A novel smoothed particle hydrodynamics formulation for thermo-capillary phase change problems with focus on metal additive manufacturing melt pool modeling

Highlights

• Smoothed particle hydrodynamics scheme for thermo-capillary phase change problems.

• Novel interface stabilization scheme reducing spurious interface flows.

• Focus on melt pool modeling in metal powder bed fusion additive manufacturing.

• Full range of relevant mechanical and thermal interface fluxes.

• Modeling of gas phase for consistent prediction of pore and spatter generation.

Abstract

Laser-based metal processing including welding and three dimensional printing, involves localized melting of solid or granular raw material, surface tension-driven melt flow and significant evaporation of melt due to the applied very high energy densities. The present work proposes a weakly compressible smoothed particle hydrodynamics formulation for thermo-capillary phase change problems involving solid, liquid and gaseous phases with special focus on selective laser melting, an emerging metal additive manufacturing technique. Evaporation-induced recoil pressure, temperature-dependent surface tension and wetting forces are considered as mechanical interface fluxes, while a Gaussian laser beam heat source and evaporation-induced heat losses are considered as thermal interface fluxes. A novel interface stabilization scheme is proposed, which is shown to allow for a stable and smooth liquid–gas interface by effectively damping spurious interface flows as typically occurring in continuum surface force approaches. Moreover, discretization strategies for the tangential projection of the temperature gradient, as required for the discrete Marangoni forces, are critically reviewed. The proposed formulation is deemed especially suitable for modeling of the melt pool dynamics in metal additive manufacturing because the full range of relevant interface forces is considered and the explicit resolution of the atmospheric gas phase enables a consistent description of pore formation by gas inclusion. The accuracy and robustness of the individual model and method building blocks is verified by means of several selected examples in the context of the selective laser melting process.

Introduction

The present work proposes a novel smoothed particle hydrodynamics (SPH) formulation for general thermo-capillary phase change problems involving solid, liquid and gaseous phases. A special focus lies on the mesoscale melt pool modeling in metal powder bed fusion additive manufacturing (PBFAM) processes, e.g. selective laser melting (SLM) or electron beam melting (EBM), requiring some additional model constituents that are specific for this application. Since the governing physics are similar, also the melt pool dynamics in laser beam welding (LBW) or electron beam welding (EBW) processes [1], [2], [3], [4], [5], [6] lie in the scope of application of the proposed model.

Basically, two main modeling approaches for surface tension effects can be distinguished in the context of SPH: formulations considering the microscale origin of surface tension in form of discrete, phase-dependent inter-particle potentials [7], [8], [9] as well as macroscale surface tension models relying on the continuum surface force (CSF) method proposed by Brackbill and Kothe [10] and widely used also in combination with other spatial discretization schemes such as finite differences, finite volumes or finite elements. The CSF approaches can be further subdivided into formulations that directly discretize the surface tension stress tensor and subsequently determine its divergence as contribution to the discrete momentum equation [11], [12] and formulations that rely on the divergence of the continuous surface tension stress tensor resulting in the well-known curvature-proportional surface tension forces in interface normal direction and tangential interface forces proportional to surface tension gradients. The present work will focus on the second category for which the first SPH discretization has been proposed by Morris [13]. Subsequently, this formulation has been extended by density-weighted color field gradients [14] as well as different interface reconstruction and smoothing techniques [15], [16], [17], [18]. There are only very few approaches to incorporate wetting effects into this type of SPH formulation as e.g. proposed by Breinlinger et al. [19] or by Das and Das [20]. One of the first SPH formulations for thermo-capillary flow, i.e. surface tension effects coupled with a thermal field, has been proposed by Tong and Browne [21] and extended by Hopp-Hirschler et al. [22]. Recently, also several SPH formulations for thermo-capillary phase change problems in the context of PBFAM melt pool modeling have been proposed [23], [24], [25], [26], [27], [28], [29]. To the best of the authors’ knowledge none of the aforementioned thermo-capillary SPH formulations have incorporated wetting effects so far, which are expected, however, to play an important role on the length scales relevant for metal PBFAM.

