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Submarine Cable 1 — Introduction
Introduction
Numerical analysis of cable systems is an active field of study. It is not only dominated by scientific insight, but also engineering practice and numerical considerations. Production cost plays an important role, and so does durability. The cable industry tends to be on the conservative side — and for good reasons; replacing a faulty submarine cable can be extremely costly1. The typical design may last for over forty years. To make sure one gets a good return on investment, the industry has been relying heavily on rules of thumb, safety margins, life-cycle analysis, and international standards such as the IEC 60287 [1].
Figure 1: The volumetric loss density in the screens and the armor of a three-core lead sheathed XLPE HVAC submarine cable at a nominal phase temperature of 90°C.
In the meantime, however, computational resources have increased drastically. When investigating research on the matter [2, 6], one cannot help but notice a strong correlation between the year of publishing and the level of detail present in the numerical models. In particular the introduction of twisted periodicity [2] and short-twisted periodicity [3] has been of great significance: Fully detailed 3D cable modeling is now possible within minutes on relatively cheap hardware (rather than hours, on cluster systems).
As a consequence, traditional methods and safety margins are being questioned: Numerical research is pushing the industry to find more cost-effective solutions.
Tutorial Series Overview
This tutorial series discusses the capacitive, inductive, and thermal properties of a three-core, lead sheathed XLPE HVAC2 submarine cable. It has a main conductor cross section of 500 mm2, a phase-to-phase voltage of 220 kV, and carries a nominal current of 655 A.
The series is written in eight chapters (as summarized on the following pages), starting with the fundamental principles in 2D. In a systematic fashion it deduces and proofs what cable experts already know — as a form of validation — and then continues to increase the complexity to a point where 2D models are no longer sufficient.
Apart from discussing cable related topics such as charging currents, bonding types, armor twists, and temperature dependency, a lot of attention is given to electromagnetism and numerical modeling in general. This includes the importance of certain material properties and geometric features, ill-posed problems, weakly coupled systems, verification, and simplification.
Note: Although this tutorial series investigates a specific kind of submarine cable, many of the discussed topics apply to terrestrial cable systems just the same. In fact, some of the treated topics (such as cross bonding) are included specifically because they are of interest for cable systems in general, not so much for submarine cables in particular.
A Note on Educational Value
Although this tutorial will be of particular use for those working within the cable industry, it is not all “just about cables”. This tutorial — and the entire series, for that matter — is about Electromagnetics, and Numerical Analysis. It is about good engineering practices, about understanding and applying theory, about result validation, and about presenting your results in an attractive and informative way.
The three-phase cable with the magnetic, twisted armor is and ideal device to illustrate and investigate various electromagnetic and numeric phenomena. Since many of these cables are standardized3, their physical properties are available from literature (allowing for validation). At the same time, they are a part of ongoing research. This makes them a suitable tool for industry professionals and academic students alike, to familiarize themselves with the numerical analysis of electromagnetic devices in general.
Note: At this point it may be tempting to skip to the 3D part of the tutorial series directly. It should be noted however, that many of the cable’s aspects can be understood and investigated perfectly fine with 2D models alone. The computational cost of a 2D model is negligible compared to 3D. When in doubt, start in 2D.
Submarine Cable 1 — Introduction
The model that results from this tutorial serves as a template for chapters 2, 4, and 6 (the Capacitive, Inductive, and Thermal Effects tutorials). It contains a detailed 2D geometry of the submarine cable, along with a mesh. Some parameters, selections, and materials are prepared as well. The corresponding instructions start in section Modeling Instructions (Introduction Tutorial).
Figure 2: The cable's 2D geometry, including the three phases (with screens) and an armor.
Experienced COMSOL users with little or no interest in these topics — geometry, mesh, selections, and such — may choose to skip this part of the series and continue with one of the following tutorials. When you are new to COMSOL Multiphysics, however, it is worthwhile to take some time for this, as it will help you to familiarize yourself with the basics.
Submarine Cable 2 — Capacitive Effects
Figure 3: The real part of the electric potential distribution at phase .
The second chapter uses the 2D geometry from the introduction tutorial to investigate the cable’s capacitive properties. It validates the assumption that for the capacitance and the charging currents, analytical approaches are sufficient (verification is included).
The influence of material properties, the cable’s length and bonding types is discussed. The model justifies the approach chosen in subsequent tutorials in this series, most notably, the Bonding Capacitive and the Inductive Effects tutorials (chapters 3, and 4).
Included Topics
•
Using the Electric Currents interface: Current Conservation, Terminal, Ground... Extracting lumped quantities using the terminal feature.
•
Comparing a numerical model to an analytical one. Comparing displacement- and conduction currents. Checking the risk of electromagnetic breakdown. Checking the conditions on the floating end of a 10 km long cable with single-point bonding applied.
•
Using expressions such as exp(-120[deg]*j) to set a phase difference in the frequency domain. Reflecting on the applicability of the used partial differential equations (whether it is safe to assume the in-plane electric fields are curl-free).
•
Reflecting on the relevance of certain details in the model. Discussing the large material contrast (conductors and insulators) and how this affects numerical stability.
Submarine Cable 3 — Bonding Capacitive
Figure 4: The charging current that builds up in the lead sheath (in case of cross bonding).
