FEA V29: Gaussian Quadrature
Summary
TLDRThis video script delves into Gaussian quadrature, a pivotal method for numerical integration in finite element analysis. It explains how Gaussian quadrature is used to calculate the stiffness matrix by integrating over an element's volume. The script highlights the importance of the Jacobian determinant in transforming integrals from global to local coordinates. It also discusses the properties of the B matrix and the challenges of integrating ratios of polynomials. The script further illustrates how Gaussian quadrature efficiently approximates integrals of polynomial functions with fewer function evaluations, providing examples of single, double, and triple integration points for increasing accuracy. Finally, it applies Gaussian quadrature to evaluate a nodal force vector, demonstrating the trade-off between accuracy and computational efficiency.
Takeaways
- đ Gaussian quadrature is a method for numerical integration used in finite element analysis.
- đ The isoparametric stiffness matrix is calculated by integrating over the volume of an element using B transpose times D times B.
- đ The Jacobian determinant is key in transforming integrals from global to local coordinates, simplifying the process.
- đ The limits for the natural coordinates s and t in numerical integration typically range from -1 to 1.
- đ B matrices depend on nodal positions, making each element's B matrix distinct in the global system.
- đą Good elements have linear terms in B and B transpose, leading to quadratic expressions when multiplied.
- đ Poor quality elements can result in complex ratios of polynomials for the Jacobian determinant, complicating integration.
- đ Gaussian quadrature is efficient for polynomial integrands, requiring fewer function evaluations for the same accuracy.
- đ The method involves evaluating the function at specific locations (integration points) and multiplying by interval widths for area approximation.
- đ Using more integration points increases accuracy but also the computational cost, as it requires more function evaluations.
- đ§ An example demonstrates the difference between single and two-point Gaussian quadrature, with the latter providing exact results for quadratic functions.
Q & A
What is Gaussian quadrature?
-Gaussian quadrature is a method for numerical integration that is particularly efficient for polynomial integrands. It approximates the integral of a function by evaluating it at specific points (integration points) and multiplying by weights, which are determined to minimize the number of evaluations needed for a given level of accuracy.
Why is Gaussian quadrature important in finite element analysis?
-In finite element analysis, Gaussian quadrature is crucial for calculating the stiffness matrix of each element in a structure. It allows for the transformation of the integration from the global coordinate system to the natural coordinate system, simplifying the process and making it computationally efficient.
What is the significance of the Jacobian determinant in Gaussian quadrature?
-The Jacobian determinant is significant because it relates the infinitesimal areas in the global coordinate system to the natural coordinate system. It is used to transform the limits of integration from global to local coordinates, which simplifies the integration process in finite element analysis.
How does the quality of an element affect the integration process?
-The quality of an element affects the integration process because it influences the complexity of the integrand. Good quality elements have simpler integrands, often polynomials, which are easier to integrate using Gaussian quadrature. Poor quality elements may have more complex integrands, such as ratios of polynomials, which are harder to integrate.
What are the properties of the B matrices in the context of Gaussian quadrature?
-The B matrices in Gaussian quadrature depend on the nodal positions and are used to map from the natural to the global coordinate system. Each B matrix is distinct for each element because it depends on the element's nodal positions. The B matrices are defined in terms of natural coordinates s and t.
How does the number of integration points affect the accuracy of Gaussian quadrature?
-The number of integration points in Gaussian quadrature directly affects the accuracy of the numerical integration. More points generally provide higher accuracy, especially for higher-degree polynomial integrands. However, it also increases the computational cost, as more function evaluations are required.
What is the role of the integration points in Gaussian quadrature?
-Integration points in Gaussian quadrature are the specific locations at which the function is evaluated. These points are chosen to optimize the accuracy for polynomial integrands. The choice of these points, along with the interval width, determines the efficiency and accuracy of the numerical integration.
Why is it beneficial to evaluate the function at the midpoint in a single integration point scenario?
-Evaluating the function at the midpoint in a single integration point scenario is beneficial because it provides an exact result for linear integrands. This is due to the property that a linear function will average out to zero over a symmetric interval, resulting in an exact integral value.
How does the width of the rectangles in Gaussian quadrature relate to the number of integration points?
-The width of the rectangles in Gaussian quadrature changes with the number of integration points. With more points, the rectangles may have different widths, with the central rectangle often being wider to give more emphasis to the central region of the integration range, which is especially important for higher-degree polynomials.
