Welding Heat Sources in Abaqus DFLUX: Models, Types & Selection Guide (2026)

Selecting the right heat source model is one of the most important steps in welding simulation with the Abaqus DFLUX subroutine. Each welding process requires a different mathematical model to represent heat input accurately. For example, laser welding often uses a Gaussian surface heat source, while arc welding commonly uses the Goldak double-ellipsoid model. Deep penetration welding usually requires volumetric heat source models. This article compares the most common DFLUX heat source models, explains their governing equations and key parameters, and helps you choose the best model based on the welding process, penetration depth, and your simulation objectives.

At Mathech, we have guided hundreds of engineers through the complexities of welding simulation. We are here to demystify the Abaqus DFLUX subroutine and provide you with a data-backed roadmap to selecting the right mathematical model for your specific process.

What Is a Welding Heat Source? Understanding Its Role in Welding Simulation

What Is a Heat Source in Welding and Why Does It Matter?

In the context of finite element analysis (FEA), a welding heat source is a mathematical representation of the thermal energy input to a workpiece. This heat source is a boundary condition applied to nodes or elements to simulate the heat generated by the arc.

The heat source drives the entire physics of the simulation that dictates the thermal cycle – the peak temperature, heating rate, and cooling rate – at every point in the material.

Specifically, the heat source in a simulation is quantified by:

• Power Input (Q): The total energy delivered to the surface (Watts).
• Efficiency (η): A value less than 1.0 accounting for energy lost to radiation, convection, and the environment.
• Spatial Distribution: How the heat is spread across the surface or volume.

Why Accurate Heat Source Modeling Is Critical in Abaqus Finite Element Analysis

In Abaqus finite element analysis, the accuracy of your welding simulation hinges entirely on the heat source. Treating it as a simple, uniform heat flux over a large area is a recipe for disaster.

Here is why precision matters:

1. Residual Stress Prediction: The thermal expansion and contraction during the weld thermal cycle generate residual stresses. If your heat source is too broad, the thermal gradients become shallow, leading to an underestimation of tensile residual stresses and potential distortion.
2. Microstructure and Material Phase: In steel, the cooling rate from the peak temperature determines the formation of martensite, bainite, or pearlite. A localized heat source results in faster cooling rates, which is critical for predicting hardenability and the risk of cold cracking.
3. Distortion (Buckling): The thermal shrinkage forces are proportional to the thermal gradient. An inaccurate heat source may cause the simulation to predict no distortion, while the real-world component warps significantly.

Relationship Between Heat Input, Temperature Distribution, and Residual Stress

To understand the physics, we need to look at the heat input formula, expressed as the net heat input per unit length:

(Where V = Voltage, I = Current, v = Travel Speed)

However, the distribution of this energy is just as critical as the total value. The Goldak double ellipsoid, for example, creates a teardrop shape; the front quadrant has a steep gradient (high heating rate), while the rear quadrant has a shallower gradient (cooling phase).

Why does this matter for residual stress?
A steep temperature gradient (sharp front) creates high thermal compression in the weld pool, leading to a state of tensile residual stress in the fusion zone (FZ) and heat-affected zone (HAZ). If you use a Gaussian distribution (which is generally wider), you might reduce the peak temperature, artificially lower the thermal gradient, and produce a residual stress profile that is completely non-conservative for the design.

Types of Welding Heat Sources Used in Abaqus DFLUX

When we discuss Abaqus DFLUX, we are discussing the user subroutine that allows us to define non-uniform, moving heat flux distributions.

Surface Heat Sources vs Volumetric Heat Sources

The distinction here is based on penetration depth:

• Surface Heat Sources: The heat is applied only to the top layer of elements (via DFLUX with a surface-based flux). This is suitable for shallow penetration processes like GTAW (TIG) on thin plates or welding with a low current.
• Volumetric Heat Sources (Body Heat Flux): The heat is distributed throughout the depth of the material. This is implemented using the DFLUX subroutine with a body flux flag. This is essential for processes with significant penetration, like GMAW (MIG), laser welding, or electron beam welding.

Moving Heat Sources in Welding Simulation

A moving heat source is defined by coordinating the flux intensity with time and coordinate position. In Abaqus DFLUX, you must program the subroutine to calculate the distance (R) from the center of the heat source to the current integration point. As the center of the source moves along the weld line (e.g., X = X₀ + v*t), the heat input at any point (x, y, z) is calculated.

Distributed Heat Sources for Thick and Thin Plates

• Thin Plates: For thicknesses < 5mm, heat flows primarily in the 2D plane. A surface Gaussian or surface double-ellipsoid is usually sufficient.
• Thick Plates: Heat flows in 3D. Volumetric sources like the Conical or Double Ellipsoid are required to capture the “keyhole” or the deep penetration effect.

