3.1. Finite Element Geometry Modeling
The solid modeling method is used to establish the electron beam weld weak magnetic detection finite element analysis model, whose structure is composed mainly of several parts, such as an aluminum alloy plate with an electron beam weld, weld defects, air layer, etc. [
27]. The finite element geometry model is shown in
Figure 3. The thickness of the 7A09 aluminum alloy in the model is 5 mm, the length is 110 mm, the width is 80 mm, the air domain size is 200 mm × 160 mm × 140 mm, and the weld width is 2 mm. The crack is simplified to a rectangular groove defect in the center of the surface of the electron beam weld of 7A09 aluminum alloy.
After establishing the solid model and setting the cell type, the material properties of the weldment are set according to the physical conditions, materials, and parameters of the test. The key components in the model are the aluminum alloy plate, weld seam, defect, and air domain, and the material properties of these components need to be defined separately in the model. In this paper, 7A09 aluminum alloy is used as the research object, and the defect is air. The relative permeability size of air is known to be 1.0, and the relative permeability of 7A09 aluminum alloy is 1.00002.
In areas where the magnetic field does not vary drastically, a larger grid spacing can shorten the computer running time; for more accurate solution results and areas where the magnetic field varies drastically, the grid density needs to be slightly tighter, but the required computational time and computational memory are greatly increased. The rationality of grid division is even more important when calculating using 3D models; otherwise, the rationality of grid division will lead to nonconvergence of the model calculation results. In this paper, we use the physical field control grid method of meshing: the cell size is hyperfine, the electron beam welding defects of the finite element 3D model of the mesh division are shown in
Figure 4, and the weld uses a uniform dense mesh to improve the accuracy of the calculation. Three mesh densities (coarse mesh of 0.5 mm, medium mesh of 0.2 mm, and ultra-fine mesh of 0.1 mm) were used to calculate the amplitude ∆
B when the crack depth was 1 mm. The results were 32.1 nT, 33.8 nT, and 34.1 nT, respectively, and the relative error was less than 3% (
Table 1). This indicates that the ultra-fine mesh (0.1 mm) has reached the saturation of calculation accuracy, and further mesh refinement has a negligible impact on the results (in line with the mesh independence principle of finite element analysis). The number of elements and element sizes in the region are shown in
Table 2. This meshing strategy balances computational accuracy and resource consumption, in line with the best practices of finite element analysis.
3.2. Parameter Settings of the Geomagnetic Field
Geomagnetic anomalies represent the deviation between the natural static magnetic field and the predicted value of the Earth’s magnetic field based on the magnetic dipole model. Static magnetic problems with negligible currents can be solved using magnetic scalar potentials. Variations in magnetic permeability cause small deviations from standard values (local geomagnetic anomalies), which can be accurately modeled using the approximate magnetic potential formulation of the “magnetic field, no current” interface. In the simulation domain, the background magnetic field is assumed to be uniform.
In the no-current region, the equation is as follows:
The magnetic scalar potential
can be defined and is obtained as follows:
This calculation is similar to the definition of the potential of an electrostatic field. Using the intrinsic relationship between the magnetic induction strength and the magnetic field, the following was obtained:
The equation for
can be derived as follows:
The approximate potential formula used in this model splits the total magnetic potential into external and approximate potentials:
, where the approximate potential
is the dependent variable:
The external magnetic potential is more easily defined as the external magnetic field; therefore, the actual equation used is as follows:
To simulate the background magnetic field, the external magnetic field components are represented by the total intensity, magnetic declination, and magnetic inclination as follows:
The magnetic inclination and magnetic declination angles are approximately Incl = 59.357° and Decl = 12.275°, respectively, and the magnitude of the natural magnetic induction is estimated to be 48.163 .
3.3. Analysis and Discussion of Simulation Results
The distribution of magnetic induction lines and magnetic induction intensity clouds for the aluminum alloy plate cross-section, both with and without defects, is shown in
Figure 5. The figure indicates that the distribution of magnetic induction intensity at the defect-free weld seam is uniform and remains essentially unchanged. The magnetic induction lines pass uniformly through the weld seam, forming a stable magnetic field, with the lines parallel to each other. In the presence of defects, the magnetic induction lines become significantly distorted, and the magnetic induction intensity cloud is unevenly distributed at the defect sites. This demonstrates that the weak magnetic detection technique is theoretically feasible and effective.
