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A comparative study of three domain-integral evaluation techniques in the boundary-domain integral equation method for transient thermoelastic crack analysis in FGMs
No full textA boundary-domain integral equation method is applied to the transient thermoelastic crack analysis in functionally graded materials. Fundamental solutions for homogeneous, isotropic and linear elastic materials are used to derive the boundary-domain integral equations. The radial integration method, the Cartesian transformation method and the cell-integration method are applied for the evaluation of the arising domain-integrals. Numerical results for dynamic stress intensity factors obtained by the three approaches are presented, compared and discussed to show the accuracy and the efficiency of the domain-integral evaluation techniques
Error Expansion of Classical Trapezoidal Rule for Computing Cauchy Principal Value Integral
No full textThe composite classical trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel 1/(x-s) is discussed. Based on the investigation of the superconvergence phenomenon, i.e., when the singular point coincides with some priori known point, the convergence rate of the classical trapezoidal rule is higher than the globally one which is the same as the Riemann integral for classical trapezoidal rule. The superconvergence phenomenon of the composite classical trapezoidal rule occurs at certain local coordinate of each subinterval and the corresponding superconvergence error estimate is obtained. Some numerical examples are provided to validate the theoretical analysis
Half Space Acoustic Problems Analysis by Fast Multipole Boundary Face Method
No full textIn this paper, a half space adaptive fast multipole boundary face method (FMBFM) is presented for solving the three-dimensional half space exterior acoustic problems. In the presented method, the Burton-Miller equation based on the conventional boundary integral equation (CBIE) and its hyper-singular boundary integral equation (HBIE) is used to deal with the fictitious eigenfrequencies problem. The half space Green’s function is employed, thus the tree structure in the fast multipole method can be used only for the real domain. The higher order elements and an adaptive tree structure are used to improve the efficiency of the FMBFM. This half space adaptive FMBFM for half space acoustic problems is an extension of the adaptive FMBFM for full space acoustic problems developed by the authors. Numerical examples for half space acoustic problems in this paper demonstrate the efficiency and validity of this method
Numerical Approximate Solutions of Nonlinear Fredholm Integral Equations of Second Kind Using B-spline Wavelets and Variational Iteration Method
No full textIn this paper, nonlinear integral equations have been solved numerically by using B-spline wavelet method and Variational Iteration Method (VIM). Compactly supported semi-orthogonal linear B-spline scaling and wavelet functions together with their dual functions are applied to approximate the solutions of nonlinear Fredholm integral equations of second kind. Comparisons are made between the variational Iteration Method (VIM) and linear B-spline wavelet method. Several examples are presented to compare the accuracy of linear B-spline wavelet method and Variational Iteration Method (VIM) with their exact solutions
Dynamic Stress Intensity Factors of Collinear Cracks under a Uniform Tensile Stress Wave
No full textAn analysis is presented for an array of collinear cracks subject to a uniform tensile stress wave in an isotropic material. An integral equation for the problem is established by modeling the cracks as distributions of dislocations. The integral equation is solved numerically in the Laplace transform domain first and the solution is then inverted to the time domain to calculate the dynamic stress intensity factors. Numerical examples of one, two, or three collinear cracks are given. The results of one or two cracks are checked to agree closely with the existing results
Thermocapillary Motion of a Spherical Drop in a Spherical Cavity
Get PDFA theoretical study of the thermocapillary migration of a fluid sphere located at an arbitrary position inside a spherical cavity is presented in the quasisteady limit of small Reynolds and Marangoni numbers. The applied temperature gradient is perpendicular to the line through the drop and cavity centers. The general solutions to the energy and momentum equations governing the system are constructed from the superposition of their fundamental solutions in the spherical coordinates originating from the two centers, and the boundary conditions are satisfied by a multipole collocation method. Results for the thermocapillary migration velocity of the drop are obtained for various cases. When the fluid sphere is at the center of the cavity, the collocation result is in excellent agreement with the available exact solution. The normalized thermocapillary migration velocity decreases with increases in the drop-to-cavity radius ratio and in the relative distance between the drop and cavity centers, vanishing as the drop surface touches the cavity wall. For a given configuration, this velocity augments with increases in the relative viscosity of the drop and thermal conductivity of the cavity phase. The boundary effects on the thermocapillary motion perpendicular to the line connecting the drop and cavity centers is significant, but in general weaker than that parallel to this line
A Coupled BEM-MLPG Technique for the Thermal Analysis of Non-Homogeneous Media
Get PDFThis paper presents a technique that couples the boundary element method (BEM) with the meshless local Petrov-Galerkin (MLPG) method, formulated in the frequency domain. It is then used to study the transient heat diffusion through a two-dimensional unbounded medium containing confined subdomains where the material properties vary from point to point.
