Computationally efficient finite difference method for metal additive

Materials and Design 132 (2017) 226–243

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Materials and Design journal homepage: www.elsevier.com/locate/matdes

Computationally efficient finite difference method for metal additive manufacturing: A reduced-order DFAM tool applied to SLM Matthew McMillan, Martin Leary ⁎, Milan Brandt RMIT Centre for Additive Manufacture, RMIT University, Melbourne, Australia

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Metal Additive Manufacturing (MAM) simulation is computationally demanding. • Geometric simplifications allow MAM to be simulated in one-dimension (1D). • A tri-diagonal system of equations is developed to represent the MAM component. • Temperature profiles qualitatively match observations in MAM components.

a r t i c l e

i n f o

Article history: Received 2 February 2017 Received in revised form 22 June 2017 Accepted 25 June 2017 Available online 28 June 2017 Keywords: Design Additive manufacture 3D printing Temperature field Simulation

⁎ Corresponding author. E-mail address: [email protected] (M. Leary).

http://dx.doi.org/10.1016/j.matdes.2017.06.058 0264-1275/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t Metal Additive Manufacturing (MAM) allows the production of complex lattice structures that exceed the manufacturing capability of traditional processes. However, the MAM process is highly complex, including: transient thermo-mechanical loading, spatially and temporally transient boundary conditions and multi-scale affects. Furthermore, MAM is subject to significant experimental and statistical uncertainties. Consequently, the MAM process is poorly understood and build-process simulations are computationally demanding; this limits the availability of Design for Additive manufacturing (DFAM) tools, and necessitates experimental validation for commercial MAM applications. This research develops a novel finite difference method (FDM) simulation of the MAM temperature field that is based on a computationally efficient 1D transient tri-diagonal system of equations, and reduces data generation effort by the reuse of existing MAM production data. The 1D simulation is particularly suited to lattice geometries due to the slenderness of lattice strut elements. The reduced-order simulation method developed in this research provides a timely and useful DFAM tool that enables qualitative design insight early in the design phase and pre-production validation. The proposed method is validated by comparison to existing analytical results, numerical results and by application to a titanium and aluminium lattice structures manufactured by a commercial Selective Laser Melting (SLM) process. © 2017 Elsevier Ltd. All rights reserved.

M. McMillan et al. / Materials and Design 132 (2017) 226–243

227

Nomenclature Symbol Units

Name

Heat source parameters P W heat source power v m/s heat source velocity d m heat source beam diameter Time parameters s tprod tpre s tman s s tpost N – n – theat s tenv s trecoat texp

s s

tcool tproc tsimp

s s s

total production time excluding post processing total time to prepare the machine for manufacture total manufacturing time total post manufacture time until part completion total number of layers current layer/polygon subscript time taken to preheat the machine prior to manufacture time required to achieve build chamber environmental conditions time for the recoater to traverse the build volume time for the heat source to traverse the tool paths on a given layer time for the part to normalise to ambient conditions time required to post process part simplified estimation of production time

Thermal parameters T K temperature Q W heat q" W/m2 heat flux across a surface Tbase K build platform temperature T0 K ambient temperature 3 ρ kg/m density k W⁄m°K thermal conductivity Cp J⁄kg°K specific heat capacity k α m2/s thermal diffusivity ðα ¼ ρCp Þ Energy parameters J/m2 surface energy density Es Ev J/m3 volumetric energy density Etotal J total energy applied to geometry throughout a simulation Geometric parameters ° θ inclination angle relative to the build platform A m2 area of layer D m strut diameter V m3 volume of geometry H m cylinder length/height Lt m simulation layer thickness p m Layers perimeter length 2 DAn m overhanging area of layer n 2 SAn m supporting area of layer n Subscript notation m – polygon number subscript for multiple items on a single layer n – layer/node location subscript p – time increment subscript

1. Introduction Additive manufacture (AM) is the layer-wise joining of materials to fabricate an object from digital model data [1]. AM enables significant design freedom in comparison with traditional manufacturing methods [2]. Utilisation of this design freedom allows the production of structurally optimal geometries with highly integrated functionality [3,4]; as has been demonstrated in aerospace [5,6], automotive [7] and medical applications [8–12]. In particular metal additive manufacturing (MAM) enables the manufacture of optimised metal engineering structures. Within the MAM families (Fig. 1), powder bed fusion (PBF) and directed energy deposition (DED) are the most commonly applied in research and industry. DED creates geometry continuously with a wire or powder feedstock that is delivered coaxially with a heat source [13]. Although the reduced-order simulation method developed in this research is compatible with DED, associated case studies are not explicitly evaluated. PBF is commercially embodied by Selective Laser Melting (SLM) and Electron

Fig. 1. MAM processes classified according to material type, ASTM family and commercial process names [1].

Beam Melting (EBM). The simulation method developed in this research is compatible with SLM and EBM; case studies are provided for the application of the method to aluminium and titanium lattice structures using SLM (Sections 8.1 and 8.2). Design for additive manufacture (DFAM) tools aid in the synthesis of shape; geometric mesostructures, such as lattice unit cells; material composition and microstructures to best utilise AM technologies [14, 15]. At present there exists a “significant lack” of suitable DFAM tools [16]. To enable the commercial application of MAM the development of custom DFAM tools are required [16–19]. The prediction of the MAM temperature field is highly complex [20,21], due to multiple interacting physical phenomena [22] that are temporally and spatially transient over multiple time and space scales [23,24]. The MAM process is also subject to significant experimental (systemic) and statistical (aleatory) uncertainties [25,26]. The identified complexity and uncertainty coupled with a lack of suitable DFAM tools necessitates experimental validation within the standard MAM workflow [27], imposing experimental costs that stymie MAM commercialisation (Fig. 2). The development of a computationally efficient simulation tool that can be readily implemented during the design phase as a manufacturability analysis DFAM tool [14] is a critical requirement for MAM commercialisation [27,28].

