Implementing a Moving Heat Source in Abaqus for Welding

1. Core Methodology: The User Subroutine Approach

Accurate welding simulation requires a moving heat flux boundary condition, which is implemented in Abaqus via the DFLUX or HETVAL user subroutines. DFLUX is typically preferred for surface-based heat flux, while HETVAL handles volumetric heat generation. Both require writing Fortran code to define the heat source’s spatial and temporal evolution.

The industry standard for arc welding is Goldak’s double-ellipsoidal heat source model, which we will implement in DFLUX. The model divides the heat source into front and rear quadrants with different geometrical parameters.

2. Step-by-Step Implementation Workflow

2.1. Goldak Model Parameters and Geometry Definition

Before coding, define the heat source parameters for your specific process. These are typically calibrated from weld bead geometry or thermal camera data.

Table 1: Goldak Double-Ellipsoidal Heat Source Parameters

2.2. DFLUX Subroutine Code Structure

Create a Fortran file (welding_dflux.for) with the following core structure:

      SUBROUTINE DFLUX(FLUX,SOL,KSTEP,KINC,TIME,NOEL,NPT,COORDS,
     1 JLTYP,TEMP,PRESS,SNAME)
C
      INCLUDE 'ABA_PARAM.INC'
C
      DIMENSION FLUX(2), TIME(2), COORDS(3)
      CHARACTER*80 SNAME
C
      DOUBLE PRECISION x,y,z,x0,y0,z0,vx,vy,vz,t_start
      DOUBLE PRECISION a_f,a_r,b,c,Q_eff,f_f,f_r
      DOUBLE PRECISION dist_front,dist_rear,heat_flux
C
C     User-defined parameters (in consistent units)
      Q_eff = 2400.0          ! Effective power (W)
      a_f = 0.003             ! Front length (m)
      a_r = 0.010             ! Rear length (m)
      b = 0.005               ! Width (m)
      c = 0.008               ! Depth (m)
      f_f = 0.6               ! Front fraction
      f_r = 1.4               ! Rear fraction
      vx = 0.002              ! Welding speed in x (m/s)
C
C     Calculate current heat source center
      t_start = 0.0
      t_current = TIME(2)
      x0 = vx * (t_current - t_start)
      y0 = 0.0
      z0 = 0.0
C
C     Current coordinates relative to moving center
      x = COORDS(1) - x0
      y = COORDS(2) - y0
      z = COORDS(3) - z0
C
C     Goldak double-ellipsoidal model
      IF (x .GE. 0.0) THEN
C        Front quadrant
         dist_front = (x**2)/(a_f**2) + (y**2)/(b**2) + (z**2)/(c**2)
         IF (dist_front .LE. 1.0) THEN
            heat_flux = (6.0*sqrt(3.0)*Q_eff*f_f)/(a_f*b*c*PI*sqrt(PI))
            heat_flux = heat_flux * exp(-3.0*dist_front)
         ELSE
            heat_flux = 0.0
         ENDIF
      ELSE
C        Rear quadrant
         dist_rear = (x**2)/(a_r**2) + (y**2)/(b**2) + (z**2)/(c**2)
         IF (dist_rear .LE. 1.0) THEN
            heat_flux = (6.0*sqrt(3.0)*Q_eff*f_r)/(a_r*b*c*PI*sqrt(PI))
            heat_flux = heat_flux * exp(-3.0*dist_rear)
         ELSE
            heat_flux = 0.0
         ENDIF
      ENDIF
C
      JLTYP = 0    ! Surface flux type
      FLUX(1) = heat_flux
C
      RETURN
      END

2.3. Abaqus/CAE Setup Procedure

1. Create Heat Transfer Step: Use a transient heat transfer analysis with appropriate time period (weld length / speed).
2. Define Surface Interaction: Create a surface-based heat flux load and associate it with the DFLUX subroutine.
3. Compile and Link: In the job module, specify the Fortran user subroutine file.
4. Mesh Considerations: Element size must be smaller than heat source radius (b, c) to capture the gradient—typically 1/3 of the heat source dimension.

