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S h e l l D i f f u s i o n

Introduction

A goal for many applications is to predict physics in thin structures, such as shells, without modeling the thickness of the structure. This is because large aspect ratios can cause meshing and geometry analysis problems. The model reported here demonstrates how to use the tangential derivative variables in COMSOL Multiphysics to solve partial differential equations in curved 3D shells and 2D boundaries without modeling their thickness.

Model Definition

The steel tank shown below has two pipe connections. One is grounded and the other connects to a dead current source. This model calculates the current density in the tank shell along with the potential distribution across the surface.

E Q U A T I O N S

The fundamental equation to solve is the current conduction, or charge conservation, equation.

(1)

Here, σ is the electrical conductivity (S/m) and V is the electric potential (V).

The material is a 1 mm thick steel sheet with a conductivity of 4.032·106 S/m. You are working with a surface in 3D so there is no thickness in the model. To account for the charge conservation in Equation 1 you must multiply the current flux expression with the shell thickness d:

(2)

0 V

400 V

∇ σ ∇V–( )⋅ 0=

∇ σ d∇V–( )⋅ 0=

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Results

Figure 1 shows the potential distribution across the surface.

Figure 1: Electric potential distribution across the surface (V).

Figure 2 adds the current field as an arrow plot, showing clearly how the current collects toward the grounded connection.

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Figure 2: Arrow plot of the local current field.

Figure 3: Local magnitude of the electric current density (A/m2).

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The plot of the magnitude of the local current density in Figure 3 is interesting because you can use it to calculate the resistive heating in the material as an extension to the model.

Notes About the COMSOL Implementation

Model Equation 2, the current conduction equation, using a Coefficient Form Boundary PDE interface, setting the diffusion coefficient c = σd. To define the current field components use tangential derivative variables, which you access in COMSOL Multiphysics by adding a T suffix to the variable name before specifying the gradient component. So, for example, the tangential derivative (∂u/∂x)T is represented by the variable uTx.

Model Library path: COMSOL_Multiphysics/Equation-Based_Models/shell_diffusion

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click Next.

3 In the Add physics tree, select Mathematics>PDE Interfaces>Lower

Dimensions>Coefficient Form Boundary PDE (cb).

4 Click Add Selected.

5 In the Dependent variables table, enter the following settings:

6 Click Next.

7 Find the Studies subsection. In the tree, select Preset Studies>Stationary.

8 Click Finish.

G L O B A L D E F I N I T I O N S

Parameters1 In the Model Builder window, right-click Global Definitions and choose Parameters.

V

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2 In the Parameters settings window, locate the Parameters section.

3 In the table, enter the following settings:

G E O M E T R Y 1

Work Plane 11 In the Model Builder window, under Model 1 right-click Geometry 1 and choose Work

Plane.

2 In the Work Plane settings window, locate the Work Plane section.

3 From the Plane list, choose yz-plane.

Bézier Polygon 11 In the Model Builder window, under Model 1>Geometry 1>Work Plane 1 right-click

Plane Geometry and choose Bézier Polygon.

2 In the Bézier Polygon settings window, locate the Polygon Segments section.

3 Find the Added segments subsection. Click the Add Linear button.

4 Find the Control points subsection. In row 2, set xw to 0.15.

5 Find the Added segments subsection. Click the Add Linear button.

6 Find the Control points subsection. In row 2, set yw to 1.

7 Find the Added segments subsection. Click the Add Linear button.

8 Find the Control points subsection. In row 2, set xw to 0.05.

9 Locate the General section. From the Type list, choose Open curve.

10 Click the Build Selected button.

Fillet 11 In the Model Builder window, under Model 1>Geometry 1>Work Plane 1 right-click

Plane Geometry and choose Fillet.

2 On the object b1, select Points 3 and 4 only.

3 In the Fillet settings window, locate the Radius section.

4 In the Radius edit field, type 0.05.

5 Click the Build Selected button.

NAME EXPRESSION DESCRIPTION

sigma 4.032e6[S/m] Conductivity

d 1[mm] Shell thickness

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Revolve 11 In the Model Builder window, under Model 1>Geometry 1 right-click Work Plane 1 and

choose Revolve.

2 Right-click Revolve 1 and choose Build Selected.

Work Plane 21 Right-click Geometry 1 and choose Work Plane.

2 In the Work Plane settings window, locate the Work Plane section.

3 From the Plane list, choose yz-plane.

4 Click the Show Work Plane button.

Circle 11 In the Model Builder window, under Model 1>Geometry 1>Work Plane 2 right-click

Plane Geometry and choose Circle.

