pat328, section 3, march 2001 s5 - 1 mar120, lecture 4, march 2001mar 120 - element formulations...

PAT328, Section 3, March 2001 S5 - 1MAR120, Lecture 4, March 2001MAR 120 - Element Formulations

Element Formulations and Accuracy:

Shear Locking, Assumed Strain, Herrmann, Fully Integrated and Reduced Integration Elements


Fundamental Characteristics Of Bending (One Element)

Linear variation of axial strain, exx, through the thickness (y direction).

No strain in the thickness direction, eyy, (if we take Poisson’s ratio as zero).

No membrane shear strain.


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.

To control this problem there are special element types called Assumed Strain (AS) and Reduced Integration (RI) elements

RI, AS, and quadratic elements will be discussed later

Shear Locking

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 the angle between the horiz. and vert. lines

Shear Locking


Shear Locking 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 this:

However a new problem develops called, Hourglassing (discussed later) which can be controlled by using Assumed Strain elements

Thus shear-locking is controlled with RI elements but RI elements introduced hourglassing, which is in turn controlled by AS elements.

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Standard Orthogonal elements


Shear Locking

Don’t use first order fully integrated elements in regions dominated by bending. Instead, try AS elements.

Linear Shape-function elements Quad4 (plane strain, plane stress) (element 3,7) Hex8 (element 11)

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)

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Skewed parallelogram elements

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Skewed trapezoidal elements

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Assumed Strain Elements


Assumed Strain Elements Conventional isoparametric four-node plane stress and plane strain, and eight-

node brick elements behave poorly in bending In reality an applied pure bending moment will create only bending stress and no

shear Because of the linear assumption of these conventional elements, they cannot

“bend” – any deformation is due to shear This additional “non-physical” shear stress tends to store more energy than

bending stress and is called “Parasitic Shear” It generates a stiffer solution in general For these elements, the shape functions

have been modified such that shear strain variation can be better represented

This modification is optional and istermed “assumed strain”



Disadvantages: The substantially improved accuracy of the solution is at the

expense of a slightly increased computational costs during the stiffness assembly

They are sensitive to distortion and cannot capture bending behavior well in distorted meshes

Advantages: These are the most cost effective continuum elements for bending

dominated problems Can model bending with only one element through the thickness They have no hourglass modes They can be used confidently with plasticity and contact

The most benefit of the assumed strain procedure is obtained for coarse meshes. The effect decreases with mesh refinement

If a fine mesh is used, then the assumed strain option could be left off (and providing a slightly faster analysis)


Assumed Strain Elements: Example

Cantilever Beam example using conventional and assumed strain fields

Result table shows the normalised tip deflection (FE:Theory)

Note the effectiveness of the 3x1 and 6x1 mesh when assumed strain is invoked

Section A-A

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Mesh Quad4(AS) Quad4(Standard)

3X1 0.932 0.025

6X1 0.952 0.093

12X2 0.983 0.291

24X4 0.994 0.621

48X8 0.998 0.868

96X16 1.004 0.963



In both Mentat and Patran, assumed strain may be invoked from: The main Jobs forms as shown here As a geometric property specifically

assigned to elements


Herrmann Elements


Volumetric Locking The element in the corner has only one node that is

not constrained. The incompressibility constraint restricts the path of motion of that node to those paths that do not violate that constraint. Unless the loading causes the node to move exactly along that path the element “locks”.


Volumetric Locking Imagine a square “hole” filled with an incompressible

material. Whether the pressure is 1 psi or 1,000,000 psi there will be no deformation of the material and we have a non-unique (singular) solution. The Herrmann variable allows a unique solution of the problem by tracking the “internal” pressure in the element.

Pressure


Herrmann Elements

Incompressible Materials cause severe numerical constraints and 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.

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 over-constrained - it “locks”.


Herrmann Elements have been formulated to handle fully or nearly incompressibility

Available for small or large strain analyses Include plane stress/strain, axisymmetric, and

three-dimensional elements Benefits:

Good (obtainable!) results for incompressible materials

Enable Poisson’s ratio of 0.5 Quadratic rate-of-convergence during

solution Used also for compressible elastic

materials, since their hybrid formulation usually gives more accurate stress prediction

Herrmann Elements


Support both the Total and Updated Lagrange framework Can be used in conjunction with other material models (e.g. elastic-plastic) Can be used in rigid-plastic flow analysis problems and coupled soil-pore pressure

analyses Pressure is treated as an independently interpolated solution variable Actually…the standard displacement formulation is modified based on the

“Herrmann” formulation Generally:

Lower-order elements have an additional node which contains the pressure (Lagrange multipliers)

Higher-order elements have the pressure at each corner node Elements 155 (plane strain triangle), 156 (axisymmetric triangle), and 157 (3-D

tetrahedron) are exceptions - they have an additional node located at the centre of the elements and have Lagrange multipliers at each corner node

Elements 155-157 (tri3/tet4) can be used for large strain plasticity

Herrmann Elements


Reduced Integration Elements


Reduced Integration And Hourglassing

First Order Reduced integrated Elements have only one integration point. These are cheaper, and often better in performance too.