Based on the results from the Capacitive Effects tutorial, it is justified to neglect the capacitive coupling between the screens and consider one single isolated phase, together with its screen. As opposed to the Capacitive, Inductive, and Thermal Effects tutorials, this tutorial uses a 2D axisymmetric geometry representing the entire 10 km of cable.
For several bonding types, the build-up of charging currents and the corresponding losses in the screen are analyzed (verification is included). The model validates the assumption that the high phase potential induces a uniform charging current — one that barely depends on the screen potential — and so justifies the approach chosen in the Capacitive and Inductive Effects tutorials (chapters 2, and 4).
Included Topics
•
Using the Electric Currents interface: Current Conservation, Electric Potential, Ground... Determining the total loss in a system by means of a volume integration.
•
Using a 2D axisymmetric geometry. Using scaled coordinate systems to make extremely long geometries manageable. Using an anisotropic conductivity to stabilize a model.
•
Discussing several bonding types: single-point bonding, solid bonding, and cross bonding, and reflecting on how they affect the build up of charging currents. Comparing a numerical model to analytical ones (one for each bonding type).
Submarine Cable 4 — Inductive Effects
Figure 5: The instantaneous magnetic flux density norm when the armor twist is included.
Results from the Capacitive Effects and Bonding Capacitive tutorials show there is only a weak coupling between the inductive and capacitive phenomena in the cable. In addition to this, research [2, 6] suggests 2D and 2.5D inductive models are able to provide a good approximation of the cable’s lumped quantities, and at only a fraction of the computational cost (as compared to long 3D twist models).
This justifies a 2D/2.5D inductive model that includes out-of-plane currents only. The model demonstrates methods suitable to approximate the armor twist, as well as certain milliken conductor designs. It serves as a basis and a reference for the Thermal Effects and the Inductive Effects 3D tutorials (chapters 6, and 8). Verification is included; the results are compared to the cable’s official specifications.
Included Topics
•
Using the Magnetic Fields interface: Ampère’s Law, Coil (group), Magnetic Insulation... Exciting a 2D/2.5D inductive model by means of the Coil feature.
•
Using a Coil group to model the armor twist (useful for cross bonding too). Using the Homogenized multiturn setting to approximate a milliken conductor.
•
Advanced postprocessing. Analyzing and verifying results, investigating the differences between the Plain 2D, 2.5D and 2.5D + milliken configurations. Reflecting on the difference between the AC- and DC resistance.
Submarine Cable 5 — Bonding Inductive
Figure 6: The magnetic flux density norm in a simplified geometry (in case of solid bonding).
In the Inductive Effects tutorial, the Coil group option is mentioned as a means to mimic the effects of cross bonding, but the exact extent of the validity of this approach is not shown. In order to investigate the different bonding types more closely, the fifth tutorial considers three sections of cable individually, represented by three separate Magnetic Fields interfaces.
The model uses a strongly simplified geometry. Even so, the results correspond very well to those from the Inductive Effects tutorial. This justifies both the simplified geometry in this tutorial, as well as the cross-bonding approach suggested in the Inductive Effects tutorial. Finally, as opposed to the other inductive models in this series, this one allows for investigating dissimilar section lengths.
Included Topics
•
Using the Magnetic Fields interface: Ampère’s Law, Coil, Magnetic Insulation... Connecting three finite element models in series, using an Electrical Circuit.
•
Discussing several bonding types: single-point bonding, solid bonding, and cross bonding, and reflecting on how they affect the potential and currents in the screens.
•
Validating if cross bonding can be modeled in 2D by means of a Coil group. Comparing the losses and lumped parameters to the detailed plain 2D model from the Inductive Effects tutorial and reflecting on what details are really important.
Submarine Cable 6 — Thermal Effects
Figure 7: The temperature distribution inside the cable at sea bed temperature Text = 20°C.
This tutorial uses the Inductive Effects model from this series as a basis and adds thermal effects, including a temperature dependent conductivity (through linearized resistivity). It shows how to achieve a multiphysics coupling between electromagnetic fields and heat transfer, using the frequency-stationary study type (induction heating).
The influence of elevated temperatures on losses in the phases, screens, and armor is investigated (verification is included). The obtained temperature values are used in the Inductive Effects 3D tutorial, to apply a first-order temperature correction in 3D. Finally, the tutorial demonstrates how to match the resulting AC resistance to the one given by the IEC series of standards.
Included Topics
•
Using the Magnetic Fields interface, the Heat Transfer in Solids interface and the Electromagnetic Heating multiphysics coupling to model induction heating.
•
Using linearized resistivity. Investigating resistive- and magnetic losses. Reflecting on the difference between current-driven and voltage-driven conductors, and how the heat affects the electromagnetic properties of the cable. Investigating the difference between the AC- and DC resistance, and the validity of stationary-electric reasoning.
•
Using a global ODE to set a phase conductivity such that the cable's AC resistance matches a certain specified (temperature dependent) value.
Submarine Cable 7 — Geometry & Mesh 3D
Figure 8: The helically twisted, swept mesh in the phases, the screens, and the armor.
The Inductive Effects 3D tutorial requires quite a lot of preparation in terms of geometry handling and meshing. In fact, this is what oftentimes represents a major part of the efforts spent on large 3D FEM models (twisted cable models in particular). In order not to overlook these important topics, they are addressed in a separate tutorial.
