FEA 30: 2-D Gaussian Quadrature
Summary
TLDRThis video script explains Gaussian quadrature in two dimensions, focusing on its application in finding the stiffness matrix for 2D quadrilateral elements in finite element analysis. It covers single and multiple integration points, illustrating how to use reduced and full integration with examples. The script also discusses the impact of integration methods on structural stiffness, noting that reduced integration can soften the structure.
Takeaways
- đ Gaussian quadrature is extended to two-dimensional elements for numerical integration.
- đ The concept of single-point integration is introduced, simplifying the process by evaluating integrals at a single point in both directions.
- đ The standard element used is a bilinear quadrilateral element, ranging from -1 to 1 in each direction with an interval width of 2.
- 𧟠Reduced integration for a linear or Q4 element is explained, which involves evaluating the function at the center of the S&T coordinate system.
- đą Full integration for a linear quadrilateral element (Q4) is achieved with 2x2 integration points, totaling 4 points.
- đ The process is further expanded to 3x3 integration points for quadratic quadrilateral elements (Q8), providing exact stiffness matrix results for well-shaped elements.
- đ The integral to resolve involves the BDB terms inside the stiffness matrix expression, which is approximated as a double sum of weights times function evaluations.
- đ The locations for integration points are detailed, including the use of square root values to determine positions in the S and T directions.
- đ The stiffness matrix for a Q4 element is found using full integration, which involves evaluating the integrand at multiple points and considering the Jacobian determinant.
- đ§ The difference between full and reduced integration is highlighted, with full integration providing a stiffer structure and reduced integration potentially softening it.
Q & A
What is Gaussian quadrature and how is it extended to two-dimensional elements?
-Gaussian quadrature is a method for numerical integration, which approximates the integral of a function by using weighted sum of the function's values at specified points. In two-dimensional elements, this concept is extended by using a double sum over the number of intervals in two directions, typically denoted as 's' and 't', to evaluate the integrals at multiple points in both directions.
What is the significance of using a single integration point in Gaussian quadrature for 2D elements?
-Using a single integration point simplifies the calculation by reducing the number of evaluations needed. It's often referred to as reduced integration and is used for linear or bilinear quadrilateral elements. It evaluates the function at the center of the 's' and 't' coordinate system, which is when both 's' and 't' are equal to zero.
How does increasing the number of integration points affect the accuracy of the stiffness matrix calculation?
-Increasing the number of integration points, such as moving from a 1x1 to a 2x2 or 3x3 grid, increases the accuracy of the stiffness matrix calculation. A 2x2 grid provides full integration for a linear quadrilateral element, while a 3x3 grid gives full integration for a quadratic element, potentially yielding exact results for the stiffness matrix.
What is the term 'reduced integration' in the context of the script?
-Reduced integration refers to the practice of using fewer integration points than the full number required for exact integration. This approach can be used to reduce computational cost and sometimes to mitigate issues with overly stiff elements in finite element analysis.
Can you explain the term 'stiffness matrix' as mentioned in the script?
-The stiffness matrix is a key component in finite element analysis that represents the structural stiffness of an element. It relates the nodal forces to the nodal displacements and is used to model the elastic properties of the element.
What is the role of the Jacobian determinant in the stiffness matrix calculation?
-The Jacobian determinant is crucial in transforming the integral from the natural coordinate system ('s' and 't') to the global coordinate system. It accounts for the area transformation between these coordinate systems and is used to scale the integration weights correctly.
What does the 'B' matrix represent in the context of the stiffness matrix calculation?
-The 'B' matrix is derived from the partial derivatives of the shape functions with respect to the coordinates ('s' and 't'). It relates the strains in the element to the nodal displacements and is a part of the stiffness matrix formulation.
How does the shape of the element affect the choice between full and reduced integration?
-The choice between full and reduced integration can depend on the element's shape. For elements with a good shape, full integration can provide exact results. However, for elements with poor shape quality, reduced integration might be used to avoid numerical issues such as locking phenomena.
What are the implications of using reduced integration on the structural model's stiffness?
-Using reduced integration can result in a softer structural model because it effectively reduces the stiffness of the elements. This can be beneficial in some cases to offset the natural tendency of finite element models to be overly stiff.
