A Geometrically Parameterized Quasi-Stationary 3D Model for High-Frequency Induction Tube Welding

1 Introduction

• High-frequency induction welding heats a continuously formed steel strip by induced eddy currents before pressure joins it into a longitudinal weld.The process involves complex electromagnetic fields, multiple current paths, and nonlinear material behavior.Source: §1 · p. 1
• The prior model could not naturally prescribe coil voltage or current or directly evaluate system impedance, and its magnetic vector potential formulation was computationally demanding for large 3D models.The paper identifies the magnetic scalar potential formulation as an alternative that can reduce computational cost.Source: §1 · p. 1
• The present work targets an alternative formulation incorporating global circuit parameters, alongside a robust approach to geometry definition and parametrization.Source: §1 · p. 1
• AISI 304 stainless steel tubes provide the validation setting because their electro-thermo-physical properties are well established across typical welding temperatures.The material is non-magnetic and has relatively high electrical resistivity, reducing localized current-density gradients and easing mesh and computational demands.Source: §1 · pp. 1–2
• The validated framework is used to compare how a soft magnetic composite and a classical ferrite impeder influence high-frequency induction welding.Source: §1 · p. 2

2.1 Problem Domain and Geometric Decomposition

• The computational domain is partitioned into a conducting region Ωc and a non-conducting complement domain ΩC_c.Source: §2.1 · p. 2
• The conducting region comprises the welded steel tube and individual turns of the multi-turn induction coil.Each coil turn carries a predefined total alternating current I at operating frequency f.Source: §2.1 · p. 2
• The non-conducting complement includes the surrounding air and the magnetic impeder core inside the tube, where macroscopic eddy currents are suppressed.Source: §2.1 · p. 2
• The domain is truncated at a symmetry plane and at the tube’s inlet and outlet boundaries.Source: §2.1 · p. 2

2.2.1 Governing Maxwell Equations

• The eddy-current model neglects displacement current and combines Maxwell’s equations with temperature-dependent constitutive laws to govern the magnetic field across the computational domain.In non-conducting regions, the current vanishes and the magnetic field is constrained to be irrotational.Source: §2.2.1 · p. 3

2.2.2 Field Decomposition, Topology of the Domain Ω, and Total Coil Current I

• Cuts restore a simply connected insulator domain, while the field decomposition represents the prescribed net current associated with each topological cut.Figure 2 identifies the cuts used to handle the insulator domain’s multi-connected topology.Source: §2.2.2 · pp. 3–4, Figure 2 · p. 4

2.2.3 Interface Continuity

• Across conductor–air interfaces, the tangential magnetic field and normal magnetic flux density remain continuous.The conditions follow from the absence of surface currents and Gauss’s law for magnetism.Source: §2.2.3 · p. 4

2.2.4 Boundary Conditions

• The model truncates the domain with vanishing tangential magnetic-field conditions at sufficiently distant outer boundaries and on the symmetry plane.The symmetry condition exploits geometric and physical symmetry to reduce computational cost.Source: §2.2.4 · p. 4

2.2.5 Time-Harmonic (Complex-Valued) Formulation

• For sinusoidal excitation, the model uses complex phasors and an effective permeability to retain a single-frequency formulation for nonlinear magnetic materials.The effective permeability is derived from the dehysterized B-H curve using co-energy conservation; it approximates the response at the fundamental frequency although the physical response is multi-harmonic.Source: §2.2.5 · p. 5

2.3.1 Governing Heat Transport Equation

• The moving workpiece’s temperature field is governed by a quasi-stationary advection–diffusion equation with heat carried by material motion and conducted along temperature gradients.Specific heat capacity and thermal conductivity depend on temperature, while mass density is constant.Source: §2.3.1 · p. 5

2.3.2 Boundary Conditions

• The thermal boundary conditions fix the incoming strip at its baseline temperature and impose a nonlinear Robin condition for convective heat loss from exposed surfaces.The inlet condition uses the prescribed entry temperature; the Robin condition uses the ambient air temperature and convective heat-transfer coefficient.Source: §2.3.2 · pp. 5–6

• The volumetric heat source drives thermal transport and represents localized macroscopic energy dissipation, computed from the current-density phasor and its complex conjugate.The mean source is evaluated directly in the frequency domain without temporal integration.Source: §2.4 · p. 6
• The tube heat-source model neglects magnetic hysteresis losses and heat generated by mechanical deformation, assuming eddy-current Joule heating dominates.Source: §2.4 · p. 6
• The model decouples the fields using steady, time-averaged power density because the tube’s macroscopic thermal timescale is much longer than the electromagnetic period.Source: §2.4 · p. 6

