- #1

feynman1

- 435

- 29

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter feynman1
- Start date

- #1

feynman1

- 435

- 29

- #2

pasmith

Homework Helper

2022 Award

- 2,591

- 1,196

Why not construct the stiffness matrix for [tex]

\frac{d^2u}{dx^2} = -f(x)[/tex] for [itex]x \in (0,1)[/itex] using elements [itex][nh, h(n+1)][/itex] with basis functions [tex]

\begin{split}

f_0 : [0,h] \to \mathbb{R} : x &\mapsto 1 - \frac{x}{h} \\

f_1 : [0,h] \to \mathbb{R} : x &\mapsto \frac{x}{h}

\end{split}[/tex] and see what happens as you reduce [itex]h[/itex]?

\frac{d^2u}{dx^2} = -f(x)[/tex] for [itex]x \in (0,1)[/itex] using elements [itex][nh, h(n+1)][/itex] with basis functions [tex]

\begin{split}

f_0 : [0,h] \to \mathbb{R} : x &\mapsto 1 - \frac{x}{h} \\

f_1 : [0,h] \to \mathbb{R} : x &\mapsto \frac{x}{h}

\end{split}[/tex] and see what happens as you reduce [itex]h[/itex]?

Last edited:

- #3

Arjan82

- 418

- 320

Euhrm... sources?

Considering that the stiffness of the construction decreasing with mesh refinement would be really silly, do you mean the stiffness of the differential equation? But even in this case I'm not sure about this, why would this be true in general? So again: sources?

- #4

hutchphd

Science Advisor

Homework Helper

2022 Award

- 5,525

- 4,705

Instead of an admonishment I will provide a tangential and perhaps incorrect annswer.

I recommend Galileo Galilei "Two New Sciences" where one of which is scale invariance and the strength of materials . To my mind these arguments point directly quantum mechanics which set the scale by which big and small are tto be measured

I believe the examination of FEA issues withion scale of the simulation are really a recapitulation of Galileo's arguments about his brave new world. Read the original!

.

I recommend Galileo Galilei "Two New Sciences" where one of which is scale invariance and the strength of materials . To my mind these arguments point directly quantum mechanics which set the scale by which big and small are tto be measured

I believe the examination of FEA issues withion scale of the simulation are really a recapitulation of Galileo's arguments about his brave new world. Read the original!

.

Last edited:

- #5

Arjan82

- 418

- 320

Instead of an admonishment I will provide a tangential and perhaps incorrect annswer.

I recommend Galileo Galilei "Two New Sciences" where one of which is scale invariance and the strength of materials . To my mind these arguments point directly quantum mechanics which set the scale by which big and small are tto be measured

I believe the examination of FMEA issues withion scale of the simulation are really a recapitulation of Galileo's arguments about his brave new world. Read the original!

.

This answer and the relation to the OP's question is completely lost to me...

- #6

Arjan82

- 418

- 320

Or does the OP mean shear locking?

- #7

hutchphd

Science Advisor

Homework Helper

2022 Award

- 5,525

- 4,705

This answer and the relation to the OP's question is completely lost to me...

My apologies. I was trying to be not very complete on purpose. I do not know the exact issues with FEA to which the OP alludes but my instinct is that they are related to issues of scale invariance vis. Galileo's discussion of scaling the size of animals and how the bones must modify shape). If one is not very careful issues of absolute scale can be changed with mesh size and subtle errors result. That Calileo fellow was pretty astute.

Last edited:

- #8

FEAnalyst

- 307

- 129

- #9

feynman1

- 435

- 29

no not shear locking in particular, though shear locking is due to stiffness being too highOr does the OP mean shear locking?

- #10

feynman1

- 435

- 29

many thanks, could you tell which chapter in Bathe's book?

- #11

FEAnalyst

- 307

- 129

It’s in the chaptermany thanks, could you tell which chapter in Bathe's book?

- #12

bob012345

Gold Member

- 1,842

- 802

I recently worked through the derivation of the FEA model (with the help of @pasmith , thanks) and I know the term stiffness matrix goes back to mechanical engineering application dealing with the literal stiffness of materials but what do you mean when you say

For example I derived the node equations for the differential equation

$$ \frac{d^2u(x)}{dx^2} = -f(x)$$

I got for linear test functions ##N_1(x) = \frac{x_2 - x}{\Delta{x}}## and ##N_2(x) = \frac{x - x_1}{\Delta{x}}##

$$\frac{1}{\Delta{x}}\begin{pmatrix}

1 & -1 \\

-1 & 1

\end{pmatrix}

\begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \\

=

\begin{pmatrix}

- u_1' \\

u_2' \end{pmatrix} +

\begin{pmatrix}

\int_{x_1}^{x_2} N_1(x)f(x)\,dx \\

\int_{x_1}^{x_2} N_2(x)f(x)\,dx

\end{pmatrix}$$

So, as ##\Delta{x}## gets smaller, in what sense is the stiffness being reduced?

- #13

Frabjous

Gold Member

- 1,048

- 1,175

Is there a thread?I recently worked through the derivation of the FEA model (with the help of @pasmith , thanks)

- #14

bob012345

Gold Member

- 1,842

- 802

No, my full derivation was done in a private chat and not a public PF forum. It was actually of the wave equation which is more complex and then the equation in this thread. There is a limited precursor thread;Is there a thread?

https://www.physicsforums.com/threads/fem-method-for-the-wave-equation.1012366/#post-6603716

However, if you wish and start a thread I can help you work through it. Or ask me in private.

- #15

berkeman

Mentor

- 64,454

- 15,829

It sounds like it could be an interesting thread...