What is the practical example given in the script for using Gaussian quadrature?
-The practical example given in the script is the evaluation of a nodal force vector in a finite element model. The script explains how Gaussian quadrature can be used to approximate the integral of a quadratic function to find the force vector, with different levels of accuracy depending on the number of integration points used.
What is the difference between one-point and two-point Gaussian quadrature integration?
-One-point Gaussian quadrature integration evaluates the function at the midpoint of the interval and is exact for linear integrands. Two-point integration evaluates the function at two specific points and is exact for cubic integrands. The two-point method is more accurate for quadratic integrands compared to the one-point method.
Outlines
đ Introduction to Gaussian Quadrature in Finite Element Analysis
The first paragraph introduces Gaussian Quadrature as a method for numerical integration, particularly crucial in finite element analysis. It explains that every element in a structure requires a stiffness matrix, which necessitates integration over the element's volume. The paragraph then delves into the isoparametric stiffness matrix, detailing how it's calculated through integration over the volume of B transpose times D times B. It highlights the use of the Jacobian determinant to transform the integral from global to local coordinates, simplifying the process by integrating over natural coordinates s and t. The discussion then shifts to the properties of the integral, emphasizing the dependency of the B matrices on nodal positions and the implications of element quality on the complexity of the integral. It concludes by setting the stage for Gaussian Quadrature as an efficient method for numerical integration, especially suited for polynomial integrands.
đ Gaussian Quadrature for Polynomial Integration
This paragraph explains how Gaussian Quadrature works in practice, focusing on its application for polynomial integrands. It describes the process of approximating an integral as the sum of rectangular areas, where the width and height of the rectangles are determined by specific mathematical formulas. The paragraph illustrates the concept with examples of single, double, and triple integration points, showing how the number and position of these points affect the accuracy of the integration. It emphasizes the importance of evaluating the function at specific locations, known as integration points, to achieve optimal results. The discussion also touches on the practical implications of function evaluation in terms of computational efficiency, especially when dealing with large matrices in finite element analysis. The paragraph concludes with an example of using Gaussian Quadrature to evaluate a nodal force vector, demonstrating the method's accuracy and efficiency.
đ Accuracy of Gaussian Quadrature in Polynomial Integration
The third paragraph provides a detailed analysis of the accuracy of Gaussian Quadrature, particularly when dealing with polynomial functions of varying degrees. It contrasts the results of single-point and two-point integration with the actual value of a quadratic function to illustrate the method's effectiveness. The paragraph explains how single-point integration can lead to significant errors for non-linear integrands, while two-point integration provides exact results for up to cubic integrands. It further discusses the implications of these findings for the evaluation of nodal force vectors in finite element analysis, emphasizing the importance of choosing the right number of integration points to ensure accuracy. The paragraph concludes by setting the stage for further discussion on the application of Gaussian Quadrature in two-dimensional elements and the calculation of stiffness matrices.
Mindmap
Keywords
đĄGaussian Quadrature
đĄNumerical Integration
đĄIsoparametric Stiffness Matrix
đĄJacobian Determinant
đĄShape Functions
đĄNodal Positions
đĄIntegration Points
đĄBilinear Quadrilateral Element
đĄPolynomial Integrands
đĄDistributed Force
đĄIsoparametric Element
Highlights
Gaussian quadrature is introduced as a method for numerical integration.
Gaussian quadrature is crucial for finite element analysis in structural engineering.
The stiffness matrix is derived through integration over the element's volume.
The Jacobian determinant is used to transform integrals from global to local coordinates.
The stiffness matrix in the global system is calculated by integrating over local coordinates.
The B matrices are defined in terms of natural coordinates s and t.
The limits for s and t in numerical integration range from -1 to 1 for a 2x2 square element.
The B matrices depend on nodal positions, making each element's B matrix distinct.
Good quality elements have linear terms in B and B transpose, resulting in quadratic expressions.
Poor quality elements can have polynomial ratios in the Jacobian determinant, complicating integration.
The determinant of the Jacobian is constant for good elements, simplifying the integration process.
Gaussian quadrature is an efficient method for polynomial integrands, requiring fewer function evaluations.
The integration points in Gaussian quadrature are chosen to optimize results for polynomials.