What is the significance of the 'D' matrix in the stiffness matrix calculation?
-The 'D' matrix, also known as the material matrix, relates the stress and strain in the element according to the material's constitutive law. It is dependent on material properties such as Young's modulus and Poisson's ratio and is independent of the element's geometry.
How does the script suggest evaluating the stiffness matrix for a bilinear quadrilateral element?
-The script suggests evaluating the stiffness matrix for a bilinear quadrilateral element using reduced integration by evaluating the 'B' matrix and the Jacobian determinant at the center of the 's' and 't' coordinate system (where 's' and 't' are both zero).
Outlines
đ Introduction to 2D Gaussian Quadrature
The script introduces the concept of Gaussian quadrature in two dimensions, specifically for elements like quadrilaterals. It explains how to find the stiffness matrix for a 2D quadrilateral element using a single integration point. This involves evaluating integrals over the element's area using the Gaussian quadrature approach, which simplifies the process by reducing the number of calculations needed. The script also discusses how increasing the number of integration points can lead to more accurate results, such as using 2x2 or 3x3 integration points for full integration of linear and quadratic quadrilateral elements.
đ Detailed Explanation of Integration Points
This paragraph delves deeper into the specifics of the integration points used in the Gaussian quadrature method. It describes how the integration points are calculated and their coordinates in the s-t coordinate system. The script explains the process of evaluating the function at these points and how these evaluations contribute to the overall stiffness matrix calculation. It also covers the different scenarios of integration points for linear (Q4) and quadratic (Q8) quadrilateral elements, highlighting the importance of accurate integration point selection for achieving exact results.
đ Stiffness Matrix Calculation for Quadrilateral Elements
The final paragraph focuses on the practical application of the Gaussian quadrature method for calculating the stiffness matrix of quadrilateral elements. It discusses the process of evaluating the B matrix and the Jacobian determinant at specific integration points for both full and reduced integration approaches. The script provides a step-by-step guide on how to perform these calculations, including the use of the plane stress D matrix and the impact of element thickness and material properties on the stiffness matrix. It concludes with a comparison of the stiffness matrices obtained from full and reduced integration, highlighting the trade-offs between accuracy and computational efficiency.
Mindmap
Keywords
đĄGaussian Quadrature
đĄStiffness Matrix
đĄIntegration Points
đĄBilinear Quadrilateral
đĄReduced Integration
đĄFull Integration
đĄJacobian Determinant
đĄShape Functions
đĄB Matrix
đĄPlane Stress
đĄPoisson's Ratio
Highlights
Introduction to extending Gaussian quadrature to two-dimensional elements.
Explanation of single integration point for 2D quadrilateral elements.
Concept of reduced integration for linear or Q4 elements.
Advancing to full integration with 2x2 integration points for Q4 elements.
Detailed exploration of the 4 integration points for a 2x2 Gaussian quadrature.
Transition to three integration points in each direction for quadratic elements (Q8).
Integral resolution using 3x3 integration points for Q8 elements.
Description of the stiffness matrix calculation using Gaussian quadrature in 2D.
How to evaluate the stiffness matrix for a Q4 element using full integration.
Process of finding the B matrix for a bilinear quadrilateral element.
Explanation of reduced integration approach for a Q4 element.
Detailed calculation of the B matrix and Jacobian determinant for reduced integration.
Construction of the stiffness matrix using reduced integration for a Q4 element.
Comparison between full and reduced integration in terms of structural stiffness.
Practical application of reduced integration to offset potential over-stiffness in FEA.
Final stiffness matrix obtained through full integration for comparison.