The tutorial is organized in four sections. The first part configures the camera settings (ensuring the geometry, mesh, and plots render as intended). Then, the geometry sequence is added, together with the selections. Finally, the mesh is constructed. Some postprocessing is done to investigate the quality of the mesh.
Included Topics
•
Using advanced geometry handling and meshing to optimize a 3D mesh to produce a high level of detail, while keeping the number of degrees of freedom (DOFs) low.
•
Using highly anisotropic meshes. Combining Swept meshes, Free Tetrahedral meshes, and Boundary Layer meshes. Reflecting on the importance of conforming meshes for periodicity planes. Demonstrating how mesh conformity can be enforced and checked.
•
Improving the accuracy of the model by compensating for geometric effects like slanted cuts and the truncation of circles (linear elements will convert them to polygons).
•
Using predefined selections of geometric entities to automate the work flow.
Submarine Cable 8 — Inductive Effects 3D
Figure 9: The volumetric loss density in the armor and the screens, for the case where a first-order temperature correction has been applied.
This last tutorial intends to give a “final answer” to 3D cable modeling. It has been developed with feedback from several experts from within the industry, and is on a par with the latest research (2020) when considering both performance and level of detail. Validation is included: The behavior of the models is analyzed within the context of refs. [2, 4, 5, and 6], and the cable’s official specifications.
The tutorial is organized in five sections: First, the cable is modeled without twist and compared to the much more detailed 2D models. Then, the twist is added and the model is solved and verified a second time. A temperature correction is added in a third step and an experimental numerical stabilization method is demonstrated in a fourth step. The tutorial finishes by switching to a different kind of periodicity, reducing the size of the model a hundredfold while still providing a similar accuracy.
Included Topics
•
The demonstration of a large (industrial scale) COMSOL model, including a highly optimized mesh, optimized solver settings, advanced postprocessing (with global evaluations, integrations, plots, and animations), and a detailed theoretical treatment of the observed physical phenomena — including references to recent numerical research.
•
Using the Magnetic Fields interface in 3D: Ampère’s Law, Coil, twisted Periodicity... Exiting a 3D inductive model by means of a Coil feature (with slanted cut), combined with a Coil Geometry Analysis preprocessing step.
•
A demonstration of twisted periodicity and short-twisted periodicity, phase- and armor lay length, first- (and higher) order temperature corrections, and the analysis of resistive and magnetic losses in the phases, the screens, and the armor.
•
A detailed comparison between the 3D twist models — with or without linearized resistivity included — and 2D/2.5D models (with or without induction heating).
•
A detailed analysis of the longitudinal and transverse eddies forming in the armor wires, and the paths taken by the magnetic field lines as they encircle the phase conductors.
•
A reflection on numerical stability and the demonstration of a numerical stabilization method that manages to improve the accuracy while keeping the solving time low.
References
1. International Electrotechnical Commission, Electric cables – Calculation of the Current Rating; IEC 60287; IEC Press: Geneva, Switzerland, 2006.
2. J.C. del-Pino-López, M. Hatlo, and P. Cruz-Romero, “On Simplified 3D Finite Element Simulations of Three-Core Armored Power Cables,” Energies 2018, 11, 3081.
3. D. de Vries, “3D Cable Modeling in COMSOL Multiphysics®” IEEE Spectrum, 2020, https://spectrum.ieee.org/webinar/3d_cable_modeling_in_comsol_multiphysics.
4. M. Hatlo, E. Olsen, R. Stølan, and J. Karlstrand, “Accurate Analytic Formula for Calculation of Losses in Three-Core Submarine Cables,” Proc. 9th Int’l Conf. on Insulated Power Cables (Jicable’15).
5. D. Willen, C. Thidemann, O. Thyrvin, D. Winkel, and V.M.R. Zermeno, “Fast Modelling of Armour Losses in 3D Validated by Measurements,” Proc. 10th Int’l Conf. on Insulated Power Cables (Jicable’19).
6. J.J. Bremnes, G. Evenset, R. Stølan, “Power Loss and Inductance of Steel Armoured Multi-Core Cables: Comparison of IEC Values with 2.5D FEA Results and Measurements,” (Cigré 2010).
Application Library path: ACDC_Module/Tutorials,_Cables/submarine_cable_01_introduction
Modeling Instructions (Introduction Tutorial)
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
In this case, we will not select any physics. This will be done in subsequent tutorials, depending on the analysis performed.
2
Click  Done.
Geometry 1
At this point, there are two options. One is to go to section Appendix — Modeling Instructions (Geometry) in this tutorial and follow the instructions for building the geometry there. In case you have limited interest in manually building geometries in COMSOL (because you intend to use CAD software for example), there is a second option. The completed geometry sequence can be inserted from a file:
1
In the Geometry toolbar, click  Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file submarine_cable_e_geom_sequence.mph.
3
In the Geometry toolbar, click  Build All.
4
Click the  Zoom Extents button in the Graphics toolbar.
You now have built the geometry. This step coincides with the last instructions in section Appendix — Modeling Instructions (Geometry). You can continue by preparing some parameters and selections.
