Solid Multilayer Shell Comparison

This example is a benchmark test showing that the physics interface gives results equivalent to those from a model using the interface based on a solid 3D representation. Figure 1 shows the solid representation of the model geometry where an aluminum shell and a copper shell are connected through a thin semiconductor layer.

The model solves for the structure both on the true solid representation as shown in Figure 1, and on the approximative shell representation in Figure 2. The shell description is simpler from the geometrical drawing and meshing perspectives, but requires specialized features that offer the option to couple different layer materials properly. A mix of several geometrical junctions and feedings, together with also showing the full 3D equivalent, makes this model suitable as an introduction to the interface.

Figure 1: The solid representation of the model.

Figure 2: The shell representation of the model.

Model Definition

The model contains a planar, 0.1 mm thick, semiconductor layer embedded between two thicker complex-shaped metallic profiles. The lower layer is made of copper, while the upper layer is made of aluminum. A current is applied on a square terminal on the upper surface of the aluminum layer, while ground is placed on a lateral surface of the copper. The electrical conductivity of the semiconductor is 1000 S/m, that is considerably lower than that of the connecting metal layers. The aim is to compute and compare the electric potential distribution in the various layers, both for the solid representation and for the layered shell approach.

Modeling approach

For easy comparison, the model is solved with interface and the interface applied on separate objects in the same geometry as shown in figure 3 and 4. In these figures, the solid representation is on the left side, while the shell representation is on the right side.

Figure 3: Side view of solid and shell geometric configurations.

Figure 4: 3D view of solid and shell geometric configurations.

Solid modeling is, from the physics point of view, the standard choice and usually the most accurate approach. It is featured in many applications including the Introduction to COMSOL Multiphysics manual. The only special thing here is that, in the solid representation, the thin semiconductor layer is meshed using a swept mesh. The geometry for the layered shell model does not require any particular attention and a free surface triangular mesh is used.

On the accuracy of using the Electric Currents in layered Shells interface

The interface represents the shell through boundaries by taking care of the details inside the shell (each Layered Material domain contains an underlying extra dimension where the volumetric 3D physics is represented). By setting the Shell Properties ‘From material’ and checking ‘Use all layers’, the properties of the shell can be automatically detected. Wherever the is applied (magenta and red regions in Figure 2) both perpendicular and tangential currents are fully taken into account. Wherever or other material with property are applied (yellow and cyan regions in Figure 2) only tangential currents are accounted for and the electric potential will be constant in the direction normal to the shell. This is physically consistent with the single layer where the current is almost exclusively tangential. The limitation of the accuracy of the shell model is then essentially within the approximation of the geometrical representation of the shell midplane and the expected discrepancy of solid versus shell models is within the order of the ratio of thickness to length. This means the thinner the shell is, the more accurate the shell model will be, in respect to the exact solution. It is however worth noting that the interface offers features which make it possible to simulate objects which are so thin that a 3D volumetric representation would very cumbersome or expensive due to meshing difficulties.

Results and Discussion

The model is solved in a stationary study. A solid/volumetric representation of the surface electric potential in the solid and shell objects is plotted in Figure 5. A special dataset enables the user to represent the shell with its appropriate physical thickness and offset.

Figure 5: Electric potential on the solid (left) and on the shell description (right).

For a more quantitative comparison, in Figure 6, the electric potential is plotted along inner (blue) and outer (red) edge profiles for the solid description (solid line) and for the shell description (symbols). The comparison shows good agreement between the 3D solid model and the layered shell model. Note that the results compare well also in the region where the blue and red lines differ substantially. This region has a significant potential drop in the normal direction that is introduced by the thin semiconductor layer. The agreement among curves and symbols in the potential drop between red and blue data sets implies that the layered shell model resolves the current perpendicular to the semiconductor layer. The agreement in the potential drop from left to right indicates that the tangential currents are properly solved too. To conclude, this benchmark example showcases the ability of the layered shell interface to describe 3D currents in generic resistive and conductive layered structures.

