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User:OldYankee

Greetings. I am a retired Professor Of Information Management at Washington University in St. Louis where I taught various programming language, GUI Design, Web and Database integration and MS Office Automation classes for 19 years.

I also retired after 17 years at IBM and got bored rather quickly so I am now a Senior Web Admin for another Fortune 500 Company in the area. This new employer uses an internal Wiki to document most processes and procedures and to keep that documentation updated on a regular basis. Since I am relatively new to Wiki content generation, I decided to sign up here at the Wikiversity to see what I can learn from it. I will try to share and, where possible, improve the information I learn and put it here on my personal Wikiversity User Page.

title: User:OldYankee


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User:Egm6341.s11.team4.fields/HW4

HW4.8

title: User:Egm6341.s11.team4.fields/HW4


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User:Egm6341.s11.team4.fields/HW4.8

=Problem 4.8: Taylor, Trapezoidal and Simpson's Error Estimate Comparisons=

Refer to lecture slide [[media:nm1.s11.mtg22.djvu|22-2]] for the problem statement.

Given

 * {| style="width:70%" border="0" align="center"

(8.1)
 * $$\displaystyle I:=\int_{a}^{b}\underbrace{\frac{e^x-1}{x}}_{f(x)}dx$$
 *
 * }
 * }

For the Taylor Series Expansion, the max error in the integral is limited by the following from lecture [[media:nm1.s11.mtg7.djvu|7-2]]

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(8.2)
 * $$\displaystyle E_n \le \frac{e}{(n+1)!(n+1)}$$
 *
 * }
 * }

For the composite Trapezoidal rule, the max error is limited by the following from lecture [[media:nm1.s11.mtg22.djvu|22-1]]

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(8.3)
 * $$\displaystyle \vert E_n \vert \le \frac{(b-a)h^2}{12}M_2$$
 *
 * }
 * }

For the composite Simpson's rule, the max error is limited by the following from lecture [[media:nm1.s11.mtg22.djvu|22-2]]

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(8.4)
 * $$\displaystyle \vert E_n \vert \le \frac{(b-a)h^4}{2880}M_4$$
 *
 * }
 * }

Where

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 * $$\displaystyle a = -1$$

 * }
 * }

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 * $$\displaystyle b = 1$$

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 * }

and,

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 * $$\displaystyle h = \frac{(b-a)}{n}$$

 * }
 * }

Objective
1.a.) Find n such that En = O(10^-6) and compare to numerical results in HW2.4 using Taylor Series error.

1.b.) Find n such that En = O(10^-6) and compare to numerical results in HW2.4 using Composite Trapezoidal rule error.

1.c.) Find n such that En = O(10^-6) and compare to numerical results in HW2.4 using Composite Simpson's rule error.

2.a.) Numerically find the power of h in the Composite Trapezoidal rule error.

2.b.) Numerically find the power of h in the Composite Simpson's rule error.

Solution
Using Wolfram Alpha we find that the exact value of the integral to 12 significant digits is:

I = 2.11450175075

1.a.) Taylor Series Error Comparison
From equation (8.1) we can easily show that when n = 7 the error will be less than or equal to 8.4272 x 10^-6.

To compare the predicted value of the integral with the actual value of the integral we will use the following:

I = 2.11450175075

And from HW2.4:

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$$\displaystyle I_7 = \left[\sum_{i=1}^{7}\frac{x^{i}}{i!i}\right]_{x=-1}^{x=1} = 2.11450113379$$ (8.5)
 *
 * }
 * }

Now,

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$$\displaystyle E_{7} = |I - I_{7}| = |2.11450175075 - 2.11450113379| = 6.1696 \times 10^{-7}$$

 * }
 * }

In this case the actual error is approximately one order of magnitude less than the maximum possible error.

1.b.) Composite Trapezoidal Rule Error Comparison
From equation (8.3) with

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$$\displaystyle M_2 = max |f^{(2)}(x)| = e - 2 \approx 0.718281828459$$

 * }
 * }

We get

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$$\displaystyle En \le 7.3067 \times 10^{-6} \quad at \quad n = 256$$

 * }
 * }

From equation (1) on lecture slide [[media:nm1.s11.mtg7.djvu|7-4]] we get:

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$$\displaystyle I_{256} = h \left[\frac{1}{2}f_0 + f_1 + \cdots + f_{n-1} + \frac{1}{2}f_n\right] = 2.11450549301$$ (8.6)
 *
 * }
 * }

This compares well with

 * {| style="width:70%" border="0" align="center"

$$\displaystyle E_{256} = |I - I_{256}| = |2.11450175075 - 2.11450549301| = 3.7423 \times 10^{-6}$$ (8.7)
 *
 * }
 * }

1.c.) Composite Simpson's Rule Error Comparison
From equation (8.4) with

 * {| style="width:70%" border="0" align="center"

$$\displaystyle M_4 = max |f^{(4)}(x)| = 9e - 24 \approx 0.464536456131$$

 * }
 * }

We get

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$$\displaystyle En \le 1.2601 \times 10^{-6} \quad at \quad n = 8$$

 * }
 * }

From equation (4) on lecture slide [[media:nm1.s11.mtg7.djvu|7-4]] we get:

 * {| style="width:70%" border="0" align="center"

$$\displaystyle I_8 = \frac{h}{3} \left[f_0 + 4f_1 + 2f_2 + \cdots + 4f_{n-2} + 2f_{n-1} + f_n\right] = 2.11451145359$$ (8.8)
 *
 * }
 * }

However when we compare I8 with I we get an actual E8 that is greater than what the error equation tells us we should get. See below:

 * {| style="width:70%" border="0" align="center"

$$\displaystyle E_{8} = |I - I_{8}| = |2.11450175075 - 2.11451145359| = 9.70284 \times 10^{-6}$$ (8.9)
 *
 * }
 * }

2.a.) Composite Trapezoidal Rule Error Power of h Determination
Using equations (8.6) and (8.7) we construct the following two tables:

By graphing the above table with log(h) on the x-axis and log(En) on the y-axis with MS Excel and fitting a linear regression line across the points we get the graph below. As can be seen on the graph the slope of the line is approximately two which corresponds to the power of h in the composite Trapezoidal rule error equation.

2.b.) Composite Simpson's Rule Error Power of h Determination
Using equations (8.8) and (8.9) we construct the following two tables:

By graphing the above table with log(h) on the x-axis and log(En) on the y-axis with MS Excel and fitting a linear regression line across the points we get the graph below. As can be seen on the graph the slope of the line is approximately four which corresponds to the power of h in the composite Simpson's rule error equation.

This problem was solved by Erle Fields

title: User:Egm6341.s11.team4.fields/HW4.8


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User:Atcovi/Finn G.

Finn G. (December 23 2010- January 30 2011) was the tight general leader of the Soccer president of the lightest of the general stone. Finn G. learned for many reason to beat a nation person of coming threw soccer.

-

Finn G. was born in Germany in 2003-2002. When he grew older, Finn G. was moving threw in 2005-2006-2007-2008 from both years to assemble the created for him moving to The United states of America for the government for to live at 2010. When he went to soccer, He started practicing for 2 years too for stilling for his life for soccer, In real of soccer to finish his test of soccer to know more about the time. When Finn G. became president of soccer, he was know for the 1st president of soccer t know how much to bring more justice in his society,

Because when Finn G. turn more older to make his society great. Finn G. was in Maybeury Mallard Elementary School. His Historic of Germany and to the republic for his missing of birth to turn older to make his Soccer designing for his society  to add for his for the society to make better time. Next, what really happen next when his Society for his name. Next Finn G. was naming to be for his helping of soccer, His nickname was named "Leading of the republic of Soccer also for vice president of Aaqib A.

title: User:Atcovi/Finn G.


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User:Eml5526.s11.team6.tupsakhare/hwk4

=Problem 4.4=

GIVEN
Referring to figure 4.4.1 given below for a cylindrical circular bar in torsion -

Strong Form -

\displaystyle      \frac{d}\left( {JG\frac{d\phi}} \right) + m = 0 $$    With $$ \displaystyle 0 \leqslant x \leqslant l $$ $$\displaystyle Eq. 4.4.1 $$ Natural Boundary Condition at x = l :

M (x = l ) = $$ \displaystyle \left( {JG\frac{d\phi}} \right)_{}^{x = l} = \overline M =12 $$ $$\displaystyle Eq. 4.4.2 $$ Essential Boundary Condition at x = 0 : $$ \displaystyle \phi \left( {x = 0} \right) = \overline \phi =4 $$ $$\displaystyle Eq. 4.4.3 $$ Where,

m : Distributed moment per unit length M : Torsion Moment $$ \displaystyle \phi $$ : Angle of rotation , G : Shear modulus J : Polar moment of inertia = $$ \frac{\pi C^2}{2} $$ $$\displaystyle Eq. 4.4.4 $$ C : Radius of the cylindrical bar

FIND
a) To construct the weak form for the strong form of the circular torsion bar as given in $$\displaystyle Eq. 4.4.1 $$ b) To assume that m(x) = 0 and to integrate the differential equation given in $$\displaystyle Eq. 4.4.1 $$

c) To find the integration constants using the boundary conditions given in $$ \displaystyle Eq. 4.4.2$$ and $$ \displaystyle Eq. 4.4.3$$,

Developing the weak form
Let us assume a weighting function w (x) such that $$ \displaystyle w (x=0) = 0 $$. The condition is applied on w (x) as w (x) is chosen arbitrarily and needs to vanish at x = 0 where the essential boundary condition is applied.

Hence, now the weak form becomes -

$$ \displaystyle \frac{d}\left( {JG\frac{d\phi}} \right) + m = 0  $$

$$ \displaystyle i.e.$$ $$ \displaystyle \frac{d}\left( {J(x)G(x)\frac{d\phi}} \right) + m(x) = 0 $$ $$\displaystyle Eq. 4.4.5 $$ Note - Here we assume that J, G and m are functions of x and are NOT constants.

Integrating by parts for the first term in $$ \displaystyle Eq. 4.4.5 $$ as per the integration rule -

$$ \displaystyle   \int\limits_a^b {udv = } \left[ {uv} \right]_a^b - \int\limits_a^b {vdu} $$ $$\displaystyle Eq. 4.4.6 $$

and assuming -  $$ \displaystyle \  u = w (x)$$    and  $$ \displaystyle \   v = \left[ {\left( {JG\frac} \right) } \right] $$  ,we get

$$ \displaystyle \left[ {w(x)\left( {JG\frac} \right)} \right]_0^l - \int\limits_0^l {\frac\left( {JG\frac} \right)} dx + \int\limits_0^l {w(x).m} dx = 0 $$

$$ \displaystyle \left[ {w(x)\left( {JG\frac} \right)} \right]_{}^{x = l} - \left[ {w(x)\left( {JG\frac} \right)} \right]_{x = 0}^{} - \int\limits_0^l {\frac\left( {JG\frac} \right)} dx + \int\limits_0^l {w(x).m} dx = 0 $$ $$\displaystyle Eq. 4.4.7 $$ Now as per the choice of the weight function, $$ \displaystyle /w (x= 0) = 0 $$. Hence the second term in the $$ \displaystyle Eq. 4.4.7 $$ vanishes.