Global Definitions
Geometric Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Geometric Parameters 1 in the Label text field.
If you have inserted the geometry sequence from a file, you might be surprised the table is already populated. This is because the parameters necessary for building the geometry are imported automatically.
Geometric Parameters 2
1
In the Home toolbar, click  Parameters and choose Add>Parameters.
2
In the Settings window for Parameters, type Geometric Parameters 2 in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file submarine_cable_b_geom_parameters.txt.
Eight items have been added to the table. The first is the main conductor’s true cross section, Acon; 500 mm2. Since these conductors are not actually solid but consist of a group of compacted strands, there is some space in between, and Acon is not identical to pi*(Dcon/2)^2 (the surface area implicitly used in the geometry). This gives rise to the parameter Ncon. Apbs is pretty straightforward. The last five parameters are related to the cable’s length.
Definitions
Next add some selections to use later on, when assigning physics features for instance. Fifteen domain selections have already been defined by the geometry sequence (as shown at the end of this tutorial). Based on the geometry-induced selections, you can add four derived selections: two unions, a difference, and an adjacent selection.
Phases
1
In the Definitions toolbar, click  Union.
2
In the Settings window for Union, type Phases in the Label text field.
3
Locate the Input Entities section. Under Selections to add, click  Add.
4
In the Add dialog box, in the Selections to add list, choose Phase 1, Phase 2, and Phase 3.
5
Click OK.
6
Click the  Zoom to Selection button in the Graphics toolbar.
Metals
1
In the Definitions toolbar, click  Union.
2
In the Settings window for Union, type Metals in the Label text field.
3
Locate the Input Entities section. Under Selections to add, click  Add.
4
In the Add dialog box, in the Selections to add list, choose Phases, Screens, Steel Helix (Fiber), and Cable Armor.
5
Click OK.
6
Click the  Zoom to Selection button in the Graphics toolbar.
By unifying the phase, screen, fiber and armor selections, you have just found an efficient strategy to determine the Metals selection. Notice that selecting the armor wires individually by hand, would have been kind of tedious and error prone. That is why the geometry sequence is used to generate the selection for the armor wires (the same operator in the geometry sequence that builds the wires, also defines their selection). For more details on this, see section Appendix — Modeling Instructions (Geometry).
Insulators (External to Phase)
1
In the Definitions toolbar, click  Difference.
2
In the Settings window for Difference, type Insulators (External to Phase) in the Label text field.
3
Locate the Input Entities section. Under Selections to add, click  Add.
4
In the Add dialog box, select Cable Domains in the Selections to add list.
5
Click OK.
6
In the Settings window for Difference, locate the Input Entities section.
7
Under Selections to subtract, click  Add.
8
In the Add dialog box, in the Selections to subtract list, choose Metals, Semi-Conductive Compound, and Cross-Linked Polyethylene (XLPE).
9
Click OK.
10
Click the  Zoom to Selection button in the Graphics toolbar.
Thermal Contact
1
In the Definitions toolbar, click  Adjacent.
2
In the Settings window for Adjacent, type Thermal Contact in the Label text field.
3
Locate the Input Entities section. Under Input selections, click  Add.
4
In the Add dialog box, select Insulators (External to Phase) in the Input selections list.
5
Click OK.
6
In the Settings window for Adjacent, locate the Output Entities section.
7
Select the Interior boundaries check box.
Now that the selections are finished, the materials will be next. To start with, you can modify the View to show the material colors. Then, the materials will be added, they will be given an appropriate label (if they do not already have it), a selection and an appearance. At the end of the instructions a table is listed that allows you to re-check the given settings.
View 1
1
In the Model Builder window, click View 1.
2
In the Settings window for View, locate the Colors section.
3
Select the Show material color and texture check box.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Air.
4
Right-click and choose Add to Component 1 (comp1).
5
In the tree, select Built-in>Water, liquid.
6
Right-click and choose Add to Component 1 (comp1).
Materials
Air (mat1)
1
Click the  Zoom to Selection button in the Graphics toolbar.
The air is applied to all domains. This is a good thing. You can use it as a default material and override it with other materials where applicable.
Water, liquid (mat2)
1
In the Model Builder window, click Water, liquid (mat2).
2
Select Domain 2 only.
3
Click the  Zoom to Selection button in the Graphics toolbar.
Gravel, saturated
1
In the Model Builder window, right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Gravel, saturated in the Label text field.
3
Select Domains 1 and 7 only.
4
Click the  Zoom to Selection button in the Graphics toolbar.
5
Click to expand the Material Properties section. In the Material properties tree, select Basic Properties>Relative Permeability.
6
Click  Add to Material.
7
Repeat these steps for Electrical Conductivity, Relative Permittivity, Thermal Conductivity, Density, and Heat Capacity at Constant Pressure.
8
Click to collapse the Material Properties section. Click to expand the Appearance section. From the Material type list, choose Rock.
Now, you have created a blank material with material properties for μr, σ, εr, k, ρ, and Cp; the material properties used for the electromagnetic and thermal equations respectively.
In a typical modeling scenario you would add the physics first however, before setting up the materials. The physics will then automatically request the material properties required to make the model solve. Setting up the materials in detail without the physics included takes a bit more effort, but it comes with the advantage that subsequent tutorials within this series will have most of the material properties preincluded. This allows the rest of the tutorial series to focus on the material properties that are of special importance to the particular kind of analysis performed.
