Review Reports - Magnetic Field Simulation of Demagnetization Process in Complex Ferromagnetic Cavity Structures

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript is an engineering study of a specific theoretical model of a ferromagnetic cavity system. Therefore, the appropriateness of this choice requires justification, and the authors' references to similar analyses in the literature.

(1) Fig. 1. In the caption below the figure and in the text, provide further explanation, including what B+ and B- are in the figure, and whether the presented plots are the result of an experiment, computer simulation, illustrative diagram, etc.

(2) Each abbreviation used in the work requires explanation. On page 3, it is CST. Explain what software it is. Provide a reference.

(3) Explain why the ferromagnetic cavity structure with the parameters given in Table 1 has the shape and dimensions it does. Provide a reference to the actual system and the actual operating conditions for such a system in practice. Do the conducted analyses of magnetic properties apply to smaller, macroscopic systems with different wall thicknesses, different electrical conductivities, etc.?

(4) How large currents are necessary to generate varying magnetic fields with the amplitudes in the results discussed?

(5) Did the field calculations (Fig. 4, 5) take into account the variation in cavity shape by selecting appropriate demagnetization fields N? How did the demagnetization field coefficients N vary with shape? Similarly, were demagnetization coefficients used in the eddy current calculations?

(6) Tables 2 and 3 explain the symbols in the legend. Additionally, Tim -> Time in Table 2.

(7) A brief comment is necessary regarding heat release and different heating in an alternating magnetic field, depending on its setting/direction. How might taking into account thermal effects modify the obtained results?

Author Response

Reviewer#1, Concern # 1: Fig. 1. In the caption below the figure and in the text, provide further explanation, including what B+ and B- are in the figure, and whether the presented plots are the result of an experiment, computer simulation, illustrative diagram, etc.

Author response: We thank the reviewer for this valuable comment. We have revised the figure caption and the corresponding text in the manuscript to provide a clearer explanation. Specifically:

We have defined the symbols B+ and B- as the positive and negative saturation magnetic flux density, respectively.
We have explicitly stated that Figure 1 is a schematic diagram illustrating the principle of the dynamic demagnetization process.

Author actions:

By applying a sufficiently strong alternating magnetic field to a magnetic body and then gradually reducing the amplitude of the alternating magnetic field to zero, a magnetic neutral state is achieved, which is also referred to as a dynamic magnetic neutral state. During the demagnetization process, the workpiece is placed in an alternating magnetic field, and demagnetization is performed via the decreasing hysteresis loop, as shown in the schematic diagram of Figure 1. In the figure, B+ and B- represent the positive and negative saturation magnetic flux density, respectively. As the amplitude of the alternating magnetic field gradually decreases, the trajectory of the hysteresis loop shrinks progressively. When the magnetic field is gradually reduced to zero, the residual magnetization in the workpiece is close to zero. Therefore, during demagnetization, changes in the direction and magnitude of the current and magnetic field must undergo both commutation and attenuation simultaneously. The number of cycles needed to attenuate to zero should be as large as possible (generally requiring more than 30 cycles), and the current amplitude for each attenuation should be as small as possible. If the attenuation amplitude is too large, the intended demagnetization effect cannot be achieved [15].

Figure 1. Schematic diagram of the dynamic demagnetization process via progressively shrinking hysteresis loops. The amplitude of the alternating magnetic field (H) is gradually reduced to zero, leading to a decrease in the magnetic flux density (B) from the saturation levels (B+ and B-) and ultimately resulting in a residual magnetization (Br) close to zero. (Note: B+ and B- denote the positive and negative saturation magnetic flux density, respectively.)

------------------------------------------------------------------------------------------------------

Reviewer#1, Concern # 2: Each abbreviation used in the work requires explanation. On page 3, it is CST. Explain what software it is. Provide a reference.

Author actions:

Thank you for this valuable comment. We have revised the manuscript to ensure that all abbreviations are defined upon their first use.

Specifically, on page 3, we have now defined "CST" as "Computer Simulation Technology (CST)" and have added the corresponding reference to the software we used, which is the CST Studio Suite from Dassault Systèmes [24].

Dassault Systèmes. CST Studio Suite [Software]. (2024). Available online: https://www.3ds.com/products/simulia/cst-studio-suite.

We have also double-checked the entire manuscript to ensure that all other abbreviations are properly explained. The relevant modifications can be found in the revised version of the manuscript.

--------------------------------------------------------------------------

Reviewer#1, Concern # 3: Explain why the ferromagnetic cavity structure with the parameters given in Table 1 has the shape and dimensions it does. Provide a reference to the actual system and the actual operating conditions for such a system in practice. Do the conducted analyses of magnetic properties apply to smaller, macroscopic systems with different wall thicknesses, different electrical conductivities, etc.?

