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error cpu

tags/CANDIDATE
gtheler 6 lat temu
rodzic
9a200940c4
commit
6fc14e32c1
3 zmienionych plików z 1027 dodań i 3 usunięć
Widok podzielony Pokaż statystyki zmian
  1. +499
    -0
      error-M-vs-cpu.svg
  2. +502
    -0
      error-MB-vs-cpu.svg
  3. +26
    -3
      nafems4.md

+ 499
- 0
error-M-vs-cpu.svg Wyświetl plik

@@ -0,0 +1,499 @@
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@@ -673,8 +673,31 @@ Two of the hundreds of different ways the infinite pressurized pipe can be solve
:::::


You can get both the exponential nature of each added bullet and how easily we can add new further choices to solve a FEM problem. And each of these choices will reveal you something about the nature of either the mechanical problem or the numerical solution. It is not possible to teach any possible lesson from every outcome in college, so you will have to learn them by yourself getting hands at them. I have already tried to address the particular case of the infinite pipe in a [recent report](https://www.seamplex.com/fino/doc/pipe-linearized/)^[<https://www.seamplex.com/fino/doc/pipe-linearized/>] that is worth reading before carrying on with this article.
You can get both the exponential nature of each added bullet and how easily we can add new further choices to solve a FEM problem. And each of these choices will reveal you something about the nature of either the mechanical problem or the numerical solution. It is not possible to teach any possible lesson from every outcome in college, so you will have to learn them by yourself getting hands at them. I have already tried to address the particular case of the infinite pipe in a [recent report](https://www.seamplex.com/fino/doc/pipe-linearized/)^[<https://www.seamplex.com/fino/doc/pipe-linearized/>] that is worth reading before carrying on with this article. The conclusions of the report are:

* Engineering problems ought not to be solved using black-boxes (i.e. privative software whose source code is not freely available)---more on the subject below in\ [@sec:two-materials].
* The pressurized infinite pipe has only one independent variable (the radius $r$) and one primary dependent variable (the radial displacement $u_r$).
* The problem has analytical solution for the radial displacement and the radial, tangential and axial stresses.
* There are no shear stresses, so these three stresses are also the principal stresses.
* Analytical expressions for the membrane and membrane plus bending stresses along any radial SCL can be obtained.
* The spatial domain can be discretized using linear or higher-order elements. In particular first and second-order elements have been used in this report.
* For the same number of elements, second-order grids need more nodes than linear ones, although they can better represent curved geometries.
* The discretized problem size depends on the number of nodes and not on the number of elements.
* The finite-element results for the displacements are similar to the analytical solution, with second-order grids giving better results for the same number of elements (this is expected as they involved far more nodes).
* The three stress distributions computed with the finite-element give far more reasonable results for the second-order case than for the first-order grid. This is qualitatively explained by the fact that first-order tetrahedra have uniform derivatives and such the elements located in both the external and external faces represent the stresses not at the actual faces but at the barycenter of the elements.
* Membrane stresses converge well for both the first and second-order cases because they represent a zeroth-order moment of the stress distribution and such the excess and defect errors committed by the internal and external elements approximately cancel out.
* Membrane plus bending stresses converge very poorly in first-order grids because the excess and defect errors do not cancel out because it is a first-order moment of the stress distribution.
* The computational effort to solve a given problem, namely the CPU time and the needed RAM depend non-linearly on various factors, but the most important one is the problem size which is three times the number of nodes in the grid.
* The error with respect to the analytical solutions as a function of the CPU time needed to compute the membrane stress is similar for both first and second-order grids. But for the computation of the membrane plus bending stress ([@fig:error-MB-vs-cpu]), first-order grids give very poor results compared to second-order grids for the same CPU time.

::::: {#fig:error-vs-cpu}
![Membrane $\text{M}$](error-M-vs-cpu.svg){#fig:error-M-vs-cpu width=48%}
![Membrane plus bending $\text{MB}$](error-MB-vs-cpu.svg){#fig:error-MB-vs-cpu width=48%}

Error in the computation of the linearised stresses vs. CPU time needed to solve the problem using FEM
:::::

The last two bullets lead to an issue that has come many times when discussing the issue of convergence with respect to the mesh size with other colleagues. There apparently exists a common misunderstanding that the number of _elements_ is the main parameter that defines how complex a FEM model is. I have seen a lot of graphs where a certain result is plotted against the number of elements

## ASME stress linearisation (not linearity!)

@@ -698,11 +721,11 @@ $$
$$
where the sign should be taken such that the resulting combined scalar stress (i.e. Tresca, Von Mises, Principal\ 1, etc.) represents the worst-case scenario.

No need to know or even understand these two integrals which for sure are not introduced to students in regular college courses. But it would be good to, as linearisation is a cornerstone subject for any serious mechanical analysis of pressurised components following the ASME code.
No need to know or even understand these two integrals which for sure are not introduced to students in regular college courses. But it would be good to, as linearisation is a cornerstone subject for any serious mechanical analysis of pressurised components following the code. So get a copy of ASME sec.\ VIII div.\ 2 annex\ 5-A and then search online for for “stress linearisation” (or “linearization”).



## Two (or more) materials
## Two (or more) materials {#sec:two-materials}

open source!
que hacen los programas? NADIE SABE

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