User:Egm5526.s11.team4.zou/HW6

=Problem 6.3 Reproduce a tutorial=

Assigned in the lecture slide Mtg33

Problem statement
Reproduce all steps in the Beede’s tutorial

Part1: Buliding Geometry and Meshing
Opening the SciTE text editor inplanted in the bConverged Calculix pacage, and paste the following content into it then save as an .fbd file, like beam.fbd

pnt p1 0 0 0 pnt p2 25 0 0 pnt p3 50 0 0 pnt p4 75 0 0 pnt p5 100 0 0 line l1 p1 p2 25 line l2 p2 p3 25 line l3 p3 p4 25 line l4 p4 p5 25 seta lines l l1 l2 l3 l4 swep lines sweplines tra 0 10 0 10 seta surfaces s A001 A002 A003 A004 swep surfaces swepsurface tra 0 0 1 1 elty all he8 mesh all plot m all

The beam created:



This beam.fbd will reproduce the same geometry property as stated in the tutorial. For detailed explanation of the lines, please refer to the the Calculix cgx online documentary or previous wikiversity tutorials by our team.

Part2: Exporting Mesh, Loads, and Boundary Conditions
Open the SciTE editor again and paste such content, then overwrite the previous beam.fbd file. The new file pretty much the same geometry and mesh as the previous but use different name for the mesh and exports the nodal information to a .msh file.

seto beam pnt p1 0 0 0 pnt p2 25 0 0 pnt p3 50 0 0 pnt p4 75 0 0 pnt p5 100 0 0 line l1 p1 p2 25 line l2 p2 p3 25 line l3 p3 p4 25 line l4 p4 p5 25 seta lines l l1 l2 l3 l4 swep lines sweplines tra 0 10 0 10 seta surfaces s A001 A002 A003 A004 swep surfaces swepsurface tra 0 0 1 1 setc beam elty beam he8 mesh beam send beam abq rot -z frame

Then followed the steps of the tutorial as it use the “interactive” style of operating Calculix.

The nodal information in beam.msh file:



Setting nodes for the fixed end:



Boundary information:



The nodal information for the load:



Part3: Writing an Input File for the CCX Solver
Use SciTE to create a new file named beam.inp and write down the following content,

*HEADING Model: Calculix Beam Input File *INCLUDE,INPUT=beam.msh *BOUNDARY *INCLUDE,INPUT=fixed_123.bou *MATERIAL,NAME=EL *ELASTIC 30000000, 0.3 *SOLID SECTION,ELSET=Ebeam,MATERIAL=EL *STEP *STATIC *DLOAD *INCLUDE,INPUT=load.dlo *NODE FILE U *EL FILE S *END STEP

Then click on the SciTE menu “Tools->Solve”, when the “Job done!” is prompt we can click on the menu Tools->Post Process thus we open the result file. Choose “Datasets->STRESS” and then “Entities->Mises” than we can see the result just same as the tutorial.



For detailed explanation on the .inp file, please refer to the Calculix ccx online documentary or our team’s previous homework. .

Part4: Post Processing Results in CGX
By manipulating the cgx module for post processing we have the following results:

The displacement distribution when the bending is seen:



The displacement distribution at a certain plain:



The Graph function of post processing:



Problem General Statement
Solving problem 6.1,6.2,6.3 from Fish and Belytschko's 'A first course in Finite Elements'

Problem 6.1(F&B) Statement:
Given a vector field:$$\displaystyle {{q}_{x}}=-{{y}^{2}}$$,$$\displaystyle {{q}_{y}}=-2xy$$ on the domain shown in figure 1 .verify the divergence theorem.



Solution:
First consider the divergence of vector $$\displaystyle \vec{q}$$

Integrating the above over the problem domain gives:

So:

Evaluating the boundary integral counterclockwise gives:

Equation (1.1) equals to equation (1.2). Thus we have verified the divergence theorem for this example.

Problem6.2(F&B) Statement:
Given a vector field $$\displaystyle {{q}_{x}}=3{{x}^{2}}y+{{y}^{3}}$$,$$\displaystyle {{q}_{y}}=3x+{{y}^{3}}$$ on the domain shown in Figure 2. Verify the divergence theorem. The curved boundary of the domain is a parabola.



Solution:
First:

So

While

Equation (2.1) equals equation (2.2), so the divergence theorem verified.

Problem6.3(F&B) Statement
Using the divergence theorem prove:

Proof:
Denote $$\displaystyle \vec{q}=[1,1........1]$$

Using the divergence theorem: