Research Papers

Implementation and Assessment of a Residual-Based r-Adaptation Technique on Structured Meshes

[+] Author and Article Information
Aniruddha Choudhary

Department of Aerospace and
Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: anirudd@vt.edu

William C. Tyson, Christopher J. Roy

Department of Aerospace and
Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24061

1Corresponding author.

Manuscript received November 29, 2018; final manuscript received April 29, 2019; published online May 13, 2019. Assoc. Editor: Kevin Dowding.

J. Verif. Valid. Uncert 3(4), 041005 (May 13, 2019) (17 pages) Paper No: VVUQ-18-1036; doi: 10.1115/1.4043652 History: Received November 29, 2018; Revised April 29, 2019

In this study, an r-adaptation technique for mesh adaptation is employed for reducing the solution discretization error, which is the error introduced due to spatial and temporal discretization of the continuous governing equations in numerical simulations. In r-adaptation, mesh modification is achieved by relocating the mesh nodes from one region to another without introducing additional nodes. Truncation error (TE) or the discrete residual is the difference between the continuous and discrete form of the governing equations. Based upon the knowledge that the discrete residual acts as the source of the discretization error in the domain, this study uses discrete residual as the adaptation driver. The r-adaptation technique employed here uses structured meshes and is verified using a series of one-dimensional (1D) and two-dimensional (2D) benchmark problems for which exact solutions are readily available. These benchmark problems include 1D Burgers equation, quasi-1D nozzle flow, 2D compression/expansion turns, and 2D incompressible flow past a Karman–Trefftz airfoil. The effectiveness of the proposed technique is evident for these problems where approximately an order of magnitude reduction in discretization error (when compared with uniform mesh results) is achieved. For all problems, mesh modification is compared using different schemes from literature including an adaptive Poisson grid generator (APGG), a variational grid generator (VGG), a scheme based on a center of mass (COM) analogy, and a scheme based on deforming maps. In addition, several challenges in applying the proposed technique to real-world problems are outlined.

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Fig. 1

Elemental step for 1D equidistribution of a function, W(x), using mesh node relocation

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Fig. 2

Local cell cluster (reproduced with permission from K.R. Laflin [11])

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Fig. 3

Exact solutions for 1D test cases. (a) 1D Burgers equation (Re = 128), and ((b)–(d)) quasi-1D nozzle flow.

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Fig. 4

Adaptation for 1D Burgers equation with Re = 128 and N =65 using different adaptation drivers. (a) TE (enlarged view: −2 ≤ x 2), (b) DE (enlarged view: −2 ≤ x 2), and (c) node spacing on adapted meshes (enlarged view: −2 ≤ x 2).

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Fig. 5

Adaptation for quasi-1D nozzle flow with N =65 nodes using TE as the adaptation driver. (a) Node spacing on the adapted mesh, (b) TE on uniform versus adapted mesh, and (c) DE (in %) obtained on uniform versus adapted mesh.

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Fig. 6

Driver and mesh size comparisons for quasi-1D nozzle flow. (a) Node spacing on adapted meshes obtained using combined TE versus TE from individual governing equations, and (b) adapted meshes for different mesh sizes using combined TE as driver.

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Fig. 7

Expansion fan case description. (a) Computational domain and boundary conditions, and (b) Mach contours for exact solution.

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Fig. 8

Density DE at expansion fan root (N =65 × 65). (a) APGG, (b) VGG, (c) COM, (d) deformation schemes, and (e) uniform grid.

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Fig. 9

DE distribution and grid size effects. (a) Density DE along x =0.5 slice (N =65 × 65). (b) Convergence of pressure DE.

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Fig. 10

Oblique shock case description. (a) Computational domain and boundary conditions, and (b) Mach contours for exact solution.

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Fig. 11

Adapted grids and pressure DE for oblique shock case (N =129 × 65). Full domain grid (left) and enlarged view of shock root (right): ((a) and (b)) uniform grid, ((c) and (d)) APGG scheme, and ((e) and (f)) COM scheme.

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Fig. 12

Convergence of pressure DE with grid refinement

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Fig. 13

Karman–Trefftz airfoil mapping

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Fig. 14

Karman–Trefftz airfoil case description. (a) Computational domain and boundary conditions, and (b) Mach contours for exact solution.

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Fig. 15

Adapted grids and u-velocity DE for Karman–Trefftz airfoil case (N =129 × 65). Leading edge (left) and trailing edge (right): ((a) and (b)) uniform grid, ((c) and (d)) APGG scheme, and ((e) and (f)) COM scheme.

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Fig. 16

Convergence of pressure DE with grid refinement for Karman–Trefftz airfoil case



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