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Proceedings of the 16th International Conference
on Computational and Mathematical Methods
in Science and Engineering, CMMSE 2016
4–8 July, 2016.
Accuracy analysis of a 2D adaptive mesh refinement method
using lid-driven cavity flow and two refinements
Zhenquan Li1and Robert Wood1
1School of Computing and Mathematics, Charles Sturt University
emails: jali@csu.edu.au,rwood@csu.edu.au
Abstract
Locating accurate centres of vortices is one of the accurate measures for computa-
tional methods in fluid flow and the lid-driven cavity flows are widely used as bench-
marks. This paper analyses the accuracy of an adaptive mesh refinement method using
2D steady incompressible lid-driven cavity flows for two refinements. The adaptive
mesh refinement method performs mesh refinement based on the numerical solutions
of Navier-Stokes equations solved by Navier2D, a vertex centred Finite Volume that
uses the median dual mesh to form the Control Volumes (CVs) about each vertex. The
accuracy of the refined meshes is demonstrated by the centres of vortices obtained in the
benchmarks being contained in the twice refined grids. The adaptive mesh refinement
method investigated in this paper is proposed based on the qualitative theory of differ-
ential equations. Theoretically infinite refinements can be performed on an initial mesh.
Practically we can stop the process of refinement based on tolerance conditions. The
method can be applied to find the accurate numerical solutions of any mathematical
models containing continuity equations for incompressible fluid, steady state fluid flows
or mass and heat transfer.
Key words: adaptive mesh refinement, finite volume method, lid-driven cavity flow
1 Introduction
Meshing is the process of breaking up a physical domain into finite smaller sub-domains
(called elements, cells or grids) in order to evaluate the discrete numerical solutions of
differential equations at the nodes. Adaptive mesh refinement is a computational technique
to improve the accuracy of the numerical solutions by starting the calculations on a coarse
initial mesh and then refining this mesh based on refinement criteria.
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CMMSE ISBN: 978-84-608-6082-2
Accuracy analysis of a 2D adaptive mesh refinement
There are a large number of publications on adaptive mesh refinements and their appli-
cations. Some refinement methods use a refinement criterion which is based on local trun-
cation errors (e.g. Almgren, Bell, Colella, Howell & Welcome [1]). Other common methods
include the so-called h-refinement (e.g. Lohner [19]), p-refinement (e.g. Bell, Berger, Saltz-
man & Welcome [3]) or r-refinement (e.g. Miller & Miller [20]), with different combinations
of these also possible (e.g. Capon & Jimack [4]). The overall aim of these adaptive algo-
rithms is to allow a balance to be obtained between accuracy and computational efficiency
in solving differential equations.
We proposed adaptive mesh refinement methods for 2D velocity fields (Li [15]) and
for 3D fields (Li [14]) based on a theorem in qualitative theory of differential equations
(Theorem 1.14, page 18, Ye et al. [22]). The theorem indicates that a divergence free vector
field has no limit cycles or one sided limit cycles, that is, the trajectories (or streamlines)
of divergence free vector fields are closed curves in bounded domains (singular points are
streamlines) that have also been shown by benchmarks (e.g. Erturk et al. [6]). The adaptive
mesh refinement methods adaptively refine meshes based on the information of evaluated
numerical velocity fields to obtain refined meshes on which the linear interpolation of the
numerical velocity fields approximates continuous divergence free vector fields. The area on
which the linear interpolation is not equivalent to a divergence free vector field reducibly
closes to zero when the number of refinements increases.
Identification of accurate locations of singular points and asymptotic lines (planes), and
drawing closed streamlines are some of the accuracy measures for computational methods.
Using numerical velocity fields obtained by taking the vectors of the analytical velocity fields
at nodes of the refined meshes, examples show the accuracy of the adaptive mesh refinement
methods include: locating the singular points and asymptotic lines for 2D [12]; the singular
points and asymptotic plane for 3D [13]; and drawing closed streamlines (Li [12], [13]). We
showed that the once refined mesh for 2D velocity fields provides accurate estimates for
the singular points of 2D steady incompressible lid-driven cavity flows using the numerical
velocity fields (Lal & Li [10]). The numerical velocity fields are obtained by solving the
Navier-Stokes equations with the boundary conditions numerically using a second order
colocated finite volume method (GSFV) with a splitting method for time discretization
(Faure, Laminie & Temam [7]). We applied the adaptive mesh refinement method to the
initial meshes and the numerical velocity fields, and take the centres of refined grids in the
vortex regions as the estimates of the singular points. The comparison of the estimates with
the benchmarks shows that the estimates for the singular points are accurate.