Figure 6: Comparison of the simulated electric potential along Path A and B for the solid and shell description, where the definitions of Path A and B are given in the section Modeling Instructions — Solid-Shell Comparison.

ACDC_Module/Layered_Shell/solid_multilayer_shell_comparison

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

In the tree, select .

Click .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Cu_thickness	1[mm]	0.001 m	Thickness of copper layer (J shaped lower profile)
Al_thickness	2[mm]	0.002 m	Thickness of aluminum layer (Z shaped upper profile)
Air_thickness	3[mm]	0.003 m	Air gap thickness
Semi_thickness	0.1[mm]	1E-4 m	Thin semiconductor layer thickness

Add Material

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Click .

In the tree, select .

Click .

In the tree, select .

Click .

In the toolbar, click to close the window.

Global Definitions

Material 4 (mat4)

In the window, under right-click and choose .

In the window for , type Semiconductive Material in the text field.

Click to expand the section. In the tree, select .

Click .

In the tree, select .

Click .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	1000	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Layered Material 1 (lmat1)

Right-click and choose .

In the window for , type Al-Air-Cu Stack in the text field.

Locate the section. Click .

Click .

In the table, enter the following settings:


Layer	Material	Rotation (deg)	Thickness (m)	Mesh elements
Al	Aluminum (mat2)	0.0	Al_thickness	2
Air	Air (mat1)	0.0	Air_thickness	2
Cu	Copper (mat3)	0.0	Cu_thickness	2

Layered Material 2 (lmat2)

Right-click and choose .

In the window for , type Cu-Semi-Al Stack in the text field.

Locate the section. Click .

Click .

In the table, enter the following settings:


Layer	Material	Rotation (deg)	Thickness (m)	Mesh elements
Cu	Copper (mat3)	0.0	Cu_thickness	2
Semi	Semiconductive Material (mat4)	0.0	Semi_thickness	2
Al	Aluminum (mat2)	0.0	Al_thickness	2

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

Click .

Work Plane 1 (wp1)>Rectangle 1 (r1)

In the toolbar, click .

In the window for , locate the section.

In the text field, type 0.15.

In the text field, type Cu_thickness.

Work Plane 1 (wp1)>Rectangle 2 (r2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.05.

In the text field, type Air_thickness.

Locate the section. In the text field, type Cu_thickness.

Work Plane 1 (wp1)>Rectangle 3 (r3)

Right-click and choose .

In the window for , locate the section.

In the text field, type Al_thickness.

Locate the section. In the text field, type Cu_thickness+Air_thickness.

Work Plane 1 (wp1)>Rectangle 4 (r4)

Right-click and choose .

In the window for , locate the section.

In the text field, type Al_thickness.

In the text field, type 0.055-Cu_thickness-Air_thickness.

Locate the section. In the text field, type 0.05.

Work Plane 1 (wp1)>Rectangle 5 (r5)

Right-click and choose .

In the window for , locate the section.

In the text field, type Cu_thickness.

In the text field, type 0.055-2*Cu_thickness-Al_thickness.

Locate the section. In the text field, type 0.15-Cu_thickness.

In the text field, type Cu_thickness.

Work Plane 1 (wp1)>Rectangle 6 (r6)

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.1-Al_thickness.

In the text field, type Al_thickness.

Locate the section. In the text field, type 0.05+Al_thickness.

In the text field, type 0.055-Al_thickness.

Work Plane 1 (wp1)>Rectangle 7 (r7)

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.05.

In the text field, type Cu_thickness.

Locate the section. In the text field, type 0.1.

In the text field, type 0.055-Al_thickness-Cu_thickness.

Work Plane 1 (wp1)>Rectangle 8 (r8)

Right-click and choose .

In the window for , locate the section.

In the text field, type Semi_thickness.

Locate the section. In the text field, type 0.055-Al_thickness-Semi_thickness.

In the toolbar, click .

Click the button in the toolbar.

Work Plane 1 (wp1)>Union 1 (uni1)

In the toolbar, click and choose .

Select the objects , , and only.

In the window for , locate the section.

Clear the check box.

Work Plane 1 (wp1)>Union 2 (uni2)

In the toolbar, click and choose .

