Grid-based volume integration for elasticity: Traction boundary integral equation

Lumsden, Ian; Gray, L. J.; Ye, Wenjing

doi:10.1016/j.engfracmech.2017.02.009

Cited by 2 publications

(3 citation statements)

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“…A regular grid volume integration algorithm for the non-homogeneous 3D Stokes equation has been presented. The key to modifying the original volume integral is to represent the Green's function (Stokeslet) as the Laplacian of a function H. This is analogous to previous volume integral treatments for the Laplace [39,40] and elasticity equations [41,42], as the Laplacian can be viewed the zero-pressure Stokes equations. With the function H, the domain integral exactly transforms to a simple boundary integral, plus a volume term wherein the modified source function is everywhere zero on the boundary.…”

Section: Resultsmentioning

confidence: 99%

“…In the former, the source F is extended by solving an exterior Dirichlet boundary integral equation, with boundary values given by F . For the Stokes algorithm herein, as well as the previous Laplace and elasticity treatments discussed in [39,40,41,42], a function F 0 is defined on Ω by means of an interior boundary integral solution (again with boundary values from F ). With B denoting the covering box, and with the help of Green's Theorem, the modified body force integral becomes…”

Section: Introductionmentioning

confidence: 99%

“…A second key aspect of the work in [39,40,41,42] is the construction (in a simple analytic form) of a function H that satisfied…”

Section: Introductionmentioning

confidence: 99%

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Boundary integral analysis for non-homogeneous, incompressible Stokes flows

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A regular-grid volume-integration algorithm is developed for the non-homogeneous 3D Stokes equation. Based upon the observation that the Stokeslet U is the Laplacian of a function H, the volume integral is reformulated as a simple boundary integral, plus a remainder domain integral. The modified source term in this remainder integral is everywhere zero on the boundary and can therefore be continuously extended as zero to a regular grid covering the domain. The volume integral can then be evaluated on the grid. Applying this method to the Navier-Stokes equations will require obtaining velocity gradients, and thus an efficient algorithm for post-processing these derivatives is also discussed. To validate the numerical implementation, test results employing a linear element Galerkin approximation are presented.

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