You can save some time by taking this blank material as a template, and duplicating it a number of times.
Polyethylene
1
Right-click Gravel, saturated and choose Duplicate.
2
In the Settings window for Material, type Polyethylene in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Polyethylene.
4
Click the  Zoom to Selection button in the Graphics toolbar.
5
Click to expand the Appearance section. From the Material type list, choose Oil.
(Admittedly, Polyethylene isn’t “oil”, but the color seems suitable).
Polypropylene
1
Right-click Polyethylene and choose Duplicate.
2
In the Settings window for Material, type Polypropylene in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Polypropylene.
4
Click to expand the Appearance section. From the Material type list, choose Plastic.
5
From the Color list, choose Gray.
Cross-linked polyethylene (XLPE)
1
Right-click Polypropylene and choose Duplicate.
2
In the Settings window for Material, type Cross-linked polyethylene (XLPE) in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Cross-Linked Polyethylene (XLPE).
4
Click to expand the Appearance section. From the Color list, choose White.
Semi-conductive compound
1
Right-click Cross-linked polyethylene (XLPE) and choose Duplicate.
2
In the Settings window for Material, type Semi-conductive compound in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Semi-Conductive Compound.
4
Click to expand the Appearance section. From the Color list, choose Black.
Bitumen compound
1
Right-click Semi-conductive compound and choose Duplicate.
2
In the Settings window for Material, type Bitumen compound in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Bitumen Compound.
The glass is readily available in the material library, you can add it as follows.
Add Material
1
Go to the Add Material window.
2
In the tree, select Built-in>Silica glass.
3
Right-click and choose Add to Component 1 (comp1).
Materials
Silica glass (mat9)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Silica glass (mat9).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Fiber Optic Core.
4
Click the  Zoom to Selection button in the Graphics toolbar.
Stainless steel
1
In the Model Builder window, under Component 1 (comp1)>Materials right-click Bitumen compound (mat8) and choose Duplicate (make sure to duplicate the bitumen, not the glass).
2
In the Settings window for Material, type Stainless steel in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Steel Helix (Fiber).
4
Click to expand the Appearance section. From the Material type list, choose Steel.
Copper
1
Right-click Stainless steel and choose Duplicate.
2
In the Settings window for Material, type Copper in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Phases.
4
Click the  Zoom to Selection button in the Graphics toolbar.
For the phases, the screens and the armor, we will investigate the effect of elevated temperatures on the cable’s electromagnetic properties in a fully coupled induction heating model (see the Thermal Effects tutorial). To this end, add linearized resistivity to the copper (and indirectly, to the lead and the steel).
5
Click to expand the Material Properties section. In the Material properties tree, select Electromagnetic Models>Linearized resistivity.
6
Click  Add to Material.
7
Click to collapse the Material Properties section. Click to expand the Appearance section. From the Material type list, choose Copper.
Lead
1
Right-click Copper and choose Duplicate.
2
In the Settings window for Material, type Lead in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Screens.
4
Click the  Zoom to Selection button in the Graphics toolbar.
5
Click to expand the Appearance section. From the Material type list, choose Lead.
Galvanized steel
1
Right-click Lead and choose Duplicate.
2
In the Settings window for Material, type Galvanized steel in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose Cable Armor.
4
Click the  Zoom to Selection button in the Graphics toolbar.
5
Click to expand the Appearance section. From the Material type list, choose Steel.
6
In the Home toolbar, click  Add Material to close the Add Material window.
Now that the materials are in place, it is a good practice to re-check their basic settings (as presented in the following table).
7
In the Model Builder window, under Component 1 (comp1)>Materials, check whether the following materials are present:
 
Label
Selection
Appearance
mat1
Air
All domains (partially overridden)
Air
mat2
Water, liquid
Domain 2
Water
mat3
Gravel, saturated
Domains 1 and 7
Concrete
mat4
Polyethylene
Polyethylene
Plastic
mat5
Polypropylene
Polypropylene
Plastic
mat6
Cross-linked polyethylene (XLPE)
Cross-Linked Polyethylene (XLPE)
Steel
mat7
Semi-conductive compound
Semi-Conductive Compound
Plastic
mat8
Bitumen compound
Bitumen Compound
Plastic
mat9
Silica glass
Fiber Optic Core
Custom
mat10
Stainless steel
Steel Helix (Fiber)
Steel
mat11
Copper
Phases
Copper
mat12
Lead
Screens
Lead
mat13
Galvanized steel
Cable Armor
Steel
Please see to it that the materials are actually put in Component 1 (comp1)>Materials, rather than Global Definitions>Materials.
Materials
What remains are the material properties. The properties that should be added are listed in the following table. Please check all of them for the correct value, even the ones that are already filled in. Note that for cases like this, a convenient option is to copy-paste the values directly from this *.pdf file to COMSOL.
Furthermore, note that material properties marked with an “x” are not used at all (they may be present, but their value is irrelevant). Material properties marked with a “-”, are deliberately left blank. They will be given a value in subsequent tutorials. Common practice is to include the unit when typing: “1e-14[s/m]”, although strictly speaking that should not be necessary in this case. The entire tutorial series assumes standard SI units.