Select the objects , , and only.

In the window for , locate the section.

Clear the check box.

In the toolbar, click .

Click the button in the toolbar.

Work Plane 1 (wp1)>Delete Entities 1 (del1)

In the window, right-click and choose .

On the object , select Boundaries 6 and 9 only.

In the window for , click .

Click the button in the toolbar.

Work Plane 1 (wp1)>Line Segment 1 (ls1)

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the text field, type 0.05.

Locate the section. In the text field, type 0.15.

Locate the section. In the text field, type 0.055.

Work Plane 1 (wp1)>Convert to Solid 1 (csol1)

In the toolbar, click and choose .

Select the objects and only.

In the window for , click .

Work Plane 1 (wp1)>Polygon 1 (pol1)

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type 0.16 0.31 0.31 0.31 0.31 0.26 0.26 0.26 0.21 0.21 0.21.

In the text field, type 0 0 0 0.052 0.052 0.052 0.052 0.052 0.052 0 0.

In the toolbar, click .

Click the button in the toolbar.

Work Plane 1 (wp1)>Fillet 1 (fil1)

In the toolbar, click .

On the object , select Point 7 only.

In the window for , locate the section.

In the text field, type 0.01.

Click .

Work Plane 1 (wp1)>Fillet 2 (fil2)

In the toolbar, click .

On the object , select Point 5 only.

In the window for , locate the section.

In the text field, type 0.01+Al_thickness.

Click .

Work Plane 1 (wp1)>Fillet 3 (fil3)

In the toolbar, click .

Click the button in the toolbar.

On the object , select Point 3 only.

In the window for , locate the section.

In the text field, type 0.01.

Click .

Work Plane 1 (wp1)

In the window, click .

Extrude 1 (ext1)

In the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Distances (m)
0.1
0.2

Click the button in the toolbar.

Click .

Work Plane 1 (wp1)

In the window, collapse the node.

Delete Entities 1 (del1)

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Domains 2, 3, and 7 only.

Click .

Delete Entities 2 (del2)

Right-click and choose .

On the object , select Boundaries 50, 52, and 56 only.

In the window for , click .

Work Plane 2 (wp2)

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

On the object , select Boundary 14 only.

Plane Geometry

In the window, click .

Work Plane 2 (wp2)>Rectangle 1 (r1)

In the toolbar, click .

In the window for , locate the section.

In the text field, type 0.01.

Locate the section. In the text field, type -0.025.

In the text field, type -0.005.

Work Plane 2 (wp2)

In the window, click .

In the window for , locate the section.

Select the check box.

In the text field, type Feeding surface in Volumetric Geometry.

Click .

Copy 1 (copy1)

In the toolbar, click and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 0.16.

In the text field, type -Cu_thickness-Air_thickness-Al_thickness.

In the text field, type Feeding surface in Shell Geometry.

Locate the section. Select the check box.

Click .

Click the button in the toolbar.

Materials

Material Link 1 (matlnk1)

In the window, under right-click and choose .

In the window for , type Solid: Air in the text field.

Select Domain 3 only.

Material Link 2 (matlnk2)

Right-click and choose .

In the window for , type Solid: Aluminum in the text field.

Locate the section. From the list, choose .

Select Domain 4 only.

Material Link 3 (matlnk3)

Right-click and choose .

In the window for , type Solid: Copper in the text field.

Locate the section. From the list, choose .

Select Domains 1 and 2 only.

Material Link 4 (matlnk4)

Right-click and choose .

In the window for , type Solid: Semiconductive Material in the text field.

Locate the section. From the list, choose .

Select Domain 5 only.

Layered Material Link 1 (llmat1)

Right-click and choose .

In the window for , type Shell: Al-Air-Cu in the text field.

Select Boundaries 50 and 51 only.

Locate the section. From the list, choose .

In the text field, type -(Cu_thickness+Air_thickness)/(Cu_thickness+Air_thickness+Al_thickness).

