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1.
R. T. Fenner J. O. Watson 《International journal for numerical methods in engineering》1988,26(11):2517-2529
Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two-dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques. 相似文献
2.
The paper deals with the numerical implementation of local integral equations for solution of boundary value problems and
interior computations of displacements and their gradients in functionally graded elastic solids. Two kinds of meshless approximations
and one element based approximation are employed in various formulations. The numerical stability, accuracy, convergence of
accuracy and cost efficiency are investigated in numerous test examples with exact benchmark solutions. 相似文献
3.
J. Sladek V. Sladek R. Van Keer 《International journal for numerical methods in engineering》2003,57(2):235-249
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. 相似文献
4.
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. 相似文献
5.
Vasilis Vavourakis 《Computational Mechanics》2009,44(6):777-790
A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional
steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution
of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm.
A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local
form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles.
Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken
into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions.
On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior
fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’
interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed,
as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically.
The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy
and convergence. 相似文献
6.
A meshless local boundary integral equation (LBIE) method for solving nonlinear problems 总被引:11,自引:0,他引:11
A new meshless method for solving nonlinear boundary value problems, based on the local boundary integral equation (LBIE)
method and the moving least squares approximation, is proposed in the present paper. The total formulation and a rate formulation
are developed for the implementation of the present method. The present method does not need domain and boundary elements
to deal with the volume and boundary integrals, which will cause some difficulties for the conventional boundary element method
(BEM) or the field/boundary element method (FBEM), as the volume integrals are inevitable in dealing with nonlinear boundary
value problems. This is the same for the element free Galerkin (EFG) method which also needs element-like cells in the entire
domain to evaluate volume integrals. The “companion fundamental solution” introduced in Zhu, Zhang and Atluri (1998) is used
so that no derivatives of the shape functions are needed to construct the stiffness matrix for the interior nodes, as well
as for those nodes with no parts of their local boundaries coinciding with the global boundary of the domain of the problem,
where essential boundary conditions are specified. It is shown that the satisfaction of the essential as well as natural boundary
conditions is quite simple, and algorithmically very efficient, in the present nonlinear LBIE approach. Numerical examples
are presented for several problems, for which exact solutions are available. The present method converges fast to the final
solution with reasonably accurate results for both the unknown variable and its derivatives. No post processing procedure
is required to compute the derivatives of the unknown variable (as in the conventional FBEM), since the solution from the
present method, using the moving least squares approximation, is already smooth enough. The numerical results in these examples
show that high rates of convergence for the Sobolev norms ∥·∥0 and ∥·∥1 are achievable, and that the values of the unknown variable and its derivatives are quite accurate. 相似文献
7.
《Engineering Analysis with Boundary Elements》2006,30(8):702-708
Three relative error measurements in the numerical solution of potential problems are firstly investigated in detail, and then an algorithm based on the proposed dual error indicators is developed for the meshless local boundary integral equation (LBIE) method. Numerical experiments show that a combined use of the two error indicators is necessary to adequately measure the error of the LBIE solutions. 相似文献
8.
An efficient numerical method is proposed for 2-d potential problems in anisotropic media with continuously variable material
coefficients. The method is based on the local integral relationships (integral form of balance equation and/or integral equations
utilizing fundamental solutions) and consistent approximation of field variable using standard domain-type elements. The accuracy
and convergence of the proposed method is tested by several examples and compared with benchmark analytical solutions. 相似文献
9.
In this paper, analytical forms of integrals in the meshless local integral equation method in the Laplace space are derived and implemented for elastodynamic problems. The meshless approximation based on the radial basis function (RBF) is employed for implementation of displacements. A weak form of governing equations with a unit test function is transformed into local integral equations. A completed set of the local boundary integrals are obtained in closed form. As the closed form of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically. Several examples including dynamic fracture mechanics problems are presented to demonstrate the accuracy of the proposed method in comparison with analytical solutions and the boundary element method. 相似文献
10.
