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Fluid–Structure Interaction in Aluminum Extrusion
Introduction
Out of all metals, the most frequently extruded is aluminum. Aluminum extrusion entails using a hydraulic ram to squeeze an aluminum bar through a die. This process will form the metal into a particular shape. Extruded aluminum is used in many manufacturing applications, such as building components for example. In massive forming processes like rolling or extrusion, metal alloys are deformed in a hot solid state with material flowing under ideally plastic conditions. Such processes can be simulated effectively using computational fluid dynamics, where the material is considered as a fluid with a very high viscosity that depends on velocity and temperature. Internal friction of the moving material acts as a heat source, so that the heat transfer equations are fully coupled with those ruling the fluid dynamics part. This approach is especially advantageous when large deformations are involved.
This model is adapted from a benchmark study in Ref. 1. The original benchmark solves a thermal-structural coupling, because it is common practice in the simulation of such processes to use specific finite element codes that have the capability to couple the structural equations with heat transfer. The alternative scheme discussed here couples non-Newtonian flow with heat transfer equations. In addition, because it is useful to know the stress in the die due to fluid pressure and thermal loads, the model adds a structural mechanics analysis.
The die design is courtesy of Compes S.p.A., while the die geometry, boundary conditions, and experimental data are taken from Ref. 1.
Note: This model requires the Heat Transfer Module and the Structural Mechanics Module. In addition, it uses the Material Library. To create the geometry from scratch, the CAD Import Module or Design Module are required.
Model Definition
The model considers steady-state conditions, assuming a billet of infinite length flowing through the die. In the actual process, the billet is pushed by the ram through the die and its volume is continuously reducing.
Figure 1 shows the original complete geometry with four different profiles. To have a model with reasonable dimensions, consider only a quarter of the original geometry. The simplification involved in neglecting the differences between the four profiles does not affect the numerical scheme proposed. Figure 2 shows the resulting model geometry.
Figure 1: Original benchmark geometry.
Figure 2: Quarter of the original geometry considered in the model.
Material Properties
The documentation for the benchmark model (Ref. 1) serves as the data source for properties of the two main materials: AISI steel for the die and the container (the ram is not considered here) and aluminum for the billet.
Structural Analysis
Because only the steel part is active in the structural analysis, consider a simple linear elastic behavior where the elastic properties are those of the material H11 mod (AISI 610) that can be found in the COMSOL Multiphysics Material Library.
Heat Transfer Analysis
The benchmark model uses the following properties for aluminum and steel:
Aluminum
Value
Description
 kal
210 N/(s·K)
Thermal conductivity
 ρal
2700 kg/m3
Density
 Cpal
2.94 N/(mm2·K)/ρal
Heat capacity at constant pressure
Steel
Value
Description
 kfe
24.33 N/(s·K)
Thermal conductivity
 ρfe
7850 kg/m3
Density
 Cpfe
4.63 N/(mm2·K)/ρfe
Heat capacity at constant pressure
Non-Newtonian Flow
The properties of the aluminum were experimentally determined and then checked using literature data for the same alloy and surface state. However the benchmark proposes an experimental constitutive law, suited for the structural mechanics codes usually used to simulate such processes, in the form of the flow stress data. For this model this requires a recalculation of the constitutive law to derive a general expression for the viscosity. The equivalent von Mises stress, σeqv, can be defined in terms of the total contraction of the deviatoric stress tensor as
or, using where is the strain rate and η is the viscosity, as
(1)
Introducing the equivalent strain rate
Equation 1 can be expressed as
The strain rate tensor is defined as (Ref. 2)
The shear rate is defined as
so that
The flow rule
states that plastic yielding occurs if the equivalent stress, σeqv, reaches the flow stress, κf. The viscosity is defined as (see Ref. 2 for further details)
The organizers of the benchmark propose specific flow-stress data expressed in terms of a generalized Zener–Hollomon function
where A = 2.39·108 s−1, n = 2.976, α = 0.052 MPa−1, and
with Q = 153 kJ/mol and R = 8.314 J/(K·mol).
Sources, Initial Conditions, and Boundary Conditions
Structural Analysis
Because the model geometry is a quarter of the actual geometry, use symmetric boundary conditions for the two orthogonal planes. On the external surfaces of the die, apply roller boundary conditions because in reality other dies, not considered here, are present to increase the system’s stiffness.
The main loads are the thermal loads from the heat transfer analysis and the total stress from the fluid dynamics analysis.
Heat Transfer Analysis
For the billet, use a volumetric heat source related to the viscous heating effect.