3.1.1 Weak Formulation

• The magnetic weak formulation seeks a curl-conforming field and uses a constrained space to enforce the outer boundary conditions.The H(curl) space permits jumps in the normal field at material interfaces while maintaining tangential continuity.Source: §3.1.1 · pp. 6–7

3.1.2 Discretization

• The discrete magnetic field combines nodal scalar-potential and edge basis functions with cohomology functions that represent prescribed coil-current linkage.The formulation uses curl-conforming Nédélec basis functions and compatible nodal contributions.Source: §3.1.2 · p. 7

3.1.3 Post-Processing of Terminal Voltages

• Coil terminal voltages are recovered in post-processing from the computed magnetic field, with the loop cut represented implicitly by cohomology basis functions.Source: §3.1.3 · p. 8

3.2.1 Weak Formulation

• The thermal weak formulation seeks temperature in H1, enforces the inlet Dirichlet condition, and incorporates the Robin condition on exposed surfaces.Source: §3.2.1 · p. 8

3.2.2 Stabilized Discrete Formulation

• The temperature field uses first-order nodal Lagrange functions, while SUPG modifies test functions along streamlines to stabilize dominant-advection regimes.The element-dependent stabilization parameter is evaluated using the local Peclet number; for linear elements, the diffusion contribution to the SUPG strong residual is omitted.Source: §3.2.2 · pp. 8–9

3.3 Numerical Implementation within the GetDP Solver Framework

• The framework uses GetDP’s declarative workflow and Gmsh integration to define geometry, finite-element spaces, formulations, and resolutions.Gmsh can automatically compute cohomology basis functions for multiply connected domains.Source: §3.3 · p. 9
• The solver fork adds an element-level function that computes localized Peclet numbers and stabilization parameters for the stabilized thermal formulation.The standard GetDP solver lacks a native mechanism for evaluating element-dependent τe in non-symmetric Petrov–Galerkin test spaces.Source: §3.3 · pp. 9–10

3.3.1 Coupling Scheme and Resolution Strategy

• An outer root-finding loop determines the coil current required to match a target temperature at the squeeze-roll weld point for a prescribed frequency.A modified secant method initializes its current estimate using the approximate quadratic dependence of Joule heating on excitation current, then uses standard secant updates.Source: §3.3.1 · p. 10
• For each current estimate, Picard iterations couple an electromagnetic solve with temperature-dependent properties to a Newton–Raphson thermal solve.Nonlinear magnetic materials add an inner fixed-point iteration to update effective permeability.Source: §3.3.1 · p. 10
• After convergence, post-processing extracts secondary quantities including coil voltage.The solver implementation supports user-defined weak formulations, conforming finite-element spaces, global quantities, and circuit constraints.Source: §3.3 · p. 9, §3.3.1 · p. 10

4.1 Tube Parametrization and Velocity Field

• The cross-section mapping preserves the outer-diameter arc length as the opening changes, determining local radius and angle for each open profile.At the welded state, the opening is zero, the radius equals the tube radius, and the opening angle is zero.Source: §4.1 · p. 12
• The three-dimensional velocity field follows the forming geometry’s streamlines, with material parameters held constant along each streamline under steady-state flow.The field is coupled to the nominal line-feed speed through the chain rule.Source: §4.1 · pp. 12–13

4.2 Parameterized Geometric Modeling in Gmsh

• A unified set of global parameters defines the tube, coil turns, impeder, their relative positions, and mesh-quality controls in the Gmsh model.The parameterized geometry is constructed in Gmsh’s script-based environment.Source: §4.2 · p. 13
• A standalone workflow turns manually placed strip-edge control points into a smooth spline, piecewise linear segments, and a one-dimensional mesh.Users specify mesh density interactively to resolve electrical current distributions, with geometric-progression node spacing smoothing element-size transitions.Source: §4.2 · pp. 13–14

5.1 Validation Case Specification

• The validation case uses operational data from a Weldac high-frequency induction welding installation, with domain length based on assumed upstream and downstream current-decay distances.The model neglects magnetic core losses within the impeder.Source: §5.1 · p. 14, §5.4 · p. 18