1
In the Model Builder window, under Component 1 (comp1)>Materials, add the following material properties:
 
mur
sigma [S/m]
epsr
k [W/(m*K)]
rho [kg/m^3]
Cp [J/(kg*K)]
mat1
1
1e-14
1
k(T)
rho(pA,T)
Cp(T)
mat2
x
x
x
k(T)
rho(T)
Cp(T)
mat3
1
1
28
1
2020
2512
mat4
1
1e-18
2.25
0.46
935
2302
mat5
1
1e-18
2.36
0.25
946
1920
mat6
1
1e-18
-
0.46
930
2302
mat7
1
2
2.25
10
1055
2405
mat8
1
3.2e-9
3.16
0.17
1062
1885
mat9
1
1e-14
3.75
1.38
2203
703
mat10
1
1.46e6
1
17.5
7920
475
mat11
-
-
1
-
8940
385
mat12
-
-
1
35.3
11340
127
mat13
-
-
1
58
7850
475
Mesh 1
The last part of this tutorial consists of checking the mesh. COMSOL includes a whole range of tools for advanced meshing. In this case however, we are dealing with a 2D geometry of little to modest complexity: Even with a coarse mesh, the model will solve quite accurately. In the results, the difference between Extra fine and Coarse will show up somewhere in the 4th or 5th significant digit. Feel free to check this in subsequent tutorials.
So instead of spending time on intelligent mesh optimization schemes, we will choose a pragmatic approach and just use the element size setting Normal (the default).
1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
This will be our mesh for the Capacitive, Inductive and Thermal Effects tutorials. it is more than fine enough to capture skin effects and small geometric features.
You have now completed this tutorial. The result is a COMSOL Multiphysics file with geometry, materials and mesh, that will be used as a template for a multitude of models investigating capacitive, inductive and thermal properties. Subsequent tutorials will refer to this file as submarine_cable_01_introduction.mph. The next tutorial in this series will include a detailed capacitive analysis.
1
From the File menu, choose Save As.
2
Browse to a suitable folder and type the filename submarine_cable_01_introduction.mph.
 
Appendix — Modeling Instructions (Geometry)
The geometry in this tutorial series is based on parameters. This is not strictly necessary in COMSOL, rather it accommodates quick adjustments and keeps things consistent. Most of the parameters are based directly upon the mayor national or international standards (it is a standardized cable). This particular case is a 500 mm2, 220 kV cable. By adjusting the parameters, other types can easily be made. In order to speed up this part of the tutorial, the parameters will be loaded from a file.
Global Definitions
Geometric Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Geometric Parameters 1 in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file submarine_cable_a_geom_parameters.txt.
Geometry 1
The geometry is constructed by means of a geometry sequence. The advantage of the sequence is that it not only saves the final result, but the entire process (the recipe, if you like). This way, it is easy to modify or parameterize the geometry. Additionally, some selections are being made. These selections will be used later on, when assigning material properties for instance.
You can start with the three phases. By setting their labels, and enabling the Resulting objects selection, you automatically set a domain selection that coincides with the shape you create. The selections will then propagate through the sequence (when a shape is copied, its selection will cover both the original and the copy). At the end of these modeling instructions, a blank material is created allowing you to check all geometry-induced selections once more.
Phase 1
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Phase 1 in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type Dcon/2.
4
Locate the Position section. In the x text field, type -(Dpha/2)/sqrt(3).
5
In the y text field, type Dpha/2.
6
Locate the Rotation Angle section. In the Rotation text field, type 120.
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
8
Click  Build Selected.
9
Click the  Zoom Extents button in the Graphics toolbar.
Phase 2
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Phase 2 in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type Dcon/2.
4
Locate the Position section. In the x text field, type Dpha/sqrt(3).
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Phase 3
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Phase 3 in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type Dcon/2.
4
Locate the Position section. In the x text field, type -(Dpha/2)/sqrt(3).
5
In the y text field, type -Dpha/2.
6
Locate the Rotation Angle section. In the Rotation text field, type -120.
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
8
Click  Build Selected.
9
Click the  Zoom Extents button in the Graphics toolbar.
Circle 4 (c4)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dcon/2+Tscc.
5
Locate the Position section. In the x text field, type Dpha/sqrt(3).
6
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Tscc
7
Click  Build Selected.
Circle 5 (c5)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dins/2-Tscc.
5
Locate the Position section. In the x text field, type Dpha/sqrt(3).
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Tins
7
Click  Build Selected.
8
Click the  Zoom Extents button in the Graphics toolbar.
Circle 6 (c6)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dins/2.
5
Locate the Position section. In the x text field, type Dpha/sqrt(3).
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Tscc
7
Click  Build Selected.
Circle 7 (c7)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dins/2+Tpbs.
5
Locate the Position section. In the x text field, type Dpha/sqrt(3).
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Tpbs
7
Click  Build Selected.
Semi-Conductive Compound
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
In the Settings window for Union, type Semi-Conductive Compound in the Label text field.
3
Select the objects c4 and c6 only.
4
Locate the Union section. Clear the Keep interior boundaries check box.
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
6
Click  Build Selected.