The external temperature of the ram and the die is held constant at 450°C (723 K). The ambient temperature is 25°C (298 K). For the heat exchange between aluminum and steel, use the heat transfer coefficient of 11 N/(s·mm·K). Also consider convective heat exchange with air outside the profiles with a fixed convective heat transfer coefficient of 15 W/(m2·K).
Apply initial temperatures as given in the following table:
Part
Value
Ram
380°C (653 K)
Container
450°C (723 K)
Billet
460°C (733 K)
Die
404°C (677 K)
Non-Newtonian Flow
At the inlet, the ram moves with a constant velocity of 0.5 mm/s. Impose this boundary condition by simply applying a constant inlet velocity. At the outlet, a normal stress condition with zero external pressure applies. On the surfaces placed on the two symmetry planes, use symmetric conditions. Finally, apply slip boundary conditions on the boundaries placed outside the profile.
Results and Discussion
The general response of the proposed numerical scheme, especially in the zone of the profile, is in good accordance with the experience of the designers. A comparison between the available experimental data and the numerical results of the simulation shows good agreement.
On the basis of the results from the simulation, the engineer can improve the preliminary die design by adjusting relevant physical parameters and operating conditions. For this purpose, the volume plot in Figure 3 showing the temperature field inside the profile gives important information. Furthermore, the combined streamline and slice plot in Figure 4 reveals any imbalances in the velocity field that could result in a crooked profile. A proper design should also ensure that different parts of the profile travel at the same speed. Figure 5 shows the von Mises equivalent stress in the steel part considering the thermal load and the pressure load due to the presence of the fluid.
Figure 3: Temperature distribution in the billet.
Figure 4: Velocity field and streamlines at the profile section.
Figure 5: Equivalent von Mises stress distribution in the container.
References
1. M. Schikorra, L. Donati, L. Tomesani, and A.E. Tekkaya, “The Extrusion Benchmark 2007,” Proceedings of the Extrusion Workshop 2007 and 2nd Extrusion Benchmark Conference, Bologna, Italy.
2. E.D. Schmitter, “Modelling massive forming processes with thermally coupled fluid dynamics,” Proceedings of the COMSOL Multiphysics User’s Conference 2005 Frankfurt, Frankfurt, Germany.
Application Library path: Structural_Mechanics_Module/Fluid-Structure_Interaction/aluminum_extrusion_fsi
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Heat Transfer > Conjugate Heat Transfer > Laminar Flow.
3
Click Add.
4
In the Select Physics tree, select Structural Mechanics > Solid Mechanics (solid).
5
Click Add.
6
Click  Study.
7
In the Select Study tree, select General Studies > Stationary.
8
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
D_alfe
1[mm]
0.001 m
Thickness of the high conductive layer
Heat_alfe
11[N/(s*mm*K)]
11000 W/(m²·K)
Aluminum-steel heat exchange coefficient
T_billet
460[degC]
733.15 K
Billet temperature
T_container
450[degC]
723.15 K
Container temperature
T_ram
380[degC]
653.15 K
Ram temperature
T_pd1
404[degC]
677.15 K
Initial temperature around thermocouple at point PD1
V_ram
0.5[mm/s]
5E-4 m/s
Ram velocity
P_init
0[bar]
0 Pa
External reference pressure
T_air
25[degC]
298.15 K
Ambient temperature
Q_eta
153000[J/mol]
1.53E5 J/mol
Parameter Q for the generalized Zener-Hollomon function
n_eta
2.976
2.976
Parameter n for the generalized Zener-Hollomon function
A_eta
2.39e8[1/s]
2.39E8 1/s
Parameter A for the generalized Zener-Hollomon function
alpha_eta
0.0521[1/MPa]
5.21E-8 1/Pa
Parameter alpha for the generalized Zener-Hollomon function
H_conv
15[W/(m^2*K)]
15 W/(m²·K)
Convective heat exchange coefficient with air
F
sqrt(1/3)
0.57735
Factor for the conversion of the shear rate to COMSOL's definition
Geometry 1
The model geometry is provided in a separate MPHBIN file. If you prefer to create it from scratch, follow the steps outlined in the Geometry Modeling Instructions. Note that a license for the CAD Import Module or the Design Module is required to create the geometry. Otherwise, you can import the geometry as follows:
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file aluminum_extrusion_fsi.mphbin.
5
Click  Import.
6
Click the  Zoom Extents button in the Graphics toolbar.
7
In the Geometry toolbar, click  Build All.
You should now see the following geometry.
Cleanup Log
1
In the Definitions toolbar, click  Local Variables.
2
In the Geometry Cleanup dialog that opens, click Clean up Automatically to automatically clean up the geometry.
