Accuracy Variations in Residual Distribution and Finite Volume Methods on Triangular Grids

Abstract

This paper presents an analytical and numerical approach in studying accuracy deterioration of residual distribution and cell-vertex finite volume methods on triangular grids. Results herein demonstrate that both methods preserve the order-of-accuracy reasonably well for uniformly skewed triangular grids and the $L_2$ errors of both second-order accurate methods behave similarly with values of the same magnitude. On the other hand, the first-order finite volume method has an $L_2$ error of about an order of magnitude higher than its residual distribution counterpart. Both first-order methods are unable to preserve the order-of-accuracy for high-frequency data when the grids are highly skewed although the residual distribution approach has a slightly better performance. Both second-order methods perform quite decently for high-frequency data on uniformly skewed grids. However, the order-of-accuracy of finite volume methods excessively deteriorate when the grids are skewed non-uniformly unlike the residual distribution methods which preserve the order-of-accuracy.

Appendix: Triangle Skewness Control

Consider an arbitrary triangle for analyzing the skewness, aspect ratio and the relation between these two with the point coordinates of the triangle.

Fig. 23

Prototype triangle for analyzing the skewness and aspect ratio

Since the scaling and rotation of the whole triangle do not affect the skewness and aspect ratio, in this analysis the longest length of the triangle considered 1 (Fig. 23). The shaded part would be the domain of (x, y) for a general triangle; because, firstly, if $x<\frac{1}{2}$ or $y<0$, that triangle could be replaced with the completely similar one in which (x, y) are in the shaded-domain by flipping. Second, if the $x^2+y^2>1$, then there will be a similar triangle such that the longest edge will be scaled into 1. Consequently, by analyzing this prototype the relation between skewness and aspect ratio with (x, y) will be determined.

Consider,

$$\begin{aligned} x^2+y^2\le 1,\quad x\ge \frac{1}{2},\quad y\ge 0 \end{aligned}$$

(48)

Since the longest edge of the triangle is 1, the aspect ratio and skewness will be

$$\begin{aligned} \text {Aspect Ratio}\,(\text {AR})=\frac{4A}{\sqrt{3}}\frac{1}{\max \left( l_i^2\right) },\quad \text {Skewness}\,(Q)=\max \left( \frac{\left| 90-\alpha _i\right| }{90}\right) \end{aligned}$$

(49)

By some calculation on the triangular element, the skewness and aspect ratio can be determined as a function of (x, y),

$$\begin{aligned} \text {AR}=\frac{2y}{\sqrt{3}},\quad Q=\frac{2}{\pi }\arccos \left( \frac{y}{\frac{1}{2}\left( \sqrt{x^2+y^2}\right) \left( \sqrt{\left( x-2\right) ^2+y^2}\right) } \right) \end{aligned}$$

(50)

1.1 Right Triangle

To control a triangular grid in a rectangular domain, the right running triangle is chosen to fit inside the domain (Fig. 24). Because, all the elements will have the same amount of skewness or aspect ratio. Recall the previously skewness (Eq. 50); assume $C=\sec ^2\left( \frac{\pi }{2}Q\right)$,

$$\begin{aligned} 4Cy^2=\left( x^2+y^2\right) \left( \left( x-2\right) ^2+y^2\right) \end{aligned}$$

(51)

Fig. 24

Right-running triangle element for analyzing the skewness and aspect ratio

Using the Pythagorean theorem, one could conclude that

$$\begin{aligned} \left( x^2+y^2\right) +\left( \left( x-1\right) ^2+y^2\right) =1\quad \rightarrow \quad x^2+y^2=x \end{aligned}$$

(52)

Combining the two above equations,

$$\begin{aligned} 4C\left( x-x^2\right) =x\left( x-4x+4\right) \quad \rightarrow \quad x=\frac{4C-4}{4C-3} \end{aligned}$$

(53)

Since $\frac{1}{2}\le x\le 1$, the solutions of the inequality are $-8\le -6$ and $C\ge \frac{5}{4}$. The first one is trivial; however, while $C\ge 1$ in general; meaning that, all the values for the skewness, [0, 1], could not be taken. Recall the relation between C and skewness which is $C=\sec ^2\left( \frac{\pi }{2}Q\right)$, the minimum skewness that will be covered by a right triangle will be $C=\frac{5}{4}$ or $Q\sim 0.2952$. In this point of view, for the best skewness which is approximately 0.2952; the element is an isosceles right triangle.