In a COMSOL geometry, all boundaries are equipped with a surface normal. The surface normal is a unit vector that is tied to what is considered the upside and downside of the boundary. For layered materials, the normal vector dictates the direction in which the layers are stacked. When an offset is specified, the layer stack is shifted in the direction of the surface normal. Although the shift will not affect the solution (as curvature is neglected), it does affect how the layers are displayed in postprocessing.

Layered Material Link 2 (llmat2)

Right-click and choose .

In the window for , type Shell: Cu-Semi-Al in the text field.

Locate the section. From the list, choose .

Select Boundary 58 only.

Locate the section. From the list, choose .

In the toolbar, click and choose .

Add Material

Go to the window.

In the tree, select and .

Click .

Materials

Aluminum (mat5)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click to expand the section. In the tree, select .

Click .

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type Al_thickness.

In the window, click .

In the window for , type Shell: Al in the text field.

Select Boundaries 53, 55, and 56 only.

Locate the section. From the list, choose .

In the text field, type 1+2*(Semi_thickness+Cu_thickness)/Al_thickness.

Copper (mat6)

In the window, click .

In the window for , locate the section.

From the list, choose .

Click to expand the section. In the tree, select .

Click .

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type Cu_thickness.

In the window, click .

In the window for , type Shell: Cu in the text field.

Select Boundaries 49, 52, 54, 57, 59, and 60 only.

Locate the section. From the list, choose .

Electric Currents (ec)

In the window, under click .

In the window for , locate the section.

In the list, select .

Click .

Select Domains 1, 2, 4, and 5 only.

Ground 1

In the toolbar, click and choose .

Click the button in the toolbar.

Select Boundaries 1 and 5 only.

Click the button in the toolbar.

Terminal 1

In the toolbar, click and choose .

In the window for , locate the section.

In the I0 text field, type 1.

Select Boundary 15 only.

Click the button in the toolbar.

Electric Currents in Layered Shells (ecis)

In the window, under click .

In the window for , locate the section.

Select the check box.

Conductive Shell 1

In the window, expand the node, then click .

Terminal 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 51 only.

Locate the section. From the list, choose .

Locate the section. In the I0 text field, type 1.

Ground 1

In the toolbar, click and choose .

Select Edges 107, 109, and 111 only.

In the window for , locate the section.

Clear the check box.

Specify the vector as


	Al
	Air
√	Cu

Ground 2

In the toolbar, click and choose .

Select Edge 105 only.

Continuity 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Continuity 2

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

In the table, enter the following settings:


	Layered material	Offset (m)
√	Shell: Cu (mat6)	0

Continuity 3

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

In the table, enter the following settings:


	Layered material	Offset (m)
√	Shell: Al (mat5)	0

Continuity 4

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Insulating Layer 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the vector as


	Al
√	Air
	Cu

Mesh 1

Free Tetrahedral 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domains 1 and 2 only.

Click .

Swept 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domain 5 only.

Click .

Free Tetrahedral 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domain 4 only.

Click the button in the toolbar.

Click .

Click the button in the toolbar.

Free Triangular 1

Right-click and choose .

Select Boundaries 49–60 only.

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

Select the check box.

In the text field, type 0.005.

Click .

For the solid representation, there is one unique expression for the electric potential: <c>V</c>. For the layered shell representation however, there are several (depending on which side of the boundary you are): <c>ecis.Vup</c>, <c>ecis.Vdown</c>, and <c>ecis.Vav</c>. As certain parts of the shell geometry show material discontinuities and varying definitions of the surface normal, the appropriate expression to use in the plot differs from place to place. In order to simplify postprocessing, we define a couple of intermediate variables.

The battery charging and discharging variables are set up as follows:

For the solid representation, there is one unique expression for the electric potential: V. For the layered shell representation however, there are several (depending on which side of the boundary you are): ecis.Vup, ecis.Vdown, and ecis.Vav. As certain parts of the shell geometry show material discontinuities and varying definitions of the surface normal, the appropriate expression to use in the plot differs from place to place. In order to simplify postprocessing, we define a couple of intermediate variables.

The battery charging and discharging variables are set up as follows:

Definitions (comp1)

Variables 1