2. Powder bed fusion processes Powder bed fusion (PBF) processes offer: high geometric resolution of complex topologies, approximately 100% part density, robust mechanical properties and high material yield [29–31]. These attributes enable the efficient production of tailored structures such as lattices and topologically optimal geometries to achieve high specific strength [32, 33], stiffness [32,33] and other engineering advantages [3,9,34–40]. The PBF process creates bulk geometry by iteration of the following steps: 1. A recoater uniformly distributes a layer of metallic powder over the previously processed layer or build platform [41]. 2. A focused heat source melts the powder in a controlled atmosphere along with surrounding solid material. The liquefied metal fuses while the heat source continues scanning the entire layer [42]. 3. The build platform is incrementally lowered within the build chamber by the desired layer thickness, in the order of 30–100 μm [43].

228


Fig. 2. Generalised MAM production workflow. The experimental validation required for this workflow is costly, and stymies MAM commercialisation.

2.1. Powder bed fusion process parameters

2.2. Slicing of AM geometry

The MAM process is controlled by heat source, geometric and time parameters including: heat source power, P, heat source velocity, v, heat source beam diameter, d, and layer thickness, Lt. The associated energy input is defined as either a surface (Eq. (1)) or volumetric (Eq. (2)) energy density. Both surface (Eq. (1)) and volumetric (Eq. (2)) energy density only determine an average value across the heat input width and does not consider overlap.

Prior to AM, geometry must be sliced into a series of polygons and associated tool paths. The slicing process generates the following data:

ES ¼

P vd

ð1Þ

EV ¼

P vLt d

ð2Þ

• Preheat time, theat – time for the machine to achieve the required build temperature. • Environmental time, tenv – time to achieve suitable atmospheric conditions (vacuum or shielding gas) for manufacture. • Recoater time, trecoat – time for the recoater to deposit a layer of powder (constant for each layer). • Exposure time, texp – time for the heat source to traverse the tool path required to generate the current layer geometry (varies according to the geometry). • Post manufacture cooling time, tcool – time required for the part to cool to safe handling conditions. • Post processing time, tproc – time associated with any finishing operations including, removal from build platform, surface modification, final machining, heat treatment and cleaning.

The time taken to complete the manufacturing stage, tman, is the sum of exposure (Eq. (4)) and recoater times for each layer (Eq. (3b)). However, for geometry with a relatively small cross-sectional area, for example lattice structures, texp, may be insignificant in comparison to trecoat, resulting in a simplified estimation of total build time, tsimp (Eq. (5)). t prod ¼ t pre þ t man þ t post

ð3Þ

t pre ¼ maxðt heat ; t env Þ

ð3aÞ

⏟

pre manufacture

n¼N ∑n¼1 ðt recoat ⏟

þ t exp

ð3bÞ manufacturing stage

t post ¼ t cool þ t proc ⏟

t exp ¼

Simulation thickness at which the input STL geometry is sliced, Lt. Number of layers within the part volume, N. Polygon of intersection of part volume with each associated layer, n. The cross-sectional area of each uniquely defined polygon, within every layer, n. • Approximate part volume, V (Eq. (19)). 3. Complexity of powder bed fusion

Total part production time, tprod, is the sum of pre-production, tpre, manufacture, tman, and post-production time, tpost, (Eq. (3)), where:

t man ¼

• • • •

ð3cÞ post manufacture stage

An;m vd

t simp ≈ max ðt heat t env Þ þ Nt recoat þ t cool þ t post for t exp ≪t recoat

ð4Þ

ð5Þ

This research develops a DFAM simulation tool to qualitatively predict the temperature profile for the Powder bed fusion (PBF) process in a computationally efficient manner. The PBF process consists of multiple interacting physical phenomena including (Fig. 3): • A transient heat source with associated fluid dynamics, including Marangoni currents, material phase change and vaporisation. • Heat transfer by conduction, convection and radiation and by interaction with unfused powder and via fused powder to the heated platform. • Dynamic interaction between recoater, unfused powder and fused material.

The behaviour of the physical MAM phenomena is determined by numerous material, process and design parameters (Sections 3.1–3.7). Heat transfer between the heat source and the powder surface causes large temperature gradients that liquefy powder at the melt pool [44]. The melt pool is a complex thermo-fluidic event that has a unique size, stability and shape depending on processing conditions and material [42,45]. Transient heat transfer then occurs between the melt pool, ambient environment, existing geometry and surrounding powder. This multi-phase transient heat transfer process is highly complex [27], and requires consideration of many input phenomena including: component geometry and orientation, energy source and scan strategy, melt pool dynamics, atmosphere interaction, support material, build platform, powder dynamics and heat transfer. 3.1. Component geometry and orientation Component geometry and specific orientation has a profound influence on AM build quality, as changes in layer cross section result in varying exposure times (Eq. (4)) and heat transfer resistance. Relevant issues associated with geometry include: • Downward facing surfaces have the potential to overheat, partially fusing powder to the downward facing surface (Fig. 4) [46,47]. This reduced surface quality can have a significant impact on mechanical properties [48]. • Transient cooling of the manufactured geometry results in residual stresses, which can be sufficient to yield the material [49–51]. • Residual stress is especially dependant on part orientation and geometry [52], and is especially challenging in bulk and thick-walled parts


229

Fig. 3. Schematic overview of the PBF process showing thermo-fluidic affects within the melt pool (inset). The SLM process uses a laser heat source, and galvanometer as the control apparatus with an argon or nitrogen atmosphere. The EBM process uses an electron beam heat source, and magnetic fields as control apparatus within a vacuum atmosphere.

and can cause cracking [51,53,54], reduce fatigue life [43,49,50,55,56], and part deformation [53,57,58]. 3.2. Heat source and scan strategy A concentrated heat source enables powder melting. The strategy used to apply this heat source to the powder bed has a significant effect on: final geometry, porosity, production rates, microstructure and mechanical properties, for example: • Thermal gradients from the heat source into surrounding geometry are extremely high, in the order of 103–1011 K/s [25,59]. • High intensity laser/electron beam powers are desired to minimise porosity [25,60], and increase material production rates; however, excessive power can result in poor surface quality [61].

