Heat energy balance

Elements tested

DC3D4

DC3D8

Features tested

This section provides basic verification tests of energy balance in heat transfer analyses of powder bed-type additive manufacturing processes where:

the material deposition is simulated by progressive element activation; and
the heating induced by a moving heat source is simulated by a moving heat flux.

Problem description

Each model simulates powder bed-type additive manufacturing processes of three disconnected cubes with dimensions 1.0 × 1.0 × 1.0. The analyses use the special-purpose techniques for powder bed-type additive manufacturing processes in Abaqus/Standard (see Thermomechanical Analysis of Powder Bed–Type Additive Manufacturing Processes Using the Trajectory-Based Method and Thermomechanical Analysis of Powder Bed–Type Additive Manufacturing Processes Using the Pattern-Based Method).

Model

Different types of progressive element activation are applied to the cubes to simulate the layer-upon-layer material deposition:


Cube 1	Progressively activated by partial element activation
Cube 2	Progressively activated by full element activation (that is, elements can be only inactive or fully activated)
Cube 3 (reference)	No element activation (that is, elements are always active)

The material deposition motion of a cube is shown in Figure 1. The line in the figure represents the path traveled by the recoater (or roller blade) as new layers of powder material are deposited. The material layer thickness is 0.05. A total of 20 layers is required to build one cube. An event series that represents the path of the recoater is used in the trajectory-based method. For the pattern-based method, the layer thickness is defined in a parameter table.

Mesh: Each cube is modeled by a mesh of 29 DC3D4 elements or a mesh of 3 × 3 × 3 DC3D8 elements.

Material

All cubes have the same constant thermal properties.


Conductivity	$k$ = 5000
Density	$ρ$ = 8000
Specific heat	$c_{p}$ = 500

(The units are not important.)

Loading

A moving heat flux is applied to each cube to simulate the progressive heating by a moving heat source (such as a laser). As shown in Figure 2, the scanning path of the heat source for each layer consists of a 74° hatching pattern with 0.075 spacing followed by a contour. There is no rotation between each layer.

The scanning is performed with a laser power P = 1 × 10⁹, with the energy absorption efficiency as $η$ = 100%. The laser beam spot size is assumed to have a radius of 0.025 at the intersection with the part surface. The penetration depth is assumed to be 0.025. Three types of energy distribution of the laser are tested: concentrated point source, uniform distribution in a finite box, and Goldak double ellipsoid distribution. An event series that represents the path of the heat source is used in the trajectory-based method. For the pattern-based method, the path of the heat source is not used explicitly. Abaqus estimates the amount of energy deposited into the structure based on the scan pattern definition. The difference between the total heat input into a layer estimated from the scan pattern definition and that estimated from the trajectory is not expected to be significant. The energy distribution is assumed to be a concentrated point source when the pattern-based method is used.

Boundary conditions and initial conditions: Each cube is insulated on all edges. The initial temperature of newly deposited material is $θ_{i}$ = 0.0.

Results and discussion

Based on the heat energy balance, energy input by the laser should be equal to the increase of the internal heat energy of the cube. Therefore, the final steady-state temperature of a cube can be computed analytically,

θ_{f} = θ_{i} + \frac{η P Δ t}{ρ c_{p} V} = 39.115,

where $Δ t$ = 0.15646 is the total time of the laser scanning and $V$ = 1.0 is the volume of the cube.

The simulation results of the final steady-state temperature were found to be within 1% of the analytical solution, as shown in Table 1 and Table 2.

For the trajectory-based method, output variable ALLHUMDFLUX, which is the total heat energy input into the whole model (consisting of three cubes), also agrees with the analytical solution $Q = 3 η P Δ t = 4.6938 \times 10^{8} .$

Abaqus estimates the amount of energy deposited into the structure based on the scan pattern definition when the pattern-based method is used. The output variable ALLHUMDFLUX for one cube indicates the total heat energy input $E$ = $1.71 \times 10^{8}$ , which is $9.3 %$ higher than the heat energy input in analyses performed using the trajectory-based method. Therefore, the final steady-state temperature of a cube can be computed analytically,

θ_{f} = θ_{i} + \frac{E}{ρ c_{p} V} = 42.75.

Table 1. The final steady-state temperature of analyses using DC3D4 elements and the trajectory-based method.
Heat source energy distribution	Concentrated		Uniform		Goldak
Heat source energy distribution	Temperature	Error	Temperature	Error	Temperature	Error
Partial element activation	39.116	0.00%	39.099	0.04%	39.433	0.81%
Full element activation	39.116	0.00%	39.099	0.04%	39.433	0.81%
No element activation (reference)	39.116	0.00%	39.099	0.04%	39.434	0.82%

Table 2. The final steady-state temperature of analyses using DC3D8 elements and the trajectory-based method.
Heat source energy distribution	Concentrated		Uniform		Goldak
Heat source energy distribution	Temperature	Error	Temperature	Error	Temperature	Error
Partial element activation	39.115	0.00%	39.115	0.00%	39.435	0.82%
Full element activation	39.115	0.00%	39.115	0.00%	39.435	0.82%
No element activation (reference)	39.115	0.00%	39.115	0.00%	39.435	0.82%

Table 3. The final steady-state temperature of analyses using DC3D4 elements and the pattern-based method.
Heat source energy distribution	Temperature	Error
Partial element activation	42.75	0.00%
Full element activation	42.75	0.00%
No element activation (reference)	42.75	0.00%

Table 4. The final steady-state temperature of analyses using DC3D8 elements and the pattern-based method.
Heat source energy distribution	Temperature	Error
Partial element activation	42.75	0.00%
Full element activation	42.75	0.00%
No element activation (reference)	42.75	0.00%

Input files

The trajectory-based method

am_heatbalance_dc3d4_concentrated.inp: DC3D4 elements, concentrated heat source.
am_heatbalance_dc3d4_uniform.inp: DC3D4 elements, heat source.
am_heatbalance_dc3d4_goldak.inp: DC3D4 elements, heat source using the Goldak distribution.
am_heatbalance_dc3d8_concentrated.inp: DC3D8 elements, concentrated heat source.
am_heatbalance_dc3d8_uniform.inp: DC3D8 elements, heat source using a uniform distribution in a finite box.
am_heatbalance_dc3d8_goldak.inp: DC3D8 elements, heat source using the Goldak distribution.

The pattern-based method

am_heatbalance_dc3d4_patternbased_full.inp: DC3D4 elements, full element activation.
am_heatbalance_dc3d4_patternbased_partial.inp: DC3D4 elements, partial element activation.
am_heatbalance_dc3d4_patternbased_iniact.inp: DC3D4 elements, elements are always active.
am_heatbalance_dc3d8_patternbased_full.inp: DC3D8 elements, full element activation.
am_heatbalance_dc3d8_patternbased_partial.inp: DC3D8 elements, partial element activation.
am_heatbalance_dc3d8_patternbased_iniact.inp: DC3D8 elements, elements are always active.

Input files containing definitions or data to be included in the input files listed above

ABQ_am_special_purpose_types.inp: Types of property tables, parameter tables, and event series used by the special-purpose techniques for the simulation of common additive manufacturing processes in Abaqus.
es_heatbalance_laser.inp: Event series data of the laser scanning path used in the trajectory-based method.
es_heatbalance_roller.inp: Event series data of the roller blade motion used in the trajectory-based method.