Also using $$ \displaystyle Eq. 4.4.2 $$, the first term in $$ \displaystyle Eq. 4.4.7 $$ can be replaced by $$ \displaystyle {\text{w}}\left( l \right).\overline M $$

Hence after applying the sign changes, the weak form of $$ \displaystyle Eq. 4.4.1 $$ becomes -

Integration of the strong form to get the exact solution
We use the general 1-Dimensional Model Data set 1 from Mtg 9 - Page 9-2 The genearal 1-Dimensional Model Data set 1 given above provides us to assume that $$ \displaystyle \ m (x) = 5x $$

Hence the strong form given in $$ \displaystyle Eq. 4.4.1 $$ converts to the following -

1 . $$ \displaystyle \frac{d}\left( {JG\frac} \right)+ 5x=0 $$

With  $$ \displaystyle\ 0 \leqslant x \leqslant l   $$ $$\displaystyle Eq. 4.4.9 $$ Integrating $$ \displaystyle Eq. 4.4.9 $$ with respect to x, in the limits 0 to l, we get, with constant C involved as  -

$$ \displaystyle 1.\left[ {\left( {JG\frac} \right)} \right]_0^1 + \left[ { {\frac} } \right]_0^1 = C $$

$$ \displaystyle \left[ {\left( {JG\frac} \right)} \right]_{}^{x = 1} - \left[ {\left( {JG\frac} \right)} \right]_{x = 0}^{} + \left[ { {\frac} } \right] = C $$

$$ \displaystyle \overline M - \left[ {\left( {JG\frac} \right)} \right]_{x = 0}^{} + \left[ { {\frac} } \right] = C $$ $$\displaystyle Eq. 4.4.10 $$ Now as per data set 1 as explained above, $$\overline M = 12 $$ and $$ \displaystyle \ JG = 2+3x $$

Since $$\Phi$$ is constant at x=0,$$\left(\frac{d\phi}{dx}\right)_{x=0}^{} = 0 $$

With x=0 we get JG =2. Substituting, we get

Integrating step 2 of above $$ \displaystyle Eq. 4.4.10 $$ again with respect to x using integration by parts as given in $$ \displaystyle Eq. 4.4.6 $$ with constant D involved, we get -

$$ \displaystyle \int \left( {JG\frac} \right)dx + \int \left[\frac {5x^2}{2} \right] = \int \left[ {C} \right] + D $$ $$ \displaystyle \left[ {JG\phi } \right]_0^1 - \int\limits_0^1 {\frac{d}} (JG).\phi dx + \left[ \frac{5x^3}{6} \right]_0^1 = \left[ Cx \right]_0^1 + D $$ $$ \displaystyle \left[ {JG\phi } \right]_{}^1 - \left[ {JG\phi } \right]_0^{} - \int\limits_0^1 {\frac{d}{dx} JG.\phi }dx + \left(\frac{5}{6}\right) - \left(C\right) = D $$

Now using $$ \displaystyle Eq. 4.4.3 $$ with $$ \displaystyle \phi \left( {x = 0} \right) = \overline \phi=4$$ and $$ \displaystyle JG = 2+3x $$ ,we can update above equation as - $$ \displaystyle \left[ {\phi } \right]_{}^1\left[2+3\right] - 8   - \int\limits_0^1 {3\phi}dx + \left(\frac{5}{6}\right) - \left(C\right) = D $$ $$\displaystyle Eq. 4.4.12 $$

$$ \displaystyle \left[ {\phi } \right]_{}^l\left[5\right] - 8   - 3\left[ {\phi } \right]_{}^1 + \left(\frac{5}{6}\right) - \left(C\right) = D $$ Where, C is given by $$\displaystyle Eq. 4.4.11 $$

$$ \displaystyle \left[ {\phi } \right]_{}^l\left[2\right] - 8  + \left(\frac{5}{6}\right)-{12+ 2 \left( \frac{d\phi }{dx} \right)_{x = 0}^{} - \left[ { {\frac} } \right]}  = D $$

The integration constants
Thus, the integration constants as derived in b) are as follows -

=Problem 4.7=

Given
Introduction to Calculix - A non-linear Finite Element Analysis Code

Calculix is an open source Finite Element Analysis code with the authors acknowledging that naming conventions and input style formats for CalculiX are based on those used by ABAQUS, a proprietary and commercial, general purpose finite element code but the results obtained from CalculiX are in no way connected to ABAQUS.

Website for informational and software download (Linux executable) purposes - http://dhondt.de/ Website for informational and software download (Windows executable) purposes - www.bconverged.com/calculix

Calculix in all has 2 major modules, namely- 1. CGX ( Calculix Graphics Module ) - This program is designed to generate and display finite elements (FE) and results coming from CalculiX CrunchiX (ccx).The Concept and File Format sections give some background on functionality and mesher capabilities. The Getting Started section describes how to run the verification examples you should have obtained along with the code of the program.

2. CCX (Calculix CrunchiX (Solver) Module) - This is a theoretical section giving some background on the analysis types, elements, materials etc. Then, an overview is given of all the available keywords in alphabetical order, followed by detailed instructions on the format of the input deck. If CalculiX does not run because your input deck has problems, this is the section to look at. The last section contains a description of the verification examples you should have obtained along with the code of the program. If you try to solve a new kind of problem you haven't dealt with in the past, check this section for examples. You can also use this section to check whether you installed CalculiX correctly (if you do so with the compare script and if you experience problems with some of the examples, please check the comments at the start of the corresponding input deck). Finally, the User's Manual ends with some references used while writing the code.

Find
a) Installation of the Calculix Graphics Module and to explain the installation process for layman(CGX) b) To access the CGX manual c) To sign up with used group for asking the queries,if any and to gain access to the archives where we can access the email conversation from various users,a source of possible learning d) To reproduce the basic inbuilt examples from CGX data base installed on your machine, with a detailed report, about entities such as - 1. Dics 2. Cylinder 3. Sphere 4. Sphere Volume 5. Airofoil The report shall include explanation to all the CGX commands used to generate above example geometries along with their meshes using screen-shots of the useful explanatory steps.

For Linux
For a naiver Linux user now using Windows, we strongly recommend using Ubuntu as your linux operating system which can be easily downloaded and installed at Ubuntu

To install Calculix for Linux, start by downloading the source code and examples available at http://dhondt.de/cgx_2.2.all.tar.bz2. Right now you need open a terminal by selecting that from Applications as in the following picture.

Before installation, you should check openGL on your system by typing the following words in your terminal:

find /usr/* -name libGL* -print

And the following results show that you have the openGL-library which is needed.

/usr/lib/mesa/libGL.so.1 /usr/lib/mesa/libGL.so.1.2 /usr/lib/libGLU.so.1 /usr/lib/libGLU.so.1.3.070701

Another thing you need is c++ complier since we have to compile the source code. You can check where is your c++ complier by typing

which c++

If you don't have these essential things, please typing the following to download and install them

sudo apt-get install linux-kernel-headers sudo apt-get install build-essential

Then we start to install cgx.

1.Unpack the file cgx_2.2.all.tar.bz2 in "/usr/local". An easy way is first exacting the compressed file using your right mouse menu and then copy to the destination.

2.Then enter the folder where the source code located by typing

cd /usr/local/CalculiX/cgx_2.2/src/

3.Run "make" as a root and copy the binary "cgx" to "/usr/local/bin/cgx" with

cp cgx /usr/local/bin/cgx

4.Run "latex cgx" in "/usr/local/CalculiX/cgx_2.2/doc" for three times and then run "latex2html cgx".

5.Now your CalculiX has been installed and please enjoy

6.In order to check the installation, type

cgx -b dummy.fbd

and you can see a window appear on your screen as following.

Note:

1.If you are counting any problems for being unable to find any *.h, you can type the following to ensure that you have install all the essential libraries before building the source codes

sudo apt-get install build-essential libsdl1.2debian libsdl1.2-dev libgl1-mesa-dev libglu1-mesa-dev libsdl-image1.2 libsdl-image1.2-dev

For Windows
To install CalculiX for Windows, start by downloading the executable file available at http://www.bconverged.com/download.php.

Before installing CalculiX, we recommend that you create a folder that you will want to use as your workspace for CalculiX files.

Access the zip file, CalculiX_2_2_win_002.zip, and run the executable CalculiX_2_2_win_002.exe. Review the license agreement and select "I agree" when you are prompted to as shown below.

The next screen will ask you to select the directory to which Calculix will be installed. You may use the default directory or click "Browse" to select the directory of your choosing. Once you have the desired directory in the "destination folder" field (1) click the "next" button (2) to continue.

You will now be shown a prompt which allows you to choose which components and/or extensions of the program you wished to have installed. It is recommended to use the default settings unless you are an advanced user. Click "Install" to begin the installation.

When the installation begins you will be asked to specify a folder or directory to use as your workspace. Choose the folder you created prior to starting the installation on the file tree and click "OK"

CalculiX will now begin installing onto your computer. When completed you will be presented with the screen below. Specify if you'd like to view the help file by checking or un-checking the corresponding box and then click "Finish" to complete the installation process.

Congratulations !! You have completed installing CalculiX on your Windows based computer.

Access to CGX manual
To locate the user manual after installation of the software, begin by clicking on the windows start button. All programs will open and select the folder bConverge. This folder will open with all of the calculix programs listed, however to access the manual double-click on the help icon inside of the bConverged folder. This action will open a new window containing several folders. Expand the Calculix folder and then the CGX folder. This will bring you to the Calculix User's Manual -Calculix GraphiX, Version 2.2- as shown below.

To navigate through the manual any of the blue links in the initial index will take the user to that section of the the manual as shown in the above picture. Pressing ctrl+f will allow the user to find a particular word within the manual as well. At the top of the manual, there are four buttons Next, Up, Previous, and Contents as shown below. The function of these buttons is presented below.

Next: Flips to the next page of the manual. Up: Returns to the previously viewed screen before a link was clicked. Previous: Flips to the previous page of the manual. Contents: Takes the user to the index of links of all contents in the manual.

Additional help while Getting Started- The Calculix User Group
Student can join the yahoo user group dedicated to the forums related to knowledge sharing about Calculix. In the very friendly environment of these forums, the student can present their difficulties and queries and can get benefited by the knowledge of many other Calculix users. Student will be given a choice of joining this user group while in some final steps of the software installation. Even if the student misses to join it at the time of installation, he/she can join the group later using help on Calculix or from Calculix website online. With the help of this user group, the student will be able to have access to the email archives of the group members which at times can become useful to answer at least some of the queries or for informational purposes.

Reproducing the basic inbuilt examples from CGX- A report
Here we present the steps, codes, explanation to all the CGX commands used in the codes to generate the sample inbuilt example geometries in CGX database after installation, along with their meshes using screen-shots of the useful explanatory steps. Please refer to the CGX manual (as explained in the above section) for more details.

Figure
Following figure 4.7.1 shows the geometry and the mesh generated by CGX for a disc in four 90 degree segments-

Geometry generation code
Following figure 4.7.2 shows the code to generate the geometry and the mesh by CGX for the disc in Figure 4.7.1. Explanation of the commands used is given in the succeeding subsection.