- #16

FEAnalyst

- 307

- 129

What the OP meant is that FEM always converges from below (displacements and stresses increase with mesh refinement). This is proven in the aforementioned books, based on strain energy which is always underestimated in FEM.what do you mean when you saystiffness of an FEA model decreases.What is decreasing?

- #17

bob012345

Gold Member

- 1,842

- 802

Like this simple example of a 1D temperature profile of a rod of length 10 units where the red is the analytic solution, the points are the FEM numerical solution and the lines are the linear interpolations between the node values?What the OP meant is that FEM always converges from below (displacements and stresses increase with mesh refinement). This is proven in the aforementioned books, based on strain energy which is always underestimated in FEM.

Last edited:

- #18

FEAnalyst

- 307

- 129

Not really, such graphs are in fact commonly used to illustrate the concept of discretization. The plot showing what we discuss here (convergence from below) may look like this (from "A First Course in the Finite Element Method" by Logan):Like this simple example of a 1D temperature profile of a rod of length 10 units where the red is the analytic solution, the points are the FEM numerical solution and the lines are the linear interpolations between the node values?

- #19

bob012345

Gold Member

- 1,842

- 802

If I pick one point on my curve and plot the extrapolated solution vs. the number of elements used I get this where the red line is the exact value and the points are the extrapolated values between the node points for different numbers of elements;Not really, such graphs are in fact commonly used to illustrate the concept of discretization. The plot showing what we discuss here (convergence from below) may look like this (from "A First Course in the Finite Element Method" by Logan):

View attachment 298814

- #20

feynman1

- 435

- 29

same question from me, how to see the stiffness being reduced?I recently worked through the derivation of the FEA model (with the help of @pasmith , thanks) and I know the term stiffness matrix goes back to mechanical engineering application dealing with the literal stiffness of materials but what do you mean when you saystiffness of an FEA model decreases.What is decreasing?

For example I derived the node equations for the differential equation

$$ \frac{d^2u(x)}{dx^2} = -f(x)$$

I got for linear test functions ##N_1(x) = \frac{x_2 - x}{\Delta{x}}## and ##N_2(x) = \frac{x - x_1}{\Delta{x}}##

$$\frac{1}{\Delta{x}}\begin{pmatrix}

1 & -1 \\

-1 & 1

\end{pmatrix}

\begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \\

=

\begin{pmatrix}

- u_1' \\

u_2' \end{pmatrix} +

\begin{pmatrix}

\int_{x_1}^{x_2} N_1(x)f(x)\,dx \\

\int_{x_1}^{x_2} N_2(x)f(x)\,dx

\end{pmatrix}$$

So, as ##\Delta{x}## gets smaller, in what sense is the stiffness being reduced?

- #21

feynman1

- 435

- 29

many thanks really helpful, though it's not explained 100%, for all it says is that the solution isIt’s in the chapter4.3. Convergence of analysis results. Especially paragraph4.3.4. Properties of the Finite Element Solution.

- #22

FEAnalyst

- 307

- 129

I didn't have time to go through the whole section carefully but I think that the explanation lies in the equations presented there and the text that you are talking about is just a comment to these equations.many thanks really helpful, though it's not explained 100%, for all it says is that the solution isunderestimatedwithout saying why.

- #23

bob012345

Gold Member

- 1,842

- 802

I'm now looking for a comprehensible 2D example of a heated plate that I can learn and teach someone using the FEM analysis. I've looked but everything is either too complex and theoretical or uses triangular elements rather than square element which I need for a simple example by hand. Any suggestions would be appreciated.I didn't have time to go through the whole section carefully but I think that the explanation lies in the equations presented there and the text that you are talking about is just a comment to these equations.

- #24

hutchphd

Science Advisor

Homework Helper

2022 Award

- 5,525

- 4,705

- #25

bob012345

Gold Member

- 1,842

- 802

Thanks. That's a good suggestion but we did a couple different 1D examples both analytically solvable and now the student wants to see a 2D version.

- #26

hutchphd

Science Advisor

Homework Helper

2022 Award

- 5,525

- 4,705

- #27

bob012345

Gold Member

- 1,842

- 802

I know. I want to avoid anything like those boundary value problems in Grad school.

- #28

hutchphd

Science Advisor

Homework Helper

2022 Award

- 5,525

- 4,705

Life is messy. Physics a little less so.

- #29

feynman1

- 435

- 29

what can we imply from this derivation?Why not construct the stiffness matrix for [tex]

\frac{d^2u}{dx^2} = -f(x)[/tex] for [itex]x \in (0,1)[/itex] using elements [itex][nh, h(n+1)][/itex] with basis functions [tex]

\begin{split}

f_0 : [0,h] \to \mathbb{R} : x &\mapsto 1 - \frac{x}{h} \\

f_1 : [0,h] \to \mathbb{R} : x &\mapsto \frac{x}{h}

\end{split}[/tex] and see what happens as you reduce [itex]h[/itex]?

- #30

FEAnalyst

- 307

- 129

Check "A First Course in Finite Elements" book by Fish. You will find such example there.I'm now looking for a comprehensible 2D example of a heated plate that I can learn and teach someone using the FEM analysis. I've looked but everything is either too complex and theoretical or uses triangular elements rather than square element which I need for a simple example by hand. Any suggestions would be appreciated.

Share:

- Last Post

- Replies
- 7

- Views
- 623

- Replies
- 2

- Views
- 556

- Last Post

- Replies
- 3

- Views
- 542

- Replies
- 2

- Views
- 380

- Replies
- 0

- Views
- 340

- Last Post

- Replies
- 9

- Views
- 580

- Replies
- 3

- Views
- 287

- Last Post

- Replies
- 12

- Views
- 708

- Last Post

- Replies
- 9

- Views
- 657

- Last Post

- Replies
- 2

- Views
- 1K