FEA 27: Isoparametric Element Example

Summary

This video script provides a detailed tutorial on deriving the B Matrix for a finite element model with a non-standard shape. It covers the process of defining the mapping between the natural and global coordinate systems, calculating the Jacobian matrix, and then using it to find the B Matrix. The tutorial emphasizes the importance of the Jacobian's inverse for transforming shape function derivatives from natural to global coordinates, crucial for stiffness matrix computation.

Takeaways

• 📐 The video discusses the process of finding the Jacobian and B Matrix for a non-standard finite element in the XY or global coordinate system.
• 🔗 The mapping from the natural coordinate system (s, t) to the global coordinate system (X, Y) is established using shape functions and nodal positions.
• 📍 Nodal positions are defined for the element, with each node's X and Y coordinates listed.
• 🧩 The transformation from the natural coordinate system to the global system involves the shape function matrix multiplied by the nodal position vector.
• 📋 The B Matrix is derived from the Jacobian matrix, which is crucial for determining the element's stiffness matrix.
• ✏️ The Jacobian matrix is calculated from the partial derivatives of the shape functions with respect to the natural coordinates s and t.
• 🔑 The determinant of the Jacobian matrix is important for finding its inverse, which is used to transform the derivatives from the natural coordinate system to the global system.
• 📉 The inverse of the Jacobian matrix is used to calculate the partial derivatives of the shape functions with respect to X and Y.
• 🔍 The video provides a detailed example of calculating the Jacobian and B Matrix for a specific element, emphasizing the need for accurate nodal position data.
• 🔗 The process of finding the B Matrix involves understanding the relationship between the shape functions in the natural coordinate system and their derivatives in the global coordinate system.

Q & A

What is the purpose of finding the Jacobian and B Matrix in finite element analysis?

The purpose of finding the Jacobian and B Matrix is to map the element's shape functions from the natural coordinate system (s, t) to the global coordinate system (x, y). The Jacobian matrix handles the transformation between these coordinate systems, while the B Matrix is used in the stiffness matrix calculation, which is key in analyzing the behavior of finite elements.

What is the significance of the nodal positions in the transformation?

The nodal positions define the shape and size of the finite element in the global coordinate system. These positions are used in the transformation equation that maps the natural coordinate system (s, t) to the global (x, y) system, which helps calculate the Jacobian matrix and eventually the B Matrix.

How are the shape functions used in the mapping between coordinate systems?

The shape functions are used to interpolate the positions of points within the element in both the natural and global coordinate systems. The mapping equation x = N * X uses the shape function matrix (N) and nodal position vector (X) to express the relationship between coordinates in the s-t system and the x-y system.

Why is the derivative of the shape functions important in this process?

The derivatives of the shape functions with respect to the natural coordinates (s, t) are used to calculate the Jacobian matrix. These derivatives are also needed to find the partial derivatives of the shape functions with respect to the global coordinates (x, y), which are essential for constructing the B Matrix.

What is the role of the Jacobian matrix in this context?

The Jacobian matrix represents the relationship between the natural coordinate system (s, t) and the global coordinate system (x, y). It provides the transformation needed to convert derivatives from the natural system to the global system, which is essential for constructing the B Matrix and for calculating the stiffness matrix in finite element analysis.

How is the inverse of the Jacobian matrix used in finding the B Matrix?

The inverse of the Jacobian matrix is used to convert the derivatives of the shape functions from the natural coordinate system (s, t) to the global coordinate system (x, y). This conversion is required to fill out the B Matrix, which contains the derivatives of the shape functions with respect to x and y.

What is the significance of calculating the determinant of the Jacobian matrix?

The determinant of the Jacobian matrix is crucial for calculating the inverse of the Jacobian, which is necessary for transforming derivatives between coordinate systems. Additionally, the determinant plays a role in the integration process when calculating the stiffness matrix, as it relates to the area or volume of the element in the global coordinate system.

How are the derivatives of the shape functions with respect to s and t calculated?

The derivatives of the shape functions with respect to s and t are calculated based on the specific shape functions defined for the bilinear quadrilateral element. These shape functions are functions of the natural coordinates (s, t), and their partial derivatives are determined analytically.

What does the final B Matrix represent, and why is it important?

The final B Matrix represents the spatial derivatives of the shape functions with respect to the global coordinates (x, y). It is a key component in the stiffness matrix formulation, which governs the relationship between forces and displacements in finite element analysis. The B Matrix is essential for determining the element's response to applied loads.

What are the next steps after constructing the B Matrix in this example?

The next steps after constructing the B Matrix involve combining it with the D Matrix (which represents the material properties) inside an integral to compute the stiffness matrix for the element. This process is crucial for solving the finite element equations and analyzing the mechanical behavior of the structure.