In metal PBFAM, a focused laser beam, typically within an inert gas atmosphere, melts pre-defined contours into thin layers of pre-applied metal powder to create the cross-section of a final solid part in a repeated layer-wise buildup procedure. Under typical processing conditions the peak temperatures on the melt pool surface exceed the boiling temperature of the liquid metal. The density jump and accompanied recoil pressure in the phase transition from liquid metal to metal vapor results in a considerable distortion and highly dynamic topology changes of the liquid–gas interface at the melt pool surface giving rise to defects such as spatter, i.e. ejection of melt drops, or pores, i.e. gas bubble inclusions [30]. Pioneering modeling approaches in this field are e.g. given by the thermo-hydrodynamics finite element model proposed by Khairallah et al. [31], [32], [33], who considered temperature-dependent surface tension and evaporation-induced recoil pressure forces, based on a phenomenological recoil pressure model [34], as primary driving forces of the process. Comparable models based on finite difference, finite volume, finite element, Lattice Boltzmann or meshfree discretizations are e.g. given by [2], [23], [24], [26], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44]. A more refined model has been proposed by [45], [46], [47], where the gas/vapor phase is explicitly resolved. Typically, the aforementioned models do not account for wetting effects at the triple line solid–liquid–gas. On the contrary, the works [48], [49], [50] specifically focus on the interplay between wetting effects and different power particle configurations, without considering however evaporation-induced recoil pressure.

The present work proposes a weakly compressible SPH formulation for thermo-capillary phase change problems involving solid, liquid and gaseous phases. Specifically, evaporation-induced recoil pressure, temperature-dependent surface tension and wetting forces are considered as liquid–gas interface fluxes in the Navier–Stokes equation. In the thermal problem, a Gaussian laser beam heat source as well as evaporation-induced heat losses are considered as liquid–gas interface fluxes, while convection boundary conditions are obsolete due to the explicit modeling of the atmospheric gas phase. All mechanical and thermal interface fluxes are modeled in a diffuse sense in analogy to the CSF approach. The following original contributions of the present work can be identified: The first SPH formulation for thermo-capillary problems is proposed that also considers wetting effects. A novel interface stabilization scheme based on viscous interface forces is proposed, which is shown to allow for a stable and smooth liquid–gas interface by effectively damping spurious interface flows well-known for the CSF approach. Moreover, different SPH discretizations for the tangential projection of the temperature gradient, as required for the discrete Marangoni forces, are reviewed. Based on a thorough analysis it is shown that standard two-sided gradient approximations are sufficient for this purpose as long as zero-order consistency is satisfied, e.g. by anti-symmetric gradient construction. In the context of metal AM melt pool modeling, the present approach is – to the best of the authors’ knowledge – the first model that (i) considers the full range of relevant interface forces consisting of evaporation-induced recoil pressure, temperature-dependent surface tension and wetting forces, and (ii) resolves the atmospheric gas phase and, thus, can consistently account for defects such as gas inclusions.

The remainder of this work is organized as follows: Section 2 presents the governing equations, i.e. continuity equation, momentum equation, energy equation and equation of state, in space-continuous form. Discretization in space, based on SPH, and in time, based on an explicit velocity-Verlet scheme, is presented in Sections 3 Spatial discretization via smoothed particle hydrodynamics, 4 Time integration scheme. In Section 5 different SPH approximations for the tangential temperature gradient are thoroughly analyzed and compared. Finally, in Section 6, the accuracy of the individual model and method components is verified by means of selected benchmark examples with analytical/numerical reference solutions. Eventually, the suitability of the proposed melt pool model for typical metal AM application scenarios is verified by means of point and line melting examples with and without resolved powder particles. Here, a special focus lies on the robustness of the computational model, i.e. the ability to represent challenging and practically relevant scenarios of dynamically changing interface topologies (e.g. generation of melt spatter or gas inclusions) without inducing spurious interface flows or instabilities of the discretization scheme.