• Alternating scan vector angle between layers decreases porosity and surface roughness [53]. • Short scan vectors reduce thermal gradients and limit residual stress [53,54]. • Increased speed leads to elongation and narrowing of the melt pool, which results in instability known as balling [59,62–65].

3.3. Melt pool dynamics The melt pool is a volume of liquefied metal on the surface of the powder bed. The melt pool is a highly dynamic event that is dependent on multiple parameters including: material thermal properties, powder morphology, heat source power, scan speed, the presence of

Fig. 4. Comparison of the downward facing (left) and upward facing (right) surfaces of a SLM plate inclined to 10°. Manufactured using Ti6Al4V on a SLM 250 machine.

230


neighbouring fused geometry and the associated temperature field. In general:

3.7. Powder dynamics and heat transfer

• The melt pool can be approximated by an elliptical or teardrop profile with major axis aligned to the scan vector, and with approximate major and minor dimensions of 100 and 200 μm [66–70]. • Temperature and surface tension gradients within the melt pool drive both natural and Marangoni convection [71,72].

Powdered metallic alloys are the raw build material for the PBF process. The size and distribution of powder particles varies depending on material and manufacturer, which in turn influences the required processing parameters. In general:

3.4. Atmosphere interaction The MAM atmosphere functions to prevent oxidisation and remove soot. Relevant effects of atmosphere on the PBF process include: • Gas flow rate is highly influential on component properties and process repeatability [73,74]. • System thermal efficiency is reduced by the atmosphere and vaporised material absorbing a portion of the heat source energy [73,75]. 3.5. Support material Support material can be utilised to reduce the distortion effects of residual stress and control local heat transfer to enhance manufacturability [23,76]. However, the use of support material is typically undesirable as it increases manufacturing time [40], post processing cost [77] and contributes to material waste [78]. 3.6. Build platform The build platform acts as heat sink to allow part cooling and must be microstructurally compatible with the associated PBF powder to enable bonding of fused powder. Relevant effects of build platform attributes on the PBF process include: • Preheating of the build platform reduces residual stress [79], improves heat source absorption, material wettability [80] and dimensional accuracy [81]. • Distortion of the build platform can occur in large parts due to residual stress [82].

• The heat source penetration depth into the powder is poorly understood, but depends on source type [78,83], powder material, morphology and packing [84,85]. • Powder size affects the amount of energy required to fully melt a powder particle. For example, for an LMD (Fig. 1) process with fixed process parameters, Ti-6Al-4V particles with 25 μm and 45 μm diameters achieve a maximum temperature of 1350 K and 900 K respectively during the LMD process with fixed parameters [86]. • Thermal cracking occurs more readily in brittle materials such as 316L or tungsten [51]. • Powder bed conductivity is significantly lower than that of solid material and is dependent on the particle morphology and atmosphere surrounding the powder particles [27]. • Unmelted powder can be recycled, with insignificant changes to the chemical composition and morphology [31]. 4. Simulation complexity The complexity of the PBF process makes experimental testing of all independent parameters infeasible [27,87], and necessitates simplification of numerical simulations (Sections 4.1–4.4). Numerical methods are an approximation that is accomplished by discretising fundamental partial differential equations into time and space steps using various methods including: the finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). Numerical analysis methods have the capability to solve non-linear and transient problems. However, as more physical phenomena are included in the fundamental differential equations, computational complexity increases. MAM phenomena including: multi-scale effects, highly dimensional multi-physics, a spatially and temporally transient heat source, and material addition add significant computational cost to numerical methods. 4.1. Multi-scale effects MAM processes utilise a concentrated heat source that creates multiple zones (Fig. 3), including: • The cyclic thermal zone – where temperature varies cyclically in response to the transient heating of the melt pool. This zone is dominated by the transient heat source which causes changes in material states and has inherently non-linear material behaviour.

Fig. 5. Three elements with unique conductivities, k, element size, Δx, node temperature, T, and nodal spacing, δ, at three nodal increments, n and time increment p. ΔE ΔQ ¼ . Δt pþ1 p ΔE ρC p An Δxn T n −T n ¼ Δt Δt ∂T Q ¼ −kA ! !! ∂x pþ1 T pþ1 −T npþ1 T pþ1 −T n−1 − −knþ1=2 Anþ1=2 nþ1 ΔQ ¼ −kn−1=2 An−1=2 n δn−1=2 δnþ1=2

Fig. 6. Geometric data available from slice data, including: overhanging area, DA; supporting area, SA.


231

• The bulk thermal zone – where temperature varies monotonically without material changing state and is dominated by conduction with relatively linear material behaviour.

However, for the part to be simulated using discrete numerical methods the entire body must be meshed, potentially resulting in a large number of discretised elements requiring significant computational resources [19,64]. To reduce computational cost, MAM simulations often dynamically remesh geometry such that small elements exist near the cyclic thermal zone to capture transient simulation details, while large elements exist in bulk geometry zones [64,88]. 4.2. Highly dimensional multi-physics MAM processes are highly dimensional, and subject to non-linear and transient physical phenomena [22]. Due to the associated computational expense, physical phenomena of MAM are routinely simplified [27]. For example: melt pool fluid dynamics [89–91]; atmospheric effects such as convection [92]; deformation during manufacture [93–95]; kinetic motion of the powder [88,96]; discrete powder particles; and, inhomogeneous powder properties [45,97].

Fig. 8. ratio of supported area to cross sectional area.

4.3. Spatially and temporally transient heating 5. Existing simulation strategies As PBF is primarily a thermal process, it is critical that transient heat sources be appropriately applied to achieve meaningful simulation outcomes [87]. However, due to associated complexities, simplification of the heat source is typical. For example, the heat source intensity profile is often approximated as: a Gaussian beam [45,64,88,91,97–103]; a uniform intensity source [90]; a quadrate rule [64,104]; or, by internal heat generation [89]. Further simplification is achieved by heating single or multiple layers concurrently with a temperature or flux field that represents layer scanning [96]. 4.4. Material addition Simulation of AM material addition adds complexity in numerical methods due to the discrete nature of mesh-based solutions [105]. Methods implemented that enable material addition are: element deactivation and reactivation [94,106]; use of varying material properties to emulate material addition [91,94]; creation of multiple simulations that map previous data to a mesh with new material added. Alternatively material addition may be neglected in simulations, for example by single track, or single layer simulation (Section 5.2).

