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1.
In this paper, a semi-analytical approach for the eigenproblem of circular plates with multiple circular holes is presented.
Natural frequencies and modes are determined by employing the null-field integral formulation in conjunction with degenerate
kernels, tensor rotation and Fourier series. In the proposed approach, all kernel functions are expanded into degenerate (separable)
forms and all boundary densities are represented by using Fourier series. By uniformly collocating points on the real boundary
and taking finite terms of Fourier series, a linear algebraic system can be constructed. The direct searching approach is
adopted to determine the natural frequency through the singular value decomposition (SVD). After determining the unknown Fourier
coefficients, the corresponding mode shape is obtained by using the boundary integral equations for domain points. The result
of the annular plate, as a special case, is compared with the analytical solution to verify the validity of the present method.
For the cases of circular plates with an eccentric hole or multiple circular holes, eigensolutions obtained by the present
method are compared well with those of the existing approximate analytical method or finite element method (ABAQUS). Besides,
the effect of eccentricity of the hole on the natural frequency and mode is also considered. Moreover, the inherent problem
of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress
the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational efficiency are
the main features of the present method thanks to the semi-analytical procedure. 相似文献
2.
Following the success of using the null-field integral approach to determine the torsional rigidity of a circular bar with
circular inhomogeneities (Chen and Lee in Comput Mech 44(2):221–232, 2009), an extension work to an elliptic bar containing
elliptic inhomogeneities is done in this paper. For fully utilizing the elliptic geometry, the fundamental solutions are expanded
into the degenerate form by using the elliptic coordinates. The boundary densities are also expanded by using the Fourier
series. It is found that a Jacobian term may exist in the degenerate kernel, boundary density or boundary contour integral
and cancel out to each other. Null-field points can be exactly collocated on the real boundary free of facing the principal
values using the bump contour approach. After matching the boundary condition, a linear algebraic system is constructed to
determine the unknown coefficients. An example of an elliptic bar with two inhomogeneities under the torsion is given to demonstrate
the validity of the present approach after comparing with available results. 相似文献
3.
A null-field approach is employed to derive the Green's function for boundary value problems stated for the Laplace equation with circular boundaries. The kernel function and boundary density are expanded by using the degenerate kernel and Fourier series, respectively. Series-form Green's function for interior and exterior problems of circular boundary are derived and plotted in a good agreement with the closed-form solution. The Poisson integral formula is extended to an annular case from a circle. Not only an eccentric ring but also a half-plane problem with an aperture are demonstrated to see the validity of the present approach. Besides, a half-plane problem with a circular hole subject to Dirichlet and Robin boundary conditions and a half-plane problem with a circular hole and a semi-circular inclusion are solved. Good agreement is made after comparing with the Melnikov's results. 相似文献
4.
Jeng-Tzong Chen Jia-Wei Lee Chine-Feng Wu Ying-Te Lee 《Engineering Analysis with Boundary Elements》2013,37(7-8):1107-1116
In this paper, the resonance of a circular harbor is studied by using the semi-analytical approach. The method is based on the null-field boundary integral equation method in conjunction with degenerate kernels and the Fourier series. The problem is decomposed into two regions by employing the concept of taking free body. One is a circular harbor, and the other is a problem of half-open sea with a coastline subject to the impermeable (Neumann) boundary condition. It is interesting to find that the SH wave impinging on the hill can be formulated by the same mathematical model. After finding the analogy between the harbor resonance and hill scattering, focusing of the water wave inside the harbor as well as focusing in the hill scattering are also examined. Finally, two numerical examples, circular harbor problems of 60° and 180° opening entrance, are both used to verify the validity of the present formulation. 相似文献
5.
Following the success of the CHEEF method [Chen IL, Chen JT, Kuo SR, Liang MT. A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method. J Acoust Soc Am 2001; 109(3):982–98] and the real-part BEM [Kuo SR, Chen JT, Huang CX. Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real-part dual BEM. Int J Numer Methods Eng 2000; 48:1401–22] for solving the membrane eigenproblem, we extend to the plate problem in this paper. The boundary integral equation method (BIEM) using only the real-part kernel instead of the complex-valued kernel is employed to solve the plate eigenproblem for saving half effort in computation. The spurious eigenvalue that resulted due to insufficient constraint is examined. To deal with this problem, a combined Helmholtz exterior integral equation formulation method (CHEEF) is employed to provide sufficient constraints to filter out spurious eigenvalues. The constraint equations of the transverse displacement, normal derivative and tangent derivative for the exterior collocating points are derived. If these constraint equations are properly chosen, one collocating point was sufficient to filter out all the spurious eigenvalues easily and efficiently, even for the repeated spurious eigenvalues. Finally, numerical experiments are performed to verify the analytical results. 相似文献
6.