These elements have the following bending behavior: The elements should detect and prevent strain at the corners, but

do not. The deformation is a spurious zero energy mode, shaped like an

hourglass, hence the name (Also called “keystoning” because of the trapezoidal shape.)

Hourglassing


Reduced Integration Elements: Example

Cantilever Beam Normalised Tip Deflection (FE:Theory) Note the effectiveness of the 3x1 and 6x1 mesh when

reduced integration is used – even compared to the assumed strain elements…

Section A-A

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Mesh Quad4(RI)

Quad4 (AS)

Quad4(Standard)

3X1 0.973 0.932 0.025

6X1 0.994 0.952 0.093

12X2 0.994 0.983 0.291

24X4 0.997 0.994 0.621

48X8 0.999 0.998 0.868

96X16 1.000 1.004 0.963


Constant Dilatation Elements


This is the remedy to the typical element locking experienced during fully plastic (incompressible) material behaviour

It is a modified variational principle that imposes a constant dilatational strain constraint on the element

It is known as a “modified volume strain integration”, “mean dilatation” or “B-bar” approach

This procedure has been implemented for a number of lower-order elements in Marc (type 7, 10, 11, 19 and 20)

The combination of these elements and the constant dilatational option is recommended for inelastic (e.g. plasticity and creep) analysis where incompressible or nearly incompressible behaviour occurs and should always be used in metal forming analysis

The use of finite strain plasticity using an additive decomposition of the strain rates obviates the need of the constant dilatation parameter

For materials exhibiting large strain plasticity with volumetric changes (for example, soils, powder, snow, wood) the use of constant dilatation (or plasticity) will enforce the incompressibility condition and, in such materials, yield incorrect and non-physical behavior

Constant Dilatation Elements


Guidelines for Selecting Elements


Guidelines for Selecting Elements Choose quad’s over tri’s Choose bricks over wedges Avoid low order tetrahedral elements wherever possible

The element exhibits slow convergence with mesh refinement This element provides accurate results only with very fine meshing This element is recommended only for filling in regions of low stress gradient in

meshes of Hex8 elements, when the geometry precludes the use of Hex8 elements throughout the model

For tetrahedral element meshes the second-order element should be used Full integration, first order elements in conjunction with the assumed strain

procedure work well for most applications Use low order quad’s and hex’s with assumed strain with reduced

integration if the mesh is refined, of high quality and will not deform very badly

Use 2nd order reduced integration quads and solids if the mesh is coarse Don’t use 2nd order elements if there are gaps in the simulation Contact fully supports lower and higher order elements


Guidelines for Selecting Elements

During plastic deformation, metals exhibit incompressible behavior The incompressible behavior can lead to certain types of elements

being over-constrained This leads to an overly stiff behavior (volumetric locking) Turn ON the “Constant Dilatation” option to correct for this

During hyperelastic deformation, rubbers exhibit incompressible behavior The incompressible behavior can lead to certain types of elements

being over-constrained This leads to an overly stiff behavior (volumetric locking) Use the corresponding Hermann element to correct for this

Second order elements are susceptible to volumetric locking when modeling incompressible materials In general, avoid using them to model hyperelasticity and plasticity

For large strain analyses, lower-order elements are recommended Second order elements should be used with caution above 20%

strain


Element Quality


AngularDistortion

UndistortedElement

Element Quality

Not good

Aspect ratiodistortion

UndistortedElement

Parallelogramdistortion

UndistortedElement

ExcessiveMidside Node

Distortion

UndistortedElement

Not good

Good

Good


Mesh Type Angle Disp %Error

Trapezoidal 15 degrees 3.098E-2 71.3

30 degrees 1.410E-2 86.9

45 degrees 9.212E-3 91.5

Parallelogram 15 degrees 5.114E-2 52.6

30 degrees 2.009E-2 81.4

45 degrees 1.167E-2 89.2

Rectangular None 1.073E-1 0.6

Beam Theory ---- 1.080E-1 ----

Element Quality: Distorted (don’t use AS)(Use high order + RI instead)

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Vertical Displacement of Tip for a 6x1 assumed strain quad mesh


Element Quality: Distorted Mesh

Mesh Type Angle Disp %Error

Trapezoidal 15 degrees 3.134E-2 71.0

30 degrees 1.134E-2 89.5

45 degrees 5.019E-3 95.4

Parallelogram 15 degrees 8.106E-2 24.9

30 degrees 5.160E-2 52.2

45 degrees 2.962E-2 72.6

Rectangular None 1.073E-1 0.6

Beam Theory ---- 1.080E-1 ----

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Vertical Displacement of Tip for a 6x1 assumed strain + reduced integration quad mesh