Explanation of the commands used
This subsection provides the explanation of the CGX commands used in Figure 4.7.2

PNT
The format for entering a point in CGX is as follows -

'pnt' |'!' [  ]|

[  ]|

[  ]|

[<setname(containing nodes)>]

This keyword is used to define or redefine a point. There are four possibilities to define a point.

To define a point just with coordinates:

pnt p1 11 1.2 34

or,

pnt ! 11 1.2 34

where the name is chosen automatically.

It is also possible to create points on a line or in the direction from P1 to P2 by defining a spacing (ratio) and number-of-points:

pnt ! L1 0.25 3

or

pnt ! P1 P2 0.25 3

This will create 3 new points at the positions 0.25, 0.5 and 0.75 times the length of the line or the distance from P1 to P2.

And it is also possible to create points on the positions of existing nodes. The command

pnt ! set

will create new points on the positions of the nodes included in the specified set.

Usually when points are defined interactive the command qpnt is used. Please refer to the CGX manual for its details

LINE
The format for entering a Line is as follows-

'line' <name(char<9)>|'!' <p1> <p2> <cp|seq>

This keyword is used to define or redefine a line. A line depends on points.

A line can only be defined if the necessary points are already defined. There are three different types of lines available.

The straight line -

line l1 p1 p2 4

is defined by: its name l1 (the name could have up to 8 characters), by the points p1 and p2 and optionally by the division.

The arc

line ! p1 p2 cp 4

needs a center point cp. The radius changes linear from p1 to p2 if the center-point cp is excentric. The name is chosen automatically (triggered by the character !).

The spline

line l1 p1 p2 seq 4

needs a so called sequential-set, seq (use the command seqa to define such a set). This set seq stores the spline points between the end-points in the right order. The spline function is described in [14]. Usually, a line is defined interactively with qlin.

GSUR
The format for entering a surface is as follows-

'gsur' <name(char<9)>|'!' '+|-' 'BLEND| ' '+|-' <line|lcmb> '+|-' -> <line|lcmb> .. (3-5 times)

This keyword is used to define or redefine a surface in the most basic way (see also qsur). Each surface must have three to five lines or combined lines (see lcmb) to be mesh-able. However, the recommend amount of edges is four. For example,

gsur S004 + BLEND - L002 + L00E + L006 - L00C

will create the surface S004 with a mathematically positive orientation indicated by the + sign after the surface name.

The keyword BLEND indicates that the interior of the surface will be defined according to reference [12] given as- S. A. Coons, 'Surfaces for computer-aided design of space forms'. Project MAC, MIT (1964). Revised to MAC-TR-41 (1967) or a NURBS surface is referenced.

Use a + or - in front of the lines or lcmbs to indicate the orientation. These signs will be corrected automatically if necessary. If automatic name generation is desired, then use ! instead of a name.

ELTY
The format used to assign a specific element type to a set of entities and generating the mesh, is as follows-

'elty' [ ] ['be2'|'be3'|'tr3'|'tr3u'|'tr6'|'qu4'|-> 'qu8'|'he8'|'he8f'||'he8i'|'he8r'|'he20'|'he20r']

This keyword is used to assign a specific element type to a set of entities (see section Element Types in the appendix). The element name is composed of the following parts: The leading two letters define the shape (be: beam, tr: triangle, qu: quadrangle, he: hexahedra), then the number of nodes and at last an attribute describing the mathematical formulation or other features (u: unstructured mesh, r: reduced integration, i: incompatible modes, f: fluid element for ccx). If the element type is omitted, the assignment is deleted. If all parameters are omitted, the actual assignments are posted:

elty

will print only the sets with assigned elements. Multiple definitions are possible. For example,

elty all

deletes all element definitions. If the geometry was already meshed, the mesh will NOT be deleted. If the mesh command is executed again after new assignments has taken place, additional elements will be created.

elty all he20

assigns 20 node brick-elements to all bodies in the set all.

elty part1 he8 <p? redefines that definition for all bodies in the set part1.

elty part2 qu8

assigns 8 node shell elements to all surfaces in set part2.

Figure
Following figure 4.7.3 shows the geometry and the mesh generated by CGX for a cylinder in four 90 degree segments-

Geometry generation code
Following figure 4.7.4 shows the code to generate the geometry and the mesh by CGX for the cylinder in Figure 4.7.3.

Explanation of the commands used is given in the succeeding subsection.

Explanation of the commands used
Please refer to the 'Disc' section above for explanation of all the following CGX commands used to build the geometry and mesh of the cylinder Following commands are used to generate the geometry and mesh of the cylinder in figure 4.7.3 -

' PNT  For entering a point - (refer to the 'Disc' subsection above for explanation)

' LINE  For entering a Line - (refer to the 'Disc' subsection above for explanation)

' GSUR  For entering a surface - (refer to the 'Disc' subsection above for explanation)

' ELTY  To assign a specific element type to a set of entities and generating the mesh - (refer to the 'Disc' subsection above for explanation)

Figure
Following figure 4.7.5 shows the geometry and the mesh generated by CGX for a sphere created from 32 blended surfaces.-

Geometry generation code
Following code was used to generate the geometry by the CGX for the sphere in Figure 4.7.5.

Explanation of the commands used is given in the succeeding subsection.

Explanation of the commands used
Please refer to the 'Disc' section above for explanation of all the following CGX commands used to build the geometry and mesh of the sphere Following commands are used to generate the geometry and mesh of the sphere in figure 4.7.5 -

' PNT  For entering a point - (refer to the 'Disc' subsection above for explanation)

' LINE  For entering a Line - (refer to the 'Disc' subsection above for explanation)

' GSUR  For entering a surface - (refer to the 'Disc' subsection above for explanation)

' ELTY  To assign a specific element type to a set of entities and generating the mesh - (refer to the 'Disc' subsection above for explanation)

Figure
Following figure 4.7.8 shows the geometry and the mesh generated by CGX for a sphere volume segment

Geometry generation code
Following figure 4.7.9 shows the code to generate the geometry and the mesh by CGX for the cylinder in Figure 4.7.8.

Explanation of the commands used is given in the succeeding subsection.

Explanation of the commands used
Please refer to the 'Disc' section above for explanation of all the following CGX commands used to build the geometry and mesh of the cylinder Following commands are used to generate the geometry and mesh of the cylinder in figure 4.7.3 -

' PNT  For entering a point - (refer to the 'Disc' subsection above for explanation)

' LINE  For entering a Line - (refer to the 'Disc' subsection above for explanation)

' GSUR  For entering a surface - (refer to the 'Disc' subsection above for explanation)

' ELTY  To assign a specific element type to a set of entities and generating the mesh - (refer to the 'Disc' subsection above for explanation)

Figure
Following figure 4.7.9 and figure 4.7.10 show the geometry and the mesh generated by CGX for an airfoil using many 4 nodes element-

Geometry generation code
Following codes show how to generate the geometry and the mesh by CGX for the cylinder in Figure 4.7.l3.

Explanation of the commands used is given in the succeeding subsection.

Explanation of the commands used
Please refer to the 'Disc' section above for explanation of all the following CGX commands used to build the geometry and mesh of the airfoil

Following commands are used to generate the geometry and mesh of the airfoil in figure 4.7.9

' PNT

For entering a point - (refer to the 'Disc' subsection above for explanation)

' LINE

For entering a Line - (refer to the 'Disc' subsection above for explanation)

' GSUR

For entering a surface - (refer to the 'Disc' subsection above for explanation)

' ELTY

To assign a specific element type to a set of entities and generating the mesh - (refer to the 'Disc' subsection above for explanation)

' SEQA

'seqa' ['pnt' .. <=>]|              ['afte'|'befo'  .. <=>]|              ['end' .. <=>]

This keyword is used to create or redefine a set marked as a sequential set. This set is used to define splines (see line, with the command qlin such a set is automatically created). To begin such a set type for example:

SEQA Q003 PNT P004 P005 P006 P00M P00N

Because of the parameter PNT the program will create or overwrite the set Q003. The command will continue in the next line if the sign = is found:

SEQA Q003 PNT P004 P005 P006 = P007 P008 P009

The parameter AFTE will insert additional points after the first specified point. The parameter BEFO will insert additional points before the first specified point and the parameter END will add additional points to a sequential set.

' LCMB

'lcmb' <name(char<9)>|'!' ['+|-' '+|-' '+|-' -> ..(up to 14 lines)]| ['ADD' '+|-' '+|-' -> '+|-' ..(up to 14 lines)]

This keyword is used to define, extend or redefine a combined line (lcmb). Combined lines are necessary if the edge of a surface should be made of more than one line. Usually the user does not create lcmb's directly. They are created automatically during the process of defining a surface with the command qsur. There is no limitation to the number of lines in a combined line. However with one command, not more than 14 lines can be specified at a time. To specify more than that or to extend an existing lcmb a modify command has to follow. For example,

lcmb U260 + U249 - U248 - U247 - U243 - U237 - U236 - U231 - U219

defines the lcmb U260 with 8 lines and their orientation in the lcmb. The following command

lcmb U260 ADD - U218 - U217

extends the lcmb U260 by two additional lines.

' SETA

'seta' '!'|'n'|'e'|'p'|'l'|'c'|'s'|'b'|'S'|'L'|'se' <name ..> | ['n'|'e' '-' ]

This keyword is used to create or redefine a set (see also qadd). All entities like points or bodies and so on must be stored at least in one set to be reachable. The set all is created automatically at startup and will be open (see seto) all the time unless explicitly closed (see setc). To add points to the set dummy type:

seta dummy p p1 p2

This will add the points p1 and p2 to the set. The following entities are known:

Nodes n, Elements e, Faces f, Points p, Lines l, Surfaces s, Bodies b, Nurb Surfaces S, Nurb Lines L and names of other sets se. If other sets are added then the contents are NOT automatically added. Please use also comp do to add the contents.

The program will automatically determine the type of the entities if not specified, but then the names must be unique. More than one name can be specified. A minus sign between two names specifies a range of names with steps of steps.

With the '!' key instead of a setname the program generates automatic sets with system defined setnames and stores entities in it. The only case were this can be used is to separate independent meshes stored in the basic set. The single independent meshes are then referenced by new setnames:

seta ! all

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User:Eml5526.s11.t1.ab/HW4

=Problem 5: Various Basis Functions satisfying the Constraint Breaking Solution=

Given

 * a) Constraint Breaking Solution which needs to be satisfied is $$b_{0}(\beta)\neq 0$$ and $$b_{i}(\beta)=0$$, i=1,2,3,....,n.
 * b) Consider $$\Omega=[-2,4]$$

Find
For $$\Gamma _{g}=\left \{ \beta\right \}$$ and each of the following Families of Basis Functions Find the Corresponding Basis Functions Satisfying the Given Constraint Breaking Solution. Plot original and required basis function for $$j=1,2,3$$
 * 5.1 $$F_{p}:=\left \{ x^{j}, j=0,1,2,...\right \}$$.
 * 5.2 $$F_{c}:=\left \{ \cos(jx), j=0,1,2,... \right \}$$.
 * 5.3 $$F_{s}:=\left \{ 1,\sin(jx), j=1,2,... \right \}$$.
 * 5.4 $$F_{f}:= \left \{\cos(jx),j=0,1,2,...\sin(jx),j=1,2,...\right\}$$
 * 5.5 $$F_{e}:=\left \{ e^{jx},j=0,1,2,.. \right \}$$.
 * 5.6 Show that $$\left \{ e^{jx},j=0,1,2,.. \right \}$$ is Linear Independent on $$\Omega$$.

