Research Papers: Techniques and Procedures

Effective Geometric Algorithms for Immersed Boundary Method Using Signed Distance Field

[+] Author and Article Information
Chenguang Zhang

Department of Chemical Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mails: zhan21@lsu.edu;

Chunliang Wu

ANSYS, Inc.,
10 Cavendish Court,
Lebanon, NH 03766;
Department of Chemical Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: chunliangwu@gmail.com

Krishnaswamy Nandakumar

Department of Chemical Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: nandakumar@lsu.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received May 4, 2018; final manuscript received October 15, 2018; published online December 10, 2018. Assoc. Editor: Samuel Paolucci.

J. Fluids Eng 141(6), 061401 (Dec 10, 2018) (13 pages) Paper No: FE-18-1321; doi: 10.1115/1.4041758 History: Received May 04, 2018; Revised October 15, 2018

We present three algorithms for robust and efficient geometric calculations in the context of immersed boundary method (IBM), including classification of mesh cells as inside/outside of a closed surface, projection of points onto a surface, and accurate calculation of the solid volume fraction field created by a closed surface overlapping with a background Cartesian mesh. The algorithms use the signed distance field (SDF) to represent the surface and remove the intersection tests, which are usually required by other algorithms developed before, no matter the surface is described in analytic or discrete form. The errors of the algorithms are analyzed. We also develop an approximate method on efficient SDF field calculation for complex geometries. We demonstrate how the algorithms can be implemented within the framework of IBM with a volume-average discrete-forcing scheme and applied to simulate fluid–structure interaction problems.

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Grahic Jump Location
Fig. 1

Summary of the immersed boundary scheme

Grahic Jump Location
Fig. 2

Example SDFs of (a) a unit circle, (b) a 3×1 rectangle, (c) an equilateral triangle of height 2, and (d) the union of (a) and (b). The geometries are shown in black curves. The axis range is [−2,2]×[−2,2], and the color range is clipped to [−1,1].

Grahic Jump Location
Fig. 3

Projection of cell centroid onto the surface (in gray). The uniform parallel lines mark SDF contours.

Grahic Jump Location
Fig. 4

The left shows the three base cases of line-square intersections when the square's bottom left vertex is within the surface. The bottom two cases are similar and considered as the same case. The right shows an enlarged view of case A, with detailed annotations for discussion in the text.

Grahic Jump Location
Fig. 5

Two-dimensional illustration of the pyramid decomposition method for volume calculation, as well as the calculation of pyramid top vertex p from diagonals

Grahic Jump Location
Fig. 6

Approximate SDF of an implicit analytic surface f(r)=0

Grahic Jump Location
Fig. 7

(a) The SDF of a two-dimensional propeller, the thick purple line marks the propeller, and (b) the adaptive quad-tree used to represent the SDF

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Fig. 8

Illustration of the error of geometric calculations caused by nonzero curvature

Grahic Jump Location
Fig. 9

A cell intersected by a sharp corner. The dashed lines represent a mesh refinement operation that subdivides this cell into four smaller cells.

Grahic Jump Location
Fig. 10

(a) the relative error in the calculated circle area versus the mesh resolution, expressed by the number of mesh cells (size δ) across the circle diameter (D=2). (b) the relative error in the calculated bump area versus the bump amplitude A. (c) the relative error in the calculated three-dimensional ellipsoid volume versus the mesh resolution, expressed by the number of cells (size δ) across the longest principal axial length (2c=2). Each panel shows the results from both the step and the present algorithm. The dashed line in (a) and (c) shows second-order convergence. The inset of each panel illustrates the problem.

Grahic Jump Location
Fig. 11

Vertical position (a) and velocity (b) of the settling sphere plotted directly on the figure from Ref. [51]. The symbols are the experimental measurement. The dashed/solid lines are simulation results using the step/present method, respectively.

Grahic Jump Location
Fig. 12

(a) an instantaneous flow field of the entire domain, visualized by line integral convolution (LIC). (b) the object's trajectory, and the definition of drift angle α. The red dot here and in (c) marks the beginning of the dynamic steady-state. The dashed line is a linear fitting of the trajectory. (c) time series of the tilt angle θ, the dashed line is the time average of θ over the period of the dynamic steady-state. (d) an expanded view around the object. The object is marked by the ε=0.5 contour.

Grahic Jump Location
Fig. 13

Angular velocity ωi of the inner cylinder. The dots are the finite difference solution, and the line is the immersed boundary solution. The horizontal dashed line plots ωi=1, which corresponds to the steady-state where the inner cylinder and the fluid rotate together as a rigid body.

Grahic Jump Location
Fig. 14

An instantaneous flow field for the two-dimensional propeller simulation, inlet velocity is U0=4. The flow field is visualized by LIC. The color encodes the magnitude of the flow velocity.

Grahic Jump Location
Fig. 15

Time series of the rotating frequency fR of the propeller. Each curve corresponds to an U0 value, with U0=(1.0,2.0,2.5,3.0,3.2,3.6,4.0) from bottom to top.

Grahic Jump Location
Fig. 16

The dependence of fR¯, A, and f on Reynolds number. The “+”in (a) are 1/4 of the data in (c). The dashed line in (a) is the estimation fR¯=U0/6π (see text).

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Fig. 17

The trajectories of the spherical particles together with snapshots of their position (a), and time series of their vertical position (b) and velocity (c). The blue/red curves in (b) and (c) correspond to the blue/red spheres in (a), respectively.



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