Due to the significant complexity and uncertainty associated with PBF materials, process and machine types, many classes of PBF simulation methods have been developed. In order of increasing geometric resolution and scale, these simulations can be broadly classified as: • Melt pool simulations – investigates the region of the melt pool, which aids understanding of the cyclic thermal zone, but not the bulk thermal zone (Section 5.1). • Single layer simulations – investigates a single layer consisting of multiple tracks, which can be used to investigate the cyclic thermal zone and effects of scan strategy (Section 5.2). • Layer-by-layer simulations – investigates key interactions between layers and provide insight into the thermally cyclical nature of the PBF processes (Section 5.3). • Reduced order simulations – simplify physical phenomena in order to achieve results for entire geometries, focussing on the bulk thermal zone, in a computationally efficient manner (Section 5.4). • Higher order simulations – attempt to match the physical process by robustly, including high order physical phenomena (Section 5.5).

5.1. Melt pool simulations Melt pool simulations typically simulate a single track to investigate the effect of material and heat source parameters on the melt pool, for example [45,87,89,97,107,108]. Melt pool simulations:

Fig. 7. Effect of simulation slice thickness, Lt, on strut continuity for an inclined cylinder of diameter, D. The ratio of overhanging area, DA, to supported area, SA, increases with Lt until geometry becomes discontinuous (right). D A ∝ . n sinðθÞ Lt DAn ∝ . tanðθÞ

D Lt − SA ∝A −DAn ¼ . n n sinðθÞ tanðθÞ D Lt b cosðθÞ

• Alleviate the complexity of multi-scale issues that are present with larger simulations. • Can be used to determine the effect of process parameters on the melt pool, thereby enabling simulation of local effects, including melt pool fluid dynamics. • Facilitate understanding of the manufacture of a single track, as is relevant to DED processes. 5.2. Single layer simulations Single layer simulations are developed to investigate the effects of scan strategy and heat sources parameters on the quality of a single layer [64,87,98,99,101,102,104]. Single layer simulations:

232


Fig. 9. Abstraction of lattice structure connectivity network showing three levels of enlargement.

• Allow greater detail regarding actual build affects than do melt pool simulations; however, a single layer is not representative of bulk PBF geometry. • Typically ignore melt pool fluid dynamics by necessity to achieve the computational efficiency required to simulate an entire layer.

• Have the ability to predict the thermally cyclical nature of PBF processes. • Often ignore or simplify melt pool fluid effects. • Use coarse mesh or dynamic remeshing to accommodate increasing geometry and the multi-scale nature of MAM. • Have the capability to generate data that approximates the PBF process at full scale.

5.3. Layer-by-layer simulations

5.4. Reduced order simulation

Layer-by-layer simulations are developed to investigate the effects of PBF properties on part quality across multiple layers [91,94,99,100]. Layer-by-layer simulations:

Reduced order simulations exclude physical phenomena in order to generate a useful prediction of full complexity geometry, within an acceptable computational cost. Reduced order simulations:

Fig. 10. Common thermal boundary conditions.


• Typically use element sizes larger than the heat source diameter in order to accommodate the increasing magnitude of the geometry [96,109]. • Require experimental validation due to the numerous inherent simplifications. • Enable prediction of final build outcomes such as transient temperature field and consequent residual stress and distortion for full scale geometry. 5.5. Higher order simulation Higher order simulations aim to include all relevant physical phenomena with the aim of robustly representing relevant physical processes, regardless of the computational expense. Higher order simulations: • Of full geometry are not currently feasible even with supercomputing facilities. • Exist for single track scale parts; however these have significant computational cost. For example, Khairallah and Anderson required over 1E5 CPU hours to simulate a very small single melt track (1000 × 300 × 50 μm3) [110,111]. • As MAM process are not fully understood or documented, higherorder simulations cannot currently accommodate the multitude of material phases and physical interactions that exist. • Have the potential to predict effects that are difficulty to determine experimentally, such as the cooling effect of evaporation [110]. 5.6. Identified gaps in simulation capability From the review of existing PBF simulation methods, it is apparent that the majority of existing simulation methods focus on small geometries, and are not useful DFAM tools for commercial MAM design. Simulation methods are required that: • Consider full scale complex geometry as produced by PBF. • Are based on reduced order models that have a lower fidelity but higher computational performance [28], this enables a new category of simulation more suited to DFAM. • Can qualitatively predict large scale effects such as the bulk zone transient temperature field and consequent residual stress and thermal distortion [27,28]. • Take advantage of simplifications that are possible in a reduced order simulation (Section 6). • Facilitate integration with the DFAM process, in particular the early design phases and pre-production validation [28]. 6. Reduced-order PBF simulation DFAM tools are required to increase MAM productivity and minimise experimental iteration; however there are limitations in

233

Table 1 Parameters implemented for the numerical validation. Material properties from [116]. Name

Value

Description

Tbase k Cp ρ trecoat texp v d P An

0 °C 6.7 W/m°C 526.3 J/kg°C 4430 kg/m3 6s (Eq. (4)) 1 m/s 0.05 mm 5W m2

Temperature of the lower node (build platform) Material conductivity Material specific heat capacity Material density Recoating time Exposure time Scan velocity Beam diameter Heat source power Area at a given layer

