1. SECTION 8
CHOICE OF ELEMENTS:
INTEGRATION METHODS

2. TABLE OF CONTENTS Section Page
Choice of Elements: Integration Methods
Overview…………………………………………………………………………………………………………….. 8-3
Fundamental Characteristics Of Bending (One Element)……………………………………………………… 8-4
Shear Locking………………………………………………………………………………………………………. 8-5
Reduced Integration And Hourglassing………………………………………………………………………….. 8-6
Eliminating Shear Locking……..………………………………………………………………………………….. 8-7
Assumed Strain Option……………………………………………………………………………………………. 8-9
Second Order Reduced Integration Elements…………………………………………………………………
Herrmann Elements………………………………………………………………………………………………
Some Guidelines For Selecting Elements………………………………………………………………………
Element Selection…………………………………………………………………………………………………

3. OVERVIEW Fundamental Character of bending Shear Locking
Performance of Standard 2D solid Elements
Cantilever Beam Case
Improvement of Quad 4
Reduced Integration and Hourglassing
Detecting and Controlling Hourglassing
Performance of Assumed Strain 2D Solid Elements
Effects of a Distorted Mesh in Assumed Strain Elements
Second Order Reduced Integration Elements
Herrmann Formulation Elements
Guidelines for Selecting Elements

4. FUNDAMENTAL CHARACTERISTICS OF BENDING (ONE ELEMENT)
Linear variation of axial strain, exx, through the thickness (y direction).
No membrane shear strain (fibers remain normal).
No strain in the thickness direction, ezz, (if we take Poisson’s ratio as zero).
Linear element wrongly computes shear strain at the integration points because fibers remain straight.
Physical Behavior
Linear Element Behavior

5. SHEAR LOCKING
Fully integrated 1st order 4 node quad elements, such as Type 3, in bending:
The element detects shear strains that are physically non-existent but are present solely because of the numerical formulation used.
The locked shear absorbs energy from the work done by external forces and so the computed deflection is less than the physical one.
To control this problem there are special element types named; Reduced Integration (RI) and Assumed Strain (AS)
RI, AS, and quadratic elements are next discussed
The axial strain can be viewed as the change in length of the horizontal lines through the integration points.
The thickness strain is the change in length of the vertical lines, and the shear strain is the change in angle between the horizontal and vertical lines.

6. REDUCED INTEGRATION AND HOURGLASSING
Reduced Integration recurs to using (for linear elements) a single integration point in the center of the element, as shown.
Because the fibers passing through the point remain normal, no shear locking occurs.
However a new problem develops, called Hourglassing because of the shape taken by pairs of elements as shown in this page. Others refer to it as “keystoning” because of the trapezoidal shape of the upper half.)
Hourglassing may be controlled by using the Assumed Strain formulation or by stacking up enough number of elements.
Hourglassing can usually be seen in deformed shape plots by simply inspecting a plot of deformations in exaggerated scale.
Hourglassing

7. ELIMINATING SHEAR LOCKING
Section A-A b a
The element cannot bend without shear.
The negative consequence-significant effort (strain energy) goes into shearing the element rather than bending it.
Leads to overly stiff behavior.
Reduced integration elements will correct shear locking on a problem like the one in this page.
Results obtained using Reduced integration with an 8x1 mesh for Linear and Nonlinear analysis in Workshop 1. Try using Standard integration and compare with these results. Create skewed elements with different skew angles and compare the different calculated deflections.

8. ELIMINATING SHEAR LOCKING (CONT.)
Thus shear-locking is controlled with RI elements but RI elements introduced hourglassing, which is in turn controlled by the AS (Assumed Strain) formulation
Assumed Strain elements will produce inaccurate results on skewed meshes.
The further away from 90 the edge angle, the worse the results
Both parallelogram or trapezoidal -shape are bad (figures)
Don’t use first order fully integrated elements in regions dominated by bending. Instead, try RI or AS elements as long as the element edges make straight angles throughout the analysis.
The Assumed Strain formulation may be turned on in MSC.Patran’s Analysis: Translation Parameters form
First Order Reduced integrated Elements (e.g., Type 114) have only one integration point. These are cheaper, and often better in performance too.
These elements have the following bending behavior:
The single element should detect strain, but it does not.
The deformation is a spurious zero energy mode.

9. ASSUMED STRAIN OPTION
These are perhaps the most cost-effective solid continuum elements for bending dominated problems.
They are a compromise in cost between the first and second order reduced integration elements. They have many of the advantages of both.
They can model bending with only one element through the thickness.
They have no hourglass modes and can be used confidently with plasticity and contact.However, they are sensitive to distortion and cannot capture bending behavior well in meshes with alternating trapezoidal shaped elements.

10. SECOND ORDER REDUCED INTEGRATION ELEMENTS
Second Order (quadratic) elements do not suffer from shear locking because they develop fiber curvature.
The results from tests and practice show that the 2nd order reduced integration elements such are very cost effective in bending
No spurious shear strain, little chance of hourglassing.
Advantages
Hourglassing is not an issue in any reasonable mesh, since the hourglass modes cannot propagate through the mesh.
They are especially effective in capturing stress concentrations because of their higher order of interpolation.

11. SECOND ORDER REDUCED INTEGRATION ELEMENTS (CONT.)
Advantages
They are almost always more effective than fully integrated second order elements
Disadvantages
Second order elements are sensitive to distortion, and therefore may not be as effective as first order elements for some finite deformation problems.
They have limitations in contact problems.

12. HERRMANN ELEMENTS
For Incompressible Materials Standard Elements perform poorly, due to Volumetric Locking.
The volume of an element of incompressible material must remain fixed, causing severe constraints on the kinematically admissible displacement fields.
The volume of an element of incompressible material must remain fixed, causing severe constraints on the kinematically admissible displacement fields.
Herrmann Elements are suitable for fully or nearly incompressible materials. In these elements, pressure is treated as an independently interpolated solution variable.
For example, in a fine mesh of standard Hex 8: Each element has, on average, 3 DOF (1 node/elem), but 8 constraints. For, the volume at each of the 8 integration point must remain fixed. Hence, the mesh is overconstrained - it “locks”.

13. SOME GUIDELINES FOR SELECTING ELEMENTS
Use 2nd order reduced integrated quads and bricks if mesh is very coarse. Caution: Don’t use 2nd order elements if there are gaps in the simulation, it’s best to avoid them for all contact problems.
For models with contact were a sudden slippage may occur, use 1st order reduced integration or assumed strain elements. Use Herrmann integration for Incompressible Materials.
Choose Quadrilaterals over Triangles.
Choose Bricks over Wedges. Avoid Tetras if possible.
Use 1st order Quads and Hexes with Assumed Strain with reduced integration if the mesh is of high quality, and will not deform very badly.

14. ELEMENT SELECTION
During plastic deformation, metals exhibit incompressible behavior
The incompressible behavior can lead to certain types of elements being over-constrained, which leads to an overly stiff behavior (Volumetric Locking). Turn ON the “Constant Dilatation” option to correct for this.
Second order elements are very susceptible to volumetric locking when modeling incompressible materials
In general, you should avoid using them to model plasticity.
If you wish to use second order elements, use hybrid formulation instead
First order elements tend to suffer from volumetric locking unless they are used with the “Constant Dilatation” option turned ON
Reduced Integration Elements
Have fewer integration points at which incompressible constraints must be satisfied
Second order elements should be used with caution above 20% - 40% strains
Fully-Integrated, first order elements work well for most applications