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) > Definitions click Variables 1.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Z_eta
F*spf.sr*exp(Q_eta/(R_const*T))
1/s
Zener-Hollomon parameter
mu_al
asinh((Z_eta/A_eta)^(1/n_eta))/(3*alpha_eta*F*spf.sr+sqrt(eps))
Pa·s
Viscosity of aluminum
Create the selections to simplify the model specification.
Outside
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Outside in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 35–38, 42, 43, 50, 51, 53, 55, 70, 71, 79–82, 87, 88, 93, 95, 102, 103, 106, and 108 only.
For more convenience in selecting these boundaries, you can click the Paste Selection button and paste the above numbers.
Interior
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Interior in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 8, 11, 14, 15, 19, 20, 24, 29–34, 41, 44, 45, 49, 52, 58–60, 64, 69, 72, 73, 76–78, 86, 89–92, 101, 104, 105, and 109 only.
Before creating the materials for the model, specify the fluid and solid domains. Using this information, the software can detect which material properties are needed.
Heat Transfer in Solids and Fluids (ht)
Fluid 1
1
In the Model Builder window, under Component 1 (comp1) > Heat Transfer in Solids and Fluids (ht) click Fluid 1.
2
Select Domains 3 and 4 only.
3
In the Settings window for Fluid, locate the Domain Selection section.
4
Click  Create Selection.
5
In the Create Selection dialog, type Billet in the Selection name text field.
6
Click OK.
Laminar Flow (spf)
1
In the Model Builder window, under Component 1 (comp1) click Laminar Flow (spf).
2
In the Settings window for Laminar Flow, locate the Physical Model section.
3
From the Compressibility list, choose Incompressible flow.
4
Locate the Domain Selection section. From the Selection list, choose Billet.
Solid Mechanics (solid)
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
In the Settings window for Solid Mechanics, locate the Domain Selection section.
3
Click  Clear Selection.
4
Select Domains 1 and 2 only.
Now, define the material for each domain.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Structural steel.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Structural steel (mat1)
1
Select Domains 1 and 2 only.
2
In the Settings window for Material, locate the Material Contents section.
3
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Heat capacity at constant pressure
Cp
4.63[N/(mm^2*K)]/rho
J/(kg·K)
Basic
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
24.33[N/(s*K)]
W/(m·K)
Basic
Poisson’s ratio
nu
0.3
1
Young’s modulus and Poisson’s ratio
The heat capacity is not used since it does not enter in the stationary heat transfer equation for solids without translational movement. The heat capacity is only provided for completeness in case you want to extend the model to perform transient simulations.
Billet
1
In the Materials toolbar, click  Blank Material.
2
In the Settings window for Material, type Billet in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Billet.
4
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Heat capacity at constant pressure
Cp
2.94[N/(mm^2*K)]/rho
J/(kg·K)
Basic
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
210[W/(m*K)]
W/(m·K)
Basic
Dynamic viscosity
mu
mu_al
Pa·s
Basic
Density
rho
2700[kg/m^3]
kg/m³
Basic
Structural steel 1 (mat3)
1
In the Model Builder window, under Component 1 (comp1) > Materials right-click Structural steel (mat1) and choose Duplicate.
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Interior.
Now, define the volume reference temperature in the solid domains. This temperature is used by materials to define the density in solids for both the Heat Transfer in Solids and Fluids and Solid Mechanics interfaces, and by the Thermal Expansion feature as the zero strain reference.
Definitions
Model Input 1
1
In the Physics toolbar, click  Shared Properties and choose Model Input.
2
Select Domain 1 only.
3
In the Settings window for Model Input, locate the Definition section.
4
Click  Select Quantity.
5
In the Physical Quantity dialog, type temperature in the text field.
6
In the tree, select General > Volume reference temperature (K).
7
Click OK.
8
In the Settings window for Model Input, locate the Definition section.
9
In the text field, type T_container.
This value overrides the value defined under Default Model Inputs for the selected domain.
Shared Properties
Model Input 2
1
In the Physics toolbar, click  Shared Properties and choose Model Input.
2
Select Domain 2 only.
3
In the Settings window for Model Input, locate the Definition section.
4
Click  Select Quantity.
5
In the Physical Quantity dialog, select General > Volume reference temperature (K) in the tree.
6
Click OK.
7
In the Settings window for Model Input, locate the Definition section.
8
In the text field, type T_pd1.
With the materials defined, you can set up the remaining physics of the model.
Laminar Flow (spf)
In the current model, the high viscosity of the fluid flow implies that the model is diffusion dominated. Pseudo time-stepping is thus not adapted to this case as it is based on the scale of the convective flux.
1
Click the  Show More Options button in the Model Builder toolbar.