Cross-Linked Polyethylene (XLPE)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
In the Settings window for Union, type Cross-Linked Polyethylene (XLPE) in the Label text field.
3
Select the object c5 only.
4
Locate the Union section. Clear the Keep interior boundaries check box.
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
6
Click  Build Selected.
Screens
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
In the Settings window for Union, type Screens in the Label text field.
3
Select the object c7 only.
4
Locate the Union section. Clear the Keep interior boundaries check box.
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
6
Click  Build Selected.
Next, will be the polyethylene parts. In order to simplify the geometry and accommodate meshing, a deliberate overlap between the circles is created (using a parameter called mfil). This avoids having infinitely narrow regions between circles touching. The sequence can be adjusted if necessary, to support a more complex combination of polymers.
Circle 8 (c8)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dpha/2+mfil.
5
Locate the Position section. In the x text field, type Dpha/sqrt(3).
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Tpe+mfil
7
Click  Build Selected.
Circular Arc 1 (ca1)
1
In the Geometry toolbar, click  More Primitives and choose Circular Arc.
2
In the Settings window for Circular Arc, locate the Radius section.
3
In the Radius text field, type Dpha3/2-mfil.
4
Locate the Angles section. In the Start angle text field, type -60.
5
In the End angle text field, type 60.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the objects c8, ca1, uni1, uni2, and uni3 only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type 0[deg], 120[deg], 240[deg].
5
Click  Build Selected.
6
Click the  Zoom Extents button in the Graphics toolbar.
Here, three different angles are used to create three copies of the same input objects (while propagating their selections).
Circle 9 (c9)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dcab/2.
5
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Dcab/2-(Dpha3/2-mfil)
Note that for longer expressions like this one, the easiest way to go, is to copy-paste them directly from this *.pdf file to COMSOL.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Circle 10 (c10)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Darm/2+Tarm/2+marm.
5
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Tarm+2*marm
6
Click  Build Selected.
Polyethylene
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
In the Settings window for Difference, type Polyethylene in the Label text field.
3
Select the objects c9, rot1(1), rot1(2), and rot1(3) only.
4
Locate the Difference section. Find the Objects to subtract subsection. Select the  Activate Selection toggle button.
5
Select the object c10 only.
6
Clear the Keep interior boundaries check box.
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
8
Click  Build Selected.
You have now created the polyethylene parts. What is more, you have created a domain selection that refers to these parts. Next, let us have a look at the polypropylene parts.
Circle 11 (c11)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type Dpha*0.241335+mfil.
4
Locate the Position section. In the x text field, type Dpha*-0.836015.
5
Click  Build Selected.
Circle 12 (c12)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type Dpha*0.118595+mfil.
4
Locate the Position section. In the x text field, type Dpha*-0.890205.
5
In the y text field, type Dpha*-0.355815.
6
Click  Build Selected.
Circle 13 (c13)
1
Right-click Circle 12 (c12) and choose Duplicate.
2
In the Settings window for Circle, locate the Position section.
3
In the y text field, type Dpha*0.355815, that is, remove the minus sign.
4
Click  Build Selected.
Perhaps you are wondering about the numbers used here. These are the result of geometrical expressions that make the circles fit (apart from the overlap margin used). The size and position of the circles scale with Dpha, and the expressions have been simplified accordingly.
Rotate 2 (rot2)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the objects c11, c12, and c13 only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type 0[deg], 120[deg], 240[deg].
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
6
From the Show in physics list, choose Off.
7
Click  Build Selected.
Circle 14 (c14)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type Dpha*(1/sqrt(3)-1/2)+mfil.
4
Click  Build Selected.
The selection for Rotate 2 is not shown in the physics, because it is only meaningful within the geometry sequence itself (it is used in Difference 2). The polypropylene domains are now created by taking the result from Rotate 2 and c14, and subtracting a hole for the fiber, together with a copy of the polyethylene.
Circle 15 (c15)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type Dfic/2+Tfih.
4
Locate the Position section. In the x text field, type Dpha*-0.836015.
5
Click  Build Selected.
Copy 1 (copy1)
1
In the Geometry toolbar, click  Transforms and choose Copy.
2
In the Settings window for Copy, locate the Input section.
3
From the Input objects list, choose Polyethylene.
4
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
5
From the Show in physics list, choose Off.
6
Click  Build Selected.
Polypropylene
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
In the Settings window for Difference, type Polypropylene in the Label text field.
3
Locate the Difference section. From the Objects to add list, choose Rotate 2.
4
In addition to Rotate 2, select the object c14; the center filler domain.
5
Find the Objects to subtract subsection. Select the  Activate Selection toggle button.
6
From the Objects to subtract list, choose Copy 1.
7
In addition to Copy 1, select the object c15; the small hole for the fiber on the left.
8
Clear the Keep interior boundaries check box.
9
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
10
Click  Build Selected.
Now that the polypropylene filler material is complete, let us add some details for the fiber. Although a single circle object would suffice, the helix and the core are made as separate objects. This will result in two separate geometry-induced material selections.
Circle 16 (c16)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Object Type section.
3
From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dfib/2.
5
Locate the Position section. In the x text field, type Dpha*-0.836015.
6
Click  Build Selected.
Steel Helix (Fiber)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Steel Helix (Fiber) in the Label text field.
