Skewness is the Angular Measure of Element quality with respect to the Angles of Ideal Element Types.
It is one of the Primary Qualities Measures of FE Mesh. Skewness determines how close to ideal (i.e., equilateral or equi-angular) a face or cell is.
There are two different methods for calculating Skewness for 2D elements.
Method-1: Calculation of Skewness for Triangular/Quadrilateral Elements (Angular Measure)
Draw a line from each node to mid point of its opposite side, Draw another line joining mid-points of other two sides measure the angles between two lines. Repeat the step for all the three nodes and find all six angles (Θ1 to Θ6).
Skewness is calculated by subtracting Minimum angle from 90 degree.
Skewness of an Equilateral Triangle:
For Equilateral Triangle, Θ1, Θ2….. Θ6 = 90 degree
Hence, Skewness = 0 degree
Skewness= 90-min(Θ1, Θ2….. Θ6)
Draw the lines joining the mid points of opposite sides and Measure the Angle between these two lines (Θ1 & Θ2).
Skewness is calculated by Subtracting the Minimum angles of Θ1 & Θ2 from 90 degrees.
Skewness for Square Shaped Element:
For Square Θ1 & Θ2 = 90 degree
Hence, Skewness for Square = 0 degree
Skewness = 90-min(Θ1 , Θ2)
Note: The acceptable Range of skewness is " 0 ̊ to 45 ̊ " beyond which results may to be close to the actual solution.
Method-2: Calculation of Skewness for Triangular/Quadrilateral Elements (Normalized angle deviation )
In the normalized angle deviation method, skewness is defined (in general) as "maximum of ratio of Angular deviation from Ideal element.
θmax = Largest Angle in the face or cell
θmin = Smallest Angle in the face or cell
θe = Angle for an equi-angular face or cell
where, θe = 60 ̊ for Equilateral Triangle and 90 ̊ for Square
θmax = 110 ̊, θmin =30 ̊, θe = 60 ̊
Skewness for Tria= max(0.42,0.50)= 0.50
θmax = 145 ̊, θmin =44 ̊, θe = 90 ̊
Skewness for Quad= max(0.61,0.51)= 0.61
Note: The acceptable Range of skewness is "0 to 0.5" beyond which results may to be close to the actual solution.
Finite Element Analysis For Design Engineers by Paul M. Kurowski (Chapter 5.3.1).
Practical Finite Element Analysis by Nitin S Gokhale, Sanjay S Deshpande, Sanjeev V Bedekar and Anand N Thite (Chapter 7.9).
The Finite Element Method : Practical Course by G. R. Liu , S. S. Quek (Chapter 11.4.2).
Ansys theory reference manual.
Hypermesh users guide.
Note (**): Ansys, Hypermesh are Registered trademarks of their respective owners.