2
In the Show More Options dialog, in the tree, select the checkbox for the node Physics > Advanced Physics Options.
3
Click OK.
4
In the Model Builder window, under Component 1 (comp1) click Laminar Flow (spf).
5
In the Settings window for Laminar Flow, click to expand the Advanced Settings section.
6
Find the Pseudo time stepping subsection. From the Use pseudo time stepping for stationary equation form list, choose Off.
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1) > Laminar Flow (spf) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the p text field, type P_init.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
Select Boundaries 9 and 114 only.
Inlet 1
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
Select Boundary 10 only.
3
In the Settings window for Inlet, locate the Velocity section.
4
Click the Velocity field button.
5
Specify the u0 vector as
0
x
0
y
V_ram
z
Wall 2
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, locate the Boundary Selection section.
3
From the Selection list, choose Outside.
4
Locate the Boundary Condition section. From the Wall condition list, choose Slip.
Outlet 1
1
In the Physics toolbar, click  Boundaries and choose Outlet.
2
Select Boundary 40 only.
3
In the Settings window for Outlet, locate the Pressure Conditions section.
4
In the p0 text field, type P_init.
Heat Transfer in Solids and Fluids (ht)
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1) > Heat Transfer in Solids and Fluids (ht) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the T text field, type T_container.
Temperature 1
1
In the Physics toolbar, click  Boundaries and choose Temperature.
2
Select Boundaries 2, 5, and 7 only.
3
In the Settings window for Temperature, locate the Temperature section.
4
In the T0 text field, type T_container.
Heat Flux 1
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
Select Boundary 10 only.
3
In the Settings window for Heat Flux, locate the Heat Flux section.
4
From the Flux type list, choose Convective heat flux.
5
In the h text field, type Heat_alfe.
6
In the Text text field, type T_ram.
Heat Flux 2
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
In the Settings window for Heat Flux, locate the Boundary Selection section.
3
From the Selection list, choose Outside.
4
Locate the Heat Flux section. From the Flux type list, choose Convective heat flux.
5
In the h text field, type H_conv.
6
In the Text text field, type T_air.
Outflow 1
1
In the Physics toolbar, click  Boundaries and choose Outflow.
2
Select Boundary 40 only.
Thin Layer 1
1
In the Physics toolbar, click  Boundaries and choose Thin Layer.
2
In the Settings window for Thin Layer, locate the Boundary Selection section.
3
From the Selection list, choose Interior.
4
Locate the Layer Model section. From the Specify list, choose Thermal resistance.
5
Locate the Heat Conduction section. In the Rs text field, type 1/Heat_alfe.
Materials
Structural steel 1 (mat3)
1
In the Model Builder window, under Component 1 (comp1) > Materials click Structural steel 1 (mat3).
2
In the Settings window for Material, locate the Material Contents section.
3
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Thickness
lth
D_alfe
m
Shell
Solid Mechanics (solid)
For faster convergence use linear elements. You can always improve the solution using the default quadratic elements.
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
In the Settings window for Solid Mechanics, click to expand the Discretization section.
3
From the Displacement field list, choose Linear.
Linear Elastic Material 1
Both Young’s modulus and Poisson’s ratio are temperature dependent. Follow these steps to verify that the temperature is defined by the Heat Transfer in Solids and Fluids interface.
1
In the Model Builder window, under Component 1 (comp1) > Solid Mechanics (solid) click Linear Elastic Material 1.
2
In the Settings window for Linear Elastic Material, locate the Model Input section.
3
Click  Go to Source for Temperature for the Temperature (T).
Global Definitions
Default Model Inputs
Locate the Browse Model Inputs section. The icon column of the table under Model input contributions shows that it is the Heat Transfer in Solids and Fluids interface that defines the temperature in the solid and fluid domains and on the thin-layer boundaries.
Solid Mechanics (solid)
Roller 1
1
In the Physics toolbar, click  Boundaries and choose Roller.
2
Select Boundaries 2, 5, and 7 only.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
Select Boundaries 1, 4, 112, and 113 only.
Multiphysics
Fluid–Structure Interaction 1 (fsi1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Boundary > Fluid–Structure Interaction.
2
In the Settings window for Fluid–Structure Interaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
Thermal Expansion 1 (te1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Domain > Thermal Expansion.
2
Select Domains 1 and 2 only.
Mesh 1
Free Triangular 1
1
In the Mesh toolbar, click  More Generators and choose Free Triangular.
2
Select Boundary 40 only.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section.
5
Select the Maximum element size checkbox. In the associated text field, type 0.0014.
6
Select the Curvature factor checkbox. In the associated text field, type 0.2.
