Learning 

is to memorize what have to memorize, comprehension of individual concepts and relations, and comprehension of the relations between parameters and how they effect each other and the whole system. 

Creativity 

can result in different comprehensions or points of view and then, different methods for solving problems. 

Skill 

also comes after becoming clear and exact about everything by practice and the result is increase in speed. 

Mastery

is Complete learning + Creativity results + Skill

What we learn? 

Theories - with the structure of 1- Definitions/Principals/Assumptions/Modeling or defining parameters 2- relations which are theorems and their results (mathematical, physical, psychological, illusions and ... ) 3- Problem solving

Operator Theory

in the name of God

الثلاثاء, فیفرییه 23, 2016

03:25 ب.ظ

Fundamentals of CFD

 

Operator Theory

• To discretize a problem we need have 3 parameters to know or to obtain: DSO (Derivative number + Stencil + Order of Truncation error )

We know the Derivative number. Should deal with Stencil and Order of error

• Operator Theory contains 3 parts:
  1. Theory:

Is to convert order of derivatives and shift on mesh to power by two means of Taylor Series and Definition of Shift Operator E.

1. ui+1 = eD Δx
2. E ui = ui+1

2. Definitions

Some definitions for differences and mean values between points exist like Shift operator E.

3. Relations

Because we need to calculate the derivatives, we are to re-write the theory in a practical form:

We also need to calculate the derivative using other difference or mean value parameters:

 

So there are two ways to write D in terms of these operators.

So

 

• Review: it needs to be reviewed

And other Taylor essential series.

 

Foreigner Universities Database Download

Foreigner Universities Database

matlabintro

خلاصه کتاب: مقدمه ای بر نرم افزار متلب - Matlab

چکیده کتاب Matlab Essentials for Engineering Students

Abstract: Matlab Essentials for Engineering Students

While Loop

• Maybe the problem is related to the "while" loop in your program. Check this structure:

Initialize Values                  

Initialize error   %(to enter the loop)

          while (error) && (logical constraints)

                   change variable

                   .....

                   .....

                   error recalculation

          end

Save the final value

• Initial Values and Initial Error and saving the final value must be into the outer loop if exists.
• % because of sensitivity to the Initial Values, the Initial values must be chosen logically or physically acceptable and meaningful
• % It also must b considered through logical constraints.

Some Gambit Hints

Some Gambit Hints:

Correction of a complex geometry with separated parts in Gambit is the honor of subduing the geometry and parts by 3 tools:

1. Tolerance

2. Simplify
3. Merge and unite

And some times (Smooth/Heal real surface)

 

1. Couldn't create a Volume from the selected faces:

It's because of a small gap between edges, edges don’t lie on each other, points aren't same or connected.

There may be 4 kinds of gap that must be filled:

1. Between two faces that are expected to form a plane in space.

2. Between two faces that must intersect and split one by another.

3. Between two vertices of two edge that are expected to form one curve in space.

4. Two faces of two parts that must become one.

 

Solutions:

1. Tolerance: maybe your problem of volume creation error, no matter what of the above diseases have been occurred, could be solved by setting a larger value for tolerance manually.     :O

2. If have problem a, review Hint 2.

3. If problem b, read Hint 3.
4. If problem c, read Hint 4.
5. If problem d, contact me.

 

2. Non-real to real/Connection/Merge/Unite problems - there is a small gap between faces -

If a face or volume couldn’t be converted to real, it's because of it's not a healthy face.

• There exist some hard edges/edges that have used twice

• There are some kind of discontinuities/sharp parts
• There is a small gap between faces

 

Non real solution

For non-real to real at first simplify. Then try again. Not solved? Then use the solution below.

 

Solution(the upper geometry solution):

1. Convert faces to real and smooth for a few times until the deflection value become s fixed.
2. Create an upper geometry by the desired faces with tolerance(change the tolerance that let the Volume be constructed). Then:
3. Simplify all faces
4. Merge or unite

5. Convert new created v-faces to real

6. Convert Volume to real again.

Another Solution: Delete upper geometry, create a new volume using the remaining face.

 

Weak solution: to fill the small gap, IF you have a healthy gap, you can use one of surface creation commands; But often because of edge intersections and complex relative positions, you can't use this comment.

Another Solution

There are some Sharp angles. Fix them by Collapse or tools.

Another Solution

Sealing Between Volumes with 1- Sweep 2- Intersect or Split or Subtract

 

3. Sealing between a face belongs to a volume and a face
1. Fill the Small gap by face- ways:
1. Use edges to create face: (Hint 6)
2. Sweep 1 or 2 or 3 or 4 edges: and then split faces with each other an delete extra parts.
3. Using Side edges: if we have a face next to our face, we can use the edge belongs to that.
4. Skin surface/Net surface/
   Vertex rows: split line II by a 
   few points, project the points on line III and split too.
   then connect by line and then use skin surface.
   to use net surface or vertex rows should do the same for 
   edges 0 and 1 too.
5. Use upper geometry:
2. Fill the Small gap by the Volume

• Sweep the face of volume that have to be common between

The face and the volume.

• Or create some wireframe according to hint 7 but for volume.

 

4. "Split": a way of connection

If you want to merge the two curves by one curve, and merge doesn't works, then:

1. You can split edge 2 by point 2 or 3. then delete edge l or l'.

2. Try connect vertices that must be joint.

 

5. Using upper geometry

Sometimes we need to use upper geometry because :

1. We want to create a geometry for example 2D geometry, but we don’t find the proper tool among face creation tools. So we need to Create an upper topology and delete but don’t delete the lower geometry.

or

2. Because of Hint 2.

 

6. Finding a wireframe

To utilize problem b solution, maybe you need to find a wireframe between the two faces.

Step 1. Find a Wireframe

Finding line I-ways:

1. Project edge 0

2. draw a line between points a and b
3. sweep edge 0 along line II or III to cut the face, then if needed delete the new face not lower       geometries.

Finding line II and III which are the sweep curve:

1. Project edge 0
2. Project points

And then draw II and III by line

or

3. Sweep points along a part of the edge they are belonging to. Then find

Step 2. Then Connect the beginning and the end points of edges if possible.

 

7. Unite and Merge

The difference of unite and merge?

 

8. Does your Face use an edge twice?

Simplify the face.

It may be because of use of the "Unite" command.

مواردی درباره گمبیت Gambit

1. اگر دو خط متقاطع را با آن صفحه درست کنیم، صفحه نیز تیز می شود. اما اگر دو خط را مرج کنیم، صفح گرد می سازد.
2. برای مش زدن گئشه های تیز و مشتق ناپذیر باید در آن نقطه بگذاریم و به چند خم مشتق پذیر جزیه کنیم و جداگانه مش بزنیم.

منحنی چند مشتق معروف

dy/dx=y/x شکلی را معرفی می کند که در نقاط آن شیب با نسبت عرض به طول یکسان است که در این صورت

 مجموعه خطوط شعاعی را تداعی می کند. به تبع، dy/dx=-x/y منحنی دایره را معرفی می کند. 

در همین راستا از آنجا که در هذلولی افقی یا تابع هموگرافیک، شیب با قرینه عرض به طول برابر است، dy/dx=-y/x بیانگر یک تابع هموگرافیک است و همچنین

 dy/dx=x/y بیانگر تابعی است که مشتق آن با مشتق معکوس آن برابر است که در این صورت مربوط به دسته خطوط |y|=|x| یا y=x و y=-x است. 

****لازم به ذکر است که در صورت وجود انتقال در مبدا مختصات، و یا وجود ضرایب آنها را اعمال کرد.