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feynman1
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As is well known, the stiffness of an FEA model decreases with a refined mesh. What's the rigorous derivation of this?
Euhrm... sources?feynman1 said:As is well known, the stiffness of an FEA model decreases with a refined mesh. What's the rigorous derivation of this?
hutchphd said: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!
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Arjan82 said:This answer and the relation to the OP's question is completely lost to me...
no not shear locking in particular, though shear locking is due to stiffness being too highArjan82 said:Or does the OP mean shear locking?
many thanks, could you tell which chapter in Bathe's book?FEAnalyst said:Check the "Finite Element Procedures" by Bathe and "The Finite Element Method" by Hughes. There are mathematical proofs of that in these books.
It’s in the chapter 4.3. Convergence of analysis results. Especially paragraph 4.3.4. Properties of the Finite Element Solution.feynman1 said:many thanks, could you tell which chapter in Bathe's book?
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 stiffness of an FEA model decreases. What is decreasing?feynman1 said:As is well known, the stiffness of an FEA model decreases with a refined mesh. What's the rigorous derivation of this?
Is there a thread?bob012345 said:I recently worked through the derivation of the FEA model (with the help of @pasmith , thanks)
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;caz said:Is there a thread?
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.bob012345 said:what do you mean when you say stiffness of an FEA model decreases. What is decreasing?
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?FEAnalyst said: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.
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):bob012345 said: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?
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;FEAnalyst said: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
same question from me, how to see the stiffness being reduced?bob012345 said: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 stiffness 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?
many thanks really helpful, though it's not explained 100%, for all it says is that the solution is underestimated without saying why.FEAnalyst said:It’s in the chapter 4.3. Convergence of analysis results. Especially paragraph 4.3.4. Properties of the Finite Element Solution.
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.feynman1 said:many thanks really helpful, though it's not explained 100%, for all it says is that the solution is underestimated without saying why.
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.FEAnalyst said: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.
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.hutchphd said:I suggest that as an example you choose a problem that can be solved analytically and compare it to various FEM approaches . Start in 1D ?. Do this several times.
I know. I want to avoid anything like those boundary value problems in Grad school.hutchphd said:So go to 2D? Rectangular should be easy. Maybe circular? Hexagonal? (I don't know how nasty this will get). Nonorthogonal axis can be tricky.
what can we imply from this derivation?pasmith said: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]?
Check "A First Course in Finite Elements" book by Fish. You will find such example there.bob012345 said: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.
The mesh resolution in FEA models refers to the density of the mesh used to represent the geometry of the structure being analyzed. It is an important factor as it affects the accuracy and reliability of the results obtained from the analysis.
The stiffness of an FEA model is directly affected by the mesh resolution. A finer mesh with smaller element sizes can capture more details and irregularities in the structure, resulting in a more accurate representation of the stiffness. On the other hand, a coarser mesh with larger element sizes may not capture these details and can lead to a less accurate stiffness value.
Yes, using an excessively fine mesh resolution can lead to unrealistic stiffness values in an FEA model. This is because the model may capture very small features or imperfections that do not exist in the actual structure, resulting in an artificially high stiffness value.
There is no one optimal mesh resolution for all FEA models. The optimal resolution depends on the complexity and size of the structure being analyzed, as well as the accuracy required for the results. It is important to perform a mesh sensitivity analysis to determine the appropriate mesh resolution for a specific FEA model.
To improve the stiffness accuracy in an FEA model, it is important to use a mesh resolution that is appropriate for the structure being analyzed. Additionally, using higher-order elements, such as quadratic or cubic elements, can also improve the accuracy of the stiffness results. Performing a validation study with experimental data can also help to improve the accuracy of the stiffness values in an FEA model.