Mathematical Heat Source Models for Welding Simulation

Gaussian Heat Source Model: Theory, Equation, and Applications

The Gaussian distribution is the foundational model for welding. It assumes the heat flux is distributed in a circular pattern on the surface.

Equation:

(Where ( r_b ) is the effective radius of the heat source, and ( r ) is the radial distance from the center).

Applications:

• GTAW (TIG) welding.
• Low-current GMAW on thin sections.
• Laser welding (albeit with modifications to account for the keyhole).

Pros: Simple to code, computationally cheap.
Cons: Does not account for the “trailing” effect of the weld pool where heat accumulates, leading to an overestimation of the peak temperature at the front and underestimation of the dwell time at the rear.

Goldak Double Ellipsoidal Heat Source Model Explained

Developed by Goldak, Akhlaghi, and others, this model is the industry standard for arc welding.

The double ellipsoid accounts for the asymmetry of the weld pool; the front half is steeper (to represent the arc pushing into the material) while the rear half is wider (representing the heat flowing back into the material).

Equation Structure (Front Half):

Equation Structure (Rear Half):

(Where ( f_f ) and ( f_r ) are the fractions of heat in the front and rear, summing to 2).

Applications:

• GMAW (MIG/MAG) welding.
• Submerged Arc Welding (SAW).
• Heavy-section arc welding.

Conical Heat Source Model for Deep Penetration Welding

For high-energy density processes like laser welding, the keyhole shape is closer to a cone. The conical heat source applies volumetric heat that decreases linearly with depth.

Equation:

(Where ( r_0(z) ) is the radius at a given depth ( z ), decreasing from the top radius to the bottom radius).

Applications:

• Laser welding.
• Electron Beam Welding.
• Hybrid welding (Laser + Arc).

Which Heat Source Model Should You Use for Each Welding Process?

This is where the engineering judgment comes in. Here is the Mathech Engineering Recommendation Matrix for 2026.

Best Heat Source Model for Arc Welding (SMAW, GMAW, GTAW)

• GTAW (TIG): Surface Gaussian. Since penetration is generally shallow, a surface flux is sufficient. If welding thick plates with multi-pass, use a Volumetric Gaussian.
• GMAW (MIG) & SMAW (Stick): Goldak Double Ellipsoid (Volumetric). The heat input is significant, and the weld pool shows the characteristic teardrop. Use this to capture the “digging” effect of the arc pressure. Ensure you calibrate ( a, b, c ) parameters relative to the weld bead geometry.
• SAW (Submerged Arc): Goldak Double Ellipsoid. Very high heat input requires the larger “rear” length to capture the slow cooling rate in heavy sections.

Best Heat Source Model for Laser Welding

• Keyhole Mode (High Power): Conical Model or a Goldak distribution with a very small width parameter “b” and a large depth “a”.
• Conduction Mode (Low Power): Volumetric Gaussian.
• Abaqus DFLUX Tip: For laser welding, you must consider the “deep penetration” effect. The Conical model is preferred because it naturally handles the variation in the beam radius with depth due to the plasma plume and multiple reflections.

Best Heat Source Model for Electron Beam Welding, Friction Stir Welding, and Hybrid Welding

• Electron Beam (EBW): Conical Source. The beam has a very high aspect ratio. Use a “triple-ellipsoid” or a very steep conical function.
• Friction Stir Welding (FSW): Surface Heat Source coupled with a volumetric heat source from the tool shoulder. FSW is a solid-state process; heat generation is frictional. Often, a surface Gaussian is combined with a volumetric source representing the pin stirring.
• Hybrid (Laser + Arc): A combination of a Conical Model (for the laser) and a Double Ellipsoid (for the arc) placed co-axially or in tandem. This is a complex setup requiring coupled DFLUX calls.

Implementing Welding Heat Sources Using the Abaqus DFLUX Subroutine

How the DFLUX Subroutine Defines a Moving Heat Source

The DFLUX subroutine is called for each integration point during the analysis. You must pass the time and the coordinates to the Fortran subroutine.

Logic Flow:

• Calculate the current position of the heat source center:

• Calculate the distance vector from the source center to the integration point:

• Calculate the heat flux intensity using the chosen mathematical model (e.g., Goldak).
• Return the flux value to Abaqus.

Required Parameters for DFLUX Heat Source Modeling

To implement this, you need to define:

1. Arc Power (Q): Watts.
2. Efficiency (η): Usually 0.75-0.95 for arc welding, 0.8-0.9 for laser.
3. Parameters: For Goldak (a, b, c_f, c_r). For Gaussian (r_b). For Conical (r_top, r_bottom, depth).
4. Travel Speed (v): mm/sec.
5. Start Time: When the arc turns on (usually 0).
6. Dwell Time: Time for the heat source to remain at the start (pre-heating) or end (post-heating).