Outlines

00:00

📐 Introduction to Isoparametric Element Mapping

The video begins with an introduction to isoparametric elements, focusing on how to find the Jacobian and B Matrix for a non-standard shaped element in the XY or global coordinate system. The element's nodal positions are given, and the process involves defining a transformation back to the natural coordinate system. The mapping is established by equating the global coordinates (x, y) to the shape function matrix multiplied by the nodal position vector. The video explains how to derive the scalar equations for this mapping and how it allows for a one-to-one correspondence between points in the natural (s, t) and global (x, y) coordinate systems. The process is specific to the element's nodal positions, which are crucial for the transformation and subsequent calculations of the Jacobian and B Matrix.

05:03

🧮 Deriving the Jacobian Matrix

In this section, the video script details the process of deriving the Jacobian matrix for the transformation from the natural coordinate system to the global coordinate system. The script explains how to calculate the partial derivatives of the shape functions with respect to the natural coordinates (s and t) and how these are used to form the Jacobian matrix. The determinant of the Jacobian is also calculated, which is essential for finding its inverse. The inverse of the Jacobian is then used to transform the derivatives of the shape functions from the natural coordinate system to the global coordinate system. This process is crucial for constructing the B Matrix, which relates the nodal displacements to the global coordinates.

10:06

🔍 Constructing the B Matrix

The final paragraph of the script outlines the construction of the B Matrix using the previously derived inverse Jacobian and partial derivatives of the shape functions. The B Matrix is composed of these derivatives, which are calculated for each shape function with respect to the global coordinates (x and y). The video emphasizes the importance of the determinant of the Jacobian in scaling these derivatives, which affects the final form of the B Matrix. The B Matrix is presented in its complete form, highlighting how it will be used in subsequent videos to calculate the stiffness matrix through integration with the D Matrix.

Keywords

💡Isoparametric Element

An isoparametric element is a type of element used in finite element analysis (FEA) where the same shape functions are used to describe both the element's geometry and its field variables (such as displacement). In the video, the example works through this concept by demonstrating how nodal positions and shape functions are used to define the element in both the natural and global coordinate systems.

💡Jacobian Matrix

The Jacobian matrix is a mathematical tool used to transform between coordinate systems. In this video, the Jacobian matrix is calculated to map the relationship between natural coordinates (s, t) and global coordinates (X, Y). It plays a key role in determining the deformation of the element and is essential for calculating the B Matrix.

💡B Matrix

The B Matrix, or strain-displacement matrix, relates the displacement of nodes to the strain in an element. It is calculated using the derivatives of shape functions with respect to global coordinates (X and Y). In the video, the B Matrix is derived after determining the Jacobian and the necessary partial derivatives, forming a crucial step in finite element analysis.

💡Shape Functions

Shape functions are mathematical expressions that describe how the displacement within an element varies based on its nodal values. In this video, shape functions are used to map nodal positions in the global coordinate system to any point within the element. The shape functions are defined in terms of natural coordinates (s, t) and play a critical role in calculating the Jacobian and B matrices.

💡Natural Coordinate System

The natural coordinate system (also referred to as the 's, t' coordinate system) is used in finite element analysis to standardize the calculation of variables within an element. This video shows how the natural coordinate system simplifies the process of mapping element geometry and performing integration for various elements like the bilinear quadrilateral.

💡Global Coordinate System

The global coordinate system, typically referred to as the 'X, Y' system in this video, is the universal frame of reference for the entire finite element model. Each element's position and deformation are calculated with respect to this system. The video explains how the transformation from natural coordinates to global coordinates is achieved using the Jacobian matrix.

💡Nodal Positions

Nodal positions are the coordinates of the nodes (corner points) of an element, defined in the global coordinate system. In the video, the positions of four nodes (e.g., (3,1), (5,2)) are used as input to calculate the transformation matrix, shape functions, and ultimately the B Matrix for the element.

💡Bilinear Quadrilateral Element

A bilinear quadrilateral element is a four-sided element commonly used in finite element analysis, where the shape functions are bilinear in terms of the natural coordinates (s, t). The video uses this type of element as an example, showing how it can be mapped from the natural coordinate system to the global coordinate system and how the B Matrix is derived.

💡Determinant of the Jacobian

The determinant of the Jacobian is a scalar value that represents the scaling factor of the transformation between coordinate systems. It is crucial for calculating the inverse of the Jacobian and for performing integrations. In the video, the determinant is derived to compute the inverse Jacobian and further applied to the B Matrix calculation.

💡Chain Rule

The chain rule is a calculus technique used to compute derivatives of functions that are composed of other functions. In this video, the chain rule is applied to find the derivatives of the shape functions with respect to global coordinates (X and Y), which are necessary for constructing the B Matrix from the derivatives in the natural coordinate system (s, t).