7
Click  Build Selected.
Swept 1
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 4 only.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 24.
4
Click  Build All.
Free Tetrahedral 1
In the Mesh toolbar, click  Free Tetrahedral.
Size 1
1
Right-click Free Tetrahedral 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section.
5
Select the Maximum element size checkbox. In the associated text field, type 0.0085.
Size 2
1
In the Model Builder window, right-click Free Tetrahedral 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 12 and 13 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 0.002.
Size 3
1
Right-click Free Tetrahedral 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 3 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Minimum element size checkbox. In the associated text field, type 5e-4.
Size
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Finer.
4
Click  Build All.
You should now see the following meshed geometry.
Study 1
Step 1: Stationary
Use two stationary study steps. Solve first for the fluid dynamics and heat transfer to determine the thermal load and the pressure load and then for the structural mechanics.
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for Solid Mechanics (solid).
4
In the Solve for column of the table, under Component 1 (comp1) > Multiphysics, clear the checkboxes for Fluid–Structure Interaction 1 (fsi1) and Thermal Expansion 1 (te1).
Step 2: Stationary 2
1
In the Study toolbar, click  Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the Solve for column of the table, under Component 1 (comp1), clear the checkboxes for Heat Transfer in Solids and Fluids (ht) and Laminar Flow (spf).
4
In the Solve for column of the table, under Component 1 (comp1) > Multiphysics, clear the checkbox for Nonisothermal Flow 1 (nitf1).
For the structural analysis, use a memory efficient iterative solver to make it possible to solve the problem also on computers with limited memory.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 2 node.
4
Right-click Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 2 and choose Iterative.
5
In the Study toolbar, click  Compute.
Results
Transparency 1
1
In the Model Builder window, expand the Results > Temperature (ht) node.
2
Right-click Domain and choose Transparency.
3
In the Settings window for Transparency, locate the Transparency section.
4
Find the Transparency subsection. In the Transparency text field, type 0.75.
The first default plot shows the temperature (Figure 3).
5
In the Temperature (ht) toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Modify the second default plot to see the velocity field and streamlines at the profile section (Figure 4).
Study 1/Solution Store 1 (sol2)
In the Model Builder window, expand the Results > Datasets node, then click Study 1/Solution Store 1 (sol2).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Billet.
Velocity (spf)
1
In the Model Builder window, under Results click Velocity (spf).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 1/Solution Store 1 (sol2).
Multislice 1
1
In the Model Builder window, expand the Velocity (spf) node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Multiplane Data section.
3
Find the x-planes subsection. In the Planes text field, type 0.
4
Find the y-planes subsection. In the Planes text field, type 0.
5
Find the z-planes subsection. From the Entry method list, choose Coordinates.
6
In the Coordinates text field, type 0.0151.
7
In the Velocity (spf) toolbar, click  Plot.
Velocity (spf)
In the Model Builder window, click Velocity (spf).
Streamline 1
1
In the Velocity (spf) toolbar, click  Streamline.
2
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Laminar Flow > Velocity and pressure > u,v,w - Velocity field (spatial frame).
3
Locate the Streamline Positioning section. From the Positioning list, choose Magnitude controlled.
4
In the Minimum density level text field, type 6.7.
5
In the Maximum density level text field, type 13.4.
6
Locate the Coloring and Style section. Find the Line style subsection. From the Type list, choose Ribbon.
7
In the Width expression text field, type 0.001.
8
Select the Width scale factor checkbox.
9
Find the Point style subsection. From the Type list, choose Arrow.
10
Select the Number of arrows checkbox. In the associated text field, type 70.
11
Click to expand the Inherit Style section.
Color Expression 1
1
In the Velocity (spf) toolbar, click  Color Expression.
2
In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Laminar Flow > Velocity and pressure > spf.U - Velocity magnitude - m/s.
Streamline 1
1
In the Model Builder window, click Streamline 1.
2
In the Settings window for Streamline, locate the Inherit Style section.
3
From the Plot list, choose Multislice 1.
4
In the Velocity (spf) toolbar, click  Plot.
Velocity (spf)
To get a better view, rotate the geometry in the Graphics window. You can preserve a view for a plot by creating a View feature node as follows:
1
In the Model Builder window, click Velocity (spf).
2
In the Settings window for 3D Plot Group, locate the Plot Settings section.
3
From the View list, choose New view.
4
In the Velocity (spf) toolbar, click  Plot.
5
Click  Go to Source.
View 3D 3
1
In the Model Builder window, under Results > Views click View 3D 3.
2
Use the Graphics toolbox to get a satisfying view.
3
In the Settings window for View 3D, locate the View section.