Common Mistakes When Programming DFLUX for Welding Simulation

Even with the right model, engineers often crash the analysis due to programming errors.

• Unit Inconsistency: Ensure your heat flux is in W/mm³ (if using mm units). Check your input parameters. If Q is in Watts and the volume is in mm³, it works. If you are using meters, it’s W/m³.
• Circular Reference: The subroutine must be robust if the integration point is exactly at ( r = 0 ) to avoid division by zero.
• Moving Mesh: If you are using ALE adaptive meshing, the coordinates of the integration points change. Your DFLUX must reference the physical coordinates, not the mesh coordinates.
• Ignoring “Jump” Conditions: Ensure the heat source does not jump if the time step is too large; use DFLUX with a stable time increment.

Best Practices for Selecting a Welding Heat Source Model

Choosing the Right Heat Source Based on Welding Process

The selection process is a trade-off between accuracy and complexity.

1. Start with the Process: Determine if the energy transfer is conductive or convective.
2. Check the Penetration: If the weld depth is > 2x the width, use a volumetric source (Goldak or Conical).
3. Check the Speed: Slow welding creates large, tear-drop pools (Goldak is best). Fast welding creates narrow, conical pools (Conical is best).

Matching Heat Source Parameters with Experimental Data

This is the “calibration” stage. We recommend performing a thermal calibration using thermocouple data from a test weld.

• Method: Run the simulation and compare the thermal cycles at specific points. Iterate the parameters (like “a” and “b” in Goldak) until the simulated temperature history matches the experimental data within +/- 10%.
• Weld Pool Geometry: If you don’t have thermocouples, you can calibrate based on the weld pool cross-section (macrograph). Adjust the power and parameters until the predicted FZ matches the actual weld bead.

Improving Simulation Accuracy While Reducing Computational Cost

A finer mesh is useless if the time step is not optimized.

• Mesh Coarsening: Use a fine mesh near the weld zone (where the heat source is active) and a coarse mesh far away. Use a “clustered” bias toward the weld line.
• Remeshing: If using DFLUX, ensure your element size is smaller than the heat source radius (at least 3-4 elements across the radius). Too coarse a mesh will “average” the flux and reduce peak temperature.
• Time Step: Use an automatically time-incrementing step. Force Abaqus/Standard to take small steps as the heat source passes a node.

Frequently Asked Questions About Welding Heat Sources in Abaqus

Which Heat Source Model Is Most Accurate for Welding Simulation?

A: For arc welding, the Goldak Double Ellipsoid is considered the “gold standard” due to its ability to represent the physical asymmetry of the weld pool. However, “accuracy” is tied to calibration. A simple Gaussian model can be as accurate as Goldak if you calibrate the power and radius carefully for a specific set of experimental data. For laser welding, the Conical model is more accurate.

Can the Goldak Model Simulate Laser Welding?

A: Yes, but with significant modifications. The standard Goldak parameters (specifically “a” and “b”) are designed for large arc welds. To simulate laser welding, you must set the width parameter (“b”) very small (e.g., 0.5 mm) and the depth parameter (“a”) large. However, because the laser produces a “keyhole” effect (where the beam penetrates by vaporizing material), the Conical model is usually a better starting point.

Do All Welding Simulations Require the DFLUX Subroutine?

A: No. If you are performing a simple analysis where the heat source is stationary (e.g., spot welding), you can define a heat flux directly in the interaction module or as a surface heat flux without a subroutine. However, for any moving heat source (GMAW, SAW, Laser welding), you must use the DFLUX user subroutine to define the position, movement, and distribution of the heat over time.

Conclusion: Selecting the Right Welding Heat Source for Reliable Abaqus Simulations

Key Takeaways for Engineers

Selecting a welding heat source is not a “guess.” It is a systematic decision based on:

1. Penetration Depth: Surface vs. Volumetric.
2. Process Physics: Goldak for arcs; Conical for lasers.
3. Calibration: Validation with experiments.

Remember, the best model in the world will fail if your material properties (thermal conductivity, specific heat, latent heat) are incorrect. Always verify your thermal conductivity and density first.

Download Welding DFLUX Examples and Tutorials from Mathech

Are you tired of re-inventing the wheel? At Mathech, we have compiled a library of proven Abaqus DFLUX subroutines and weld simulation templates.

• Download our “Abaqus Weld Simulation Pack” which includes ready-to-use Fortran code for:
  • Gaussian Surface Heat Source
  • Goldak Double Ellipsoid (Volumetric)
  • Conical Volumetric Source
• Access step-by-step video tutorials on calibrating these models against real-world weld data.
• [Download the Free Weld Simulation Toolkit Here]