predicting the transient thermal field for complex AM geometry. To address this identified limitation, a reduced-order computationally efficient PBF focused simulation is developed based on the following procedure: 1. Generate slice data for the desired PBF geometry. 2. From the slice data, calculate the connectivity network (Section 6.3), cross sectional area and heat input parameters (Section 6.2). 3. For each unique polygon area on every layer, define a representative 1D thermal simulation (Section 6.1) based on the current polygons layer-by-layer flow path to the build platform (Section 6.3). 4. Apply each elements unique cross-sectional area and simplified load and boundary conditions (BC) (Section 6.4). 5. Solve each flow path as a 1D thermal simulation. 6. Map the simulation results onto the 3D geometry to present the final result. This reduced-order simulation method achieves a computationally efficient result by: • The reuse of existing slice data to calculate relevant geometric parameters, combined with machine settings to determine heat source (Section 6.2, Fig. 6). • The simplification of 3D structures as a series of representative 1D elements based on the overhanging, DA, and supported areas, SA, for each simulation thickness. • The combination of individual 1D elements to form a network of 1D elements. This representation is highly compatible with the slender geometries of lattice structures and enables a significant increase in computational efficiency (Section 6.3). • Boundary conditions and loads are applied as concentrated points, i.e. the entire cross-sectional area of the layer is heated simultaneously (Section 6.4). • As the simulation is intentionally reduced order, and designed to determine qualitative data of relatively large geometries, phase change and temperature dependant material properties are not implemented. • No conduction into the powder is a conservative assumption that enables the representation of the geometry as a series of 1D thermal flow paths (Section 3.7). • Considering the heat source as a concentrated heat flux, q". • Consideration of the base platform as a constant temperature, Tbase.

6.1. Method derivation

Fig. 11. Analytical and FDM results for semi-infinite plane with constant boundary heat flux [115].

The finite difference method will be used to implicitly derive the thermal diffusion equation in one dimension for the general MAM focused case that allows uneven node spacing and variable conductivity with no internal heat generation. The resulting equations are based on nodal energy balance (Fig. 5, Eq. (6)), where the rate of change of energy within a single element, n (Eq. (7)), is equal to the net heat flow into the element as defined by

234


Fig. 12. Maximum temperature profile with respect to time for constant area validation case.

Fourier's law (Eq. (8)). Fourier's law can then be used to determine the difference between the heat flowing into the element at n − 1/2, and the heat flowing out of the element at n + 1/2 for a given time increment, p (Eq. (9)). The variables affecting heat transfer between adjacent nodes are: conductivity, k, cross-sectional area, A, and distance between nodes, δ. Both k and A are derived using the harmonic mean (Eq. (10)) to ensure more consistent heat flow at element boundaries [112]. Two adjacent element lengths, Δx, are used to calculate δ (Eq. (11)). These three terms are then reduced to a single combined weighting factor, C (Eq. (12)). An1=2 ¼

δn1=2

2 1 1 þ An An1

Δxn þ Δxn1 ¼ 2

kn1=2 ¼

2 1 1 þ kn kn1

C n1=2 ¼

kn1=2 An1=2 δn1=2

pþ1 pþ1 −C n−1=2 T pþ1 ΔQ ¼ C nþ1=2 T pþ1 n −T n−1 nþ1 −T n

ð12Þ

ð13Þ

The rate of change of energy within a single element (Eq. (7)), can then also be simplified using a weighting term, B (Eq. (14)). Eq. (13) and Eq. (15) can then be combined to obtain the discretised heat diffusion equation (Eq. (16)). Δt ρC p An Δxn

ð10Þ

B¼

ð11Þ

ΔE ¼ Δt

T npþ1 −T pn

ð14Þ

B

Fig. 13. Maximum temperature profile with respect to time for increasing area validation case.

ð15Þ


235

Fig. 14. Maximum temperature profile with respect to time for reducing area validation case.

p T pþ1 n −T n B

pþ1 pþ1 ¼ C nþ1=2 T pþ1 −C n−1=2 T pþ1 n −T n−1 nþ1 −T n

ð16Þ

Rearranging the equation in terms of current p, and future p + 1 temperatures (Eq. (17)) enables easy conversion into a matrix form (Eq. (18)). pþ1 p T pþ1 1 þ BC nþ1=2 þ BC n−1=2 þ T pþ1 n−1 −BC n−1=2 þ T n nþ1 −BC nþ1=2 ¼ T n

ð17Þ W ¼ 1 þ BC nþ1=2 þ BC n−1=2 2 3 2 pþ1 3 2 p 3 T n−1 W −BC nþ1=2 0 T n−1 7 p 4 −BC n−1=2 W −BC þ 5 6 4 T npþ1 5 ¼ 4 T n 5 pþ1 0 −BC n−1=2 W T pnþ1 T

ð18Þ

nþ1

6.2. Use of slice data for simulation From the basic slice data (Section 2.2), geometric and connectivity data relevant to the development of the reduced-order simulation can be derived (Fig. 6): • The overhanging area, DAn – area that is absent on the preceding layer, n − 1, i.e. the set complement of successive areas (Eq. (20)). A positive DA implies increasing cross-sectional area.

• Supporting area, SAn – area that overlaps between the current and layer n − 1, i.e. set intersection of current and preceding layer areas (Eq. (21)). • Connectivity network – a list of connections between uniquely defined areas across neighbouring layers (Fig. 9).

N

V ≈ ∑ An L t

ð19Þ

DAn ¼ An −An−1

ð20Þ

SAn ¼ An ∩An−1

ð21Þ

n¼1

Increasing simulation slice thickness, Lt, improves computational performance by reducing the number of simulated elements, but reduces simulation resolution. The proposed method can be applied to geometries of any scale, provided that the value of Lt is not so large that the geometry becomes discontinuous. For example, for a cylinder of diameter, D, inclined to the build platform by inclination angle, θ (Fig. 7, Eqs. (21)–(23)), Eq. (25) must be satisfied for the geometry to be continuous. As θ decreases and the slice thickness increases the ratio of the supported area, SAn, to the applied heat area, An, decreases (Fig. 8). This effect indicates that the outcomes of the proposed method are dependent

Table 2 Inclined cylinder DOE and initial physical conditions. Control factor

Unit

Level

Legend

Inclination angle, θ Strut diameter, D

deg° mm

0–90, every 5° 0.5

x-axis every 5°

2 4 Simulation layer thickness, Lt

mm

Strut length, L/build height, H Recoat time, trecoat Scan speed, v Base temperature, Tbase beam diameter, d

mm s mm/s ° C mm

0.03 0.1 0.5 18 6 1000 0 0.05

236


7. Steps 4–7 are iterated until all nodes associated with the current thermal flow path have been added to the implicit transient thermal simulation. Every thermal flow path that constitutes the 3D geometry is individually simulated and then combined to achieve a final transient thermal temperature profile for 3D geometry (Fig. 9). 6.4. Loads and boundary conditions The developed FDM is a numerical implementation of the thermal diffusion equation which is a partial differential equation stating the rate of heat transfer through a solid (Eq. (26)). 2

Fig. 15. Schematic of DOE variables for inclined cylinder heating (Table 2). Simulation results for constant build height and constant strut length are summarised in Fig. 16.

on local geometry and slice thickness; and are further discussed in Section 7.3.