2 In the Circle settings window, locate the Size and Shape section.

3 In the Radius edit field, type 0.05.

4 Locate the Position section. In the yw edit field, type 0.2.

5 Click the Build Selected button.

Extrude 11 In the Model Builder window, under Model 1>Geometry 1 right-click Work Plane 2 and

choose Extrude.

2 In the Extrude settings window, locate the Distances from Plane section.

3 In the table, enter the following settings:

4 Click the Build Selected button.

5 Click the Zoom Extents button on the Graphics toolbar.

Convert to Surface 11 In the Model Builder window, right-click Geometry 1 and choose Conversions>Convert

to Surface.

2 From the Edit menu, choose Select All, then right-click in the Graphics window to confirm the selection.

3 Click the Build Selected button.

DISTANCES (M)

0.2

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Delete Entities 11 In the Model Builder window, right-click Geometry 1 and choose Delete Entities.

Next, select all boundaries of the extruded cylinder. This is most easily done as follows:

2 Click the Go to ZX View button on the Graphics toolbar.

3 Click the Wireframe Rendering button on the Graphics toolbar.

Click the Zoom Box button on the Graphics toolbar, then click and drag in the Graphics window to enclose the small cylinder, and finally right-click to confirm the selection.

4 Click the Build Selected button.

5 Click the Go to Default 3D View button on the Graphics toolbar.

6 Click the Wireframe Rendering button on the Graphics toolbar to return to the default rendering state.

7 Click the Zoom Extents button on the Graphics toolbar.

The model geometry is now complete.

C O E F F I C I E N T F O R M B O U N D A R Y P D E

1 In the Model Builder window, under Model 1 click Coefficient Form Boundary PDE.

2 In the Coefficient Form Boundary PDE settings window, locate the Units section.

3 Find the Dependent variable quantity subsection. From the list, choose Electric

potential (V).

4 Find the Source term quantity subsection. In the Unit edit field, type A*m^-2.

Coefficient Form PDE 11 In the Model Builder window, expand the Coefficient Form Boundary PDE node, then

click Coefficient Form PDE 1.

2 In the Coefficient Form PDE settings window, locate the Diffusion Coefficient section.

3 In the c edit field, type sigma*d.

4 Locate the Source Term section. In the f edit field, type 0.

These settings specify the charge conservation equation Equation 2 for the shell surface.

Go on to set the values of the potential at the pipe connections by adding Dirichlet boundary conditions.

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Dirichlet Boundary Condition 11 In the Model Builder window, right-click Coefficient Form Boundary PDE and choose

Dirichlet Boundary Condition.

2 Select Edges 14, 15, 25, and 29 only.

3 In the Dirichlet Boundary Condition settings window, locate the Dirichlet Boundary

Condition section.

4 In the r edit field, type 400.

Dirichlet Boundary Condition 21 In the Model Builder window, right-click Coefficient Form Boundary PDE and choose

Dirichlet Boundary Condition.

2 Select Edges 40–43 only.

M E S H 1

1 In the Model Builder window, under Model 1 click Mesh 1.

2 In the Mesh settings window, locate the Mesh Settings section.

3 From the Element size list, choose Finer.

4 Click the Build All button.

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S T U D Y 1

In the Model Builder window, right-click Study 1 and choose Compute.

R E S U L T S

3D Plot Group 1The default plot shows the potential distribution.

1 Click the Zoom Extents button on the Graphics toolbar.

2 Rotate the geometry so that you see both pipe connections.

Compare the result with the plot in Figure 1.

Add an arrow surface plot of the current field as follows:

3 In the Model Builder window, right-click 3D Plot Group 1 and choose Arrow Surface.

4 In the Arrow Surface settings window, locate the Expression section.

5 In the x component edit field, type -sigma*VTx.

6 In the y component edit field, type -sigma*VTy.

7 In the z component edit field, type -sigma*VTz.

8 Select the Description check box.

9 In the associated edit field, type Current field (-sigma*VTx, -sigma*VTy, -sigma*VTz).

10 Locate the Coloring and Style section. From the Arrow length list, choose Normalized.

11 Click the Plot button.

The plot in the Graphics window should now look like that in Figure 2.

To visualize the magnitude of the local current density, follow the steps given below.

3D Plot Group 21 In the Model Builder window, right-click Results and choose 3D Plot Group.

2 In the 3D Plot Group settings window, locate the Title section.

3 From the Title type list, choose Manual.

4 In the Title text area, type Current density (A/m²).

5 Right-click Results>3D Plot Group 2 and choose Surface.

6 In the Surface settings window, locate the Expression section.

7 In the Expression edit field, type sigma*sqrt(VTx^2+VTy^2+VTz^2).

8 Click the Plot button.

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9 Click the Zoom Extents button on the Graphics toolbar.

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