4
Select the Lock camera checkbox.
Velocity (spf)
For a clearer visualization, you can duplicate this plot group and remove the geometry edges and the velocity slice, and add instead a temperature surface at the outside part.
Velocity and Outside Temperature
1
In the Model Builder window, right-click Velocity (spf) and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Velocity and Outside Temperature in the Label text field.
3
Right-click Velocity and Outside Temperature and choose Move Up.
4
Right-click Velocity and Outside Temperature and choose Move Up.
Multislice 1
1
In the Model Builder window, expand the Velocity and Outside Temperature node.
2
Right-click Multislice 1 and choose Delete.
3
Click Yes to confirm.
Surface 1
1
In the Results toolbar, click  More Datasets and choose Surface.
2
Select Boundaries 55, 95, and 108 only.
Velocity and Outside Temperature
In the Model Builder window, under Results click Velocity and Outside Temperature.
Surface 1
1
In the Velocity and Outside Temperature toolbar, click  Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Surface 1.
4
Locate the Coloring and Style section. From the Color table list, choose HeatCameraLight.
Velocity and Outside Temperature
1
In the Model Builder window, click Velocity and Outside Temperature.
2
In the Settings window for 3D Plot Group, locate the Color Legend section.
3
From the Position list, choose Right double.
4
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
5
In the Velocity and Outside Temperature toolbar, click  Plot.
Stress (solid)
The next plot shows the von Mises stress and deformation distribution in the container. To reproduce the Figure 5, apply the View 3D 3.
1
In the Model Builder window, click Stress (solid).
2
In the Settings window for 3D Plot Group, locate the Plot Settings section.
3
From the View list, choose View 3D 3.
Volume 1
1
In the Model Builder window, expand the Stress (solid) node, then click Volume 1.
2
In the Settings window for Volume, locate the Expression section.
3
From the Unit list, choose GPa.
4
In the Stress (solid) toolbar, click  Plot.
Geometry Modeling Instructions
Follow these steps to create the geometry for the aluminum extrusion model. Note that a license for the CAD Import Module or the Design Module is required for some geometric operations.
From the File menu, choose New.
New
In the New window, click  Blank Model.
Root
1
From the File menu, choose Save As.
2
Browse to a suitable folder, enter the filename aluminum_extrusion_fsi_geom_sequence.mph, and then click Save.
Add Component
In the Home toolbar, click  Add Component and choose 3D.
Geometry 1
1
In the Settings window for Geometry, locate the Advanced section.
2
From the Geometry representation list, choose CAD kernel.
3
Select the Design Module Boolean operations checkbox.
4
Click the  Wireframe Rendering button in the Graphics toolbar.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 90[mm].
4
In the Height text field, type 297[mm].
5
Locate the Position section. In the z text field, type -257[mm].
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
40[mm]
7
Clear the Layers on side checkbox.
8
Select the Layers on top checkbox.
Cylinder 2 (cyl2)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 73[mm].
4
In the Height text field, type 257[mm].
5
Locate the Position section. In the z text field, type -257[mm].
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
Block 1 (blk1)
1
In the Geometry toolbar, click  Block.
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type 90[mm].
4
In the Depth text field, type 90[mm].
5
In the Height text field, type 297[mm].
6
Locate the Position section. In the x text field, type -90[mm].
7
In the y text field, type -90[mm].
8
In the z text field, type -257[mm].
Intersection 1 (int1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Intersection.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
Cone 1 (cone1)
1
In the Geometry toolbar, click  Cone.
2
In the Settings window for Cone, locate the Size and Shape section.
3
In the Bottom radius text field, type 8.25[mm].
4
In the Height text field, type 15[mm].
5
In the Top radius text field, type 2.25[mm].
6
Locate the Position section. In the x text field, type -42.59[mm].
7
In the y text field, type -13.25[mm].
Cone 2 (cone2)
1
Right-click Cone 1 (cone1) and choose Duplicate.
2
In the Settings window for Cone, locate the Position section.
3
In the x text field, type -13.25[mm].
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose yz-plane.
4
From the Offset type list, choose Through vertex.
5
On the object cone1, select Point 5 only.
Work Plane 1 (wp1) > Plane Geometry
In the Model Builder window, click Plane Geometry.
Work Plane 1 (wp1) > Cross Section 1 (cro1)
1
In the Work Plane toolbar, click  Cross Section.
2
In the Settings window for Cross Section, locate the Cross Section section.
3
From the Intersect list, choose Selected objects.
4
Select the object cone1 only.
Extrude 1 (ext1)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Work Plane 1 (wp1) and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
From the Specify list, choose Vertices to extrude to.