2

2

∂T ∂ T ∂ T ∂ T þ þ ¼α ∂t ∂x2 ∂y2 ∂z2

! ð26Þ

The representation of loads and boundary conditions (BC) within the developed FDM is limited to the PDE order, allowing temperature and flux BCs to be applied (Fig. 10). 7. Numerical validation

6.3. Reduction to a single dimension A 3D FEM simulation requires a n × n system of equations to be solved using matrix inversion, whereas a 1D simulation requires a tridiagonal matrix of order n to be solved. The computational cost of solving an asymmetric n × n system and a tri-diagonal system of equations has asymptotic upper bound of O(n3) and O(n), respectively [113]. O(n3) methods have cubic running time and are “practical for use only on small problems” [114]. This insight enables significant reduction in computational cost by 1D simulation, and is core to the computational efficiency of the implicit FDM method developed in this research (Section 6.1). To enable simulation of a 3D geometry using 1D simulation tools, the AM slice data (Section 6.2) is leveraged to create a series of 1D thermal flow paths (Fig. 9). Each thermal flow path within the geometry is identified and solved by a layer-by-layer approach: 1. For each thermal flow path the cross-sectional area, An, supported area, SAn, and any existing temperature information is extracted from the 3D slice data (Section 6.2). 2. Each layer is represented by a node with area SAn and initial conditions set as either previously calculated or an ambient temperature. 3. A node is place at the uppermost and lowermost surface to represent the surface heat flux and build platform temperature control respectively. 4. The node representing the upper surface is used to determine exposure time, texp. 5. Heat is then applied as a flux boundary condition (Section 6.4) for texp, to the uppermost node to represent the moving heat source supplying energy to the PBF system. 6. An adiabatic boundary condition is applied to the uppermost node (Section 6.4) for the recoating period, trecoat, to represent the time taken for the build platform to lower and a new layer of metallic powder to be applied (Section 2).

Table 3 Heat source application methods. Scenario

Input power, P

Exposure time, texp

Constant Es Constant Ev

5W Proportional to Lt (Eq. (37))a

Proportional to A (Eq. (4)) Proportional to A (Eq. (4))

a Ev is calculated using Lt =0.1 mm and P=5 W which is then use as constant value for scenario 2.

To ensure that the developed reduced-order simulation's results are physically realistic several validation cases are performed: • Comparison with analytical solution for transient heat transfer in semi-infinite plane (Section 7.1). • Simulation of increasing, decreasing and constant area geometries and comparison with commercial FEM results (Section 7.2). • Simulation of an inclined cylinder with varied geometric and simulation parameters (Section 7.3).

7.1. Semi Infinite plane Due to the complexity of the PBF process, no analytical solutions are available. However, the temperature field of a semi-infinite plane with a constant flux BC can be analytically derived [115]. A plane is considered semi-infinite if Eq. (27) is satisfied, and the error associated with the analytic prediction (Eq. (28)) is at most 1.7% [115]. Fo ¼

αt L2

≤0:19

pffiffiffiffiffiffiffiffiffiffiffi 2q αt=π − x2 x e 4αt − 1− erf pffiffiffiffiffi T ðx; t Þ ¼ T i þ o k 2 αt

ð27Þ ð28Þ

Fig. 11 confirms that the FDM developed in this research correctly simulates a semi-infinite plane with constant boundary heat flux [115]. 7.2. Validation of transient varying simulations Standard analytical solutions do not exist for complex geometries with moving boundary conditions (Section 4). To provide a comparative reference case, results are compared with equivalent data generated by commercial FEM software for transient simulations with parameters matching the intended use of the method in simulating MAM (Table 1). Figs. 12–14 demonstrate that the reduced-order simulation provides equivalent results to commercial FEM code for cases with constant, increasing and decreasing cross-sectional area, with the following observations: • For constant cross-sectional area, maximum temperature increases proportionally to the build height (Fig. 12). • Increasing cross-sectional area results in significantly faster temperature increase due to increasing texp and smaller conduction paths relative to the current layer (Fig. 13).


237

Fig. 16. Peak cylinder temperature versus inclination angle for various geometry (Table 2) and, simulation conditions (Table 3). (1,5) peak temperatures occur for peak D. (2) peak temperatures are inversely proportional to Lt, for constant Es, due to increase in the total layers. (3) peak temperatures decrease due to a reduction in the total number of layers and discontinuous geometry. (4) peak temperatures proportional to Lt, for constant Ev, Due to increasing power input.

Fig. 17. Peak temperature intensity for BCC (left) and FBCZ (middle) with 1 mm strut diameter and 10 mm unit cell, and FBCZ with a 2 mm strut diameter and a 15 mm unit cell (right). Simulation parameters in Table 1.

238


Fig. 18. MAM specimens matching the analysed geometry (Fig. 17). Specimens manufactured using SLM with Al10Si6Mg with parameters defined in [117]. Dashed box indicates region of enlaged views (Fig. 19).

• Decreasing cross-sectional area results in a lower maximum temperature as layers are sequentially added to the simulation (Fig. 14); this is expected as smaller cross-sectional areas will have a reduced texp and a larger conduction path through preceding layers to facilitate heat removal.