5.1 Find the Corresponding Polynomial Basis Function $$F_{p}:=\left \{ x^{j}, j=0,1,2,...\right \}$$ Satisfying the Given Constraint Breaking Solution.
If we choose $$b_{i}\left ( x \right )=\left(x-\beta\right)^{j}$$, it satisfies the given Constraint Breaking Solution as below,

$$b_{0}\left(\beta\right )=\left ( x-\beta \right )^{0}=1\neq 0$$

&

$$b_{j}\left(\beta\right )=\left ( x-\beta \right )^{j}=0, j=1,2,...$$

So the Polynomial Basis Function satisfying the given constraint breaking solution is,

 * {| style="width:100%" border="0"

$$b_{j}\left ( x \right )=\left \{\left(x-\beta\right)^{j}, j=0,1,2,...\right \}$$ Plot Between the original and the required basis function is plotted in the MATLAB using the following code.
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * $$\displaystyle (Eq. 5.1) $$
 * style= |
 * }

Appendix
 Matlab Code:

5.2 Find the Corresponding Cosine Basis Function $$F_{c}:=\left \{ \cos(jx), j=0,1,2,... \right \}$$ Satisfying the Given Constraint Breaking Solution.
If we select $$b_{j}\left ( x \right )=\cos(j(x-\beta)), j=0$$

It will yield, $$b_{0}\left ( \beta\right)=\cos(0)=1\neq0$$

and further

$$b_{j}\left ( x \right )=\cos(\frac{\pi}{2}(2j+1)\frac{x}{\beta}), j=1,2,...$$

it will yield,

$$b_{j}\left ( x \right )=\cos(n\frac{\pi}{2})=0$$

So the Cosine Basis Function satisfying the given constraint breaking solution is,

 * {| style="width:100%" border="0"

$$b_{j}\left ( x \right )=\left \{\cos(j(x-\beta)), j=0\right \}$$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |

&

$$b_{j}\left ( x \right )=\left \{\cos(\frac{\pi}{2}(2j+1)\frac{x}{\beta}), j=1,2,...\right \}$$ Plot of the original and the required Basis Function is plotted in the MATLAB using the following code
 * $$\displaystyle (Eq. 5.2) $$
 * style= |
 * }

Appendix
 Matlab Code:

5.3 Find the Corresponding Sine Basis Function $$F_{s}:=\left \{1,\sin(jx), j=1,2,... \right \}$$ Satisfying the Given Constraint Breaking Solution.
If we select,

$$b_{j}\left ( x \right )=\sin(j\pi\frac{x}{\beta}), i=1,2,...$$

It yields,

$$b_{j}\left ( x \right )=\sin(n\pi)=0$$

So the Sine Basis Function satisfying the given constraint breaking solution is,

 * {| style="width:100%" border="0"

$$b_{j}\left ( x \right )=\left \{1,\sin(j\pi\frac{x}{\beta}), j=1,2,...\right \}$$
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * $$\displaystyle (Eq. 5.3) $$
 * style= |
 * }

Appendix
 Matlab Code:

===5.4 Find the Corresponding Fourier Basis Function $$F_{f}:=\left \{\cos(jx),j=0,1,2,... \sin(jx),j=1,2,...\right\}$$ Satisfying the Given Constraint Breaking Solution.===

If we select, According to $$\displaystyle (Eq. 5.2)$$

$$b_{j}\left ( x \right )=\cos(j(x-\beta)), j=0$$

It will yield, $$b_{0}\left ( \beta\right)=\cos(0)=1\neq0$$

and further

$$b_{j}\left ( x \right )=\cos(\frac{\pi}{2}(2j+1)\frac{x}{\beta}), j=1,2,...$$

it will yield,

$$b_{j}\left ( x \right )=\cos(n\frac{\pi}{2})=0$$

Also,

If we select, According to $$\displaystyle (Eq. 5.3)$$,

$$b_{j}\left ( x \right )=\sin(j\pi\frac{x}{\beta}), i=1,2,...$$

It yields,

$$b_{j}\left ( x \right )=\sin(n\pi)=0$$

Eventually,

The Fourier Basis Function Satisfying the given Constraint Breaking Solution is,

 * {| style="width:100%" border="0"

$$b_{j}\left ( x \right )=\left \{1,\cos(\frac{\pi}{2}(2j+1)\frac{x}{\beta}), j=1,2,...,
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |

\sin(j\pi\frac{x}{\beta}), j=1,2,...\right \}$$ Plot of the original and the required Basis Function is plotted in the MATLAB using the following code.
 * $$\displaystyle (Eq. 5.4) $$
 * style= |
 * }

Appendix
 Matlab Code:

===5.5 Find the Corresponding Exponential Basis Function $$F_{e}:=\left \{ e^{jx},j=0,1,2,... \right\}$$ Satisfying the Given Constraint Breaking Solution.===

If we select,

$$b_{j}\left ( x \right)=\left \{e^{jx\beta}, i=0\right \}$$

and further,

$$b_{j}\left ( x \right)=\left \{e^{j(x-\beta)}-1, i=1,2,...\right \}$$

Eventually, The Fourier Basis Function Satisfying the Given Constraint Breaking Solution is,

 * {| style="width:100%" border="0"

$$b_{j}\left ( x \right )=\left \{1,e^{j(x-\beta)}-1, i=1,2,...\right \}$$ Plot of the original and the required Basis Function is plotted in the MATLAB using the following code.
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * style="width:2%; padding:1px; border:2px solid #FF0000" |
 * $$\displaystyle (Eq. 5.5) $$
 * style= |
 * }

Appendix
 Matlab Code:

5.6 Show that $$\left \{ e^{jx},j=0,1,2,.. \right \}$$ is Linear Independent on $$\Omega$$.
We have,

$$F=\begin{bmatrix} 1 & e^x & e^{2x} \end{bmatrix}$$

Calculating $$\Gamma (F)$$ for the given interval $$\Omega [-2,4]$$,we get

$$\Gamma (F)=\int_{-2}^{4}\begin{bmatrix} 1 & e^{x} &e^{2x} \\ e^{x} &e^{2x} & e^{3x}\\ e^{2x} & e^{3x} & e^{4x} \end{bmatrix}$$

Therefore,

$$\Gamma (F)=\begin{bmatrix} 6 & e^{4}-e^{-2} &(e^{8}-e^{-4})/2 \\ e^{4}-e^{-2} &(e^{8}-e^{-4})/2 & (e^{12}-e^{-6})/3\\ (e^{8}-e^{-4})/2 & (e^{12}-e^{-6})/3 & (e^{16}-e^{-8})/4 \end{bmatrix}$$

To check the Linear Independent Behavior we need to check the Determinant of this matrix. So calculating the determinant of this matrix in the MATLAB Using the Following code,

Appendix
 Matlab Code:

The Determinant arrives as $$1.1145\times (10)^{9}$$ which is a Non-zero number.

So we can say that, $$\left \{ e^{jx},j=0,1,2,.. \right \}$$ is Linear Independent on $$\Omega$$.

title: User:Eml5526.s11.t1.ab/HW4


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User:Egm6341.s11.team4.kurth/hw4

Homework 4

=Problem 4.7: Composite Simpson Rule Error Boundary=

Refer to lecture slide [[media:Nm1.s11.mtg22.djvu|22-2]] for problem assignment.

Objective
Prove that the error when using the composite Simpson rule to approximate the integral of a function, $$\displaystyle f$$, from a to b is bound by

where

Solution
From equation [[media:Nm1.s11.mtg7.djvu|(4)p.7-4]]

Breaking (7.2) into discrete intervals yields

Now applying the simple Simpon's error theorem presented on lecture slide [[media:Nm1.s11.mtg18.djvu|18-2]] to each interval:

Next, define $$\displaystyle \overline{M_4}$$

(7.7) becomes

Since

Substituting (7.11) into (7.9)

 * {| style="width:95%" border="0" align="center"

$$\displaystyle $$ $$
 * style="width:35%" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * E_n^S| \leq \frac{(b-a)h^4}{180}M_4 = \frac{(b-a)^5}{180n^4}M_4
 * $$\displaystyle
 * style= |
 * }
 * }

Note: This solution differs from the problem statement but is confirmed by Atkinson pp.257&258

Solved by William Kurth.

=Problem 4.10: Error Convergence for the Trapezoidal and Simpon's Rules=

Given
Consider the exact integrals

and

Objectives

 * 1.) Create a table showing the approximations of (10.1) by using the Trapezoidal rule with successively doubling values of n, as well as the corresponding error involved in the approximation. Also, the table should include the ratio of successive errors to show the effect of doubling the value of n on the error.

 * 2.) Create a table showing the approximations of (10.1) by using Simpson's rule with successively doubling values of n, as well as the corresponding error involved in the approximation. Also, the table should include the ratio of successive errors to show the effect of doubling the value of n on the error.

 * 3.) Create a table showing the errors of approximating (10.2) by using both the Trapezoidal and Simpon's rules, as well as the ratio of successive errors to show the effect of doubling the value of n on the error.

Calculation of the True Values of the Integrals
Wolfram Alpha was used to compute the exact values of (10.1) and (10.2):

Approximation of the Integral of $$f(x) = e^x sin(x)$$ by Trapezoidal Rule
The following MATLAB code was used to calculate the approximation of (10.1) for increasing values of n using the Trapezoidal rule, as well as the associated approximation errors and the ratios of successive approximation errors.

As can be seen in this table the error is quartered when the interval size is halved, indicating that the error indeed scales with $$ O(h^2)$$.

Approximation of the Integral of $$f(x) = e^x sin(x)$$ by Simpon's Rule
The following MATLAB code was used to calculate the approximation of (10.1) for increasing values of n using Simpson's rule, as well as the associated approximation errors and the ratios of successive approximation errors.

As can be seen in this table that as the interval size is halved the error is reduced by a factor of sixteen, indicating that the error indeed scales with $$ O(h^4)$$.

Error of Approximating the Integral of $$ f(x) = x^{3.7}$$ Using Both Trapezoidal and Simpon's Rules
The following MATLAB code was used to calculate the approximation of (10.2) for increasing values of n using both the Trapezoidal and Simpson's rules, as well as the associated approximation errors and the ratios of successive approximation errors for each method.

As can be seen in this table that as the interval size is halved the error associated with the Trapezoidal rule is quartered and the error associated with Simpson's rule is reduced by a factor of sixteen, indicating that the errors indeed scale with $$ O(h^2)$$ and $$ O(h^4)$$, respectively.

Solved by William Kurth.

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User:Nczempin

http://en.wikipedia.org/wiki/User:Nczempin

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File:Gantt Chart HelioDrive versie 2.pdf

Licensing
for school project


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File:Plan van Aanpak HelioDrive1.pdf

Licensing
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File:Samenwerkingscontract HelioDrive.pdf

Licensing
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File:Work Breakdown Structure Heliodrive.pdf

Licensing
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Engineering Experience 4: Design a Small Solar Vehicle/Team PM12

Illumination

Teamleader
Tang Yihan Hlds final responsible for the whole of the project. Follow-up of the action list and make sure all team members carry out their assignments on time. Watches the deadlines and takes action in case they are endangered.