Transcripts
this video extends the concept of
Gaussian quadrature to two-dimensional
elements and then concludes with an
example with finding the stiffness
matrix for a 2d quadrilateral element
when we do 2d integration with a single
integration point that's basically like
evaluating first one integral in with a
single point and then the second
integral with a single point so we just
evaluate the two integrals applying the
Gaussian quadrature approach so instead
of having a single summation we're going
to have two summations where we sum over
the number of intervals in the S
direction and then over the number of
intervals in the Y I'm sorry in the t
direction so for this particular case if
I'm using the standard element in for
bilinear quadrilateral it's going to go
from negative 1 to 1 in each direction
so the width of each interval is going
to be 2 and I'm evaluating the function
right in the middle of the S&T
coordinate system so we're s and T are
both equal to 0 that gives me my single
point integration for a 2d quadrilateral
this is called reduced integration for a
linear or q for element now if we want
to ramp things up a bit we can go to to
integration intervals in each direction
or 2 by 2 integration points it's going
to give us a total of 4 integration
points because we have 2 in each
direction this gives us full integration
for a linear quadrilateral element or q
4 in other words it's going to give us
exact stiffness matrix results when we
have a good element it will give us
reduced integration so in complete
results for a quadratic quadrilateral
element or q 8 element so here's the
integral that we're trying to resolve
where the integrand in there is the the
B DB terms inside the stiffness matrix
expression we're going to approximate
this as a double sum of the weights
times the function evaluation at for
total locations two different locations
in s in two different locations in T
that looks like this where we have the 1
in the tool okay
for S&T we plug in the actual numbers
where we're evaluating at plus and minus
one over the square root of three and
then let's explore this a little bit
this first term corresponds to what we
call our first integration point so
we're evaluating the function at
negative one over the square root of
three in the S direction and negative
one over the square root of three in the
T direction the second term is for the
next second integration point we're
evaluating in the S direction at a
positive 1 on the square root of three
and in the T direction at a negative one
over the square root of three go to the
third term here that corresponds to our
third integration point where both the s
and T terms are positive and then lastly
the fourth integration point where now
the S is negative and the T is positive
so that explores our 4 integration
points those are the four terms that
have to be added up together to give us
the approximation for a integrand using
the 2 by 2 integration points so we're
going to ramp it up one more time we can
do three integration points in each
direction or three intervals across the
total negative one to one range for each
the s and the T variable this three
point integration in each direction
gives us full integration for the
quadratic element that's so in other
words for a q8 element that has a good
shape we will get exact results for the
stiffness matrix using 3 by 3
integration here's the integral that we
want to resolve we're going to break it
into three widths in the S direction and
three widths in the T direction we're
going to evaluate each direction at s
and T equal to zero but also at plus or
minus the square root of 0.6 or the
square root of 3/5 let's take a quick
look at the first three integration
points so it at integration point 1 the
width in the s and the T Direction is
five ninths and we're evaluating at the
position of negative square root of 0.6
for F and for T
moving up the next integration point we
have a width in the in the s direction
that's still 5/9 but in the t direction
that's our center one that has the wider
width so that's our 8/9 width in the T
direction and then we're evaluating at
negative square root of 0.6 in the S
direction and 0 in the T direction and
then continuing the Third Point again
we're back down to five 9s five ninths
and we're doing it at negative square
root of 0.6 in the S direction and
positive square root of 0.6 in the T
direction then we have three more terms
corresponding to the middle row there
where s is equal to zero at each term
and then finally for the the row on the
right where s is equal to the square
root of three fifths or the square root
of 0.6 at each of those so we get a
total of nine individual integration
points for this 3 by 3 integration so
now that we know how to do Gaussian
quadrature in 2d let's take a look at
that stiffness matrix we want to
evaluate so this is in the S&T space at
this point we're going from negative 1
to 1 so it's a 2 by 2 square and we have
B be transpose and the Jacobian
determinant all that are defined in
terms of s and T if we are evaluating
this for a Q for element or by linear
quadrilateral with 4 nodes the stiffness
matrix can be found by either using full
integration full integration is going to
be evaluating the four points shown
let's walk through how we evaluate it at
the first point so the width for each of
these rectangles is going to be one by
one so one in the s and one in the T
direction so one times one times the H
it comes from up front then we're going
to evaluate the transpose of the B
matrix at this location one so where s
