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Research Papers: Techniques and Procedures

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

[+] Author and Article Information
Sofia Sarraf

Instituto de Investigación en Tecnologías y
Ciencias de la Ingeniería IITCI (UNCo–CONICET),
Buenos Aires 1400,
Neuquén Q8300IBX, Argentina
e-mail: sofia.sarraf@fain.uncoma.edu.ar

Ezequiel López

Instituto de Investigación en Tecnologías y
Ciencias de la Ingeniería IITCI (UNCo–CONICET),
Buenos Aires 1400,
Neuquén Q8300IBX, Argentina
e-mail: ezequiel.lopez@fain.uncoma.edu.ar

Laura Battaglia

Centro de Investigación de Métodos
Computacionales CIMEC (UNL–CONICET),
Predio CONICET-Santa Fe, Colectora RN 168,
El Pozo, Santa Fe 3000, Argentina;
Facultad Regional Santa Fe (UTN),
Lavaisse 610,
Santa Fe 3000, Argentina
e-mail: lbattaglia@santafe-conicet.gob.ar

Gustavo Ríos Rodríguez

Centro de Investigación de Métodos
Computacionales CIMEC (UNL–CONICET),
Predio CONICET-Santa Fe, Colectora RN 168,
El Pozo, Santa Fe 3000, Argentina
e-mail: gusadrr@santafe-conicet.gov.ar

Jorge D'Elía

Centro de Investigación de Métodos
Computacionales CIMEC (UNL–CONICET),
Predio CONICET-Santa Fe, Colectora RN 168,
El Pozo, Santa Fe 3000, Argentina
e-mail: jdelia@cimec.unl.edu.ar

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 15, 2017; final manuscript received August 7, 2017; published online October 4, 2017. Assoc. Editor: Praveen Ramaprabhu.

J. Fluids Eng 140(1), 011401 (Oct 04, 2017) (10 pages) Paper No: FE-17-1037; doi: 10.1115/1.4037690 History: Received January 15, 2017; Revised August 07, 2017

In the boundary element method (BEM), the Galerkin weighting technique allows to obtain numerical solutions of a boundary integral equation (BIE), giving the Galerkin boundary element method (GBEM). In three-dimensional (3D) spatial domains, the nested double surface integration of GBEM leads to a significantly larger computational time for assembling the linear system than with the standard collocation method. In practice, the computational time is roughly an order of magnitude larger, thus limiting the use of GBEM in 3D engineering problems. The standard approach for reducing the computational time of the linear system assembling is to skip integrations whenever possible. In this work, a modified assembling algorithm for the element matrices in GBEM is proposed for solving integral kernels that depend on the exterior unit normal. This algorithm is based on kernels symmetries at the element level and not on the flow nor in the mesh. It is applied to a BIE that models external creeping flows around 3D closed bodies using second-order kernels, and it is implemented using OpenMP. For these BIEs, the modified algorithm is on average 32% faster than the original one.

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Figures

Grahic Jump Location
Fig. 1

Left: a rigid, closed, piecewise smooth surface A with an exterior domain Ωe: field point x, source point y, relative position r=x−y, unit normals n(x),n(y), and differential areas dAx,dAy. Right: master triangles p and q for the simplex coordinates.

Grahic Jump Location
Fig. 2

The element matrices for the steady Stokes equation in 3D domains using T1 elements (top): (i) double-layer kernel K̃(p,q)(x(ξ),y(η)), symmetric in each 3 × 3 block (bottom left); (ii) single-layer kernel S̃(p,q)(x(ξ),y(η)): symmetric with respect to the main diagonal (bottom right)

Grahic Jump Location
Fig. 3

Steady Stokes flow. Assemblies A0–A2 and CO, as functions of the DOF number M=3N: (i) the reciprocal of the condition numbers of the system matrices rcond (left), (ii) the relative errors |er%| in the Stokes force (middle left), (iii) the wall times required for assemblies A0–A2 and CO (middle right), and (iv) the relative wall times between A1 and A0 assemblies, and between A2 and A1 ones (right).

Grahic Jump Location
Fig. 4

Oscillatory Stokes flow. Assemblies A0–A1 and CO, as functions of the DOF number M=3N: (i) the reciprocal of the condition numbers of the system matrices rcond (left), (ii) the relative errors |er%| in the Stokes force (middle left), (iii) the wall times required for the assemblies A0–A1 and CO (middle left), and (iv) the relative wall times between A1 and A0 assemblies (right).

Grahic Jump Location
Fig. 5

Bodies with edges and corners: an HC (left), an SS (middle), and a PP (right) [30]

Grahic Jump Location
Fig. 6

Steady Stokes flow. Comparison between GBEM with assemblies A0–A2 and CO in the unit sphere and using a Q22 quadrature, as functions of the DOF number M=3N or the number of elements E: (i) the total wall time per DOF (left), (ii) the main memory in Mbytes (middle left), (iii) the reciprocal of the condition numbers of the system matrix rcond (middle right), and (iv) the relative errors |er%| in the Stokes force (right).

Grahic Jump Location
Fig. 7

Oscillatory Stokes flow. Comparison between GBEM with assemblies A0–A1 and CO in the unit sphere and using a Q22 quadrature, as functions of the DOF number M=3N or the number of elements E: (i) the total wall time per DOF (left), (ii) the main memory in Mbytes (middle left), (iii) the reciprocal of the condition numbers of the system matrix rcond (middle right), and (iv) the relative errors |er%| in the Stokes force (right).

Grahic Jump Location
Fig. 8

Relative error |er%| of the Stokes force in the unit sphere case as a function of the total wall time, and using GBEM with assemblies A0–A2 and CO: (i) steady creeping flow (left), and (ii) oscillatory one (right)

Grahic Jump Location
Fig. 9

Rankine closed body. Assemblies A0, A1, and A3 and CO, as functions of the DOF number M = N: (i) the reciprocal of the condition numbers of the system matrix rcond (left), (ii) the mean-squared-error (MSE%), in the dipolar density λ (middle left), (iii) the wall times required for assemblies A0, A1, and A3 and CO (middle right), and (iv) the relative wall times between A1 and A0 assemblies, and between A3 and A1 ones (right).

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