Integral Equation Methods for Stokes Flow and Isotropic Elasticity in the Plane

Greengard, Leslie; Kropinski, Mary Catherine A.; Mayo, Anita

doi:10.1006/jcph.1996.0102

Cited by 113 publications

(147 citation statements)

References 10 publications

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“…The ShermanLauricella equation does not satisfy (17,18). It also has three undesirable properties: First, it involves the arbitrary placement of points which affects its stability, see Figure 2 of [8] and [13]. Second, it can not easily be extended to setups involving cracks and inclusions.…”

Section: One Hole In An Infinite Domainmentioning

confidence: 99%

“…where the last term on the left hand side of (57) only is necessary for the special case of an inclusion being a hole, and where the weight ´Þµ of (7,8) has been added as to prepare these equations for the incorporation of cracks, see Section 6. With the help of (14,15,(25)(26)(27)(28)(29)(30)(31)(32)(33)(34) it is easy to verify that the uniqueness conditions (12,40) hold for the solution to (56,57).…”

Section: Several Holes or Inclusions In Infinite And In Finitementioning

confidence: 99%

“…For the large scale computations in Subsection 7.8 we use the fast multipole method [2,10,35], previously used in elastostatic contexts in Refs. [7,8,9,12], and the complexity and the storage requirement is proportional to AE.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Parallel to this development there has been a steady improvement of general purpose finite element packages which, among other things, can solve for stress on multiply connected domains. There has also been progress on fast solvers for large-scale problems [8].…”

Section: Introductionmentioning

confidence: 99%

“…where ª´ µ and ¤´ µ are unknown layer densities on , and where ´ µ is a weight which on contours is given by ´ µ ½ ¾ ¼ ½ AE (7) and on the cracks is given by ´ µ ´´ × µ´ µµ ½ ¾ ¾ AE · ½ AE · AE (8) In (8) the weight ´ µ is the limit from the right (relative to the orientation of the crack) of the branch given by a branch cut along and Ð Ñ ½ ´ µ ½ and ÔÖ ÁÁÁ ´¼ ¼ ½µ. They are obtained by choosing « ½ ¾ and ¬ ½ ¾, « ½ ¾ and ¬ ½ ¾, and « ¼ and ¬ .…”

Section: Introductionmentioning

confidence: 99%

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Stress Calculations on Multiply Connected Domains

Helsing¹,

Jönsson

2002

Journal of Computational Physics

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The outstanding problem of finding a simple Muskhelishvili-type integral equation for stress problems on multiply connected domains is solved. Complex potentials are represented in a way which allows for the incorporation of cracks and inclusions. Several numerical examples demonstrate the generality and extreme stability of the approach. The stress field is resolved with a relative error less than ½¼ ½¼ on a large, yet simply reproducible, setup with a loaded square plate containing 4096 holes and cracks. Comparison with previous results in the literature indicates that general purpose finite element software may perform better than many special purpose codes.

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Section: One Hole In An Infinite Domainmentioning

confidence: 99%

Section: Several Holes or Inclusions In Infinite And In Finitementioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stress Calculations on Multiply Connected Domains

Helsing¹,

Jönsson

2002

Journal of Computational Physics

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Stokes flow around cylinders in a bounded two-dimensional domain using multipole-accelerated boundary element methods

Mammoli

Ingber

1999

Int. J. Numer. Meth. Engng.

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SUMMARYThe multipole technique has recently received attention in the ÿeld of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N 2 ) (iterative solvers) orwhere N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes ow results in Fredholm equations of the second kind, which generally produce a well-posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the ÿrst kind, which, upon discretization, produce illconditioned systems of equations. The model system here is a two-dimensional wide-gap couette viscometer, where particles are suspended in the uid between the cylinders. This is a typical system that is e ciently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is su ciently well-conditioned to allow the use of fast multipole methods. The direct approach results in severe ill-conditioning, to a point where application of the multipole method leads to non-convergence of the solution iteration.

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The general boundary element method and its further generalizations

Liao

1999

Int. J. Numer. Meth. Fluids

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In this paper, the basic ideas of the general boundary element method (BEM) proposed by Liao [in Boundary Elements XVII, Computational Mechanics Publications, Southampton, MA, 1995, pp. 67–74; Int. J. Numer. Methods Fluids, 23, 739–751 (1996), 24, 863–873 (1997); Comput. Mech., 20, 397–406 (1997)] and Liao and Chwang [Int. J. Numer. Methods Fluids, 23, 467–483 (1996)] are further generalized by introducing a non‐zero parameter $\hbar$. Some related mathematical theorems are proposed. This general BEM contains the traditional BEM in logic, but is valid for non‐linear problems, including those whose governing equations and boundary conditions have no linear terms. Furthermore, the general BEM can solve non‐linear differential equations by means of no iterations. This disturbs the absolutely governing place of iterative methodology of the BEM for non‐linear problems. The general BEM can greatly enlarge application areas of the BEM as a kind of numerical technique. Two non‐linear problems are used to illustrate the validity and potential of the further generalized BEM. Copyright © 1999 John Wiley & Sons, Ltd.

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Integral Equation Methods for Stokes Flow and Isotropic Elasticity in the Plane

Cited by 113 publications

References 10 publications

Stress Calculations on Multiply Connected Domains

Stress Calculations on Multiply Connected Domains

Stokes flow around cylinders in a bounded two-dimensional domain using multipole-accelerated boundary element methods

The general boundary element method and its further generalizations

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