and T are both equal to minus one over
the square root of three then we're
going to have our D matrix which usually
does not depend on position but it might
and then we'll have another b matrix
where we're again we're evaluating the
same location remember everything along
this line is evaluated
at this location negative one over
square root of three and negative 1 over
square root of three then we evaluate
our Jacobian at that location and we
continue with location to location 3 and
location 4 that would be our approximate
solution using full integration for aq
for element and if that was a good
element ie an element that has a
constant Jacobian then this would be the
exact result as well
if the Jacobian is not constant then
this would be an approximate solution
for reduced integration is very similar
except now it's a single rectangle with
a width of 2 in each direction so 2
times 2 times H and then we simply
evaluate everything in the middle of the
range so we're S&T both equals 0 so
let's put this into practice this is an
element that we've looked at before we
found the B matrix for it what I want to
do now is use that B matrix to find the
stiffness matrix for it to use numerical
integration that means we've got to
evaluate the integrand at the
integration points for this element for
full integration that's going to be for
different points plus and minus 1 over
square root of 3 for both s and T for
reduced integration we're going to
evaluate that integrand at a single
location
SNT each equal to 0 for right now let's
take a look at the reduced integration
approach for bilinear quadrilateral in
other words a q 4 element the b matrix
always looks like this it's made up of
partial derivatives of each of our four
shape functions with respect to s x and
y now remember we had to go through the
process of finding the Jacobian and then
using the chain rule expansions to find
the partial derivatives with respect to
x and y because the shape functions are
defined in terms of s and T now we did
that all in a previous video in that
video we found that the partial
derivatives of state function 1 with
respect to X and y are as shown with
respect for shape function 2 for shape
function 3 and for shape function for in
addition in that process we also had to
find what the determinant of the
Jacobian was so what we want to do is
use all
this information from the prior video to
find K using for now reduced integration
so for reduced integration we want to
evaluate this B matrix and the
determinant of jacobian where s and T or
equal to 1 so we just take our
expressions that I just showed you for
the partial derivatives and we evaluate
them by setting s and T equal to 1 so in
this case we get that the departure
derivative for shape function 1 with
respect to x and y are respectively
negative 1 14 and negative 3/14 so I can
now plug those right into my B matrix as
shown and we're going to find that
there's a 14 in the bottom of everything
so I'm sticking it out front for shape
function 2 we find the partial
derivative with respect to X is 414 and
with respect to Y is negative 2 14 again
I can plug that right into my B matrix
evaluating it at S&T equal to 0 with
root 4 straight function 3 we get 114
and 314 that goes in there and for shape
function for we get negative 4 14 and 2
14 and those go in there so that gives
me my B matrix evaluated at s and T
equal to 0 and the determinant of
jacobian that's even easier that's going
to give me 14 over 8 or 7 over 4 so now
to find K I have to evaluate B and
determinant of jacobian at 0 0 which
I've just done and then I multiply them
so I'm going to have the width of each
interval so 2 times 2 within the S that
width in the T Direction times H the
thickness of my element times the B
matrix transpose evaluated at 0 0 times
D times B evaluated at 0 0
and lastly times the determinant of the
Jacobian evaluated at 0 0 so the first
piece that remains the same then the the
B matrix evaluate at 0 0 transpose that
I just found then let's assume plane
stress for right now that's what my D
matrix looks like for plane stress
I've got my B matrix again not transpose
this time and finally I have the
determinant of my Jacobian let's
multiply these terms out first of all
we'll assume that I have a thickness of
0.1 inch
I've got young's modulus of 30 times 10
to the 6 psi and a poissons ratio of
0.25 plug those into my plane stress D
matrix and then multiply out the terms I
end up with 10 to the fifth over 28
times most of the B transpose times most
of the D matrix and then B now this is
just a matrix multiplication it's
time-consuming by hand but it's pretty
straightforward multiply that out and we
will get K so this is the stiffness
matrix we end up with when we multiply
those three this is the reduced
integration stiffness matrix for the Q
for element shown there I went ahead and
followed this process for full
integration so evaluating the integrand
the integrand at four different points
and this is the stiffness matrix that I
obtained that way it is in many ways
similar but if we just grab a little
two-by-two section there we can see
something typical full integration
usually has a higher stiffness as you
can see there so that means reduced
integration effectively is going to
soften the structure and this is a
common result that we see sometimes it
softens it too much sometimes it softens
it just enough to offset the fact that
Fe a primarily is more stiff than
necessary so sometimes people will use
reduced integration in order to
deliberately reduce the effect of a to
stiff structure okay
5.0 / 5 (0 votes)