A new LBIE method for solving elastodynamic problems

A new local boundary integral equation (LBIE) method for the solution of elastodynamic problems in both frequency and time domain is proposed. Non-uniformly distributed points covering the analyzed domain are used for the interpolation of the involved fields. The key-point of the proposed methodology is that the support domain of each point is divided into parts with the aid of cells formed by connecting the point of interest with the nearby points. Then an efficient radial basis functions (RBF) interpolation scheme is exploited for the representation of displacements in each cell, while on the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary elements. For each point the corresponding LBIE is written in terms of displacements only, since on the boundary of support domains tractions are eliminated with the aid of the elastostatic companion solution. The integration in support domains is performed easily and with high accuracy, while due to cells the extension of the method to three dimensions is straightforward. Transient solutions are obtained after inversion of frequency domain results with the inverse fast Fourier transform (FFT). Two representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

A meshless local boundary integral equation method for solving transient elastodynamic problems

A new local boundary integral equation (LBIE) method for solving two dimensional transient elastodynamic problems is proposed. The method utilizes, for its meshless implementation, nodal points spread over the analyzed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the global boundary, displacements and tractions are treated as independent variables. The local integral representation of displacements at each nodal point contains both surface and volume integrals, since it employs the simple elastostatic fundamental solution and considers the acceleration term as a body force. On the local boundaries, tractions are avoided with the aid of the elastostatic companion solution. The collocation of the local boundary/volume integral equations at all the interior and boundary nodes leads to a final system of ordinary differential equations, which is solved stepwise by the -Wilson finite difference scheme. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

A semi-analytic method for solving some 2-D elastodynamic problems in semi-infinite media

 A new semi analytic method for solving two dimensional elastodynamic problems in semi-infinite medium is proposed. The elastodynamic field equations are Fourier transformed in the infinite space dimension(s), while time derivatives are approximated via finite difference, leading to a set of ordinary differential equations in the semi infinite direction(s), which are solved analytically. The method is inherently non-reflecting and no artificial boundaries are used. 1 and 2-D examples of a rigid body impact are studied and the non-oscillating characteristic of the solution, usually obtained by other methods, is examined. The accuracy is examined by comparing the results with solutions obtained by previous methods. Received 12 August 1998     

A new MLPG method for elastostatic problems

This paper introduces a novel meshless local Petrov-Galerkin (MLPG) method by presenting a new test function as a schema to solve the elasto-static problems. It is seen that the four ordinary MLPG methods can also be approached using the present test function. Both the moving least square (MLS) and the direct method have been applied to the method to estimate the shape function and to impose the essential boundary conditions. The results of three studied elasto-static cases; “two dimensional cantilever beam”, “first mode fracture of a center-cracked strip” and “edge-cracked functionally graded strip” show that by using less number of nodes, the present work gives sufficiently more accurate results. Meanwhile the method can also unify various kinds of MPLGs and one may conclude that the model is a good replacement for other common approaches.     

The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics

 The meshless local Petrov-Galerkin (MLPG) approach is an effective method for solving boundary value problems, using a local symmetric weak form and shape functions from the moving least squares approximation. In the present paper, the MLPG method for solving problems in elasto-statics is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method, as the essential boundary conditions can not be enforced directly when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present MLPG method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.     

Weight-adaptive isogeometric analysis for solving elastodynamic problems based on space-time discretization approach

In this paper, a novel adaptive isogeometric analysis (IGA) is introduced and its application in the numerical solution of two-dimensional elastodynamic problems based on the space-time discretization (STD) approach is studied. In the STD approach, the time is considered as an additional dimension and is discretized the same as the spatial domain. The weights of control points play the main role in the proposed method. In the conventional IGA, the same set of weights is used in the modeling of geometric and solution spaces. The idea is to define two groups of weights: geometric and solution weights. Geometric weights are known and can be determined based on the position of control points, but the solution weights are considered to be unknown and can be determined using a proper strategy so that the accuracy of the solution is optimized. This strategy is based on the minimization of an error function. The results obtained from the proposed method are compared with those obtained from the conventional IGA.     