Highlights

Introduction to isoparametric elements and the process of finding the Jacobian and B Matrix.

Explanation of mapping from natural coordinates to global coordinates using shape functions.

Description of the element's nodal positions and their role in defining the transformation.

Derivation of the shape function matrix and its multiplication with nodal position vectors.

Detailed calculation of the Jacobian matrix for the given element.

Importance of the Jacobian determinant in the transformation process.

Derivation of the inverse of the Jacobian matrix and its significance.

How the B Matrix is constructed using the inverse of the Jacobian matrix.

Calculation of partial derivatives of shape functions with respect to global coordinates.

Explanation of the role of natural coordinate system in simplifying the definition of shape functions.

The impact of the Jacobian determinant on the stiffness matrix calculation.

Detailed steps to find the partial derivatives of each shape function.

Construction of the B Matrix using the derivatives of shape functions.

Final expression for the B Matrix and its components.

Discussion on the next steps involving the B Matrix and stiffness matrix calculation.

Emphasis on the importance of integration in the context of finding the stiffness matrix.

Transcripts

this video gives an isoparametric

element example working through finding

finding the Jacobian and then

determining the B Matrix so here's the

element that we're going to find a b

Matrix for this element is in the XY or

global system and it has not a standard

shape but not a typical shape for

elements in finite element codes so we

have the nodal positions for the nodes

in this element and what we're going to

do is use those to define the

transformation back to the Natural

coordinate system and then use that

transformation to determine the Jacobian

matrix and from that the B Matrix so

let's go start by defining that mapping

the mapping is where we say x is equal

to the the X Vector is equal to the

shape function Matrix times the x

capital x Vector which is the nodal

position so you see in the upper right

the capital x Vector is X1 y1 the

positions or the X and Y position of

node one so that's 31 and then 52 for

node 2 55 for node 3 and 23 for node 4

that's how the capital x Vector is set

up the lowercase x Vector is the

position of a point where the s and t

coordinates are in the natural

coordinate system and the X and Y are in

the global system so this mapping gives

us the one: one correspondence between a

point in the St system and the same

point in the XY system system when we

write this Matrix equation out as a

scalar set of scalar equations we get

that the little X so the EXP position of

a point in the XY system is the shape

Function One multiplied by the EXP

position of node one plus shape function

2 multiplied by the X position of node 2

and so on similar for the Y component or

the yes the Y component of a point in

the XY system so now we plug in the

specific numbers for this this element

here's where this transformation becomes

specific to this element the formulation

looks the same for every single element

in our model but each element has

different nodal positions and those are

reflected here in the expressions for X

and Y what we're going to do is use this

expression along with our known shape

functions defined in terms of s and t in

order to determine what the B Matrix is

along the way we're going to have to

find the Jacobian matx Matrix for this

transformation so again here's a quick

overview of what we're doing we have

this element in the XY system and we

have a master element in the natural

coordinate system that every single

bilinear quadrilateral element

references what we're doing by defining

this mapping with little x = n * big X

is we're creating a one toone

correspondence between points that are

in the S&T coordinate system we're

linking them to corresponding points in

in the X and Y system so effectively

what we're doing is we're taking that St

coordinate system and we're mapping it

into the XY space on our

element so for a bilinear quadrilateral

we know that the B Matrix looks like

this it has partial derivatives of each

of the shape functions with respect to X

and Y but we also know that the shape

functions themselves are defined in the

S&T coordinate system this is the power

of using natural coordinate system is we

can do all of our definition in that St

system but it does introduce this

complexity when we have to get to the B

Matrix it will get simple again when we

want to find the stiffness Matrix

because it means that the integration

will occur in the S&T

system so we've got this relationship

that we previously developed where what

we're looking for is the derivative of

each shape function with respect to X

and Y and that's going to be equal to

some Matrix we're calling the inverse of

the Jacobian or J minus one multiplied

by the derivative of the shape functions

with respect to s and t now the

derivative of the shape functions with

respect to S and T is

known the derivatives with respect to X

and Y is what we're looking for the

Jacobian matrix itself is given by the

transformation that we just developed so

we know what x is in terms of s and t so

therefore we can calculate DX DS and

dxdt and similar we know for y that um

what how Y is defined in terms of s and

t so we can find dyds and

dydt the shape functions themselves just

a reminder here that's the U the ones

shown here on the screen in order to use

this um chain rule approximation so to

find DN DX and DN Dy we need the

inverses of the Jacobian but we also

need the partial derivatives of each of

the shape functions so that's what we're

developing here the derivative of N1

with respect to S derivative of N2 with

respect to S and so on so we're going to