2.4. Motion Control and Path Definition

For non-linear weld paths, implement path interpolation in the subroutine:

fortral

C     For predefined weld path points (x_i, y_i, z_i, t_i)
      CALL INTERPOLATE_WELD_PATH(x0,y0,z0,t_current)

3. Critical Modeling Strategies for Accuracy

3.1. Parameter Calibration Methodology

Never use literature values directly. Calibrate using:

1. Macro-graph matching: Adjust a_f, a_r, b, c until simulated fusion zone matches experimental cross-section.
2. Thermocouple validation: Tune η and distribution factors to match measured thermal histories at HAZ locations.

3.2. Element Birth and Death Technique

For multi-pass welding, use *MODEL CHANGE, REMOVE for filler material activation:

• Initially deactivate weld mesh elements
• Activate elements when heat source center approaches (within 2×a_r)
• Apply initial temperature slightly above ambient to avoid thermal shock

3.3. Adaptive Meshing Considerations

While Abaqus/Standard doesn’t support adaptive remeshing, you can:

• Use fine mesh only in weld and HAZ region
• Apply mesh bias to transition from fine (0.5 mm) to coarse (5 mm) elements
• For very deep penetration, consider using HETVAL for volumetric heating instead of surface DFLUX

4. Results Interpretation and Validation Metrics

4.1. Key Output Verification

• Peak temperature history: Should match typical weld thermal cycles (rapid heating > 1400°C, cooling rates 50-200°C/s).
• Fusion zone geometry: Compare width/depth to macrograph measurements.
• Thermal gradient: Check that dT/dx at solidus temperature matches expected values (typically 10-100°C/mm).

4.2. Common Post-Processing Operations

1. Temperature field animation: Verify smooth heat source movement without numerical artifacts.
2. Path plots along weld centerline: Extract cooling rate between 800°C and 500°C (t₈₅₀ time).
3. Heat affected zone (HAZ) identification: Create time-at-temperature contours for microstructural prediction.

5. Convergence Errors and Solutions

5.1. Typical Error Messages and Fixes

5.2. Numerical Stability Best Practices

• Use second-order elements (DC3D10) for better thermal gradient capture
• Implement smooth heat source start-up/ramp-down in first/last 0.1s
• Apply artificial thermal conductivity at very high temperatures (>2000°C) to stabilize
• Use *CONTROLS, ANALYSIS=DISCONTINUOUS for severe nonlinearities

6. Advanced Extensions and Hybrid Approaches

6.1. Combined DFLUX and HETVAL Implementation

For processes with both surface heating and volumetric effects (laser welding):

• Use DFLUX for surface absorption (Gaussian distribution)
• Use HETVAL for keyhole volumetric heating
• Synchronize motion parameters between subroutines via common blocks

6.2. Integration with Mechanical Analysis

For fully coupled thermomechanical analysis:

1. Run pure thermal analysis first to generate temperature history
2. Use *PREPRINT, MODEL=YES to save .fil results
3. Import as predefined field in mechanical step with *IMPORT
4. Consider *COUPLED TEMPERATURE-DISPLACEMENT for small-scale models

6.3. 3D vs. 2D Modeling Decision Guide

7. Conclusion and Next Steps

Implementing a moving heat source via DFLUX provides the foundation for accurate welding simulation in Abaqus. Success depends on careful parameter calibration, appropriate mesh design, and numerical stabilization techniques. The Goldak model implementation shown here serves as a robust starting point for arc welding processes.

For production-level analysis, develop a subroutine library with multiple heat source types (Gaussian, conical, cylindrical) and integrate with material property subroutines (UMATHT, USDFLD) that account for temperature-dependent conductivity, latent heat, and phase transformations. Validation against experimental thermo-couple data remains non-negotiable for credible simulation results.

Consider extending this framework to simulate additive manufacturing processes by incorporating layer activation, multiple heat source paths, and powder-to-solid transition models, which follow similar computational patterns but with additional complexity in material state tracking.