Introduction:
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The ratio between largest and smallest characteristic dimension of an element is known as the Aspect Ratio.
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Large aspect ratios increase the inaccuracy of the finite element representation and have a detrimental effect on convergence of Finite Element Solutions.
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An aspect ratio of 1 is ideal but cannot always be maintained. In general the aspect ratios are maintained in between 1 to 5 at critical areas in a domain where derivatives of field variable are significant.
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Along with solution accuracy the poor element shapes will often cause convergence problems in nonlinear analyses.
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Aspect ratio is one of the other mesh quality parameters like Skewness, Warping, Parallel deviation, Maximum corner angle, Jacobian and Orthogonal Quality.
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Maintaining element quality is always a challenge for analyst during mesh convergence studies. Though element size is minimized it is of no use if they are violating quality requirements.
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So, it is important to know how to calculate these parameters for different Element Shapes.
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This Presentation deals with how to calculate Aspect ratio for 2D elements.
Example Mesh with Aspect Ratios (A.R):
Quad elements with Aspect Ratio: 1
Quad elements with Aspect Ratio: 2
Ideal Shapes for Triangular and Quadrilateral Elements
Following procedures illustrate how Ansys* Software calculates Aspect Ratios for Triangular and Quadrilateral elements.
1. Aspect Ratio Calculation for Triangular Elements:
Finding The Length Ratio of Rectangle-1 Formed with L1 as Axis:
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From the triangle "ABC" join the mid point and opposite corner of each side (L1). Join two mid points of other adjacent sides (L2).
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Fixing L1 as axis, draw parallel lines to L1 at mid points of the adjacent sides at E & F.
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Draw perpendicular lines to L1 at A & D.
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Find the Length Ratio between the Max. Side to Mini. side of the Rectangle-1 Formed using L1 (The procedure is illustrated in the following figure).
Triangular Element with L1 & L2
Rectangle-1 formed with L1
Finding The Length Ratio of Rectangle-2 Formed with L2 as Axis:
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Similarly, Fixing L2 as axis, draw parallel lines to L2 at opposite corner and mid point of opposite side i.e. at A & D.
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Draw perpendicular lines to L2 at E & F.
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Find the Length Ratio between the Max. side to Mini. Side of the Rectangle-2 formed using L2 (The Procedure is illustrated in the following figure).
Rectangle-2 formed with L2
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Repeat these steps for the three corners and construct all “6” Rectangles (formed with L3 & L4, L5 & L6) and find the length ratios.
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Find the Maximum Length Ratio out of the Six Rectangles formed.
Aspect Ratio Calculation for Equilateral Triangle:
Each Side of the Triangular Element : 50mm (AB=BC=CA)
Each corner angle : 60 degree (∠A=∠B=∠C=60 degree)
Equilateral Triangle
Construct rectangle with Ref. L1
Construct rectangle with Ref. L2
Similarly, by constructing rectangles for all the other corners, the values of maximum and minimum side lengths will be same as measured above.
2. Aspect Ratio Calculation for Quadrilateral Elements:
Finding The Length Ratio of Rectangle-1 formed with L1 as Axis:
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From the Quadrilateral "ABCD" join the mid points of opposite sides. These two lines form the axis lines L1 & L2.
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Fixing L1 as axis, draw parallel lines to L1 at mid points of the other opposite sides at E & G.
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Draw perpendicular lines to L1 at F & H.
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Find the Ratio between the Max. Side to Mini. side of the rectangle-1 formed using L1 (Procedure is illustrated in the following figure).
Quadrilateral Element
Rectangle-1 formed with L1
Finding The Length Ratio of Rectangle-2 formed with L2 as Axis:
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Similarly, Fixing L2 as axis, draw parallel lines to L2 at mid points of the other opposite sides at F & H.
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Draw perpendicular lines to L2 at E & G.
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Find the ratio between the Max. side to Mini. Side of the Rectangle-2 formed using L2 (Procedure is illustrated in the following figure).
Rectangle-2 formed with L2
So, The Aspect Ratio is Maximum of the two length ratios formed by L1 & L2.
Aspect Ratio Calculation for Square:
Each Side of the Square Element : 50mm (AB=BC=CA)
Each corner angle : 90 degree (∠A=∠B=∠C=∠D=90 degree)
Square Element with L1 & L2 Axis
Since, all sides are equal (AB=BC=CD=DA) and L1 is Perpendicular to L2, the rectangles constructed by fixing L1 & L2 as Axes are same as "ABCD". So, The aspect ratio is Maximum of the two length ratios formed by L1 & L2.
Aspect Ratio Calculation for Rhombus:
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Rhombus is a quadrilateral all of whose sides have the same length (AB=BC=CD=DA).
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Assuming an element in the shape of a Rhombus with side length of 10mm each.
Rhombus shaped Element with L1 & L2 Axis
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By following the procedure iterated for the Quadrilateral Element, construct rectangles about EF and GH as shown in below.
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Find the ratio of Max. to Min. Side lengths of the rectangles formed.
Rectangle-1 formed with GH
Rectangle-2 formed with GH
Parametric Study:
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By changing the angle Θ from 10 degree to 90 degree variation in the aspect ratio is studied and tabulated below.
References:
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A First Course in the Finite Element Method, 4th Edition, by Logan, D. L. Boston (Page no: 351)
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Finite element procedures, Klaus-Jurgen Bathe, (Page no: 381)
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Fundamentals of Finite Element Analysis by Hutton DV, (Page no: 194)
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Finite Element Modeling for Stress Analysis - Robert D. Cook (page no: 110)
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Ansys, Inc. Theory Reference, Ansys Release 9.0, 2004 (Chapter: 13.7.3)
Note: * Ansys is the registered trademark of Ansys Inc.