To exploit the advantages of each method, the BEM is used for the homogeneous unbounded domain and the MLPG method is used for the non-homogeneous confined subdomains. The nodal points placed at the interface between the confined subdomains and the unbounded homogenous medium are used to couple the BEM and the MPLG method. The MLPG method is formulated using the moving leastsquares (MLS) approximation as the trial function and the Heaviside step function as the test function in local integral equations defined over small local sub-domains. The coupled BEM-MLPG approach is verified against the results provided by an analytical solution developed for a circular confined subdomain, in which the thermal diffusivity within the circular non-homogeneous region is assumed to vary in the radial direction. The proposed model is finally used to solve the case of a pair of non-homogeneous confined subdomains for which analytical solutions are not known. The analysis of time domain temperature responses is presented, which illustrates the applicability of the model
Calculation of Nearly Singular Boundary Element Integrals in Thin Structures Using an Improved Exponential Transformation
Get PDFIn this work, an improved exponential transformation is presented for nearly singular boundary element integrals in problems of thin structures. Accurate evaluation of nearly singular integrals is an important issue in the implementation of boundary element method (BEM) for thin structures. In this paper, the exponential transformation, which was firstly developed to evaluate nearly singular integrals arising in 2D BEM, is extended into 3D BEM to deal with nearly singular integrals. Firstly, a novel (α,β) coordinate system is introduced. Then, the conventional distance function is modified into a new form in (α,β) coordinate system. Based on the refined distance function, finally, an improved exponential transformation is employed in the new coordinate system. Furthermore, to perform integrations on irregular elements, an adaptive integration scheme considering both the shape of element and the projection point associated with the improved transformation is proposed. Numerical examples are presented to verify the proposed method. Results demonstrate the accuracy and efficiency of our method. Moreover, the accuracy of our method is less sensitive to the position of the projection point than that of the traditional methods
On Macroscopic Behaviors of Shape Memory Alloy Thick-walled Cylinder Under Combined Internal Pressure and Radial Temperature Gradient
Get PDFAnalytical solutions are derived for the macroscopic behaviors of a Shape Memory Alloy (SMA) thick-walled cylinder subjected to internal pressure and radial temperature gradient. The Tresca transformation criterion and linear hardening are used. Equations are given for the radial and circumferential stresses, transformation strains and martensite volume fractions at both the elastic step and the transformation step. Numerical results are presented and in good agreement with the finite element simulations
Numerical Integration with Constraints for Meshless Local Petrov-Galerkin Methods
Get PDFWe propose numerical integration rules for meshless local Petrov- Galerkin methods (MLPG) employed to solve elliptic partial different equations (PDE) with Neumann boundary conditions. The integration rules are required to satisfy an integration constraint condition of Green’s formula type (GIC). GIC was first developed in [Babuska, Banerjee, Osborn, and Zhang (2009)] for Galerkin meshless method, and we will show in this paper that it has better features for MLPG due to flexibility of MLPG in choosing different trial and test function spaces. A general constructive algorithm is presented to design the integration rules satisfying GIC. We also present a useful situation, where GIC holds automatically for Gaussian rules. According to this, we conclude that the conical weight is suggested to adopt in MLPG from viewpoint of reducing integration complexity. Approach to extending GIC in [Babuska, Banerjee, Osborn, and Zhang (2009)] to more general elliptic PDE, such as elasticity equation, is discussed. The 1D and 2D numerical results illuminate that GIC reduces the errors in the approximate solutions of MLPG significantly. Keywords: Meshless Petrov-Gale