A single integration point is used for linear integrands, providing an exact result.
Two integration points are used for quadratic and cubic integrands, also providing exact results.
Three integration points are used for higher accuracy, particularly for fifth degree polynomial integrands.
The number of function evaluations is critical due to the computational cost of evaluating matrices.
An example is provided where Gaussian quadrature is used to evaluate a nodal force vector.
Single point integration is exact for linear integrands but not for quadratic integrands.
Two point integration is exact for cubic integrands and should be accurate for quadratic functions.
The video concludes with an example of using Gaussian quadrature for 2D isoparetric elements.
Transcripts
this video introduces gaussian
quadrature as a method for numerical
integration
gaussian quadrature is an important tool
for numerical integration in finite
element analysis because every every
single element in a structure requires a
stiffness Matrix and to get to that
stiffness Matrix you need to integrate
over the volume of the element so
focusing on the isoparametric stiffness
Matrix we know that it's equal to the
integral over the volume of B transpose
times D times B
but in terms of the isoparametric we
have the B's defined in terms of s and t
and the element in global coordinates in
terms of X and Y
what we can do though is
use the property of the Jacobian
determinant which is that it's equal to
the ratio of the areas Global to local
so in in infinitesimal areas that just
means that the determinant of the
Jacobian is equal to dxdy divided by DS
DT or in other words we can substitute
dxdy for Jacobian times dsdt
so that's what we do here that gives us
an equation for the stiffness Matrix
which is going to be the stiffness
Matrix in the global system directly but
we get to integrate over the local
coordinate system the natural
coordinates of s and t and the nice
thing about that is first of all the B's
and the J are all defined in terms of s
and t but secondarily the limits for S
and T both go from negative one to one
because it's this two by two square
element this is the integral we now want
to solve using numerical integration
before we get into the integration
itself let's talk about a few properties
of this integral first off the B
matrices in here depend on the nodal
positions the B Matrix we developed in a
prior video we used the shape functions
times the nodal positions in order to
give us a mapping from the natural to
the global system and in that process we
were able to calculate the B Matrix so
every B Matrix is distinct for each
element in the global system because it
depended on those element Noble
positions
for a good element a good bilinear
quadrilateral element or Q4 both B and B
transpose will have linear terms if s
and t so when you multiply B and B
transpose you're going to get a
quadratic expression but poor quality
elements because the determinant of the
Jacobian enters into B so we're going to
be dividing by that twice and then
multiplying it by it again
um
we're going to end up with ratios of
polynomials so poor quality element will
have a polynomial for the determinant of
the Jacobian so we're going to be
dividing by that
this means that we're going to have an
issue potentially doing the integration
because a ratio of polynomials is much
more complex to integrate than just a
single polynomial however for good
elements the determinant of the Jacobian
is constant
for lower quality it's going to be a
quadratic term so our real question here
is what can we do to allow the Fe code
for every single element to go through
this integral and evaluate it it will be
a different integral for every single
one what do we do well we turn to
numerical integration this is what the
background is the reason that we want to
do gaussian quadrature
so what is gaussian quadrature well
first off quadrature is just another
term for numerical integration it is a
way of approximating the integral or the
area under a curve of a function with
the sum of rectangular areas like what's
shown here gaussian quadrature in
particular is designed to be unlike the
Vermont sums which are generic designed
to be a most efficient approach for
polynomial integrands so you evaluate
the function fewer times to get the same
level of accuracy and remember that good
quality quadrilateral elements have
quadratic integrands so they are
polynomials in gaussian quadrature what
we do is we change the number and the
width of the intervals in order to get
the accuracy that we desire for a given
polynomial also the height of the
rectangle is not evaluated right at the
midpoint such as in shown in the figure
above it's actually going to be
evaluated at a specific location chosen
to give us optimal results for
polynomials and that location is called
the integration point for the interval
and finally the choice of the interval
width and that evaluation point is what
gives us a very efficient minimizing the
number of evaluations a method of
getting an accurate result for a
polynomial
let's see how this works in practice
let's imagine for instance that we have
a function such as shown on the screen
that's a function of s where s is going
to vary from negative 1 to 1 and we find
want to find the integral of this
function on that range
so this is what we're trying to solve
for what we do with gaussian quadrature
is that we approximate this as the sum
of a series of rectangles where W
represents the width of the rectangle
and F of s i is the height of the
rectangle at a specific location s i
so if we want to do a single integration
point then what we're going to do is
evaluate the function smack dab in the
middle and multiply by the width of the
rectangle which is just going to be 2.