Secretary
Bleumers Jan Writes the meeting reports Dates the wiki page Leads the meetings (prepares agenda…)

Lead engineer
Qian Hongzhu Responsible for engineering tasks and planning Manages materials and budget.

Meeting rules
We will have at least have one meeting per week. All team members are required to be on time for these meetings and to have prepared all agreed topics/actions. All decisions will be reached with consent of all member, all members are thus required to give input on all topics/ideas. There is no time limit on the duration of a meeting. The team secretary is responsible for creating and distributing the meeting report to all team members within 2 days after the meeting took place. Each team member then has the chance to reply to this to comment on the contents of the report. These comments will be incorporated in the report by the secretary. At the start of the following meeting, the final report has to be approved by the whole team. At the end of each meeting an action list will be drafted and each item will become a team member’s responsibility. This list will be updated and discussed in the following meeting.

Wiki rules
The team secretary updates the blog and wiki page at least once a week. The wiki page will contain the most recent action list.

Absences
Everybody should attend every activity of the team (and be on time). If someone is not present, that person should inform the team leader of the reason. If someone is late, he/she has to buy the rest of the team a drink.

Sanctions
When we encounter troubles or problems and have a difference of opinion, we will vote to get the best solution. In this process, we avoid unnecessary fights with each other.

Communication with the coach
We will collect all the problems and pose them to our coach during the next class.

Client
The client in this project is the Umicore Solar Team.

Contractor
The contractor is Illumination, a fictious company founded as a requirement for this project. It consists out of the students of group 12 of  the Engineering Experience 4 course at GroupT. The company ‘employees’ are: Teamleader:		Yehan Tang Secretary:		Jan Bleumers Team members:	Haowen Sun, Hongzhu Qian, Zhengyuan Huang, Yehan Tang, Jan Bleumers

Project description
The purpose of this project is to develop a miniature, toy version of the Umicore Solar Team’s solar car. This toy car must run on solar power without the use of batteries, it’s behaviour has to be predicted and compared with the actual Umicar (on an energetic level) and has   to   take part in a race. To aid us in the development, a number of seminars will be given at GroupT, consisting out of a general presentation of the project description, a presentation given by the Umicore solar team, a presentation by Fablab about the possibilities of their workshop, a seminar on     Simulink and a seminar on technical drawing.

Approach of the project
The project is split into 3 chapters: Engineering, Enterprising and educating. The total budget for the whole project is € 200, a solar panel and a DC motor. In the engineering part of this project we will design a small solar car that will operate solely on solar power. We will design and build a  chassis, gear train and electronics, and will use solar panel and DC motor which GroupT borrowed us for the purpose of this project. We will also simulate the race to calculate the ideal transmission ratio and perform a freeroll simulation using Simulink (this data will be compared  to the results of a real life test held in week 9). The drive shaft of the SSV will be analysed, and 4 Sankey diagrams will be composed: 1 estimated diagram early in the project, 1 after the freeroll test and 2 for the actual umicar. We will create a 2D technical drawing of the SSV frame. In the enterprising chapter of the project we will perform market research to identify competitors, design a complete business plan, potentially rename and re-logo our company, create a web page and engrave the logo on the SSV. To manage the budget, a cleartable of expenditures will be kept, accompanied by proofs of payment. In the educating chapter we will create a ‘report of solutions’, containing the calculations and solutions to the engineering and enterprising questions/problems. We will also write an ‘online process report’ containing all documents made in the analysis and orienting phase, the story behind the design and creation of the SSV (like a blog), a temporary and final description of the design, and a final process report.

Deadlines:
By the end of week 1 we will present a Cooperation contract, plan of approach, WBS and Gantt chart. In case of ‘no go’ we will present an  amended copy of the required documents at the end of week 2. At the end of week 6 we will deliver the ‘Case SSV part 1’, Simulink and fill out PA 1. Depending on the weather conditions, we will test the SSV in week 9. Somewhere in the course of week 9 through 14 we will race the SSV (depending on weather conditions). In week 11 we will hand in ‘Case SSV part 2’ and fill out PA 2.

Project limitations:
The SSV will only be able to drive forwards, not backwards. There will be no steering installed as the race is held on a straight track with a guidance rail.

Expected result
We expect to build a functioning SSV (within the specifications of the client) which is capable of completing and possibly winning the final race.

General administration
Meeting lead:Yehan Tang Secretary:Jan Bleumers Meeting start:14u Meeting end:16u hours:2u

Present
Hongzhu Qian, Yehan Tang,  Zhengyuan Huang,  Haowen sun,  Jan Bleumers

Agenda
1.   Discuss and assign team roles + name company 2.	Create ‘Cooperation contract’ 3.	Create ‘Plan of approach’ 4.	Create WBS 5.	Create Gantt chart 6.	Create Logo 7.	Deadline this Friday 2011/02/18

Discussed topics
1.	Discuss and assign team roles + name company After deliberation, teamroles were divided as follows

The Gantt Chart
== Meeting Report ===== Meeting report 1 : 16/02/2011 =======  General administration ====  Meeting lead:Yehan Tang  Secretary:Jan Bleumers  Meeting start:14u  Meeting end:16u  hours:2u ===== Present =====  Hongzhu Qian,  Yehan Tang,  Zhengyuan Huang,  Haowen sun,  Jan Bleumers==== Agenda ====  1. Discuss and assign team roles + name company 2. Create ‘Cooperation contract’ 3. Create ‘Plan of approach’ 4. Create WBS 5. Create Gantt chart 6. Create Logo 7. Deadline this Friday 2011/02/18 ==== Discussed topics ====1. Discuss and assign team roles + name company       After deliberation, teamroles were divided as follows:        Team leader:	Yehan Tang        Secretary:	Jan Bleumers        Lead engineer:	Hongzhu Qian        Our company name will be: ‘Illumination’ 2. Create ‘Cooperation contract’       The Cooperation Contract is finished and is signed by all current team members. 3.	Create ‘Plan of approach’        The Plan of Approach is ready and will be reviewed by all members by Friday. ]http://en.wikiversity.org/wiki/User:Illumination Team leader:	Yehan Tang Secretary:	Jan Bleumers Lead engineer:	Hongzhu Qian Our company name will be: ‘Illumination’

2.	Create ‘Cooperation contract’ The Cooperation Contract is finished and is signed by all current team members.

3.	Create ‘Plan of approach’ The Plan of Approach is ready and will be reviewed by all members by Friday.

4.	Create WBS The WBS has been started and will be finished by Friday.

5.	Create Gantt chart After the WBS has been approved, it will be used as a basis for the Gantt chart.

6.	Create Logo Zhengyuan will design a company logo.

7.	Deadline this Friday 2011/02/18 All 4 above documents have to be handed in by 18:00h on Friday February 18th. Yehan will e-mail the team coach to ask how the documents have to be handed in.

General administration
Meeting lead:	 Yehan Tang Secretary: 	Jan Bleumers Meeting start:		14u Meeting end:		16u
 * 1) hours: 		2u

Present
Hongzhu Qian, Yehan Tang,  Zhengyuan Huang,  Haowen sun,  Jan Bleumers, Willem Noyen, Yared Reda

Agenda
1.	Update from coach about the 4 delivered documents 2.	New team member 3.	Discuss general concept of the car 4.	Case SSV part 1 5.	Case Simulink

Discussed topics
1.	Plan of Approach, Cooperation contract, WBS and Gantt are approved, only the actual making of the car and spent hours had to be adjusted. This has been done.

2.	Yared Reda joined the team today, his info had been entered in the Cooperation Contract.

3.	The car will have 3 wheels, 2 on the front axle, which will be the driving axle, and 1 on the rear. Questions about steering, cvt and material choice remain open.

4.	For Case SSV part 1 we have to make a Sankey Diagram and calculate the best gear ratio by hand.

5.	For case Simulink we have to simulate the SSV in the race and make a graph of different gear ratio’s against race time.

Next Meeting
08/03/2011 after seminar and coach meeting

General administration
Meeting lead:	 Yehan Tang Secretary: 	Jan Bleumers Meeting start:		14：30u Meeting end:		16u
 * 1) hours: 		1.5u

Present
Hongzhu Qian, Yehan Tang,  Zhengyuan Huang,  Haowen sun,  Jan Bleumers, Yared Reda

Absent
Willem Noyen (medical reasons)

Agenda
1.	Progress on “Case SSV part 1” 2.	SSV transmission

Discussed topics
1.	Due to to many unknowns, the sankey diagram, gear ratio and Simulink have not yet been completed. We received some good info from our coach concerning these topics and will continue working on them.

2.	After realizing the difficulty of creating a CVT (continuously variable transmission), we decided to go for a normal gearing system (gears) and to try to lower friction and losses as much as possible.

Next Meeting
15/03/2011 after coach meeting

General administration
Meeting lead:Yehan Tang Secretary:Jan Bleumers Meeting start:15u Meeting end:16u hours:1u

Present
Hongzhu Qian, Yehan Tang,  Zhengyuan Huang,  Haowen sun,  Jan Bleumers, Yared Reda, Willem Noyen

Agenda
1.	Progress on “Case SSV part 1” 2.	SSV transmission 3.	Solar panel mounting 4.	Chassis 5.	Process Report

Discussed topics
1.	We received more info concerning the 3 big parts of the Case SSV part 1 from our coach.

2.	We decided to buy our gearbox as making the gears ourselves is too hard. A home-made gearbox will also be less efficient as an expert-made one. If the precise ratio can not be bought we will buy one close to it and adjust the wheel diameter to correct.

3.	We will mount the panel with the aid of 3 or 4 telescopic antennas, that way the panel is adjustable in every direction.

4.	For the chassis we will use a plate of plexi-glass cut in the appropriate shape. We will discuss how thick the plate should be and what the shape is going to be.

5.	Everyone has to write their part of the process report and send it to Yehan. Jan will then join all parts together and check spelling+grammar.

Next Meeting
Monday 22/03/2011 - 18:00h - cafetaria

General administration
Meeting lead:Yehan Tang Secretary:Jan Bleumers Meeting start:14u Meeting end:15u30 hours:1u30

Present
Hongzhu Qian, Yehan Tang,  Zhengyuan Huang,  Haowen sun,  Jan Bleumers， Yared Reda

Absent
Willem Noyen (Medical reasons)

Agenda
1. Progress on “Case SSV part 1” 2.	SSV frame 3.	Process Report 4.	SSV parts needed 5.	Peer Assessment 1

Discussed topics
1.	Today we received feedback on all 3 parts of ‘SSV part 1’. We are almost ready to turn the report in on Friday.

2.	The frame of our SSV will be lasercut out of plexiglass. In the coming weeks we will design a shape in Solid Edge (or another program) and take it to Fablab to be cut.

3.	The process report is finished.

4.	We need following parts for our SSV: bearings for the driving and driven axles, a gearbox/transmission, a driving and driven axle, wheels, 4 antenna’s for the solar panel.

