FEA 30: 2-D Gaussian Quadrature

Summary

This 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

• 📐 Gaussian quadrature is extended to two-dimensional elements for numerical integration.
• 🔍 The concept of single-point integration is introduced, simplifying the process by evaluating integrals at a single point in both directions.
• 📏 The standard element used is a bilinear quadrilateral element, ranging from -1 to 1 in each direction with an interval width of 2.
• 🧮 Reduced integration for a linear or Q4 element is explained, which involves evaluating the function at the center of the S&T coordinate system.
• 🔢 Full integration for a linear quadrilateral element (Q4) is achieved with 2x2 integration points, totaling 4 points.
• 🔄 The process is further expanded to 3x3 integration points for quadratic quadrilateral elements (Q8), providing exact stiffness matrix results for well-shaped elements.
• 📑 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.
• 📐 The locations for integration points are detailed, including the use of square root values to determine positions in the S and T directions.
• 🔑 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.
• 🔧 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

00:00

📐 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.

05:02

🔍 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.

10:03

📐 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.

Keywords

💡Gaussian Quadrature

Gaussian Quadrature is a method for numerical integration, specifically for approximating the integral of a function. In the video, it is extended to two-dimensional elements, which is crucial for calculating the stiffness matrix of 2D quadrilateral elements in finite element analysis. The script describes how this method is applied to evaluate integrals over a quadrilateral element using specific points and weights, which simplifies the computation.

💡Stiffness Matrix

The Stiffness Matrix is a fundamental concept in structural mechanics and finite element analysis, representing the resistance of a structure to deformation. In the video, the process of finding this matrix for a 2D quadrilateral element is discussed. It is calculated using Gaussian quadrature, with different levels of integration affecting the accuracy and computational cost.

💡Integration Points

Integration Points are the locations within the element at which the function to be integrated is evaluated. The video explains how using a single integration point simplifies the process but reduces accuracy, while using multiple points (like 2x2 or 3x3) increases accuracy but requires more computation.

💡Bilinear Quadrilateral

A Bilinear Quadrilateral is a type of element used in finite element analysis, defined by its ability to have linear shape functions in two directions. The video discusses how Gaussian quadrature is applied to this element type, particularly in the context of reduced and full integration schemes.

💡Reduced Integration

Reduced Integration is a technique where fewer integration points are used to approximate the integral, thus reducing computational cost. The video explains that for a linear or Q4 element, reduced integration can lead to a softened structure, which is sometimes used intentionally to offset potential overstiffness in FEA models.

💡Full Integration

Full Integration refers to using the optimal number of integration points to achieve an accurate approximation of the integral. The video contrasts this with reduced integration, noting that full integration typically results in a stiffer model, which is closer to the exact solution for elements with a constant Jacobian.

💡Jacobian Determinant

The Jacobian Determinant is a scalar value resulting from the transformation between coordinate systems and is crucial in the change of variables for integration in finite element analysis. The video describes how it is calculated and used in the context of Gaussian quadrature for 2D elements.

💡Shape Functions

Shape Functions are mathematical functions used to describe the shape of an element in the finite element method. They are used to interpolate field variables within the element. The video mentions that the partial derivatives of these functions with respect to natural coordinates s and t are needed to form the B matrix.

💡B Matrix

The B Matrix is derived from the partial derivatives of shape functions and is used in the formulation of the stiffness matrix. The video explains how the B matrix is constructed and evaluated at specific integration points for both reduced and full integration schemes.

💡Plane Stress

Plane Stress is a state of stress assumed in certain two-dimensional problems where the stress on one face of the material is zero. The video uses this assumption to simplify the calculation of the D matrix, which is a part of the stiffness matrix calculation.

💡Poisson's Ratio

Poisson's Ratio is a measure of the ratio of lateral strain to axial strain in a material under axial load. It is used in the calculation of the D matrix for plane stress or plane strain conditions, as mentioned in the video when constructing the stiffness matrix.

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