Lecture 20: Isoparametric Formulations. - Engineeringmech420/Lecture20.pdf · Lecture 20:...

MECH 420: Finite Element Applications

Lecture 20: Isoparametric Formulations.

Chapter #10 Isoparametric Formulation.Isoparametric formulations help us solve two problems.

Help simplify the definition of the approximate displacement field for more complex planar elements (4-sided elements, elements with curved edges, …).Significantly reduce the integration process by ensuring that wealways have integrals in terms of natural coordinates that are taken over fixed bounds.

We start by applying the isoparametric formulation to a bar element.Move on to derivation of the stiffness matrix and ee’sfor a “plane element” that can be used in plane stress and plane strain problems.The derivations will show the set up of the integral expression.We will apply Guassian quadrature to evaluate these integrals.



§10.1 Isoparametric Formulation of the Bar Element Euations.Step 1: Set the element type.Step 1 now includes the definition of a natural (or curvilinear coordinate), s.We must create a mapping between the curvilinear coordinate s and the cartesian coordinate x.

This mapping will match the element boundaries/keypoints/nodes between the natural coordinate plane and the Cartesian coordinate system.

Element nodes

Element centroid



The mapping procedure is very similar to the creation of shape functions as in Logan Ch.’s 2, 3, 4, 5, and 6.Set the order of the approximation:

Apply some kinematic constraints on the approximation.

1 2x a a s= +

1 1 2

2 1 2

( 1)( 1)

x a ax a a= + −= + +

11 1

2 2

1 11 1

a xa x

−−⎧ ⎫ ⎧ ⎫⎡ ⎤=⎨ ⎬ ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ ⎩ ⎭

1 2

1

2 2 1

2

2

x xaa x x

+⎧ ⎫⎪ ⎪⎧ ⎫ ⎪ ⎪=⎨ ⎬ ⎨ ⎬−⎩ ⎭ ⎪ ⎪⎪ ⎪⎩ ⎭

( ) ( )1 21 1 12

x s x s x= − + +⎡ ⎤⎣ ⎦



Defining the shape functions that are the core of the mappingbetween s and x…

The shape functions define the variation of a quantity (now a coordinate x) over some domain of interest (now a natural coordinate s).We can apply the same shape functions to define the variation ofother values over the domain…

[ ] 11 2

2

xx N N

x⎧ ⎫

= ⎨ ⎬⎩ ⎭

( )

( )1

2

12

12

sN

sN

−=

+=

( )1.. 1s∈ − +



Step 2: Select a displacement function:

[ ] [ ] 111 2 1 2

2 2

ˆ

ˆx

x

duu N N N N

u d

⎧ ⎫⎧ ⎫ ⎪ ⎪= =⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎪ ⎪⎩ ⎭



Step 3: Define the stress strain relationships.Regardless of what natural coordinates we introduce, we do not change the governing physics of the structural problem:

xdudx

dudu ds

dxdxds

ε =

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

Governing physics/Constitutive equations.

How we realize the differential term(s) in the constitutive equations.

( )( )

2 1

2 1

12 1

2

2

2 2ˆ 1 1 ; ˆ

xx

x

u ududs

x xdx Lds

du udu B Bdx L L Ld

ε

−=

−= =

⎧ ⎫− ⎪ ⎪ ⎡ ⎤∴ = = = = −⎨ ⎬ ⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭

Again, we use the B notation to refer to a matrix that relates displacements to strains.



Step 4: Derive the element equations:

( ) ( ) ( )

ˆ ˆ ˆ0 T T T TS

e e eV V S

B DB dV d N b dV N T dS f d⎡ ⎤

= ⋅ + ⋅ + ⋅ +⎢ ⎥⎢ ⎥⎣ ⎦∫∫∫ ∫∫∫ ∫∫

For line elements the integrals reduces to integration in 1-D: x. (uniform cross sectional area assumed in Step 1.)

0 0 0

ˆ ˆ ˆ0L L L

T T T TS

x x x

A B DB dx d A N b dx N T dx fP d= = =

⎡ ⎤= ⋅ + ⋅ + ⋅ +⎢ ⎥

⎣ ⎦∫ ∫ ∫

1 2 3

Perimeter around the bar cross section.P ≡



Consider the first term in the functional :(the strain energy term that produces the stiffness matrix)

1

0 1

2 1

1

1

ˆ

2 2

2

ˆ

L sT T

x s

sT

s

k A B DB dx A B DB dx

x xdx Lds

Ldx ds

k A B DB Jds

=+

= =−

=+

=−

⎡ ⎤ ⎡ ⎤= ⋅ = ⋅⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦−

= =

∴ =

⎡ ⎤∴ = ⋅⎢ ⎥

⎣ ⎦

∫ ∫

∫ 2J L=

Jacobianmatrix.