A new meshless method based on MLPG for elastic dynamic problems

The basic concept and numerical implementation of a new local Petrov-Galerkin method for solving a dynamic problem are presented in this paper. It uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function, and uses the Heaviside function as a test function of the weighted residual method. The shape function has the Kronecker Delta properties for the trial-function-interpolation, and so no additional treatment to impose essential boundary conditions. The method does not involve any domain and singular integrals to generate the global effective stiffness matrix except for the mass and damping matrice; it only involves a regular boundary integral. It possesses a great flexibility in dealing with the numerical model of the elastic dynamic problem under various boundary conditions with arbitrary shapes. The Newmark family of methods is adopted in computation. The numerical results also show that using a multiquadrics (MQ) function with the polynomial basis function as the interpolation function can give quite accurate numerical results. The a_Q and a_S are investigated which are parameters of the radii of the sub-domain and influence domain, respectively.     

A regularized boundary integral equation method for elastodynamic crack problems

This paper presents a double layer potential approach of elastodynamic BIE crack analysis. Our method regularizes the conventional strongly singular expressions for the traction of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. The manipulation is systematized by the use of the stress function representation of the differentiated double layer kernel functions. This regularization, together with the use of B-spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elastodynamics.     

Meshless local boundary integral equation method for 2D elastodynamic problems

A new meshless method for solving transient elastodynamic boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation (MLS), is proposed in this paper. The LBIE with the MLS is applied to both transient and steady‐state (Laplace transformed) elastodynamics. Applying the MLS approximation for spatially dependent terms in the first approach, the LBIEs are transformed into a system of ordinary differential equations for nodal unknowns. This system of ordinary differential equations is solved by the Houbolt finite difference scheme. In the second formulation, the time variable is eliminated by using the Laplace transformation. Unknown Laplace transforms of displacements and traction vectors are computed from the LBIEs with the MLS approximation. The time‐dependent values are obtained by the Durbin inversion technique. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerically solving non-linear problems by the homotopy analysis method

In this paper, the Homotopy Analysis Method (HAM) proposed by Liao (1992a, 1992b, 1992c, 1992e, 1995a, 1997a) is greatly improved by introducing a nonzero variable ℏ. Based on the HAM, a new numerical approach for strongly non-linear problems is proposed and applied to solve, as an example, a non-linear heat transfer problem, i.e. microwave heating of an unit plate, so as to verify its validity and great potential. Our numerical experiments show that, by the proposed approach, iteration is not absolutely necessary for solving non-linear problems. This fact may deepen our understanding about numerical techniques for non-linear problems and widen our field of vision. Moreover, the basic ideas proposed in this paper may afford us a great possibility to greatly improve our current numerical techniques.     

An adaptive spacetime discontinuous Galerkin method for cohesive models of elastodynamic fracture

This paper describes an adaptive numerical framework for cohesive fracture models based on a spacetime discontinuous Galerkin (SDG) method for elastodynamics with elementwise momentum balance. Discontinuous basis functions and jump conditions written with respect to target traction values simplify the implementation of cohesive traction–separation laws in the SDG framework; no special cohesive elements or other algorithmic devices are required. We use unstructured spacetime grids in a h‐adaptive implementation to adjust simultaneously the spatial and temporal resolutions. Two independent error indicators drive the adaptive refinement. One is a dissipation‐based indicator that controls the accuracy of the solution in the bulk material; the second ensures the accuracy of the discrete rendering of the cohesive law. Applications of the SDG cohesive model to elastodynamic fracture demonstrate the effectiveness of the proposed method and reveal a new solution feature: an unexpected quasi‐singular structure in the velocity response. Numerical examples demonstrate the use of adaptive analysis methods in resolving this structure, as well as its importance in reliable predictions of fracture kinetics. Copyright © 2009 John Wiley & Sons, Ltd.     

The Trefftz method for solving eigenvalue problems

For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.     