4
On the object cone2, select Point 5 only.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the objects cone1 and ext1 only.
3
In the Settings window for Rotate, locate the Input section.
4
Select the Keep input objects checkbox.
5
Locate the Rotation section. In the Angle text field, type 90[deg].
6
Locate the Point on Axis of Rotation section. In the x text field, type -13.25[mm].
7
In the y text field, type -13.25[mm].
Union 2 (uni2)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Select the objects cone1, cone2, ext1, rot1(1), and rot1(2) only.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
5
In the Geometry toolbar, click  Defeaturing and Repair and choose Replace Faces.
6
In the Tools window for Replace Faces, in the Graphics window toolbar, clicknext to  Select Boundaries, then choose Group by Continuous Tangent.
7
On the object uni2, select Boundaries 3, 4, 6–9, 12, 14–24, and 26–29 only.
8
Locate the Replace Faces section. From the Heal method list, choose Create capping faces.
9
Click Replace Selected.
Fillet 1 (fil1)
1
In the Geometry toolbar, click  Editing and choose Fillet.
2
In the Settings window for Fillet, locate the Type of Fillet section.
3
From the Type list, choose Variable radius.
4
On the object rfa1, select Edge 15 only.
5
Locate the Radii section. In the table, enter the following settings:
Edge
Parameter
Radius
Clamp
15
0
2[mm]
 
15
1
1[mm]
 
Fillet 2 (fil2)
1
In the Geometry toolbar, click  Editing and choose Fillet.
2
In the Settings window for Fillet, locate the Edges section.
3
Select the Group by continuous tangent checkbox.
4
On the object fil1, select Edges 4, 5, 8, 10, 18–20, 23, 25, and 27 only.
5
Locate the Radius section. In the Radius text field, type 1[mm].
Work Plane 2 (wp2)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 15[mm].
Work Plane 2 (wp2) > Plane Geometry
In the Model Builder window, click Plane Geometry.
Work Plane 2 (wp2) > Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 30.6[mm].
4
In the Height text field, type 30.6[mm].
5
Locate the Position section. In the xw text field, type -42.35[mm].
6
In the yw text field, type -42.35[mm].
Work Plane 2 (wp2) > Rectangle 2 (r2)
1
Right-click Component 1 (comp1) > Geometry 1 > Work Plane 2 (wp2) > Plane Geometry > Rectangle 1 (r1) and choose Duplicate.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 27.6[mm].
4
In the Height text field, type 27.6[mm].
Work Plane 2 (wp2) > Difference 1 (dif1)
1
In the Work Plane toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object r1 only.
3
In the Model Builder window, click Difference 1 (dif1).
4
In the Settings window for Difference, locate the Difference section.
5
Click to select the  Activate Selection toggle button for Objects to subtract.
6
Select the object r2 only.
Work Plane 2 (wp2) > Fillet 1 (fil1)
1
In the Work Plane toolbar, click  Fillet.
2
On the object dif1, select Points 1–3 and 5 only.
3
In the Settings window for Fillet, locate the Radius section.
4
In the Radius text field, type 0.5[mm].
Work Plane 2 (wp2) > Fillet 2 (fil2)
1
In the Work Plane toolbar, click  Fillet.
2
On the object fil1, select Point 10 only.
3
In the Settings window for Fillet, locate the Radius section.
4
In the Radius text field, type 1[mm].
Work Plane 2 (wp2) > Fillet 3 (fil3)
1
In the Work Plane toolbar, click  Fillet.
2
On the object fil2, select Point 6 only.
3
In the Settings window for Fillet, locate the Radius section.
4
In the Radius text field, type 2[mm].
Extrude 2 (ext2)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Work Plane 2 (wp2) and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (m)
55[mm]
5[mm]
Work Plane 3 (wp3)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane type list, choose Face parallel.
4
On the object ext2, select Boundary 6 only.
Work Plane 3 (wp3) > Plane Geometry
In the Model Builder window, click Plane Geometry.
Work Plane 3 (wp3) > Cross Section 1 (cro1)
1
In the Work Plane toolbar, click  Cross Section.
2
In the Settings window for Cross Section, locate the Cross Section section.
3
From the Intersect list, choose Selected objects.
4
Select the object ext2 only.
Work Plane 3 (wp3) > Offset 1 (off1)
1
In the Work Plane toolbar, click  Offset.
2
Select the object cro1 only.
3
In the Model Builder window, click Offset 1 (off1).
4
In the Settings window for Offset, locate the Options section.
5
In the Distance text field, type 1.2[mm].
6
Locate the Input section. Clear the Keep input objects checkbox.