Some notes on the ever-present Jacobian matrix:Main Entry: Ja·co·bi·anPronunciation: j&-'kO-bE-&n, yä-

Function: nounEtymology: K. G. J. Jacobi died 1851 German mathematician: a determinant defined for a finite number of functions of the same number of variables in which each row consists of the first partial derivatives of the same function with respect to each of the variables Main Entry: de·ter·mi·nantPronunciation: di-'t&r-m&-n&ntFunction: noun1 : an element that identifies or determines the nature of something or that fixes or conditions an outcome



Some more notes on the ever-present Jacobian matrix:Referred to as a “Jacobian Matrix” or just “Jacobian”.Any matrix that defines the conversion from one ‘rate’ to another.

Like a matrix of currency conversions.J in this case is a scalar entity:

Robotics: defines how a collection of joint rates maps over a translational and rotational velocity of an end effector.

Basis for kinematic and dynamic robotics problems.Dynamics: how do changes in the state of a body affect the dynamic equilibrium of that body.

How we measure the ‘stiffness’ of a set of dynamic equations.Has consequence in how we apply numerical integrators to solve dynamics problems.

change in the cartesian coordinate ' 'change in the natural coordinate ' '

dx xJds s

= ≡



Completing the evaluation of k.

So what’s the big deal about introducing s?

1

1

1

1

ˆ

11 1

12

1 11 1

sT

s

s

s

k A B DB Jds

AL L E dsL L

LAEL

=+

=−

=+

=−

⎡ ⎤= ⋅⎢ ⎥

⎣ ⎦−⎡ ⎤⎢ ⎥ −⎡ ⎤= ⋅⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦−⎡ ⎤

= ⎢ ⎥−⎣ ⎦

∫

∫



Look at the second term in the original functional…

0

1

1

ˆ

12

1 22

LB T

x

s

xs

f A N b dx

sLA b ds

s

=

=+

=−

= ⋅

−⎡ ⎤⎢ ⎥ ⎛ ⎞= ⋅⎢ ⎥ ⎜ ⎟+ ⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦

∫

∫

Still pretty easy to evaluate but we saw for the LST how the integrals get a lot

more complicated.But now we know the limits of the

integration and the shape functions are well defined over the -1 to +1 interval



§10.4. Guassian Quadrature (Numerical Integration).Numerical integration methods provide a value for an integral I:

If we turn to numerical approaches it is because the integrand is to complex to get a useful expression for I.Numerical methods sample the integrand, y(x), and approximate the integral, I, with a weighted summation of the sample values.The choice of the weighting factors is what defines the particular integration scheme.Numerical methods effectively replace the original function, y(x),

with a polynomial approximate.

2

1

( )x

x

I y x dx= ⋅∫



y(xi) = value of the function at the sample point xi.Wi = weighting applied to this sample.

2

11

( ) ( )x n

i iix

I y x dx W y x=

= ⋅ ≈ ⋅∑∫

Guassian Quadrature uses ‘n’ sample values to create a polynomial of order ‘2n-1’ which replaces the original function.

For MECH 420 we are integrating over domains:

( )

( )

1.. 1and/or

1.. 1

s

t

∈ − +

∈ − +



Guassian quadrature:Uses a symmetric distribution of sample points over the ‘-1’ to ‘+1’ s domain.Symmetric points carry the same weight.Any number of points can be chosen.

One-point Guassian quadrature,Two-point Guassian quadrature, …

If an odd number of points is chosen then one of the points is the element centroid.

If we consider how the two-point quadrature formula is obtained then we can repeat this procedure to obtain quadrature formulas that are not tabulated.



Using the two point formula we could quickly discretize the body fixed load we looked at earlier:

Note: we would have to evaluate bx at the points:

( ) ( )

1

0 1

0.5773 0.5773

12ˆ

1 22

1 12 21.000 1.000

1 12 22 2

L sB T

xx s

x x

s s

sLf A N b dx A b ds

s

s sAL ALb b

s s

=+

= =−

=− =+

−⎡ ⎤⎢ ⎥ ⎛ ⎞= ⋅ = ⋅⎢ ⎥ ⎜ ⎟+ ⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦

⎛ − ⎞ ⎛ − ⎞⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟= +⎢ ⎥ ⎢ ⎥

+ +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠

∫ ∫

( ) ( )

( ) ( )

1 2

1 2

1 1 0.57735 1 0.577352

and1 1 0.57735 1 0.577352

x x x

x x x

= + + −⎡ ⎤⎣ ⎦

= − + +⎡ ⎤⎣ ⎦

Lecture 20: Isoparametric Formulations. - Engineeringmech420/Lecture20.pdf · Lecture 20:...

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