5.2.1 Electromagnetic properties

• AISI304 is modeled as non-magnetic with relative permeability near one, while the impeder’s nonlinear permeability makes the magnetic-field solution nonlinear.The FeNiZnV ferrite’s nonlinear B-H curve is mapped to effective permeability as a function of field intensity.Source: §5.2.1 · pp. 14, 16

5.2.2 Thermo-physical properties

• The thermal model uses temperature-dependent AISI304 density, specific heat, and conductivity, along with calculated thermal diffusivity for SUPG.Source: §5.2.2 · p. 16

5.3 Approach to Meshing

• Mesh elements are sized to resolve electromagnetic penetration depth, with at least three linear-basis elements per depth typically sufficient.The stated penetration depths are 0.8–1.3 mm for AISI304 and approximately 0.15 mm for copper at 193 kHz.Source: §5.3 · p. 16
• The mesh resolves the full tube length and welding apex, using structured meshes for conducting domains and an unstructured mesh for air.The air domain is represented by a half-cylinder with diameter eight times the tube outer diameter.Source: §5.3 · p. 17

5.4 Simulation Results

• Both simulated total power and weld-point temperature fall within their specified benchmark confidence intervals, validating the framework against the reference case.The reference values are approximately 60 kW with a 58–62 kW confidence interval and 1350 °C ± 3.7%, respectively.Source: Table 3 · p. 18, §5.4 · p. 18
• The temperature trace shows a drop beneath the induction coil and nearly indistinguishable profiles through the strip thickness; the bottom-boundary current-density components convey magnitude and phase.The temperature maximum exceeds the weld-point target, indicating non-zero downstream current density.Source: §5.4 · pp. 18–19

6.1 Motivation and design of the numerical experiment

• At 12–13 kA/m, Ferrotron 559H’s B–H curve crosses above FeNiZnV’s, giving it higher flux density and permeability and reducing impeder reluctance.The crossover corresponds to approximately 0.4 T on the unmodified curves.Source: §6.1 · p. 20, Figure 11 · p. 22
• The comparison uses AISI304, a 25.4 mm tube diameter, and a 3 mm wall to provide computational advantages, limited impeder space, and higher current and power demand.The design parameters are compiled in Table 4.Source: §6.1 · p. 20, Table 4 · p. 21

6.2 Simulation results

• At 100 m/min, Ferrotron 559H cuts coil current by 24.61% and power demand by 26.38% versus FeNiZnV ferrite.Current falls from 5122.13 A to 3861.65 A, and power from approximately 314.12 kW to 231.25 kW.Source: §6.2 · p. 20, Table 5 · p. 21
• At 40 m/min, FeNiZnV requires less current and power than Ferrotron 559H; the trend reverses at 70 and 100 m/min.The 40 m/min result indicates higher ferrite permeability at the required coil magnetomotive force.Source: §6.2 · p. 20, Table 5 · p. 21
• At 100 m/min, the two impeders produce almost indistinguishable strip-edge temperature profiles despite their large current and power differences.The current is set automatically to achieve the prescribed welding-point temperature; only a slight profile difference appears upstream of the coil.Source: §6.2 · p. 22, Figure 12 · p. 22
• Ferrotron 559H reduces currents at both tube surfaces; its lower inner-to-outer surface current ratio indicates greater impeder efficiency at high MMF and limited space.The FeNiZnV case's extra power is dissipated along the strip circumference where outer and inner surface currents are higher.Source: §6.2 · pp. 22–23, Figure 13 · p. 23, Figure 14 · p. 23

7 Conclusions

• The framework couples a time-harmonic magnetic scalar potential formulation with stabilized quasi-stationary advection–diffusion heat transport using SUPG.Implemented in GetDP and Gmsh, it dynamically evaluates element-wise grid Peclet numbers and stabilization parameters.Source: §7 · p. 23
• The model's accuracy and robustness were validated against operational data from a commercial Weldac installation processing AISI 304 stainless steel tubes.Source: §7 · p. 24
• The study compares conventional FeNiZnV ferrite and Ferrotron 559H soft magnetic composite impeders under demanding production conditions.Source: §7 · p. 24
• Future work includes systematic studies of setup parameters and advanced impeder geometries, including return-path, slotted, and hybrid-core designs.Source: §7 · p. 24
• Further development includes impeder-loss and heat-transfer modeling, shallow-penetration material meshing, and tailored iterative solvers and preconditioning.Source: §7 · p. 24