FEA V29: Gaussian Quadrature

Summary

This 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

• 📐 Gaussian quadrature is a method for numerical integration used in finite element analysis.
• 🌐 The isoparametric stiffness matrix is calculated by integrating over the volume of an element using B transpose times D times B.
• 🔍 The Jacobian determinant is key in transforming integrals from global to local coordinates, simplifying the process.
• 📏 The limits for the natural coordinates s and t in numerical integration typically range from -1 to 1.
• 📈 B matrices depend on nodal positions, making each element's B matrix distinct in the global system.
• 🔢 Good elements have linear terms in B and B transpose, leading to quadratic expressions when multiplied.
• 📉 Poor quality elements can result in complex ratios of polynomials for the Jacobian determinant, complicating integration.
• 📋 Gaussian quadrature is efficient for polynomial integrands, requiring fewer function evaluations for the same accuracy.
• 📊 The method involves evaluating the function at specific locations (integration points) and multiplying by interval widths for area approximation.
• 📉 Using more integration points increases accuracy but also the computational cost, as it requires more function evaluations.
• 🔧 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

00:00

📐 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.

05:00

📊 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.

10:01

🔍 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.

Keywords

💡Gaussian Quadrature

Gaussian Quadrature is a method used for numerical integration, which is a technique to calculate the integral of a function. It is particularly useful for finite element analysis where elements require integration over their volume to form stiffness matrices. The video explains that Gaussian Quadrature is efficient for polynomial integrands, meaning it can approximate the area under a curve with fewer function evaluations. It is used to evaluate integrals in the context of calculating the stiffness matrix of elements in a structure.

💡Numerical Integration

Numerical Integration is the process of calculating an integral using numerical methods, which is a key concept in the video. It is necessary when analytical methods are not feasible. The video discusses how Gaussian Quadrature is used to perform numerical integration in the context of finite element analysis, specifically for calculating the integral over the volume of an element.

💡Isoparametric Stiffness Matrix

The Isoparametric Stiffness Matrix is a matrix used in finite element analysis to represent the stiffness of an element. The video mentions that this matrix is calculated by integrating over the volume of an element. Gaussian Quadrature is used to perform this integration numerically, which is crucial for constructing the stiffness matrix in the global coordinate system.

💡Jacobian Determinant

The Jacobian Determinant is a scalar value resulting from the determinant of the Jacobian matrix, which relates the infinitesimal volumes in local and global coordinate systems. The video explains that the Jacobian determinant is used to transform differential areas from global to local coordinates, which simplifies the integration process in the context of finite element analysis.

💡Shape Functions

Shape Functions are mathematical functions used in finite element analysis to describe the shape of an element. They are used to map coordinates from the local to the global system. The video discusses how the B matrix, which is part of the stiffness matrix calculation, is derived from the shape functions and nodal positions.

💡Nodal Positions

Nodal Positions refer to the coordinates of the nodes in a finite element model. The video explains that each element's B matrix is distinct because it depends on the nodal positions of the element. These positions are crucial for mapping from the natural to the global coordinate system.

💡Integration Points

Integration Points are the specific locations at which the function is evaluated in Gaussian Quadrature. The video illustrates that choosing the right integration points is essential for achieving accurate results with fewer function evaluations. It shows how single, two, and three-point integration can be used for different levels of accuracy.

💡Bilinear Quadrilateral Element

A Bilinear Quadrilateral Element is a type of element in finite element analysis that uses bilinear shape functions. The video mentions that for such elements, both B and B transpose will have linear terms in s and t, leading to a quadratic expression when multiplied. This is important for understanding the complexity of the integrals involved in the stiffness matrix calculation.

💡Polynomial Integrands

Polynomial Integrands are functions that are polynomials, which are integrals of polynomial functions. The video highlights that Gaussian Quadrature is particularly efficient for polynomial integrands. It shows how the method can achieve exact results for up to a certain degree of polynomial by choosing the right number of integration points.

💡Distributed Force

Distributed Force refers to a force that is distributed over an area or volume. The video provides an example of how Gaussian Quadrature can be used to evaluate a nodal force vector due to a distributed force. It shows the calculation process and how numerical integration can be applied to find the force at a node.

💡Isoparametric Element

An Isoparametric Element is a type of element in finite element analysis that uses the same shape functions for both geometry and field variables. The video discusses how the stiffness matrix for an isoparametric element is calculated using numerical integration, specifically Gaussian Quadrature.

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