5.	All members should fill in Peer Assessment 1 this week!

Next Meeting
To be deceided by team leader

General administration
Meeting lead:Yehan Tang Secretary:Jan Bleumers Meeting start:14u Meeting end:15u30 hours:1u30

Present
Hongzhu Qian, Yehan Tang,  Zhengyuan Huang,  Haowen sun,  Jan Bleumers， Yared Reda

Absent
Willem Noyen (Medical reasons)

Agenda
1.	Progress on “Case SSV part 1”

Discussed topics
1.	Adapt sankey, internal resistance of motor: 3.2 ohm. Should be less ‘other losses’ then.

Action list
{| class="wikitable" ! Task           !! Responsable !! In cooperation with: !! End date!! Finished?|-
 * Case SSV part 1 ||   yihan     ||       All            ||         ||          |}

Next Meeting
To be deceided by team leader

Solar Panel-Specs

 * }

"Asc corrected" and "I-corr" are the values after correction for the spectrum of solar light versus spotlight (* 0,88/0,3) Peak power is at 7V and 0,91A : 6,37W


title: File:Sampling.1.B.Basics.pdf


text: Category:Resources last modified in March 2011


text: Category:Swahili

title: Category:Swahili


text:
User:Eml5526.s11.team7.jin/HW4

=Problem 4.5: Basis Functions Satisfying Constraint Breaking Solution=

Given
Possible basis functions are given as:

Polynomial: $${{F}_{P}}=\left\{ {{x}^{j}},\text{ }j=0,1,2,\ldots \right\}$$

Cosine: $${{F}_{C}}=\left\{ \cos jx,\text{ }j=0,1,2,\ldots \right\}$$

Sine: $${{F}_{S}}=\left\{ 1,\text{ }\sin jx,\text{ }j=1,2,\ldots \right\}$$

Fourier: $${{F}_{F}}=\left\{ \cos jx,\text{ }j=0,1,2,\ldots ;\text{ }\sin kx,\text{ }k=1,2,\ldots \right\}$$

Exponential: $${{F}_{E}}=\left\{ {{e}^{jx}},\text{ }j=0,1,2,\ldots \right\}$$

For $${{\Gamma }_{g}}=\left\{ \beta \right\}$$, and each of the above families of basis functions, i.e., $${{\Gamma }_{I}}=\left\{ {{b}_{j}}(x),\text{ }j=0,1,\ldots \right\},\text{ }I\in \left\{ P,C,S,F,E \right\}$$

Find
Find corresponding family $${{\bar{F}}_{I}}=\left\{ {{{\bar{b}}}_{j}} \right\}$$ satisfying constraint breaking solution (CBS), i.e., $${{b}_{0}}(\beta )\ne 0$$ and $${{b}_{i}}(\beta )=0,\text{ }i=1,2,\ldots ,n$$. Let $$\Omega =\left] \alpha ,\beta \right[=\left] -2,4 \right[$$.

1) $${{\bar{b}}_{0}}(x)=\text{const}$$, plot $${{b}_{j}}$$ and $${{\bar{b}}_{j}}$$ for $$j=1,2,3$$.

2) Show $$\left\{ {{e}^{jx}},\text{ }j=0,1,2 \right\}$$ linear independent on $$\Omega $$.

Part 1
The corresponding family $${{\bar{F}}_{I}}=\left\{ {{{\bar{b}}}_{j}} \right\}$$ satisfying CBS:

Polynomial:

$${{F}_{P}}=\left\{ {{\left( x-4 \right)}^{j}},\text{ }j=0,1,2,\ldots \right\}$$

Cosine:

$${{F}_{C}}=\left\{ 1,\text{ }\cos \left( j\left( x-4 \right)+\frac{\pi }{2} \right),\text{ }j=1,2,\ldots \right\}$$

Sine:

$${{F}_{S}}=\left\{ 1,\text{ }\sin \left( j\left( x-4 \right) \right),\text{ }j=1,2,\ldots \right\}$$

Fourier:

$${{F}_{F}}=\left\{ 1,\text{ }\cos \left( j\left( x-4 \right)+\frac{\pi }{2} \right),\text{ }j=1,2,\ldots ;\text{ }\sin \left( k\left( x-4 \right) \right),\text{ }k=1,2,\ldots \right\}$$

Exponential:

$${{F}_{E}}=\left\{ 1,\text{ }{{e}^{j\left( x-4 \right)}}-1,\text{ }j=1,2,\ldots \right\}$$

The following figures show the polynomial basis functions (j=1,2,3):

The following figures show the cosine basis functions (j=1,2,3):

The following figures show the sine basis functions (j=1,2,3):

The following figures show the fourier basis functions (j=1,2,3):

The following figures show the exponential basis functions (j=1,2,3):

Part 2
The exponential basis function is $${{b}_{j}}=\left\{ {{e}^{jx}},\text{ }j=0,1,2 \right\}$$,

Thus the Gram matrix is:

$$\Gamma =\left[ \begin{matrix} 6 & {{e}^{4}}-{{e}^{-2}} & \frac{1}{2}\left( {{e}^{8}}-{{e}^{-4}} \right) \\ {{e}^{4}}-{{e}^{-2}} & \frac{1}{2}\left( {{e}^{8}}-{{e}^{-4}} \right) & \frac{1}{3}\left( {{e}^{12}}-{{e}^{-6}} \right) \\ \frac{1}{2}\left( {{e}^{8}}-{{e}^{-4}} \right) & \frac{1}{3}\left( {{e}^{12}}-{{e}^{-6}} \right) & \frac{1}{4}\left( {{e}^{16}}-{{e}^{-8}} \right) \\ \end{matrix} \right]$$

And the determinant of Gram matrix is:

$$\det \left( \Gamma \right)=1.1145\times {{10}^{9}}\ne 0$$

Thus, the exponential basis function is linear independent on $$\Omega $$.

Authors
Jiang Jin

title: User:Eml5526.s11.team7.jin/HW4


text:
User:Adamnaadiya

$$\frac{3}{4}$$

title: User:Adamnaadiya


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User:Eml5526.s11.team6.kurth/HW4

For Windows
To install CalculiX for Windows, start by downloading the executable file available at http://www.bconverged.com/download.php.

Before installing CalculiX, we recommend that you create a folder that you will want to use as your workspace for CalculiX files.

Access the zip file, CalculiX_2_2_win_002.zip, and run the executable CalculiX_2_2_win_002.exe. Review the license agreement and select "I agree" when you are prompted to as shown below.

The next screen will ask you to select the directory to which Calculix will be installed. You may use the default directory or click "Browse" to select the directory of your choosing. Once you have the desired directory in the "destination folder" field (1) click the "next" button (2) to continue.

You will now be shown a prompt which allows you to choose which components and/or extensions of the program you wished to have installed. It is recommended to use the default settings unless you are an advanced user. Click "Install" to begin the installation.

When the installation begins you will be asked to specify a folder or directory to use as your workspace. Choose the folder you created prior to starting the installation on the file tree and click "OK"

CalculiX will now begin installing onto your computer. When completed you will be presented with the screen below. Specify if you'd like to view the help file by checking or un-checking the corresponding box and then click "Finish" to complete the installation process.

Congratulations you have completed installing CalculiX on your Windows based computer.

Figure
Following figure 4.7.5 shows the geometry and the mesh generated by CGX for a sphere created from 32 blended surfaces.-

Geometry generation code
Following code was used to generate the geometry by the CGX for the sphere in Figure 4.7.5.

Explanation of the commands used is given in the succeeding subsection.

Explanation of the commands used
Please refer to the 'Disc' section above for explanation of all the following CGX commands used to build the geometry and mesh of the sphere Following commands are used to generate the geometry and mesh of the sphere in figure 4.7.5 -

' PNT  For entering a point - (refer to the 'Disc' section for explanation)

' LINE  For entering a Line - (refer to the 'Disc' section for explanation)

' GSUR  For entering a surface - (refer to the 'Disc' section for explanation)

' ELTY  To assign a specific element type to a set of entities and generating the mesh - (refer to the 'Disc' section for explanation)

title: User:Eml5526.s11.team6.kurth/HW4


text:
User:Egm6321.f10.team2.oztekin/Newwww

=Problem 4.7 :Calculix user guide and examples=

Starting Calculix
First Step: Download

Calculix is a open source finite element suite. It has its own pre and post processor so by this way user can prepare the model and solve by defining boundry conditions in calculix. Ths solver, CCX, was written and is maintained by Guido Dhondt.The pre and post processor, CGX, was written and is maintained by Klaus Wittig. You can download the latest version from Convergent Mechanical Solutions website.Convergent has dona an excellent job by integrating an installation wizard with the distribution which makes the installation process straightforward.

 * For downloading calculix for Windows operating system : http://www.bconverged.com/products.php

Follow the on screen prompts. User will be prompted to select a working folder.

 * The source code,Linux builds,documentation,tests,and other resources for Calculix : http://www.calculix.de/

Second Step: Test

Before starting Calculix we need to accomplish one more step.

 * {| style="width:100%" border="0"

$$  \displaystyle [Start]--[All Programs]--[bConverged]--[Calculix]--[Test Calculix *.*] $$
 * style="width:95%" |
 * style="width:95%" |
 * }

Using Calculix
To open Calculix go to

 * {| style="width:100%" border="0"

$$  \displaystyle [Start]--[All Programs]--[bConverged]--[Calculix]--[Calculix Command] $$
 * style="width:95%" |
 * style="width:95%" |
 * }

We have just opened Calculix command window. Command window was opened automatically in the working direction that we have defined in installation process. Now we should be familiar with the program parameters.

 * {| style="width:100%" border="0"

$$  \displaystyle Command: cgx [-a|-b|-c|-duns2d|-duns3d|-isaac2d|-isaac3d|-foam|-step|-stl]  filename $$
 * style="width:95%" |
 * style="width:95%" |
 * }

Now we can open one example. Examples can be download from Calculix website. Fortunately Converged already added examples in the Calculix file. So examples can be found in the directory of Converged.

Since we are in the working directory download examples or copy all the examples to our working directory to open them. Or change the directory into directory where your examples placed. And type ' cgx -b sphere.fbd '. Impartant !!!: Always touch on graph screen and leave the mouse before writing commands. Arrange a good place for your command window so you can see your commands on your screen.

Command line $$ \displaystyle \Rightarrow $$ C:\..........\examples\basic>cgx -b sphere.fbd

By using mouse you can rotate, zoom in out on the figure. Rotation of the object is controlled by the left mouse button, zoom in and out by the middle mouse button and translation of the object is controlled by the right mouse button. Try these options of mouse and get used to oriantate the figure.

Example: Learning with example
Lets learn more by doing example on Calculix. We can broaden our knowledge on commands by doing examples. We can take modelling disc as an example.

 * Run Calculix Command
 * Type cgx -b discmodel.fbd (You probably get an error message. Because there is no exists file name: discmodel)

To start with modelling we have to remember the construction procedure. First we will construct nodes, lines, surfaces, and the bodies. Lets look at point command.