Etotal ¼ 4N

Es vdA

2

3

πvd

¼ 4N

Es A2

ð34Þ

2

πd

For a constant strut length (L) 7.3. Inclined cylinder with varying geometry and BCs The reduced-order simulation method has been applied to obtain the peak temperature for a full factorial design of experiments (DOE) (Table 2) of an inclined cylinder (Fig. 15). Three unique layer-by-layer simulation scenarios are compared, with different methodologies in heat source application (Table 3), but a constant recoat time, trecoat. For all scenarios, the heat flux, q" is a function of heat source power, P, and beam diameter, d (Eq. (29)). Constant strut length and constant build height simulations have the number of layers as defined by Eq. (30) & Eq. (31) respectively. For all scenarios the cross-sectional area, An, is constant for every layer, n. Due to the consistent cross–section, A, the total energy can also be derived (Eq. (32)). P 4P q} ¼ 2 ¼ 2 πd π 2d

ð29Þ

Etotal ¼ 4

Etotal ¼ 4

ð30Þ

Scenario 2, constant Ev: To maintain constant volumetric energy density, Ev (Eq. (2)), heat source power, P, is varied proportional to layer thickness, Lt (Eq. (37)). Unlike the constant Es scenario Etotal (Eq. (38)), is independent of Lt (Eq. (39), Eq. (40)). P ¼ Ev Lt vd; Ev ¼ const

ð31Þ An PA2 ¼ 4N ¼ N 2 An 3 vd πd πvd 4P

ð32Þ

Scenario 1, constant Es: To maintain a constant surface energy density, Es (Eq. (1)), the heat source power, P, is varied (Eq. (33)). The total energy, Etotal (Eq. (34)), that enters the part becomes dependent on A which is summed over the number of layers for a constant strut length (Eq. (35)) or a constant build height (Eq. (36)). P ¼ Es vd; Es ¼ const

Ev Lt vdA 3

πvd

2

¼ 4N

ð37Þ Ev Lt A2 2

πd

L sinðθÞ Ev Lt A2 4L sinðθÞEv A2 ¼ 2 2 Lt πd πd For a constant strut height (H)

Etotal ¼ 4

H N ¼ floor Lt

n¼1

ð36Þ

ð38Þ

For a constant strut length (L)

For a constant strut height (H)

Etotal ¼ ∑ q An t exp

H Es A2 4HEs A2 ¼ 2 Lt πd2 Lt πd

Etotal ¼ 4N

L sinðθÞ N ¼ floor Lt

}

ð35Þ

For a constant strut height (H)

For a constant strut length (L)

N

L sinðθÞ Es A2 4L sinðθÞEs A2 ¼ 2 2 Lt πd Lt πd

ð33Þ

Etotal ¼ 4

H 4Ev Lt A2 4HEv A2 ¼ 2 2 Lt πd πd

ð39Þ

ð40Þ

For all the performed simulations (Fig. 16), temperatures rise for the entire exposure time, texp. Consequently, geometry with a larger crosssectional area will reach a higher peak temperature (Fig. 16, label 1 and label 5). For a constant strut length scenario (Fig. 16 Left), N decreases with decreasing θ and increasing Lt (Eq. (30)). This reduces the Etotal that the part receives which results in decreased peak temperatures compared to constant build height simulations (Fig. 16 Right). The reducing number causes a limitation that is apparent for the Lt = 0.5 mm simulations, where only 6 layers are present at 10° and discontinuous geometry can occur (Fig. 16, label 3) (Section 6.2). However, in practice such trivial small parts do not occur.

M. McMillan et al. / Materials and Design 132 (2017) 226–243 Table 4 Case study lattice structures.

239

texp (Eq. (4)) which coupled with the largest Lt caused the most significant peak temperatures (Fig. 16, label 5). Computational performance is improved by increasing Lt. However, regardless of the simulation scenario employed (Fig. 16) different Lt results in a variation in the peak temperature profiles. Therefore, despite the associated performance increases offered by using a larger Lt experimental calibration is required.

8. Experimental validation Due to the simplifications applied to the reduced-order simulation; case-by-case experimental validation is required to calibrate the quantitative results for specific: processes, materials and geometries. For large scale lattice structures [117] the size of the heat source relative to geometry is very small resulting in strong multi-scale effects (Section 4.1) For smaller scale geometries such as micro-lattice structures [8,118–120], the heat source can be a relatively large compared to the geometry. This significant difference in scale that is possible when using the MAM process can result in very large variations in build conditions, which necessitates experimental validation for all classes of geometry, materials and process. Within this research two sets of experimental geometries are qualitatively aligned with the reduced –order simulation results: • Large scale aluminium lattice structures (Section 8.1). • Small scale Titanium micro lattice struts (Section 8.2).

8.1. Analysis of geometry and diameter on MAM lattice manufacturability To demonstrate the effectiveness of the simulation method as a DFAM tool, MAM lattice structures were analysed (Fig. 17) and compared to SLM manufactured parts [117] (Fig. 18). The selected lattice structures represent complex MAM structures that are commercially relevant, and include an upper-cap structure to allow compressive testing [117] (Table 4). Comparison of the reduced-order simulation results (Fig. 17) and SLM manufactured parts (Fig. 18) indicates that:

Scenario 1, constant Es: Peak temperatures increase with increasing diameter, D, and decreasing layer thickness, Lt (Fig. 16, label 1). As the inclination angle, θ, decreases the cross sectional area, A, increases (Section 6.2) which increases the total energy, Etotal (Eq. (35)), that the geometry receives; coupled with an increased resistance to heat transfer (Section 6.2) significant increases in peak temperatures occur. For the constant Es scenario Etotal is dependent on the total number of layers, N (Eq. (35)), therefore decreasing Lt also increases the peak temperatures (Fig. 16, label 2). Scenario 2, Constant Ev: For a constant volumetric energy density, Ev, peak temperatures increase with increasing D, and Lt (Fig. 16, label 5). To maintain a constant Ev heat source power, P, is set proportional to Lt (Eq. (37)) resulting in a constant Etotal (Eq. (39)). The use of a variable P results in variation of q" which splits peak temperatures proportional to Lt (Fig. 16, label 4). Larger diameter geometries have an increased

• The addition of face and z-struts in the FBCZ unit cell (Fig. 17, middle) reduces predicted peak temperature during manufacture in comparison to the BCC unit cell (Fig. 17, left). This outcome is significant as it indicates that the addition of z-struts can act as thermal conduction paths that assist heat flow, enabling reduced maximum temperatures during manufacture. This predicted result is compatible with the observed, manufactured component, as the FBCZ surface finish is superior to that of the BCC unit cell (Fig. 18, left, middle). • The FBCZ unit cell with small diameter struts (Fig. 17, middle) has significantly lower predicted peak temperature than the FBCZ unit cell with larger struts and cell size (Fig. 17, right). This prediction suggests that the larger geometry of the upper-cap introduces additional heat source in comparison to the increase in thermal conduction paths, and is compatible with the observed, manufactured component, as a catastrophic failure due to local overheating is observed at the predicted location of peak temperature (Fig. 18, right).