need these in a moment I went ahead and

developed them here we'll come back to

these after we find the Jacobian so

let's find that Jacobian again here are

our shape functions let's work through

the definition of the mapping so we

already said that X is equal to the

nodal the x coordinate of the nodal

positions multiplied by each of each of

the shape functions so 3 N1 Plus 5 N2 +

5 n 3 + 2 N4 you can look over at the

sketch of our element and see that

3552 are the four expositions of the

four notes now I plug in the shape

functions from the top of the screen

there and I get this full expression I'm

now going to multiply out the terms and

then I'm going to gather like terms

which gives us this final expression 4X

now that's a good thing to hang on to

this is going to give us that mapping

back and forth

between X and the s and t coordinate

system but what we're looking for here

is the derivative of x with respect to S

and the derivative of x with respects to

T so I've done those derivatives I've

developed those derivatives these are

now two of the terms that we need to get

to the Jacobian the other two come from

y so let's go back to Y so again we know

that Y is equal to the Y coordinates of

each of the nodes multiplied by their

respective shape function so 1 and 1 + 2

and 2 + 5 and 3 + 3 and 4 if you look at

1 2 5 and 3 you see those are the nodal

the y coordinate of the noal positions

of each of the notes plugging in what

each of those shape functions is equal

to and then multiplying out the terms

Gathering like terms we get this

expression for y so again this is an

important equation but it's not quite

the one I need for the Jacobian I need

to go take the derivative of y with

respect to S and then with respect to T

all right so these four partial

derivative expressions are the ones that

go into the

Jacobian so here are those four terms

that we've defined now we want to plug

them in to get the Jacobian matrix

itself when I do that we end up with

this for a Jacobian matrix now we also

want to find the determinant of the

Jacobian now careful when you make the

determinant of when you take the

determinant of a matrix if there's a

coefficient out front that coefficient

is multiplied by every term in The

Matrix which means that if we leave it

out front we're going to have to square

it for a 2X two two matrix it's probably

safer if you bring it inside before you

find the determinant then you won't

accidentally make a mistake and not

square or Cub a term for a 2x2 or 3x3

Matrix this is our expression for the

determinant of the Jacobian we simplify

the terms and we get 1/8 s + 3T + 14

that allows us then to calculate the

inverse of the Jacobian where for the

inverse we are swapping the two diagonal

terms putting a negative sign in front

of each of the off diagonal terms and

then dividing out front by the by the

determinant of that Matrix so that is

the inverse of the Jacobian which we're

now going to use to find the partial

derivatives of all of the shape

functions with respect to X and Y so

here's the expression we're using for

each one of our shape functions we'll

apply this shown Matrix equation so

we're going to take the derivative of

the shape function with with respect to

s and t put that into the vector on the

right hand side we'll use the inverse of

the Jacobian which we just found

pre-multiply that shape function

derivative matrix by the inverse of the

Jacobian and the result will be the

shape the the shape function derivatives

with respect to X and Y so for shape

function 1 here's what happens we've got

dn1 DX dn1 Dy that's the vector we're

looking for is equal to the inverse of

the jobian Matrix multiplied by the

derivative of shape function one with

respect to S and with respect to T now

we've done those derivatives already the

derivative of shape function 1 with

respect to S is - one4 1 minus t and the

derivative with respect to T is -14 1 -

s so we multiply that Vector times the

inverse of the Jacobian matrix but we'll

leave the coefficient out front for

right now so we get now a vector this is

what we expect because we're getting a

vector on the left hand side so we have

a vector with two rows and simplify that

a little bit and we get this for our

Vector defining the shape function

derivative with respect to X and with

respect to Y so this is giving us two

specific terms that will go into our B

Matrix dn1 DX and dn1 Dy so that will

fill out the left two columns of the B

Matrix so we repeat this process for

shape function 2 3 and four so again

this is what the b b Matrix looks like

as you can see the first two columns

depend on derivatives of shape function

one with respect to X and Y those are

the two terms we just found the rest of

the Matrix depends on the other shape

functions I've gone ahead and did that

analysis separately I'm not showing you

that process here it just repeats the

same process we just saw for shape

function one so shape function 2

derivatives look like this shape

function three looks like that and

finally shape function four looks like

those so you can see we're dividing all

of them by the same factor as you might

expect because that came from the

determinant of the Jacobian that's

pretty typical that term will show up in

all of the B Matrix

terms so now if I plug each of those

terms into our B Matrix it gets a little

bit large but this is the B Matrix that

we have as a result the B Matrix is

equal to 1 over the quantity s + 3T + 14

and then it has each one of these terms

inside the Matrix it will have a total

of eight columns and three

rows that wraps up this Example The Next

Step here would be to take this B Matrix

and put it with the D Matrix inside of

an integral in order to find the

stiffness Matrix we'll look at that in a

couple of videos because we need to talk

about how we're going to do that

integration first