is described also. The method has a benefit to prevent fracture failure and it is supplement to the traditional strength design method also

Research on the Fracture Mechanism of Coal with Three Collinear Cracks Filling with Gas

Coal usually contains many cracks which affect the stability of mining zone. In order to research the fracture mechanism of cracked coal, a special case of three collinear cracks filling with gas is considered. A set of complex stress functions which satisfy all the required conditions are first derived, based on Muskelishvili’s results. The least square boundary collocation method is used to enforce the external boundaries approximately, and the stress intensity factors (SIF) can then be computed according to the related formulae. The value of SIFs for different crack orientations, different crack distance under different loading are calculated. The results shows that the most unfavorable crack orientation is in the 45° angle and the confining stress and the distance between cracks have a considerable effect on the SIF value.

An Artificial Antigen-presenting Cell with Paracrine Delivery of IL-2 Impacts the Magnitude and Direction of the T Cell Response

Artificial antigen-presenting cells (aAPCs) are an emerging technology to induce therapeutic cellular immunity without the need for autologous antigen-presenting cells (APCs). To fully replace natural APCs, an optimized aAPC must present antigen (signal 1), provide costimulation (signal 2), and release cytokine (signal 3). Here we demonstrate that the spatial and temporal characteristics of paracrine release of IL-2 from biodegradable polymer aAPCs (now termed paAPCs) can significantly alter the balance in the activation and proliferation of CD8+ and CD4+ T cells. Paracrine delivery of IL-2 upon T cell contact with paAPCs induces significant IL-2 accumulation in the synaptic contact region. This accumulation increases CD25 (the inducible IL-2 Rα chain) on responding T cells and increases proliferation of CD8+ T cells in vitro to levels 10 times that observed with equivalent amounts of bulk IL-2. These CD8+ T cell responses critically depend upon close contact of T cells and the paAPCs and require sustained release of low levels of IL-2. The same conditions promote activation-induced cell death in CD4+ T cells. These findings provide insight into the response of T cell subsets to paracrine IL-2.

Flow and Heat Transfer Through a Polygonal Duct filled with a Porous Medium

The fully developed flow and H1 heat transfer in a polygonal duct filled with a Darcy–Brinkman medium is studied. The efficient method of boundary collocation is used. The problem is governed by the duct shape and a non-dimensional parameter s which characterizes the inverse square root of permeability. Asymptotic formulas for small and large s are derived.

A Failure Criterion for Cracked Rock Mass under Seepage Pressure and Confining Pressure

The existence of water affects the mechanical properties of cracked rock mass. Taking into account the friction generated by the crack closure, the stress intensity factor of the center cracked plate subjected to compression and seepage pressure was obtained through the theoretical analysis and numerical calculation of the boundary collocation method. The results show that crack tip stress intensity factor increases with the increasing of the seepage pressure, but decreases with the increasing of crack surface friction and the confining pressure. Finally a failure criterion for cracked rock mass under seepage pressure and confining pressure is developed.

Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method

An overdeterministic method was used for calculating the Notch Stress Intensity Factors (NSIFs) as well as the coefficients of the higher order terms for structures containing sharp notches. The series solution of displacement fields around the notch tip was fitted to a large number of nodal displacements obtained from finite element analysis. An over-determined set of simultaneous linear equations was then derived and the nodal displacements reduced to a small set of unknown coefficients by employing the concept of the least-squares method. The efficiency of the proposed method was assessed through analyzing several notched specimens under pure mode I, pure mode II and mixed modes I/II loading. The accuracy of the NSIFs and the coefficients of higher order terms was evaluated by comparing them with the results available in the literature, or the results obtained by the boundary collocation method. While the presented method is simple, it yields very good results.

Analytical and experimental analysis of stress concentration in notched multilayered composites with finite outer boundaries

A solution method for stress concentration problems of fibre- and textile-reinforced multilayered composites with account of the influence of a circular or elliptical cut-out and of the finite outer boundary of a composite plate is presented. The method is based on complex-valued displacement functions and conformal mappings in combination with the boundary collocation and least squares methods. This allows a layer-by-layer calculation of full stress, strain, and displacement fields in a generally multilayered anisotropic plate. To verify the calculation model, extensive experimental studies have been carried out. For all the combinations of multilayered GF/PP plates, laminate lay-ups, and notch and specimen dimensions investigated so far, a very good agreement between the analytical calculations and experimental results is found to exist.

Creeping-flow rotation of a slip spheroid about its axis of revolution

The problem of rotation of a rigid spheroidal particle about its axis of revolution in a viscous fluid is studied analytically and numerically in the steady limit of negligible Reynolds number. The fluid is allowed to slip at the surface of the particle. The general solution for the fluid velocity in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of separation of variables. The slip boundary condition on the surface of the rotating particle is applied to this general solution to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary collocation method or explicit formulas derived analytically. The torque exerted on the spheroidal particle by the fluid is evaluated for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic torque results and the available analytical solutions in the limiting cases is good. It is found that the torque exerted on the rotating spheroid normalized by that on a sphere with radius equal to the equatorial radius of the spheroid increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip or finite-slip spheroid and vanishes for a perfectly slip spheroid. For a spheroid with a specified aspect ratio, the torque is a monotonically decreasing function of the slip capability of the particle.