These theoretical and experimental outcomes confirm that the proposed method provides practically useful insight into the temperature field of MAM lattice structures, and can provide a qualitative DFAM tool that enables design insight early in the design phase and preproduction validation.

240


Fig. 19. Enlarged views of the lattice where failure occurred in the largest unit cell (Fig. 18). Left – BCC, Middle – FBCZ with a 1 mm strut diameter and 10 mm unit cell, Right – FBCZ with a 2 mm strut diameter and a 15 mm unit cell.

8.2. Analysis of incline and diameter on MAM micro strut manufacturability To demonstrate the effectiveness of the simulation method on micro-scale structures, a series of inclined rods analysed and compared to SLM manufactured parts (Fig. 20). The selected validation geometry (Table 5) shows the initiation of failure in single strut geometries with respect to variation of diameters and inclines (Fig. 20). Comparison of the reduced-order simulation method and SLM manufactured parts indicates that physical observations align with the simulated temperature profiles (Fig. 20): • The 60° incline is robustly manufacturable for all tested diameters (Fig. 20a).

• At a 20° incline the 0.9 mm and 1.0 mm diameter rods have failed with the 0.8 mm rod compromised (Fig. 20b). • At a 10° incline 0.7–1.0 mm diameter rods have failed and the 0.6 mm rod is compromised (Fig. 20c). • Simulation results have a small cool region at the tip of a strut (Fig. 20d). This phenomenon is caused by a decreasing cross-sectional area in the final slice layers of the structure. 9. Conclusions Metal Additive Manufacturing (MAM) offers unique advantages particularly in its ability to create tailored topology such as lattice structures with minimal material waste. Despite these commercial

Fig. 20. Comparison of experimental and simulation results for micro struts at three different inclinations (images are of the downward facing surfaces). Top, Ti6Al4V inclined rod specimens manufactured on a SLM solutions 250 machine. Bottom, Associated simulation results with matching colour contours. Legend: (a) Robust manufacture for θ = 60°. (b) Failure for larger diameter rods for θ = 20°. (c) Multiple failures for θ = 10°. (d) Low temperature region due to reducing cross sectional area. (e) Simulation results showing excessive heat for the multiple failures observed at θ = 10°.

M. McMillan et al. / Materials and Design 132 (2017) 226–243 Table 5 Parameters implemented in experimental geometry and simulation validation tests. Control factor

Unit

Level

Inclination angle, θ Strut diameter, D Strut length Simulation layer thickness, Lt P Recoat time, trecoat Scan speed, v Beam diameter, d k C ρ

deg° mm mm mm W s mm/s mm W/m°C J/kg°C kg/m3

10,20,60 0.3,0.4,0.5,0.6,0.7,0.8,0.9,1 10 0.1 5 6 1000 0.05 6.7 526.3 4430

opportunities there is a lack of design for additive manufacture (DFAM) tools for the powder bed fusion (PBF) process (Section 3). This is a function of both process uncertainty and computational complexity (Section 4). An array of simulation methods exist to promote understanding of the PBF process (Section 5). A simulation that represents all the relevant physical phenomena within the PBF process and is applied to the entire geometry would be ideal, but is computationally infeasible (Section 5.5). Due to the complexity of the PBF process, the majority of existing simulations focus of only specific element of the process such as melt pool, single tracks or single layer scan strategy (Section 5). Consequently, there exists a significant lack of simulation methods that can be applied to production size geometry, and even fewer with sufficient efficiency to enable simulation during the design phase. A reduced-order simulation was developed to addresses the identified limitations of existing DFAM tools. The developed simulation contains the dominant physical processes including (Section 6): conduction, constant flux heat source and material addition, while providing the computationally efficiency required of a DFAM tool for qualitatively predicting the AM temperature field. The developed reducedorder simulation was then validated with reference to relevant numerical (Section 7) and experimental scenarios (Section 10). The reduced-order simulation method developed in this research is applied to the powder bed fusion process but could also be extended to directed energy deposition (Section 3). The simulation of the PBF process is computationally complex. In order to provide a usable DFAM tool, the simulation method maximises efficiency by implementing: • The reuse of slice data to generate a simulation, this enables the reduction of processing time as slice data generation is already required for the PBF process. • Simplification of the PBF process as a 1D transient problem is a suitable simplification for AM lattice structures since lateral thermal diffusion into the powder is insignificant, resulting in a tri-diagonal system of equations that is significantly more computationally efficient, O(n), then equivalent three dimensional simulation methods, O(n3) (Section 6.3). • The derivation of geometric parameters from existing slice data as simulation input is a novel contribution that enables efficient generation a 1D simulation. • Simplified boundary conditions, which align with the 1D finite difference method implemented (Section 6.4). • By assuming the powder bed thermal diffusivity to be negligibly small in comparison to the solidified powder is compatible with experimental observation and existing simulations, while providing a conservative assumption that is computationally efficient (Section 3.7). • System thermodynamics assumed to be dominated by conduction not radiation or convection, thereby representing the fundamental thermodynamic process with simplified fundamental diffusion equation (Section 6.1).

241

Due to the numerous simplifications present the simulation method must be calibrated with experimentation for differing classes of geometry. At present the simulation method provides meaningful qualitative DFAM results in a computationally efficient manner, as shown for lattice unit cells and micro-scale struts manufactured with commercial SLM processes (Section 8.1).

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