《Engineering Analysis with Boundary Elements》2006,30(8):650-661
The meshless local Petrov–Galerkin (MLPG) method is used to solve stationary and transient heat conduction inverse problems in 2-D and 3-D axisymmetric bodies. A 3-D axisymmetric body is generated by rotating a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduce the original 3-D boundary value problem to a 2-D problem. The analyzed domain is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIEs) on the boundaries of the chosen subdomains. The time integration schemes are formulated based on the Laplace transform technique and the time difference approach, respectively. The local integral equations are non-singular and take a very simple form. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. Singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions. 相似文献
11.
A boundary integral equation method is applied to the solutions of three dimensional piezoelectric solids. Based on the reciprocal relations, a pair of boundary integral formulae were formulated for evaluation of the fields in the medium. The Green's functions and their first partial derivatives employed in the formulations are evaluated numerically from the line integral solutions derived from the Fourier transform. By constructing some augmented matrices, we show that the topic can be treated systematically as that in the uncoupled elastic and dielectric problems. In illustration, we present results for the internal fields of a spherical cavity in an infinite piezoelectric medium loaded by a uniform traction on its boundary. Two piezoelectric ceramics, PZT-6B and gallium arsenide, are considered in the calculations. Some comparisons are made with solutions of purely elastic solids and with our recent calculations based on the finite element method.This work was supported by the National Science Council, Taiwan, under contract NSC 83-0410-E006-041. 相似文献
12.
A weakly singular, symmetric Galerkin boundary element method (SGBEM) is established to compute stress and electric intensity
factors for isolated cracks in three-dimensional, generally anisotropic, piezoelectric media. The method is based upon a weak-form
integral equation, for the surface traction and the surface electric charge, which is established by means of a systematic
regularization procedure; the integral equation is in a symmetric form and is completely regularized in the sense that its
integrand contains only weakly singular kernels of (hence allowing continuous interpolations to be employed in the numerical approximation). The weakly singular kernels which
appear in the weak-form integral equation are expressed explicitly, for general anisotropy, in terms of a line integral over
a unit circle. In the numerical implementation, a special crack-tip element is adopted to discretize the region near the crack
front while the remainder of the crack surface is discretized by standard continuous elements. The special crack-tip element
allows the relative crack-face displacement and electric potential in the vicinity of the crack front to be captured to high
accuracy (even with relatively large elements), and it has the important feature that the mixed-mode intensity factors can
be directly and independently extracted from the crack front nodal data. To enhance the computational efficiency of the method,
special integration quadratures are adopted to treat both singular and nearly singular integrals, and an interpolation strategy
is developed to approximate the weakly singular kernels. As demonstrated by various numerical examples for both planar and
non-planar fractures, the method gives rise to highly accurate intensity factors with only a weak dependence on mesh refinement. 相似文献
13.
J. O. Adeyeye M. J. M. Bernal K. E. Pitman 《International journal for numerical methods in engineering》1985,21(5):779-787
The eigenvalue problem for the Laplace operator is numerical investigated using the boundary integral equation (BIE) formulation. Three methods of discretization are given and illustrated with numerical examples. 相似文献
14.
Meng-Cheng Chen 《International Journal of Fracture》2003,121(3-4):133-148
This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack. 相似文献
15.
Within the assumptions of linear elastic fracture mechanics, dynamic stresses generated by a crack growth event are examined for the case of an infinite body in the state of plane strain subjected to mode I loading.The method of analysis developed in this paper is based on an integral equation in one spatial coordinate and in time. The kernel of this equation, i.e., the influence or Green's function, is the response of an elastic half-space to a concentrated unit impulse acting on its edge. The unknown function is the normal stress distribution in the plane of the crack, while the free term represents the effect of external loading.The solution for the stresses is obtained with the assumption that its spatial distribution contains a square root singularity near the tip of the crack, while its intensity is an unknown function of time. Thus, the orginal integral equation in space and time reduces to Volterra's integral equation of the first kind in time. The equation is singular, with the singularity of the kernel being a combined effect of the singularity of the influence function and the singularity of the dynamic stresses at the tip of the crack. Its solution is obtained numerically with the aid of a combination of quadrature and product integration methods. The case of a semi-infinite crack moving with a prescribed velocity is examined in detail.The method can be readily extended to problems involving mode II and mixed mode crack propagation as well as to problems of dynamic external loadings. 相似文献
16.