The last step will be finding the relation between stretching parameter $s=\frac{k}{h}$ and the skewness Q.

Recall, $x=\frac{4C-4}{4C-3}$ for a right triangle element one could conclude $y=\sqrt{\frac{4C-4}{(4C-3)^2}}$. Consider, $x^2+y^2=k^2=h^2s^2$ and $(x-1)^2+y^2=h^2$; therefore,

$$\begin{aligned} s=\sqrt{\frac{x^2+y^2}{(x-1)^2+y^2}}=2\sqrt{C-1}=2\tan \left( \frac{\pi }{2}Q\right) \end{aligned}$$

(54)

which clearly is the relation between skewness Q and stretching parameter s.

Fig. 25

Arbitrary triangle element for analyzing the skewness and aspect ratio

1.2 Arbitrary Triangle

In order to control the skewness for an arbitrary triangle element (Fig. 25), the following steps should be performed.

• 1. Choose skewness and number of the points in x-direction.

• 2. Choose aspect ratio based on the skewness: For each skewness the aspect ration could vary between minimum and maximum values which are,

  $$\begin{aligned} \frac{y_\text {min}}{\frac{\sqrt{3}}{2}}\le \text {AR}\le \frac{y_\text {max}}{\frac{\sqrt{3}}{2}} \end{aligned}$$

  (55)

  where,

  $$\begin{aligned} y_\text {min}=\sqrt{C-\frac{1}{4}}-\sqrt{C-1},\quad y_\text {max}=\frac{1}{2C}\left( \sqrt{4C-1}-\sqrt{C-1}\right) \end{aligned}$$

  (56)

  and,

  $$\begin{aligned} c=\sec ^2\left( \frac{\pi }{2}Q\right) \end{aligned}$$

  (57)

  Note that the minimum aspect ratio will construct an isosceles element.

• 3. Find the x coordinate of the prototype triangle shown in Fig. 23 based on the chosen skewness and aspect ratio.

  $$\begin{aligned} x=1-\sqrt{C-\left( \frac{\sqrt{3}}{2}\text {AR}+\sqrt{C-1}\right) } \end{aligned}$$

  (58)

• 4. Find the largest length of the triangle element based on the aspect ratio and number of the points.

  $$\begin{aligned} k=\frac{h}{\frac{\sqrt{3}}{2}\text {AR}} \end{aligned}$$

  (59)

• 5. Find all the points coordinates.

  $$\begin{aligned} k_1=(1-x)k,\quad k_2=xk \end{aligned}$$

  (60)

1.3 Non-uniform Anisotropic Grids

After generating a grid, it will be randomized(disturbed) in a way that different quality of the grids in terms of skewness could be built. This is the place that we could check the solidness and ability of a numerical method during changes in skewness within the domain. It should be mentioned that each randomization constructs a different skewness distribution.

Fig. 26

The randomize grid element area with radius of the minimum distance of the each point from the surrounding edges

Fig. 27

Skewness (Q) distribution based on the number of cells for randomized grid in 10,000 total cells

Fig. 28

Randomized grid. a 50 %. b 80 %

According to Fig. 26, each point will move in fully randomize direction with a finite maximum distance (R) which avoids grid overlapping.

• The disturbed percentage: the maximum distance that a point can move from its original place is R which we can be controlled in terms of percentage defined as $\alpha \times R$. A suitable value for $\alpha$ is chosen to implement grid irregularity. Larger values of $\alpha$ denote a higher percentage grid randomization.

• Disturb number: To build a much more realistic unstructured grid one could perform the whole process (n) times, to build even more randomized grid.

The two options above might be written as $(\alpha ,n)$. It should be mentioned that in this study, we are using two different combination of grid disturbance to cover the possibilities in engineering problems which are (50 %, 5) and (80 %, 8). The distribution of skewness for two different sets are demonstrated in Fig. 27; moreover, the grid itself is shown in Fig. 28.