Author response: We sincerely thank the reviewer for these important questions. The shape and dimensions of the ferromagnetic cavity in Table 1 are representative of a large-scale, cylindrical pressure hull structure, typical of naval vessels like submarines. This configuration is designed for structural integrity and internal compartmentalization, which includes housing sensitive systems such as missile launch tubes.

However, due to confidentiality constraints related to this specific defense platform, we are unable to provide a detailed public reference or exact operational parameters for the actual system. We sincerely apologize for this limitation and kindly ask for your understanding. The operational condition analyzed—exposure to low-frequency alternating magnetic fields during degaussing procedures—is a standard practice for such vessels to reduce their magnetic signature, and our study specifically investigates the resulting internal magnetic fields and their potential impact on internal electrical systems.

Regarding the broader applicability, the fundamental physical principles governing our magnetic analysis (based on Maxwell's equations and ferromagnetic material behavior) are universal. The phenomena studied, such as magnetic flux distribution and shielding effectiveness, are scalable. Therefore, our methodology and conclusions are directly applicable to smaller macroscopic systems with different wall thicknesses or electrical conductivities. The key relationships, for instance between shielding performance and the ratio of wall thickness to skin depth, would remain valid, and our model can be adapted by adjusting these parameters accordingly.

--------------------------------------------------------------------

Reviewer#1, Concern # 4: How large currents are necessary to generate varying magnetic fields with the amplitudes in the results discussed?

Author response: We thank the reviewer for raising this important point regarding the excitation currents required to generate the reported magnetic fields.

The magnetic field amplitudes discussed in our results, specifically the initial demagnetization field of 12.27 Oe, are calculated based on the material properties and system geometry, not directly from the driving current. As derived and referenced [15], this initial amplitude is determined by the critical magnetic properties of the marine steel (coercivity H_c = 7.1 Oe, maximum susceptibility χ_m = 780) and the structural demagnetizing factor (N=0.0003), following the relation H_m = 1.4H_c(1 + χ_mN).

The current required to generate a specific field is intrinsically linked to the specific design and calibration of the field generator (e.g., coil geometry, number of turns, and core material). In our simulation setup, which uses a modeled generator, an initial pulse field of 20 Oe at the coil center is applied [11]. Figure 3 shows the normalized waveform of this excitation current, with key timing parameters (0.3s rise/fall edges, 10s pulse duration, 20s period).

Therefore, while the absolute current value is a function of the generator's design, our study provides the crucial magnetic field input (20 Oe initial pulse) and its normalized current waveform, enabling the reproduction of the core physical process—the system's magnetic response to a defined, decaying magnetic excitation. The exact current needed for a different physical setup can be calculated from this field requirement using standard electromagnet principles tailored to that specific generator's structure.

---------------------------------------------------------------------------

Reviewer#1, Concern # 5: Did the field calculations (Fig. 4, 5) take into account the variation in cavity shape by selecting appropriate demagnetization fields N? How did the demagnetization field coefficients N vary with shape? Similarly, were demagnetization coefficients used in the eddy current calculations?

We thank the reviewer for this insightful question regarding the role of demagnetization factors in our model.

Yes, the field calculations in Figures 4 and 5 did account for the complex cavity shape through the use of the demagnetization factor NN. As stated in the manuscript, a longitudinal demagnetization factor of N=0.0003N=0.0003 was used in the key calculation for the initial demagnetizing field amplitude:

H_m = 1.4H_c(1 + χ_mN) = 12.27 Oe [15].

The value N=0.0003N=0.0003 is not a universal constant but was selected as a fixed, effective value representative of this specific, elongated ferromagnetic cavity structure. This low value is consistent with the very small demagnetizing factors typically associated with long, slender bodies magnetized along their long axis, which is a standard and justified engineering approximation for systems of this general geometry. We acknowledge that the demagnetization factor is inherently dependent on shape and would vary for significantly different geometries; a discussion on the potential impact of this factor on the analysis of other complex systems will be included in a subsequent section of the paper.

Regarding the eddy current calculations, the demagnetization factor NN was not explicitly used in that specific part of the model. The eddy current simulations were primarily governed by the material's electrical conductivity, the temporal variation of the applied magnetic field, and the geometry's boundary conditions as defined in the FEM software. The chosen value of NN, however, was critical in determining the correct initial amplitude (Hm) of the applied demagnetizing field, which served as the driving input for the entire process, including the generation of eddy currents.

We hope this clarification addresses the reviewer’s concerns.