A new method for Stokes problems with circular boundaries using degenerate kernel and Fourier series
Jeng‐Tzong Chen Chia‐Chun Hsiao Shyue‐Yuh Leu 《International journal for numerical methods in engineering》2008,74(13):1955-1987
This study is concerned with the Stokes flow of an incompressible fluid of constant density and viscosity with circular boundaries. To fully capture the circular boundary, the boundary densities in the direct and indirect boundary integral equations (BIEs) are expanded in terms of Fourier series. The kernel functions in either the direct BIE or the indirect BIE are expanded to degenerate kernels by using the separation of field and source points. Consequently, the improper integrals are transformed to series sum and are easily calculated. The linear algebraic system can be established by matching the boundary conditions at the collocation points. Then, the unknown Fourier coefficients can be easily determined. Finally, several examples including circular and eccentric domains are presented to demonstrate the validity of the present method. Five gains were obtained: (1) meshless approach; (2) free of boundary‐layer effect; (3) singularity free; (4) exponential convergence; and (5) well‐posed model. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
7.
S. G. Mogilevskaya S. L. Crouch 《International journal for numerical methods in engineering》2001,52(10):1069-1106
The problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
8.
A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann
viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded
elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can,
in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic
plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching
strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included
to demonstrate the accuracy and efficiency of the method. 相似文献
9.
In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral
equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability
of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen.
For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches,
continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically
for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace)
and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations
is also determined.
Tel.: 886-2-2462-2192-ext. 6140 or 6177 相似文献
10.
This paper deals with the interaction problem of a row of elliptical inclusions under uniaxial tension. The body force method
is used to formulate the problem as a system of singular integral equations with Cauchy--type and logarithmic singularities,
where the unknowns are densities of body forces distributed in infinite plates that have the same elastic constants as those
of the matrix and inclusion. In order to satisfy the boundary conditions along the elliptical boundaries, eight kinds of fundamental
density functions, proposed in a previous paper, are applied. In the analysis, the number, shape, and position of inclusions
are varied systematically; after which the magnitude and position of the maximum stress are examined. For any fixed shape
and size of inclusions, the maximum stress is shown to be linear with the reciprocal of the number of inclusions. The present
method is found to yield rapidly converging numerical results for various geometrical conditions of inclusions.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
11.
This paper describes a numerical procedure for solving two-dimensional elastostatics problems with multiple circular holes and elastic inclusions in a finite domain with a circular boundary. The inclusions may have arbitrary elastic properties, different from those of the matrix, and the holes may be traction free or loaded with uniform normal pressure. The loading can be applied on all or part of the finite external boundary. Complex potentials are expressed in the form of integrals of the tractions and displacements on the boundaries. The unknown boundary tractions and displacements are approximated by truncated complex Fourier series. A linear algebraic system is obtained by using Taylor series expansion without boundary discretization. The matrix of the linear system has diagonal submatrices on its diagonal, which allows the system to be effectively solved by using a block Gauss-Seidel iterative algorithm. 相似文献
12.
Elizaveta Gordeliy Sofia G. Mogilevskaya Steven L. Crouch 《International journal for numerical methods in engineering》2009,80(11):1437-1462
A two‐dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non‐perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so‐called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two‐ and three‐dimensional problems demonstrates the effect of the dimensionality. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
13.
Songying Chen Leqin Wang Lei Jiao 《International journal for numerical methods in engineering》2004,59(8):1137-1146
The solutions of the displacement boundary integral equation (BIE) are not uniquely determined in certain types of boundary conditions. Traction boundary integral equations that have unique solutions in these traction and mixed boundary cases are established. For two‐dimensional linear elasticity problems, the divergence‐free property of the traction boundary integral equation is established. By applying Stokes' theorem, unknown tractions or displacements can be reduced to computation of traction integral potential functions at the boundary points. The same is true of the J integral: it is divergence‐free and the evaluation of the J integral can be inverted into the computation of the J integral potential functions at the boundary points of the cracked body. The J integral can be expressed as the linear combination of the tractions and displacements from the traction BIE on the boundary of the cracked body. Numerical integrals are not needed at all. Selected examples are presented to demonstrate the validity of the traction boundary integral and J integral. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
14.
Y.Z. Chen 《Engineering Analysis with Boundary Elements》2013,37(7-8):1089-1094
This paper provides a solution for Eshelby's elastic inclusions in a finite plate based on the complex variable boundary integral equation (CVBIE) method. In the formulation, an inclusion with Eshelby's eigenstrains is embedded in an elliptic plate, and the exterior boundary is applied by some static loading. Two BIEs are suggested in the present study. One of BIEs is used for the finite inclusion region, and the other is used for region bounded by interface and the exterior boundary. After the discretization of BIEs, a numerical solution is suggested. In the solution, an inverse matrix technique is suggested which can eliminate one unknown vector in advance. Three numerical examples under different generalized loadings are provided. Interaction between the prescribed eigenstrains and the static loading along the exterior boundary is studied in detail. 相似文献
15.