Mesh refinement is necessary for producing accurate numerical solutions. Li [16] con-
siders 2D lid-driven cavity flows using finer meshes 99×99 for Re = 1000, 121×121 for
Re = 2500 and 139×139 for Re = 5000. The results show that the different sizes of vortices
(primary, secondary, tertiary and quaternary vortices) require different densities of mesh
nodes in the separated-flow regions for similar relative errors of centre locations. The same
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CMMSE ISBN: 978-84-608-6082-2
Z. Li and R. Wood
conclusion is derived in Armaly et al. [2]. An investigation starting from coarser initial
meshes and demonstrating the centres of vortices are contained in the refined grids of once
refined meshes has been done (Li et al. [18]).
This paper reports the accuracy analysis of the same adaptive mesh refinement method
for 2D proposed by Li [15] for more refinements using the benchmarks for 2D lid-driven
cavity flows. We show that the centres of vortices obtained in the benchmarks are contained
in the twice refined grids for Re = 100 and Re = 1000. We conclude that more accurate
centres of vortices can be achieved when more refinements are performed.
2 Review of algorithm of adaptive mesh refinement
This section summarizes the adaptive mesh refinement method proposed by Li [15] based
on Theorem 1.14 of [22].
Assume that Vl=AX +Bis a vector field obtained by linearly interpolating the
vectors at the three vertexes of a triangle, where
A=a11 a12
a21 a22
is a matrix of constants,
B=b0
1
b0
2
is a vector of constants, and X= (x1, x2)T. The vector Vlis unique if the area of the triangle
is not zero [11]. The continuity equation for Vland a steady flow or an incompressible fluid
is
∇ · Vl= trace(A)=0.(1)
Let fbe a scalar function depending only on spatial variables. We assume that fVlis
divergence free and then calculate the expressions of f. Li [15] derives the expressions of
ffor the four different Jacobian forms of the coefficient matrix Aas shown in Table 1.
Variables y1and y2in Table 1 are the components of (y1, y2)T=V−1Xwhere Vsatisfies
AV =VJ and Jis one of the Jacobian matrices in Table 1. Vectors Vland fVlproduce
same streamlines if f6= 0,∞(refer to Section 2.2 of [12]). Therefore, if fVlis divergence
free, Vlproduces divergence free streamlines. The functions fare calculated by solving
differential equations [17]. Scalar functions freduce the number of refined grids in refined
meshes [17]. The conditions (MC)(MC is the abbreviation of mass conservation) are the
functions fin Table 1 not equaling zero or infinity at any point on the triangular domains
when fVlis divergence free on these triangular domains.
We review the algorithm of adaptive mesh refinement for quadrilateral meshes [18].
The algorithm is also applicable to a triangular mesh after a subdivision of a triangle to
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Accuracy analysis of a 2D adaptive mesh refinement
Table 1: Jacobian matrices and corresponding expressions of f(C6= 0)
Case Jacobean f
1r10
0r2(0 6=r16=r26= 0) C
y1+b1
r1y2+b2
r2
2r10
0 0(r16= 0) C
y1+b1
r1
3r10
0r1(r16= 0) C
y1+b1
r12
4µ λ
−λ µ(µ6= 0, λ 6= 0) C
y1+µb1−λb2
µ2+λ22+y2+λb1+µb2
µ2+λ22
a number of small triangles is defined. The following grid refinement algorithm describes
how to use the conditions (MC) to refine a quadrilateral grid in a given mesh. To avoid an
infinite refinement of the mesh, we choose a pre-specified threshold number of refinements
Tbased on the accuracy requirements. The algorithm of grid refinement is:
Step 1 Subdivide a quadrilateral grid into two triangles. If Vlsatisfies Eq. (1) on both
triangles, no refinement for the grid is required. Otherwise, go to Step 2;
Step 2 Apply the conditions (MC) to both of the triangles. If the conditions (MC) are
satisfied on both triangles, no refinement for the grid is required. Otherwise, we
subdivide the grid into a number of small grids such that the lengths of all sides of
the small grids are truly reduced (e.g. connecting the mid-points of opposite sides of
a quadrilateral by line segments produces four small quadrilaterals and the lengths of
the sides of the four small quadrilaterals are truly reduced).