Work Plane 4 (wp4)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Work Plane 3 (wp3) and choose Duplicate.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
Click to select the  Activate Selection toggle button for Planar face.
4
In the tree, select ext2.
5
On the object int1, select Boundary 7 only.
6
In the Model Builder window, expand the Work Plane 4 (wp4) node.
Work Plane 4 (wp4) > Offset 1 (off1)
1
In the Model Builder window, expand the Component 1 (comp1) > Geometry 1 > Work Plane 4 (wp4) > Plane Geometry node, then click Offset 1 (off1).
2
In the Settings window for Offset, locate the Options section.
3
In the Distance text field, type 2.25[mm].
Work Plane 4 (wp4)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click Work Plane 4 (wp4).
2
In the Settings window for Work Plane, click  Build Selected.
Loft 1 (loft1)
1
In the Geometry toolbar, click  Loft.
2
Select the objects wp3 and wp4 only.
3
In the Settings window for Loft, locate the General section.
4
From the Face partitioning list, choose Columns.
Extract 1 (extract1)
1
In the Geometry toolbar, click  Extract.
2
In the Settings window for Extract, locate the Entities or Objects to Extract section.
3
From the Geometric entity level list, choose Domain.
4
On the object ext2, select Domain 1 only.
Union 3 (uni3)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Select the objects extract1 and loft1 only.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object int1 only.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the object uni3 only.
Fillet 3 (fil3)
1
In the Geometry toolbar, click  Editing and choose Fillet.
2
In the Settings window for Fillet, locate the Edges section.
3
Select the Group by continuous tangent checkbox.
4
On the object dif1, select Edges 25, 26, 29, 35, 38, 50, 56, 57, 64, 67, 76, and 79 only.
5
Locate the Radius section. In the Radius text field, type 0.5[mm].
Fillet 4 (fil4)
1
In the Geometry toolbar, click  Editing and choose Fillet.
2
In the Settings window for Fillet, locate the Edges section.
3
Select the Group by continuous tangent checkbox.
4
On the object fil3, select Edges 14, 16, 18, 36, 50, 51, 53, 75, 89, 91, and 108 only.
5
Locate the Radius section. In the Radius text field, type 1[mm].
Extract 2 (extract2)
1
In the Geometry toolbar, click  Extract.
2
In the Settings window for Extract, locate the Entities or Objects to Extract section.
3
From the Geometric entity level list, choose Domain.
4
On the object ext2, select Domain 1 only.
5
On the object fil4, select Domain 3 only.
6
From the Input object handling list, choose Create remainder object.
Union 4 (uni4)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Select the objects extract2(1), extract2(3), and fil2 only.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
Cylinder 3 (cyl3)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 1.5[mm].
4
In the Height text field, type 64[mm].
5
Locate the Position section. In the x text field, type -19.86[mm].
6
In the y text field, type -19.86[mm].
7
In the z text field, type 20.44[mm].
8
Locate the Axis section. From the Axis type list, choose Cartesian.
9
In the x text field, type -1.
10
In the y text field, type -1.
11
In the z text field, type 0.
Rotate 2 (rot2)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the object cyl3 only.
3
In the Settings window for Rotate, locate the Rotation section.
4
From the Axis type list, choose Cartesian.
5
In the x text field, type -1.
6
In the y text field, type 1.
7
In the z text field, type 0.
8
In the Angle text field, type -12[deg].
9
Locate the Point on Axis of Rotation section. In the x text field, type 20.44[mm].
10
In the z text field, type 20.44[mm].
11
In the x text field, type -19.86[mm].
12
In the y text field, type -19.86[mm].
Difference 2 (dif2)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object extract2(4) only.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the object rot2 only.
Difference 3 (dif3)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object dif2 only.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the object uni4 only.
6
Select the Keep objects to subtract checkbox.
Union 5 (uni5)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
Convert to COMSOL 1 (ccom1)
1
In the Geometry toolbar, click  Conversions and choose Convert to COMSOL.
2
Click the  Select All button in the Graphics toolbar.
Collapse Faces 1 (clf1)
1
In the Geometry toolbar, click  Virtual Operations and choose Collapse Faces.
2
On the object fin, select Boundary 92 only.
Ignore Edges 1 (ige1)
1
In the Geometry toolbar, click  Virtual Operations and choose Ignore Edges.
2
On the object clf1, select Edges 18 and 20 only.
3
In the Settings window for Ignore Edges, click  Build Selected.
4
In the Geometry toolbar, click  Export.
5
In the Export[noun] window for Geometry, locate the Export section.
6
In the Filename text field, type aluminum_extrusion_fsi.mphbin.
7
Click  Export.