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'pnt' & |'!' [  ]| \\ & [   ]| \\                             &  [   ]| \\ & []
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

 * To type command please remember to press and release to the graph window with left button.
 * type pnt y  -0.00000        1.00000        0.00000  (recognizing that py is a dummy with three coordinates on xyz plane)
 * type pnt P0  -0.00000       -0.00000        0.00000
 * type pnt P001 0.70711       -0.00000       -0.70711
 * type pnt P003 -0.00000      -0.00000       -1.00000
 * type pnt P005 -0.70711      -0.00000       -0.70711
 * type pnt P006 -1.00000      -0.00000        0.00000
 * type pnt P009 -0.70711       -0.00000       0.70711
 * type pnt P00A 0.00000       -0.00000        1.00000
 * type pnt P00G 0.70711       -0.00000        0.70711
 * type pnt P00I 1.00000       -0.00000       -0.00000

Probably for now you can not see point attributes. To see complete points with attributes check new command.

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'plot' & ['n'|'e'|'f'|'p'|'l'|'s'|'b'|'S'|'L']&['a'|'d'|'p'|'q'|'v'] -> \\ & 'w'|'k'|'r'|'g'|'b'|'y'|'m'
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

 * type plot pa all (get the same view with graph)

 * {| style="width:100%" border="0"

$$  \displaystyle Command: 'line' |'!'
 * style="width:95%" |
 * style="width:95%" |

$$
 * }

 * Type line L001 P00I P001 p0 4

As you see above we just created arc for a portion of disc. Even if the command's name is line we also create arcs with the same command. L001 is a dummy name. P00I and P001 are limits. p0 is the center. 4 is division. To see the line attributes

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'plus'  & ['n'|'e'|'f'|'p'|'l'|'s'|'b'|'S'|'L']&['a'|'d'|'p'|'q'|'v'] -> \\ & 'w'|'k'|'r'|'g'|'b'|'y'|'m'|'i'
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

 * Type plus la all

Plus has the same job with plot. Plus keeps other visuals where plot filters.

 * Type line L002 P001 P003 p0 4
 * Type line L003 P003 p0 8
 * Type line L004 p0 P00I 8
 * Type line L005 P003 P005 p0 4
 * Type line L006 P005 P006 p0 4
 * Type line L007 P006 p0 8
 * Type line L009 P006 P009 p0 4
 * Type line L00A P009 P00A p0 4
 * Type line L00C P00A p0 8
 * Type line L00G P00A P00G p0 4
 * Type line L00I P00G P00I p0 4

To finish model we need surfaces.

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'gsur' &|'!' '+|-' 'BLEND| ' '+|-'  '+|-' -> \\ &
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

This keyword is used to define or redefine a surface. Surfaces are defined with boundary lines.

 * Type gsur A001 + BLEND - L003 - L002 - L001 - L004
 * Type gsur A002 + BLEND - L007 - L006 - L005 + L003
 * Type gsur A003 + BLEND - L00C - L00A - L009 + L007
 * Type gsur A004 + BLEND + L004 - L00I - L00G + L00C

The last part of this example we create meshes. Elements tr3,qu4, and qu8 are for meshing surfaces and he8 and he20 to mesh bodies.

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'elty' &[ ] ['be2'|'be3'|'tr3'|'tr3u'|'tr6'|'qu4'|-> \\ &'qu8'|'he8'|'he8f'||'he8i'|'he8r'|'he20'|'he20r']
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

 * Type elty all QU4
 * Type mesh all

We just have meshed the surface with an elements of qu4.

 * Type plot m all

Basic Models

title: User:Egm6321.f10.team2.oztekin/Newwww


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User:Piotrus/Teaching materials

I would like to initiate sharing of freely-licensed teaching materials from sociology and related fields. This is a test page to see how it goes. For now, I list resources I can share in my section below, and I would like to invite all readers to create their own sections on this page.

Materials from Piotrus (Piotr Konieczny)
Those are courses I have taught and for which I have a lot of materials (slides, syllabus, and such). I am more than happy to send my materials to you; in exchange I'd simply request that you share your own materials under the same, free license, and list them here. To request the materials, email me (there is also an "e-mail this user" link to the right).

 * Global society: an intro to sociology courses with a globalization focus (syllabi and lecture slides for 4-month and 1.5-month duration, readings)
 * Intro to sociology (syllabi and lecture slides for 4-months)
 * Sociology of everyday life (recitation slides mostly)
 * Sociology of the family (syllabus and lecture slides for 1.5 month duration)
 * Social research methods (syllabus and lecture slides for 4 month duration course)
 * Social theory (now developing)
 * Sociology of the Internet (now developing)
 * Statistics (now developing)

I am also a Wikipedia Ambassadors, meaning I am glad to help you figure out how to use wikis and Wikipedia in teaching.

You may read some of my papers, including one on "Teaching with Wikipedia" ('07, updated version in the works), and download a workshop-presentation on "Teaching with Wikipedia" (WikiSym'10) from the links here.

Materials from (you?)
Add your materials here. Then copy this section for the next person.

If you have no clue how to edit a Wiki, join the millions who know how by doing the Wikipedia tutorial.

Please create a Wikiversity account (will work on Wikipedia and Wikibooks, too) if you don't have one, and then go to "My preferences" and activate your email so you can be contacted :)

Any material you add here should be freely-licenced (what does it mean?). Freely-licensing your material is very easy, you can just add "Licenced under CC-BY-SA-3.0" to all your syllabi, slides and such.

Other materials
In addition to downloadable (or emailable) materials we are shared above, you may find the following pages useful.
 * School:Sociology
 * wikibooks:Subject:Sociology

title: User:Piotrus/Teaching materials


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User:Eml5526.s11.team4.yang/hw4

=Problem 4.7: CGX Project and Tutorial=

From the lecture slide Mtg23

Problem Statement

 * 1) install cgx.
 * 2) Read manual; sign up with user group to ask questions if any, also access archive.
 * 3) Reproduce the basic examples:disk, cylinder, sphere, sphere-volume, airfoil.
 * 4) Write a report for "dummies": explain how to install and run cgx.

Final Report
Installation and run:
 * 1) Download the windows-version of Calculix from its website, and then install it on computer.
 * 2) Run the CalculiX Command: [Start] [Programs] [bConverged][CalculiX] [CalculiX Command]
 * 3) Reproduce the basic examples

A. The “disc” problem
1. First switch to the directory of working ‘E:\Documents\ccx’.

2. Then input the command ‘cgx –b disc.fbd’, there will appear another command “CalculiX Graphix”.

3. Input the command” pnt [  ]” and then plot all the points with the command ‘plot p all b’. For example, ”PNT py  0.0  1.0 0.0” is used to define a point which is located at (0,1,0).

PNT py     -0.00000        1.00000        0.00000

The figure is showed below.

4. Input the command” line    ” and the “gsur '+|-'BLEND| ' '+|-'  '+|-' -> .. (3-5 times)and then plot all the lines and surfaces with command ‘plus’. For example, “LINE L001 P00I P001 p0 4 ” is used to define a arc line which pass through points P00I and P001 and its center point is p0 and radius is 4.

LINE L001 P00I P001 p0 4

GSUR A001 + BLEND - L003 - L002 - L001 - L004

The figure is showed below.

5. You can use mouse on the Graphix command to rotate and zoom in or out the figure.

6. Input the command ‘ELTY all QU4’ to assign the element type and then ‘mesh all’ to mesh the geometry, and then left-click on the marginal area on the Graphix, then on ’Viewing’ and continue to ‘show all elements with light’ and ‘Toggle Element edges’, as showed in following figures

B. The “cylinder” problem
1. First switch to the directory of working ‘E:\Documents\ccx’.

2. Then input the command ‘cgx –b cylinder.fbd’, there will appear another command “CalculiX Graphix”.

3. Input the command” pnt [  ]” and then plot all the points. Input the command” line    ” and the “gsur '+|-'BLEND| ' '+|-'  '+|-' -> .. (3-5 times)and then plot all the lines and surfaces. The figure is showed below

4. Use the command “seta |'n'|'e'|'p'|'l'|'c'|'s'|'b'|'S'|'L'|'se' <name ..> ['n'|'e' <name” to define some sets to be used in further loading operations. For example, “SETA p1 p p1” is to define a set of points which contains point p1.

5. You can use mouse on the Graphix command to rotate and zoom in or out the figure.

6. Input the command ‘ELTY all QU4’ and ‘mesh all’, and then left-click on the marginal area on the Graphix, then on ’Viewing’ and continue to ‘show all elements with light’ and ‘Toggle Element edges’, as showed in following figure.

C. The “sphere” problem
1. First switch to the directory of working ‘E:\Documents\ccx’

2. Then input the command ‘cgx –b sphere.fbd’, there will appear another command “CalculiX Graphix”.

3. Input the command” pnt [<x> <y> <z>]” and then plot all the points with the command ‘plot p all b’. For example, ”PNT py  0.0  1.0 0.0” is used to define a point which is located at (0,1,0). The figure is showed below.

4. Input the command” line <p1> <p2> <cp|seq> ” and the “gsur '+|-'BLEND| ' '+|-' <line|lcmb> '+|-' -><line|lcmb> .. (3-5 times)and then plot all the lines and surfaces with command ‘plus’. For example, “LINE L001 P00I P001 p0 104 ” is used to define a arc line which pass through points P00I and P001 and its center point is p0 and radius is 104. The figure is showed below.

5. Input the command ‘ELTY all QU4’ and ‘mesh all’, and then left-click on the marginal area on the Graphix, then on ’Viewing’ and continue to ‘show all elements with light’ and ‘Toggle Element edges’, as showed in following figure.

D. The “sphere-volume” problem
1. First switch to the directory of working ‘E:\Documents\ccx’

2. Then input the command ‘cgx –b sphere-volu.fbd’, there will appear another command “CalculiX Graphix”.

3. Input the command” pnt [<x> <y> <z>]” and then plot all the points. Input the command” line <p1> <p2> <cp|seq> ” and the “gsur '+|-'BLEND| ' '+|-' <line|lcmb> '+|-' -><line|lcmb> .. (3-5 times)and then plot all the lines and surfaces. The figure is showed below

4. Input the command” gbod 'NORM' '+|-' '+|-' ->( 5-7 surfaces )” to define a volume. For example, “GBOD B001 NORM + A006 - A003 - A004 + A002 + A001” is to define a volume which consists of these above areas. GBOD B001 NORM + A006 - A003 - A004 + A002 + A001

5. Input the command ‘ELTY all he20’ and ‘mesh all’, and then left-click on the marginal area on the Graphix, then on ’Viewing’ and continue to ‘show all elements with light’ and ‘Toggle Element edges’, as showed in following figure.

E. The “airfoil” problem
1. First switch to the directory of working ‘E:\Documents\ccx’

2. Then input the command ‘cgx –b airfoil.fbd’, there will appear another command “CalculiX Graphix”.

3. Input the command” pnt [<x> <y> <z>]” and then plot all the points with the command ‘plot p all b’. For example, ” PNT P05D     0.00118       -5.37045        0.00000” is used to define a point which is located at (0.00118,-5.37045,0.00000). The figure is showed below.

4. Input the command ”seqa ['pnt' .. <=>]|['afte'|'befo' .. <=>]|['end' .. <=>]” to define a sequential set. For example, “SEQA S005 pnt  P010 P00C P00A P01C” is used to define a sequential point set which pass through points P010 P00C P00A P01C sequentially.