《Engineering Analysis with Boundary Elements》2001,25(2):77-91
In this paper, the conventional boundary integral equation (BIE) formulation for piezoelectric solids is revisited and the related issues are examined. The key relations employed in deriving the piezoelectric BIE, such as the generalized Green's identity (reciprocal work theorem) and integral identities for the piezoelectric fundamental solution, are established rigorously. A weakly singular form of the piezoelectric BIE is derived for the first time using the identities for the fundamental solution, which eliminates the calculation of any singular integrals in the piezoelectric boundary element method (BEM). The crucial question of whether or not the piezoelectric BIE will degenerate when applied to crack and thin shell-like problems is addressed. It is shown analytically that the conventional BIE for piezoelectricity does degenerate for crack problems, but does not degenerate for thin piezoelectric shells. The latter has significant implications in applications of the piezoelectric BIE to the analysis of thin piezoelectric films used widely as sensors and actuators. Numerical tests to show the degeneracy of the piezoelectric BIE for crack problems are presented and one remedy to this degeneracy by using the multi-domain BEM is also demonstrated. 相似文献
17.
《Engineering Analysis with Boundary Elements》2003,27(1):81-90
A new meshless method for solving boundary value problems in micropolar elasticity is presented. The method is based on the local boundary integral equation (LBIE) method with the moving least squares approximation of physical quantities. Randomly scattered nodes are utilized for interpolation of field data. Every node is surrounded by a simple surface centered at the collocation point in the LBIE method. On the surface of subdomains the LBIEs are written. Fundamental solutions corresponding to uncoupled governing equations are derived. To eliminate the traction vector in the LBIE, the modified fundamental solution is introduced. 相似文献
18.
R. J. Gu 《Computational Mechanics》1990,6(3):191-196
A boundary integral equation algorithm for the contact analysis of elastic beams is presented in this paper. The analysis of this sort is complicated by the unknown and moving boundary points. The new algorithm incorporates the interface compatibility equation, derived from the principle of minimum potential energy, into the general boundary integral equation so that the locations of boundary points may be correctly identified. The incremental and iterative solution procedure is presented. Accuracy and efficiency of the new algorithm are demonstrated using examples whose classical solutions exists. 相似文献
19.
Jan Sladek V. Sladek Ch. Zhang P. Solek E. Pan 《International Journal of Fracture》2007,145(4):313-326
A contour integral method is developed for computation of stress intensity and electric intensity factors for cracks in continuously
nonhomogeneous piezoelectric body under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip
vicinity in a continuously nonhomogeneos medium is the same as in a homogeneous one. A meshless method based on the local
Petrov-Galerkin approach is applied for computation of physical fields occurring in the contour integral expressions of intensity
factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations
(LBIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for
approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the stress intensity
factors (SIF) and electrical displacement intensity factors (EDIF) are discussed by comparison with available analytical or
numerical solutions. 相似文献
20.
Xiaogang Zeng Jacobo Bielak 《International journal for numerical methods in engineering》1994,37(5):777-792
This paper is concerned with the development of an exterior domain segmentation method for the solution of two- or three-dimensional time-harmonic scattering problems in acoustic media. This method, based on a variational localized, symmetric, boundary integral equation formulation leads, upon discretization, to a sparse system of algebraic equations whose solution requires only O(N) multiplications, where N is the number of unknown nodal pressures on the scatterer surface. The new procedure is analogous to the one developed recently1 except that in the present formulation we avoid completely the use of the hypersingular operator, thereby reducing the computational complexity. Numerical experiments for a rigid circular cylindrical scatterer subjected to a plane incident wave serve to assess its accuracy for normalized wave numbers, ka, ranging from 0 to 30, both directly on the scatterer and in the far field, and to confirm that, contrary to standard boundary integral equation formulations, the present procedure is valid for critical frequencies. 相似文献