so height of or S1 is evaluated at zero
so F of 0 and then the width of the
rectangle is 2.
that gives us an exact result for any
linear integrand as you might expect you
evaluate at one point it's going to
average out negative on one side
positive and the other or above and
below we will get an exact result
however obviously for the function shown
here it's not a great approximation
well what if we want something more
accurate well we can go to two
integration points so we have the same
function here and again we're going to
approximate it as the sum of rectangles
but now we're going to apply two
rectangles as shown here and we're going
to evaluate those rectangles not smack
in the middle of the range but at
specific locations in particular when we
go from -1 to 1 the first location is is
at minus 1 over the square root of 3 and
the second is as plus 1 over the square
root of 3. so when I evaluate at those
locations that's minus 0.577 and plus
0.577 then the width of each rectangle
is going to be equal to 1 and that's
what gives me the equation shown here
where I'm evaluating each the function
at two locations plus and minus 0.577
this gives me an exact result for up to
a cubic integrand so quadratic and cubic
would both give exact results for two
integration points over the range of
negative one to one
taking it one step further if you really
if you want to get even more accurate
how about three integration points in
other words evaluating with three
rectangles here's where we're going to
see that we want to actually change the
width of the rectangles you can see here
the middle rectangle ends up being extra
emphasized it's a wider rectangle again
this is for a polynomial that's what
you'd want to have is the center section
to be more important than the outer two
and what we're going to do is evaluate
the left hand section which has a width
of 5 9. we're going to evaluate at a
position again from the negative one to
one range its first position is minus
0.774 and then the middle section is a
wider width so that's 8 9 and it's
evaluated right where the
s is equal to zero
and then the last piece is again back to
5 9 and it's evaluated at a positive
0.774
this gives me an exact result for up to
a fifth degree polynomial integround so
it's a pretty darn accurate result
especially if you've got polynomial
integrounds for just three evaluations
of your function now you notice I'm keep
talking about this evaluation of the
function and that's important because it
takes computer time to evaluate the
function remember the function here is
actually a big Matrix that we're
evaluating and we're doing it on every
single element so if you have to
evaluate that Matrix one location that's
a lot less time than say three locations
here well it's a third of the time
let's wrap up this video with a quick
example of using gaussian quadrature to
evaluate a nodal Force Vector in a prior
video I showed you how to find the force
Vector for the distributed Force shown
the t y here and we ended up with an
expression that looked like this where
we had shape function 1 and shape
function four each evaluated at the
position where s is equal to negative 1
and Y is equal to 2. so here I'm
actually evaluating with T is my
variable but again the range is from
negative one to one
so when I
um multi when I plug in the value for or
the expression for N1 and N4 we end up
with an integral at Node 1 in the y
direction that looks like this
um that's the integral that we now want
to use gaussian quadrature to evaluate
so first of all let's go back to if I
evaluate that integral directly because
obviously I can it's a simple polynomial
I get 83.3 pounds as my Force so if now
I decide to approximate that with a
single integration point which is going
to be exact up to a linear integrand
notice this is a quadratic integrand so
it will not be exact
so at one point integration means I take
the full width of the integral from
negative 1 to 1 in the T Direction so
that's 2 and then I multiply it by the
function evaluated in the middle of the
range so that's at location 0.
so that gives me this expression when I
plug in t equal to 0 in the inside the
integrand and it gives me a final total
of 62.5 pounds that's a 25 error by
using a single point integration
let's try the two point integration
which should be exact up to cubic and
since this is a quadratic function it
should give me the exact result
so again here I'm evaluating it with two
rectangles each rectangle has a width of
one and I'm evaluating at negative one
over the square root of 3 and positive 1
over the square root of 3. where T is my
my variable that is plus or minus 1 over
square root of T of 3. plugging that
into my expression with the integrand I
get 83.3 pounds so I have no error
present when I do the two point
integration and that's what I expect
because this is just a quadratic
function so that's just a small example
of using this in the next video I'll go
through what we do for two dimensional
elements and then we'll take to take
that to an example to find the stiffness
Matrix using both one point and two
point integration on a 2d isoparetric
element
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