Galerkin Boundary Elements for a Computation of the Surface Tractions in Exterior Stokes Flows

Abstract

In the computation of a three–dimensional steady creeping flow around a rigid body, the total body force and torque are well predicted using a boundary integral equation (BIE) with a single concentrated pair Stokeslet- Rotlet located at an interior point of the body. However, the distribution of surface tractions are seldom considered. Then, a completed indirect velocity BIE of Fredholm type and second-kind is employed for the computation of the pointwise tractions, and it is numerically solved by using either collocation or Galerkin weighting procedures over flat triangles. In the Galerkin case, a full numerical quadrature is proposed in order to handle the weak singularity of the tensor kernels, which is an extension for fluid engineering of a general framework (Taylor, 2003, “Accurate and Efficient Numerical Integration of Weakly Singulars Integrals in Galerkin EFIE Solutions,” IEEE Trans. on Antennas and Propag., 51(7), pp. 1630–1637). Several numerical simulations of steady creeping flow around closed bodies are presented, where results compare well with semianalytical and finite-element solutions, showing the ability of the method for obtaining the viscous drag and capturing the singular behavior of the surface tractions close to edges and corners. Also, deliberately intricate geometries are considered.

An Improved Assembling Algorithm in Boundary Elements With Galerkin Weighting Applied to Three-Dimensional Stokes Flows

“…In the first and second examples, the wall times for the three assembling techniques are compared as functions of the degrees of freedom number M. The remaining examples are introduced to show the benefits of applying the assembling strategies in problems with intricate geometries. The numerical solutions computed with GBEM are validated with other numerical methods, as in [23]. In all the examples, a Q 22 quadrature rule is employed, where the first subindex denotes the number of Gauss-Legendre (GL) quadrature points in each surface coordinate used for the self-integral and for the first layer of neighboring elements, and the second subindex is the number of GL quadrature points used for the remaining layers [23].…”

“…The numerical solutions computed with GBEM are validated with other numerical methods, as in [23]. In all the examples, a Q 22 quadrature rule is employed, where the first subindex denotes the number of Gauss-Legendre (GL) quadrature points in each surface coordinate used for the self-integral and for the first layer of neighboring elements, and the second subindex is the number of GL quadrature points used for the remaining layers [23]. The examples are computed on an I7-3930K processor with six cores, using real or complex double precision, GFortran compiler, and main optimization flags -Ofast -march=native -fwhole-file -fwhole-program -Warray-temporaries.…”

“…This is a classical test in the literature about Stokes flow simulation [1], and the numerical results computed with the GBEM technique, e.g. the traction field on the sphere surface, were validated in [23]. The absolute value of the relative error |e r | for the Stokes force is computed as |e r | = |F num /F analytic − 1|, with F analytic = F S and F S = 6πµUR is the (steady) Stokes force.…”

Validation of a Galerkin technique on a boundary integral equation for creeping flow around a torus

“…At the discrete level, nodal values and elements are denoted with super-and subscripts, respectively. Using matrix notation and reordering terms, it is obtained (D'Elía et al 2012)…”

“…In previous works (D'Elía et al 2008(D'Elía et al , 2012, the CIV-BIE alternative in the Hebeker (1986) scheme was chosen and solved with a BEM based on collocation and Galerkin weighting procedures, where the validation test case was restricted to the unit sphere under three inflow types. In the present work, another validation, in particular of the Galerkin technique, is performed through the axisymmetric and steady creeping flow around a three-dimensional torus.…”