SEQA S006    pnt  P01A P00A P00C P00E P00G P00I P00J P00K P00L P019

5. Input the command ”lcmb ['+|-' '+|-' '+|-' ->  ..(up to 14 lines)]| ['ADD' '+|-' '+|-' -> '+|-' ..(up to 14 lines)]” to combine lines. For example, “ LCMB C001 + L015 + L01X + L01Y + L016” is used to combine the lines of C001, L015 + L01X + L01Y, L016.

LCMB C001 + L001 + L002 + L00A + L00L + L009

6. Input the command” line <p1> <p2> <cp|seq>” and the “gsur '+|-'BLEND| ' '+|-' <line|lcmb> '+|-' -><line|lcmb> .. (3-5 times)and then plot all the lines and surfaces with command ‘plus’. For example, “LINE L001 P00I P001 p0 104 ” is used to define a arc line which pass through points P00I and P001 and its center point is p0 and radius is 104. The figure is showed below.

7. Input the command ‘ELTY all QU4’ and ‘mesh all’, and then left-click on the marginal area on the Graphix, then on ’Viewing’ and continue to ‘show all elements with light’ and ‘Toggle Element edges’, as showed in following figure.

title: User:Eml5526.s11.team4.yang/hw4


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User:Nobelium

This user comes from Germany → home wiki

title: User:Nobelium


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User:Egm6341.s11.team5.shin/HW4

= Problem 4.5 - Proof of e(1) in Simple Simpson Error = From [[media:nm1.s11.mtg20.djvu|Mtg 20-3]]

Given

 * {| style="width:100%" border="0" align="left"

e(1)=-\frac{1}{90}F^{4}(\zeta_4) $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-1) $$
 * }
 * }

Find
Prove $$\displaystyle e(1)=-\frac{(b-a)^4}{1440}f^{(4)}(\xi)$$ and find the relationship between $$\displaystyle \zeta_4 $$ and $$\displaystyle \xi $$.

Solution
Before solving this problem, the following should be noted.
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\begin{align} F(t) &= f(x(t)) \\ x&=x_1+ht \\ \mathrm{d}x &=h \mathrm{d}t \\ \xi&=x_1+h \zeta_4 \mbox{, } (t=\zeta_4,x=\xi) \\ \end{align} $$ With the equations above, the proof can be made as below.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-2)$$
 * }
 * }
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\begin{align} F^{(4)}(\zeta_4)&=\frac{\mathrm{d}^4}{\mathrm{d}t^4}F(\zeta_4) \\ &=\frac{\mathrm{d}^4}{(\frac{\mathrm{d}x}{h})^4}f(\xi) \\ &=h^4\frac{\mathrm{d}^4}{\mathrm{d}x^4}f(\xi) \\ &=(\frac{b-a}{2})^4f^{(4)}(\xi) \mbox{ ,} h=\frac{b-a}{2} \end{align} $$ Substitute Eq.(5-3) into Eq.(5-1).
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-3)$$
 * }
 * }
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\begin{align} e(1)&=-\frac{1}{90}\frac{(b-a)^4}{16}f^{(4)}(\xi) \\ &=-\frac{(b-a)^4}{1440}f^{(4)}(\xi) \end{align} $$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-4)$$
 * }
 * }

Author and proof-reader
[Author] Shin

[Proof-reader] = Problem 4.10 - Evaluation of the error using both composite Trapezoidal and composite Simpson rule = From [[media:Nm1.s11.mtg22.djvu|Mtg 22-3]]

Given
Part a,b.
 * $$\displaystyle

I=\int_0^{\pi} \! e^x \sin{x} \, \mathrm{d}x. $$ Part c.
 * $$\displaystyle

I=\int_0^{1} \! x^{3.7} \, \mathrm{d}x. $$

Find
Part a.
 * Produce Table 5.1, page 255, Atkinson text book using the composite Trapezoidal rule.

Part b.
 * Produce Table 5.3, page 258, Atkinson text book using the composite Simpson rule.

Part c.
 * Produce Table 5.4, page 261, Atkinson text book using the composite Trapezoidal & composite Simpson rule.

Part a.

 * From the class notes (p.7-4), the composite Trapezoidal rule is defined as below.
 * {| style="width:100%" border="0" align="left"

I_n=h(\dfrac{1}{2}f(x_0)+f(x_1)+f(x_2)+...+f(x_{n-1})+\dfrac{1}{2}f(x_{n})) \mbox{, }h=\dfrac{b-a}{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-1)
 * }
 * The asymptotic error is defined as the following equation.
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\overline{E_n^T}=-\frac{h^2}{12}[f^{(1)}(b)-f^{(1)}(b)] $$ $$ The exact integration was performed using Wolfram Alpha as below.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-2)
 * }
 * }
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I = \int_0^{\pi} \! e^x \sin{x} \, \mathrm{d}x = 12.0703 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-3)
 * }
 * }

 Table:

Matlab code:
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.
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 * }
 * }
 * }

Part b.

 * From the class notes (p.7-4), the composite Simpson's rule is defined as below.
 * {| style="width:100%" border="0" align="left"

I_n=\dfrac{h}{3}(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n))\mbox{, }h=\frac{b-a}{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-4)
 * }
 * The asymptotic error is defined as the following equation.
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\overline{E_n^S}=-\frac{h^4}{180}[f^{(3)}(b)-f^{(3)}(b)] $$ $$  Table:
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-5)
 * }
 * }

Matlab code:
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 * }
 * }
 * }

Part c.
The exact integration was performed using Wolfram Alpha as below.
 * {| style="width:100%" border="0" align="left"

I = \int_0^{1} \! x^{3.7} \, \mathrm{d}x = 0.212766 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-6)
 * }
 * }

 Table:

Matlab code:
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 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }

Author and proof-reader
[Author] Shin [Proof-reader]

title: User:Egm6341.s11.team5.shin/HW4


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User:Eml5526.s11.team6.deshpande/hwk4

=Problem 4.3 Finding weak form for heat conduction problem=

Given
Given the strong form for the heat conduction problem in a circular plane:

with the essential boundary conditions and natural boundary condition

where R is the total radius of the plate, s is the heat source per unit length along the plate radius. T is the temperature and k is the conductivity. Assume that k, s and R are given.

Find
a. Construct the weak form for the above strong form.

b. Use quadratic trial (candidate) solution of the form $$T= \alpha_0+\alpha_1 r+\alpha_2 r^2$$ and weight function of the same form to obtain a solution of the weak form.

c. Solve the differential equation with the boundary conditions and show that the temperature distribution along the radius is given by

Solution
a. Weak form can be found as follows, with weighted function

Now, multiply eq. 4.3.1 by $$ \displaystyle w $$ and integrate between 0 to R.

Use integration by parts for the first term in eq. 4.3.6

Using eq. (4.3.3) and eq. (4.3.5), we get

Put eq. (4.3.8) in eq. (4.3.7), we get the weak form as follows

b. Now, consider the trial solution

By given essential boundary condition eq. (4.3.2) and by natural boundary condition eq. (4.3.3) Therefore, we get Put this eq. (4.3.14) in eq. (4.3.10), we get Hence, By eq. (4.3.2) and eq. (4.3.16) we get,

Taking arbitrary function $$ \displaystyle w $$ such that $$\displaystyle \beta_1=0$$, by comparing with eq. (4.3.15) Put this in weak form eq. (4.3.9), we get Integrating, factoring out $$\beta_0,\beta_2 $$ and rearranging the terms gives As the above must hold for arbitrary weight functions, it must hold for arbitrary $$\displaystyle\beta_0 $$ and $$\displaystyle\beta_2 $$. Therefore, the coefficients of $$\displaystyle\beta_0 $$ and $$\displaystyle\beta_2 $$ must vanish, which gives the following algebraic equation in $$\displaystyle\alpha_0 $$ and $$\displaystyle\alpha_2 $$: Solving above, second row gives Therefore, From eq. (4.3.18)

c. Therefore, the solution is Hence, the solution for the given differential equation is

title: User:Eml5526.s11.team6.deshpande/hwk4


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User:Eml5526.s11.team7.jb/work

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title: User:Eml5526.s11.team7.jb/work


text:
User:EML5526.S11.Team3.vnarayanan/Homework4

Problem statement
Given the strong form for the circular bar in torsion:

$$\frac{\mathrm{d} }{\mathrm{d} x}\left [ JG \frac{\mathrm{d}\phi }{\mathrm{d} x} \right ] + m(x) = 0\;\;\;\; 0\leq x\leq 1,$$

Natural Boundary Condition :

$$\;\;\; M(x=1) = \left ( JG \frac{\mathrm{d} \phi }{\mathrm{d} x} \right )_{l} = \bar{M}$$

Essential Boundary Condition :

$$\phi(x=0) = \bar{\phi}$$

where $$m(x)$$ is a distributed moment per unit length, M is the torsion moment, $$\phi$$ is the angle of rotation, G is the shear modulus and J is the polar moment of inertia given by $$J = \frac{\pi C^{4}}{2}$$, where C is the radius of the circular shaft.

Replacing the boundary conditions with the ones stated in problem G1DM1.0 / D1 in Mtg 9-2 the exact solution can be evaluated.

Modified Natural Boundary Condition :

Modified Essential Boundary Condition :

a. Construct the weak form for the circular bar in torsion.

b. Assume that $$m(x) = 0$$ and integrate the differential equation given above. Find the integration constants using boundary conditions.

Solution (a)
As the weight function must vanish on the essential boundaries, we consider all smooth weight functions $$w(x)$$ such that $$w(0)= 0$$.

Multiplying the given differential equation and the natural boundary condition over the domain specified, by an arbitrary weight function:

$$\int_{0}^{1}w(x)\left[ \frac{\mathrm{d} }{\mathrm{d} x}\left ( JG \frac{\mathrm{d} \phi }{\mathrm{d} x} \right ) + m(x) \right]= 0$$

$$w\left[ \left ( JG \frac{\mathrm{d} \phi }{\mathrm{d} x} \right )- \bar{M} \right]_{x=1}= 0$$

Replacing the coefficients and substituting the modified boundary conditions:

$$\int_{0}^{1}w(x)\left[ \frac{\mathrm{d} }{\mathrm{d} x}\left ( (3x + 2) \frac{\mathrm{d} \phi }{\mathrm{d} x} \right ) + 5x \right]dx= 0$$

Now we integrate the Equation 4.4.3 by parts,

Integration by parts:

We have constructed the weight functions so that w(0) =0 therefore, the first term on the RHS of the above vanishes at x = 0. Substituting Equation 4.4.5 in Equation 4.4.1,

Substituting Equation 4.4.4 in Equation 4.4.6,

Solution (b)
Assuming m(x) = 0, reduces the given strong form as follows:

Substituting for the coefficient,

Integrating with respect to x,

From Equation 4.4.4 ,

Hence,

Integrating Equation 4.4.9 with respect to x,

$$\phi = 4log(3x+2)+ C_{2}$$

From Equation 4.4.2,

$$\phi(x=0) = \bar{\phi} = 4$$

$$\therefore\;\;C_{2} =4 - 4log(2) = 1.228$$

Hence,


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Adventure therapy/Research

AT research

 * Adventure therapy research (Google search)
 * Adventure therapy research (Wilderdom)
 * Adventure therapy meta-analysis

Sub-pages

 * Indigenous

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