Orthogonal meshless finite volume method applied to elastodynamic crack problems

An orthogonal meshless finite volume method has been presented to solve some elastodynamic crack problems. An orthogonal weighted basis function is used to construct shape function so there is no problem of singularity in this new form. In this work, for three-dimensional dynamic fracture problems, a new displacement function is used at the tip of the crack to give a new OMFVM. When the new OMFVM is used, the singularity of the stresses at the tip of the crack can be shown to be better than that in the primal OMFVM. High computational efficiency and precision are other benefits of the method. Solving some sample crack problems of thin-walled structures show a good performance of this method.     

A semi-analytic method for solving rib-type waveguide problems

A new semi-analytic method for solving optical rib-type waveguide problems is presented. In the method, the cross-section of a rib-type waveguide is divided into several regions. In each region, the refractive index profile and field distribution are expanded into Fourier cosine series, and then are substituted in the wave equation. A second-order differential matrix equation is then derived for each region, with a closed-form solution obtainable. With the boundary conditions used, an eigenmode equation for the rib waveguide can be derived and solved numerically to give the modal indices. Here, the presented method is used to deal with two rib waveguides in different geometric dimensions and/or compositions, respectively. Computational results show that the presented method is quite efficient, in terms of CPU time, in finding the modal indices accurately. The relative error in computing the modal index with the method is about 10^??5–10^??6.     

Semi-inverse method for solving circular arc crack problems

A new approach to a certain class of circular arc crack problems is proposed in this paper. The problems are reduced to the problem of linear relationship of the boundary value (the Hilbert problem). By means of the harmonic measure and conformal mapping technique, the closed solutions are given for a plane with a single circular arc crack subjected to the uniform normal and tangential loads on the part near one end of the crack and for the same plane under concentrated normal and tangential loads at an arbitrary point on the arc crack surface. The stress intensity factors at the crack tips under consideration are determined. Some known results for the plane with the central straight crack are shown as the limiting cases of the present solutions.     

A meshless method for solving nonhomogeneous Cauchy problems

In this paper the method of fundamental solutions (MFS) and the method of particular solution (MPS) are combined as a one-stage approach to solve the Cauchy problem for Poisson's equation. The main idea is to approximate the solution of Poisson's equation using a linear combination of fundamental solutions and radial basis functions. As a result, we provide a direct and effective meshless method for solving inverse problems with inhomogeneous terms. Numerical results in 2D and 3D show that our proposed method is effective for Cauchy problems.     

Decomposition method for solving optimal material orientation problems

The strain energy density is considered as a measure of the stiffness/flexibility of the composite structure. A methodology for determining the stationary points of the strain energy density in anisotropic solids is developed. The methodology proposed is based on new problem formulation, derivation and analysis of optimality conditions, and decomposition method. The optimal material orientation problem is formulated in terms of strains. The optimality conditions derived cover different material symmetries, linear and also some non-linear material models. The complexity analysis of the optimality conditions has been performed. The proposed approach allows to divide the solution of the optimal material orientation problem into less complicated subtasks.     

Boundary-integral equation method for elastodynamic scattering by a compact inhomogeneity

The set of singular integral equations which relates unknown fields on the surface of the scatterer to a time-harmonic incident wave is solved by the boundary element method. The general method of solution is discussed in some detail for scattering by an inclusion. Results are presented for a spherical cavity, and for a soft and a stiff spherical inclusion. Fields on the surface of the scatterer are compared with previous results obtained by different methods. Back-scattered and forward-scattered displacement fields are presented, both as a function of position at fixed frequency, and as a function of frequency at fixed position. The quasi-static approximation is briefly discussed.     

Fundamental solutions for variable density two-dimensional elastodynamic problems

Fundamental solutions in the form of free-space Green's functions are developed for a class of two-dimensional, variable density elastodynamic problems. These fundamental solutions are evaluated by means of a coordinate transformation based on conformal mapping in conjunction with wave decomposition, which allows for both vertical and lateral heterogeneities, and can be used within the context of a boundary integral equation formulation analogous to that originally proposed by Cruse and Rizzo (J Math Anal Appl 22 (1968) 244). Finally, a numerical example serves to illustrate the methodology developed herein.