Three-dimensional flow through large numbers of spheroidal inhomogeneities

Janković, I.; Barnes, Randal J.

doi:10.1016/s0022-1694(99)00141-9

Cited by 71 publications

(47 citation statements)

References 3 publications

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“…This is the reason why R = ρ is not included in the expression of degenerate kernels for T (s, x) and L(s, x) in Eqs. (12) and (13).…”

Section: Degenerate Kernels For Fundamental Solutionsmentioning

confidence: 99%

“…Temperature in case of steady state heat conduction [1][2][3][4][5][6], electrostatic potential [7][8][9][10][11], velocity potential in a steady flow of an ideal fluid [12][13][14], the displacement of an infinite medium under remote uniformly shear [15][16][17][18][19], and the pure torsion of an elastic bar by equilibrated end torques [20,21] are examples in which the Laplace equation is satisfied. Circular geometries often appear in engineering structures.…”

Section: Introductionmentioning

confidence: 99%

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Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

Chen

Shen

2008

Numerical Methods Partial

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In this article, a semi-analytical method for solving the Laplace problems with circular boundaries using the null-field integral equation is proposed. The main gain of using the degenerate kernels is to avoid calculating the principal values. To fully utilize the geometry of circular boundary, degenerate kernels for the fundamental solution and Fourier series for boundary densities are incorporated into the null-field integral equation. An adaptive observer system is considered to fully employ the property of degenerate kernels in the polar coordinates. A linear algebraic system is obtained without boundary discretization. By matching the boundary condition, the unknown coefficients can be determined. The present method can be seen as one kind of semianalytical approaches since error only attributes to the truncated Fourier series. For the eccentric case, vector decomposition technique for the normal and tangential directions is carefully considered in implementing the hypersingular equation in mathematical essence although we transform it to summability to divergent series. The five advantages, well-posed linear algebraic system, principal value free, elimination of boundary-layer effect, exponential convergence, and mesh free, are achieved. Several examples involving infinite, half-plane, and bounded domains with circular boundaries are given to demonstrate the validity of the proposed method.

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“…This is the reason why R = ρ is not included in the expression of degenerate kernels for T (s, x) and L(s, x) in Eqs. (12) and (13).…”

Section: Degenerate Kernels For Fundamental Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

Chen

Shen

2008

Numerical Methods Partial

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“…The number of equations N eqn (associated with the number of control points) is larger than the number of unknowns N ukn (associated with the number of Fourier terms and Chebyshev polynomials). The ratio between N eqn and N ukn is called the overspecification fold [18,19]. For simplicity, the control points are distributed uniformly along the element.…”

Section: Reduction To a Linear Algebraic Systemmentioning

confidence: 99%

“…is defined by Eq (19). by replacing τ with ζ , and ζ can be point on the boundary of any hole, or any straight line.…”

mentioning

confidence: 99%

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

2007

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This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov-Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values.Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (AN-SYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori-Tanaka schemes.

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“…Most analytical and semianalytical (e.g., analytic element) solution approaches require discretization of the problem domain into zones of piecewise constant hydraulic conductivity. Notable examples include the solutions of Obdam and Veling [1987], Janković and Barnes [1999a], and Barnes and Janković [1999]. A number of special closed-form solutions, including flow through circular and elliptical inclusions, are documented in Strack [1989] and there are a number of one-dimensional solutions available for both steady state and transient flow in piecewise-homogeneous domains [e.g., Yeh and Kuo, 2010;Liang and Zhang, 2013;Hu and Jiao, 2014].…”

Section: Introductionmentioning

confidence: 99%

A general analytical solution for steady flow in heterogeneous porous media

Craig

2015

Water Resources Research

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A novel analytical solution approach for problems of steady flow in two-dimensional heterogeneous porous media is presented, where the hydraulic conductivity may be represented as an arbitrary polynomial in space. The solution approach uses Wirtinger calculus and the Bers-Vekua theory of elliptical functions. The final form of the solution comprises an arbitrary complex polynomial solution to the Laplace equation and additional nonholomorphic terms which are determined directly from the coefficients of this polynomial. The arbitrary polynomial coefficients may be chosen to satisfy general flow conditions along system boundaries. The approach is also extended to singular flow, such as that induced by pumping wells. The solution is demonstrated to be effectively exact for a number of test cases; the problems are solved to machine precision.

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Three-dimensional flow through large numbers of spheroidal inhomogeneities

Cited by 71 publications

References 3 publications

Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

A general analytical solution for steady flow in heterogeneous porous media

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