FEA 27: Isoparametric Element Example
Summary
TLDRThis 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
- ๐ 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.
- ๐ The mapping from the natural coordinate system (s, t) to the global coordinate system (X, Y) is established using shape functions and nodal positions.
- ๐ Nodal positions are defined for the element, with each node's X and Y coordinates listed.
- ๐งฉ The transformation from the natural coordinate system to the global system involves the shape function matrix multiplied by the nodal position vector.
- ๐ The B Matrix is derived from the Jacobian matrix, which is crucial for determining the element's stiffness matrix.
- โ๏ธ The Jacobian matrix is calculated from the partial derivatives of the shape functions with respect to the natural coordinates s and t.
- ๐ 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.
- ๐ The inverse of the Jacobian matrix is used to calculate the partial derivatives of the shape functions with respect to X and Y.
- ๐ 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.
- ๐ 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
๐ 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.
๐งฎ 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.
๐ 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.
Mindmap
Keywords
๐กIsoparametric Element
๐กJacobian Matrix
๐กB Matrix
๐กShape Functions
๐กNatural Coordinate System
๐กGlobal Coordinate System
๐กNodal Positions
๐กBilinear Quadrilateral Element
๐กDeterminant of the Jacobian
๐กChain Rule
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
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