A volume integral equation method (VIEM) is used to calculate the elastostatic field in an isotropic elastic half-plane containing circular inclusions subject to remote loading parallel to the traction-free boundary. The material of the inclusions may be either isotropic or anisotropic and they are assumed to be distributed in square or hexagonal array. A detailed analysis of the stress field at the interface between the matrix and one of the inclusions is carried out for different distances between the inclusion and the surface of the half-plane. The results of the calculations are compared with available results. The VIEM is shown to be very accurate and effective for investigating the local stresses in the presence of multiple inclusions. The method can be applied to multiple inclusions of arbitrary geometry and elastic properties embedded in extended isotropic elastic media. 相似文献
16.
A convenient and efficient numerical method is presented for the treatment of Cauchy type singular integral equations of the second kind. The solution is achieved by splitting the Cauchy singular term into two parts, allowing one of the parts to be determined in a closed-form while the other part is evaluated by standard Gauss-Jacobi mechanical quadrature. Since the Cauchy singularity is removed after this manipulation, the quadrature abscissas and weights may be readily available and the placement of the collocation points is flexible in the present method. The method is exact when the unknown function can be expressed as the product of a fundamental function and a polynomial of degree less than the number of the integration points. The proposed algorithm can also be extended to the case where the singularities are complex and is found equally effective. The proposed algorithm is easy to implement and provides a shortcut for programming the numerical solution to the singular integral equation of the second kind. 相似文献
17.
Jose E. Capilla David Pulido‐Velazquez Andrés Sahuquillo Joaquín Andreu 《International journal for numerical methods in engineering》2009,79(2):179-204
A method to solve steady linear groundwater flow problems using generalized Fourier Series is developed and particularized for multiple Fourier series in two‐dimensional domains. It leads to a linear vector equation whose solution provides a finite number of generalized Fourier coefficients approximating the hydraulic head field. Its implementation is shown and two relevant properties are found for the system matrix. It is always symmetric and, once computed, if additional Fourier terms are needed for a better approximation of the hydraulic head field, previously computed matrix elements remain invariant, i.e. only new rows and columns are added to the system matrix. The method is demonstrated in three simple cases with different geometries and transmissivity fields, where solutions are compared with analytical and finite element method results. Thus, the method is verified as an alternative to other flow solvers. Additionally, it provides a direct way to obtain the spectral form of the flow equation solution, given a spectral representation of transmissivity, and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
18.
In this paper, a real-part BEM for solving the eigenfrequencies of plates is proposed for saving half effort in computation instead of using the complex-valued BEM. By employing the real-part fundamental solution, the spurious eigenequations in conjunction with the true eigenequation are obtained for free vibration of plate. To verify this finding, the circulant is adopted to analytically derive the true and spurious eigenequations in the discrete system of a circular plate. In order to obtain the eigenvalues and boundary modes at the same time, the singular value decomposition (SVD) technique is utilized. For the continuous system, mathematical analysis for the spurious eigenequation was done by using the degenerate kernel and Fourier series. Good agreement of the analytical solutions (continuous and discrete systems) is made. Three cases, clamped, simply-supported and free circular plates, are demonstrated analytically and numerically to see the validity of the present method. SVD updating technique is adopted to suppress the ocurrence of the spurious eigenvalues, and a clamped plate is demonstrated analytically for the discrete system in this paper. 相似文献
19.
Thomas Götz 《Journal of Engineering Mathematics》2002,42(2):203-221
A collocation method is presented for solving a singular integral equation of the second kind arsing in the slender-body approximation of viscous flow past slender bodies. When the spectral representation of the integral operator is explicitly known, the collocation method is shown to recover the spectrum of the continuous operator. The approximation error is estimated for two discretizations of the integral operator, and convergence is proved. The collocation scheme is validated for several test cases and extended to situations where the spectrum is not explicitly known. 相似文献
20.
Y. Z. Chen 《International Journal of Fracture》1984,25(3):155-168
Presented is an elementary solution, which is a particular solution of the circular plate containing one crack. The solution consists of two parts and satisfies the following conditions: (i) the first part corresponds to a pair of normal and tangential concentrated forces acting at a prescribed point on both edges of a single crack; (ii) the second part corresponds to some distributed tractions along both edges of the crack; (iii) the obtained elementary solution, i.e. the sum of the first and second parts, satisfies a traction free condition on the circular boundary. Using this elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations of multiple crack problems can always be obtained. The multiple crack problems of an infinite plate containing a circular hole can be solved in a similar way. Several numerical examples are given in this paper. 相似文献