---------------------------------------------------------------------------

Reviewer#1, Concern # 6: Tables 2 and 3 explain the symbols in the legend. Additionally, Tim -> Time in Table 2.

Author response: We thank the reviewer for raising this important point。

able 2. Surface magnetic field changes in complex ferromagnetic cavity structures

Time(s)	Internal magnetic fields in complex ferromagnetic cavity structures	legend
0.3		Magnetic (T)
0.5
1
3.5
10.3
10.5
10.9

Table 3. Internal magnetic field changes in complex ferromagnetic cavity structures

Time(s)	Internal magnetic fields in complex ferromagnetic cavity structures	legend
0.3		Magnetic (T)
0.5
1
3.5
10.3
10.5
10.9
13
160

---------------------------------------------------------------------------

Reviewer#1, Concern # 6: A brief comment is necessary regarding heat release and different heating in an alternating magnetic field, depending on its setting/direction. How might taking into account thermal effects modify the obtained results?

We thank the reviewer for raising this important point regarding thermal effects. The comment is well-taken, and we acknowledge that alternating magnetic fields can indeed induce heating, particularly through hysteresis and eddy current losses.

In the context of our specific demagnetization process, the duration of the applied field is relatively short. The total process time is 160 s, with each full pulse period lasting 20 s and the effective high-amplitude excitation periods being even shorter. The primary material involved is a good electrical conductor. Under these transient conditions, the temperature rise due to Joule heating from induced eddy currents is minimal.

Consequently, the associated thermal effects, such as a change in the material's electrical resistivity, are expected to be negligible. These minor resistivity variations would have an insignificant impact on the distribution of surface currents and eddy currents. Therefore, while we agree that thermal analysis can be crucial in other scenarios involving sustained AC excitation, we believe that neglecting thermal effects in this specific short-duration demagnetization study is a valid simplification that does not materially alter the obtained results.

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, the study of the internal magnetic field of complex ferromagnetic structures in the cavity during the demagnetization process is proposed. The effects of hysteresis and eddy currents are described. Unfortunately, the complex structures are not sufficiently described, and the simulation of the magnetic field in the three-dimensional space of the structures is not presented in detail. The mathematical model underlying the simulations should be added. Explanations should be added on how the curves are obtained. New figures should be included to better describe the complex structure and observation points.

Author Response

Reviewer#2: In this paper, the study of the internal magnetic field of complex ferromagnetic structures in the cavity during the demagnetization process is proposed. The effects of hysteresis and eddy currents are described. Unfortunately, the complex structures are not sufficiently described, and the simulation of the magnetic field in the three-dimensional space of the structures is not presented in detail. The mathematical model underlying the simulations should be added. Explanations should be added on how the curves are obtained. New figures should be included to better describe the complex structure and observation points.

Author response:

We sincerely thank the reviewer for these insightful comments and the opportunity to further clarify our work.

As noted in our response to Comment #3, this is due to confidentiality constraints related to the specific defense platform. The structure is representative of a large-scale, cylindrical pressure hull. Its key characteristics relevant to the electromagnetic simulation—namely its overall elongated shape, major dimensions, and material properties—were provided to the simulation environment. We acknowledge that a more detailed geometrical description would be beneficial, and within the constraints of public disclosure, a schematic diagram in Figure 2 illustrates the general structure and the specific internal observation points where the magnetic field was monitored (e.g., near critical internal electrical systems).

Regarding the simulation methodology, all field results were obtained using the commercial software CST Studio Suite [24], which is a industry-standard tool for 3D electromagnetic simulation. The underlying mathematical model is based on the full-wave formulation of Maxwell's equations, solved in the frequency domain for the harmonic excitation of the degaussing coils. The software directly computes the magnetic field distribution throughout the entire 3D volume of the structure. The effects of hysteresis are incorporated by defining the material's non-linear B-H curve for the marine steel. The eddy currents are a direct result of solving Maxwell's equations in the conductive structure.

The curves in Figures 4 and 5 are not post-processed analytical solutions but are the direct output of the 3D field solver. They represent the simulated magnetic field values over time at the predefined internal observation points. The time-domain response was obtained by simulating the system's reaction to the excitation waveform shown in Figure 3.

In summary, while the exact geometry is simplified for public release, the simulation itself was a full 3D analysis that rigorously accounted for the complex geometry, material non-linearity (hysteresis), and wave phenomena (eddy currents).

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I received clarification regarding my reservations, and where necessary, the authors made corrections. The work meets the requirements for publication.

Reviewer 2 Report

Comments and Suggestions for Authors

Congratulation for this paper. Maybe a correction for title of Table 1 is needed.