To determine the distribution of magnetic induction intensity at the weld defect, we set the defect length as 1 mm, width as 0.2 mm, depth as 1 mm, and height along the weld surface as 0.1 mm for path extraction and derived the distribution curves of magnetic induction intensity
at the points 4 mm from each side of the defect center point, as shown in
Figure 6. From the curve, the radial component of the magnetic induction
distribution curve at the defect shows two peaks centered at the origin, first positive and then negative, and symmetrically distributed on both sides of the defect center, with a maximum at the edge of the defect length.
shows a clear convex peak at the defect with positive polarity, and its value increases, i.e., the magnetic induction intensity increases at the defect, because when there is a discontinuous defect inside the weld, the defect is filled with air, and these defects will cause a change in magnetic permeability. The magnetic permeability of the air at the defect is less than the permeability of the weld itself. The magnetic field hindrance at the defect site will become larger, the magnetic induction lines at the defect will be repelled to bend, the magnetic induction line density at the defect will decrease, and the magnetic induction line density above the defect will become larger, which leads to the defect. The magnetic induction line density at the weld surface above increases; therefore, the magnetic induction intensity
curve appears as a convex magnetic anomaly, and the simulation results are consistent with the theory of weak magnetic detection.
In actual production, the width of the electron beam weld is generally small, the crack width is extremely small, and larger width cracks do not easily appear; therefore, the preset crack widths are taken as 0.2 mm to simulate real fine crack defects, while the length and depth direction crack expansion is larger. To compare and analyze the effect of crack depth and length on the magnetic field distribution, cracks with depths of 0.4 mm, 0.8 mm, 1.2 mm, and 1.6 mm were established at a crack length of 1 mm for comparative analysis, and the magnetic induction intensity
and
distribution curves for cracks of different depths were obtained. From the results in
Figure 7, the magnetic induction intensity
also shows two typical peaks, one positive and one negative, and is symmetrically distributed on both sides of the crack center, and the magnetic induction intensity
shows a typical convex peak. As the crack depth increases, the weak magnetic signal is enhanced, and the peaks of magnetic induction intensity
and
both increase rapidly at first, and then the increasing trend becomes slower; however, the peak increase in
is smaller, the peak spacing of one positive and one negative peak of curve
does not change significantly, and the wave width
(the distance between the lowest points of the signal curve around the peak) of curve
changes less. As summarized in
Table 3, the wave width
and amplitude
(the difference between the peak value
and the minimum value
of the magnetic anomaly) of the magnetic induction intensity curve
at different depths of cracks are shown in
Figure 8. The amplitude
first increases rapidly with increasing crack depth, and then the increasing trend becomes slower, and the wave width
also increases, although the overall change is small.
Cracks with lengths of 1 mm, 2 mm, 3 mm, and 4 mm were established at a crack depth of 1 mm for comparative analysis, and the magnetic induction intensity
and
distribution curves for different lengths of cracks were obtained (
Figure 9). From the results in
Figure 9, the peak value of magnetic induction
varies less at the same depth with different lengths; however, as the crack length increases, the peak spacing of the two peaks of the curve
positively and negatively increases, the wave width
of magnetic induction
also gradually becomes larger, and the peak value of
increases first and then decreases with the increasing crack length.
As summarized in
Table 4, the wave width
and amplitude
of the magnetic induction intensity curve
for different lengths of cracks are shown in
Figure 10. The amplitude
increases and then decreases as the crack length increases, and the wave width
increases approximately linearly.
In order to verify the versatility of the method, a model of pore defects was established, which is actually shown in
Figure 11. The pore model is simplified to spherical defects with diameters of 0.5 mm, 1.0 mm, 1.5 mm, and 2.0 mm. The relative permeability of pores is
μr = 1.0, and the weld material
μr = 1.00002 (consistent with the crack defect model).
Table 5 shows the magnetic anomalies of pores with different diameters, and it can be seen that the aperture is positively correlated with the amplitude of the magnetic induction intensity, which is consistent with the crack defect law, which verifies the versatility of the method.