In this paper, we subdivide a quadrilateral grid by connecting the mid-points of two
opposite sides of a quadrilateral.
The algorithm of adaptive mesh refinement is:
Step 1 Evaluate the numerical velocity field for a given initial mesh;
Step 2 Refine all grids of the initial mesh one by one using the above algorithm of grid
refinement;
Step 3 Take the refined mesh as initial mesh and go to Step 1 until a satisfactory numerical
velocity field is obtained or the threshold number Tis reached.
The abbreviations BR, BL and TL refer to bottom right, bottom left and top left
corners of the cavity, respectively. The number following these abbreviations refer to the
vortices that appear in the flow, which are numbered according to size (for example, BR1
refers to bottom right secondary vortex).
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Z. Li and R. Wood
3 Accuracy analysis by comparisons with benchmarks
In this section, we use Navier2D, an open source MATLAB CFD Codes by Darren Engwirda
for evaluating the numerical velocity field on triangular meshes [5]. The results reported in
this paper have the residuals for both xand ybeing less than 10−6in the evaluation of the
numerical velocity fields.
One of the possible comparisons is the adaptive mesh refinement which enforces refine-
ment criteria
k∇u(k)k ≤ kuhk1
everywhere in the mesh, where k·k is the L2norm, k·k1is the H1norm, is the discretization
tolerance, uhis finite-dimensional approximation for u, and kin k∇u(k)kis the number of
subdomains (Henderson [9], 293–299). Even though there might be some relations between
the refined meshes and the vorticity field as decreases, no information is provided on the
pattern of the flow field such as locations of the centres of vortices [18].
We take the case for Re = 100 as an example to show how we switch between triangular
meshes to quadrilateral meshes. Fig. 1 is the initial mesh with size 25×25 uniform grids.
The triangulated initial mesh is obtained by connecting bottom left vertex to top right
vertex by a line segment in each grid of the initial mesh and then is loaded it into Navier2D
for the first evaluation of the numerical velocity field.
Fig. 2 shows the once refined mesh using the evaluated numerical velocity field at the
nodes. Fig. 3 shows the triangulated mesh from the once refined mesh shown in Fig. 2.
The triangulated mesh is loaded into Navier2D for the second evaluation of the numerical
velocity field.
We take the results for Re = 100 [8, 21], and the results for Re = 1000 [6] as the bench-
marks for accuracy analysis. We consider the accuracy of the adaptive mesh refinement
method using the inclusion of the centres of vortices identified in the benchmarks in refined
grids.
3.0.1 Re = 100
Fig. 4 shows the twice refined mesh using the second evaluated numerical velocity field at
the nodes shown in Fig. 3 and the centres of vortices (black dots) given by the benchmarks
[8]. All three centres (primary vortex, BL1 and BR1) are contained in the twice refined
grids of the two refined mesh.
3.0.2 Re = 1000
This section shows the figures for Re = 1000 generated from an initial mesh with 35×35
uniform grids. Fig. 5 shows the twice refined mesh and the centres of vortices (black dots)
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Accuracy analysis of a 2D adaptive mesh refinement
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 1: Initial mesh for Re = 100 with size 25×25.
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Z. Li and R. Wood
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 2: Once refined mesh for Re = 100.
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Accuracy analysis of a 2D adaptive mesh refinement
1098 Nodes, 1995 Triangles
Figure 3: Triangulated mesh based on the mesh shown in Fig. 2
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Z. Li and R. Wood
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4: Twice refined mesh for Re = 100 with initial mesh size 25×25.
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Accuracy analysis of a 2D adaptive mesh refinement
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5: Twice refined mesh for Re = 1000 with initial mesh size 35×35.
given by the benchmarks [6]. All five centres (primary vortex, BL1, BR1, BL2 and BR2)
are contained in the twice refined grids.
4 Conclusion
We applied the adaptive mesh refinement method twice to the initial meshes based on
the information of numerical solutions of 2D lid-driven cavity flows using Navier2D. We
demonstrate the accuracy of the adaptive mesh refinement method by the inclusion of the
centres of vortices in twice refined grids of twice refined meshes. If we refine the initial
meshes more times, we obtain more accurate estimates for the centres of vortices. We are
able to achieve the required accuracy for the centres of vortices by selecting an appropriate
threshold number T.
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