3
Locate the Object Type section. From the Type list, choose Curve.
4
Locate the Size and Shape section. In the Radius text field, type Dfic/2+Tfih.
5
Locate the Position section. In the x text field, type Dpha*-0.836015.
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
Tfih
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
8
Click  Build Selected.
Fiber Optic Core
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Fiber Optic Core in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type Dfic/2.
4
Locate the Position section. In the x text field, type Dpha*-0.836015.
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
6
Click  Build Selected.
Next, are the armor wires. For the armor, we have the restriction that there should be an integer amount of wires in the cable’s circumference. This number, Narm, is limited by the armor wire’s diameter, and the overall size of the cable. Using the Rotate transform and the range() operator, the wires are equally distributed in the interval [0,360] degrees. In-between the armor wires, some space is left open to allow for electric insulation and a low permeability gap for the magnetic flux density.
Cable Armor
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Cable Armor in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type Tarm/2.
4
Locate the Position section. In the x text field, type Darm/2.
5
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
6
Click  Build Selected.
Rotate 3 (rot3)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
In the Settings window for Rotate, locate the Input section.
3
From the Input objects list, choose Cable Armor.
4
In the Settings window for Rotate, locate the Rotation section.
5
In the Angle text field, type 360[deg]*range(1/Narm,1/Narm,1).
6
Click  Build Selected.
This finishes the cable armor. The domain for the bitumen compound is now created by duplicating object c10, and subtracting from that, a copy of the armor.
Circle 20 (c20)
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Circle 10 (c10) and choose Duplicate.
2
In the Settings window for Circle, locate the Selections of Resulting Entities section.
3
Select the Resulting objects selection check box.
4
From the Show in physics list, choose Off.
5
Click  Build Selected.
Copy 2 (copy2)
1
In the Geometry toolbar, click  Transforms and choose Copy.
2
In the Settings window for Copy, locate the Input section.
3
From the Input objects list, choose Cable Armor.
4
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
5
From the Show in physics list, choose Off.
6
Click  Build Selected.
Bitumen Compound
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
In the Settings window for Difference, type Bitumen Compound in the Label text field.
3
Locate the Difference section. From the Objects to add list, choose Circle 20.
4
Find the Objects to subtract subsection. Select the  Activate Selection toggle button.
5
From the Objects to subtract list, choose Copy 2.
6
Clear the Keep interior boundaries check box.
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
8
Click  Build Selected.
Now that the bitumen is finished, add a circle that covers the whole cable. This is primarily to get the Cable Domains selection.
Cable Domains
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Cable Domains in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type Dcab/2.
4
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
5
Click  Build Selected.
What remains are the domains exterior to the cable. Part of this, is the circle that covers the electromagnetic domains. Outside this circle, you will only be solving for heat transfer (electromagnetic fields are assumed to have dropped to zero). Rectangular domains are added for the sea bed and the water.
Electromagnetic Domains
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, type Electromagnetic Domains in the Label text field.
3
Locate the Size and Shape section. In the Radius text field, type 5*Dcab/2.
4
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
5
Click  Build Selected.
6
Click the  Zoom Extents button in the Graphics toolbar.
Thermal Domains
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Thermal Domains in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type 10[m].
4
In the Height text field, type 8[m].
5
Locate the Position section. In the x text field, type -5[m].
6
In the y text field, type -4[m].
7
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
5[m]
8
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
9
Click  Build Selected.
10
Click the  Zoom Extents button in the Graphics toolbar.
This completes the geometry. Notice that the size of the rectangle does not depend on the cable’s parameters. A wise choice for the size of the thermal domains strongly depends on the thermal properties and conditions. For a discussion on this, see the Thermal Effects tutorial.
Before finishing, feel free to check that the geometry-induced selections exists, and are according to your expectations. You can do so by adding a blank material.
Materials
Material 1 (mat1)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Thermal Domains.
4
Click the  Zoom to Selection button in the Graphics toolbar.
5
From the Selection list, choose Electromagnetic Domains.
6
Click the  Zoom to Selection button in the Graphics toolbar.
7
From the Selection list, choose Cable Domains.
8
Click the  Zoom to Selection button in the Graphics toolbar.
9
From the Selection list, choose Bitumen Compound.
10
From the Selection list, choose Cable Armor.
11
From the Selection list, choose Fiber Optic Core.
12
From the Selection list, choose Steel Helix (Fiber).
13
From the Selection list, choose Polypropylene.
14
From the Selection list, choose Polyethylene.
15
From the Selection list, choose Screens.
16
From the Selection list, choose Cross-Linked Polyethylene (XLPE).
17
From the Selection list, choose Semi-Conductive Compound.
18
From the Selection list, choose Phase 3.
19
From the Selection list, choose Phase 2.
20
From the Selection list, choose Phase 1.
If anything seems out of order, please retrace your steps. Alternatively, check out the completed geometry sequence included in this tutorial series as a reference (submarine_cable_e_geom_sequence.mph).
21
Right-click Material 1 (mat1) and choose Delete.
 
1
Up to several hundred million US dollars for a replacement, and tens of millions of dollars for a repair.

2
Cross-linked polyethylene, high-voltage alternating current.

3
They are typically based on IEC 60287 or similar standards.