Numerical Simulation of Casting Filling Process Based on SPH-FEM Coupling Method

Abstract

The coordinated optimization of free-surface dynamics tracking and solid deformation computation remains a persistent challenge in casting filling simulations. While the traditional smoothed particle hydrodynamics (SPH) method suffers from prohibitive computational costs limiting practical applications, the delayed interface updates of the finite element method (FEM) compromise simulation fidelity. This study proposes a symmetric SPH-FEM coupling algorithm that integrates real-time particle-grid data exchange, and validation through ring filling simulations demonstrated close agreement with Schmid’s benchmark experiments, confirming flow field reconstruction reliability. Furthermore, bottom-injection plate experiments verified the method’s thermal modeling stability, achieving fully coupled flow–thermal–stress simulations with enhanced computational efficiency. The proposed symmetric coupling framework achieves engineering-ready simulation speeds without compromising accuracy, and this advancement establishes a novel computational tool for predicting casting defects including porosity and hot tears, significantly advancing the implementation of high-fidelity numerical simulation in foundry engineering applications.

1. Introduction

As a historically significant metal forming technique, casting remains crucial in modern manufacturing industries [1]. Casting components find extensive applications in strategic sectors, including aerospace, automotive, and machinery manufacturing. However, persistent challenges in conventional casting processes—including melt flow regulation, defect formation prediction, and process parameter optimization—have emerged as critical constraints impeding technological advancement. To address these limitations, numerical simulation methodologies have been progressively implemented in casting research, establishing themselves as indispensable analytical tools [2,3].

Recent advances in casting simulation reveal complementary advantages between the SPH method and the FEM. The SPH approach inherently circumvents mesh distortion via particle-based discretization, demonstrating superior capabilities in free-surface flow tracking [4] and fluid–structure interaction modeling [5]. This methodology proves particularly effective for liquid metal filling processes involving abrupt geometric transitions, such as cavity expansions from narrow runners. While SPH’s meshless nature enables precise interface evolution tracking, persistent limitations include constitutive model inaccuracies in solid mechanics and computational inefficiency, coupled with underdeveloped thermodynamic analysis frameworks. Conversely, FEM has established a robust theoretical foundation for solid mechanics through its structured mesh framework. FEM’s topology optimization enables precise simulation of casting temperature fields during mold filling while maintaining efficient memory management for large-scale simulations [6,7,8]. However, its grid dependency induces numerical diffusion artifacts during free-surface flow simulations. The SPH-FEM coupling algorithm synergistically integrates FEM’s strengths in complex boundary handling and stress analysis with SPH’s meshless dynamic interface tracking, enabling cross-scale modeling through particle–mesh hybridization; this hybrid strategy offers comprehensive solutions for multi-physics simulations in casting processes.

Xiaofeng N et al. [9] investigated a numerical simulation approach that couples smooth particle hydrodynamics (SPH) with the finite element method (FEM). This method was employed to simulate the variations in flow, temperature, and stress fields during the squeeze casting process.

HJ Park et al. [10] introduced an innovative SPH-FEM coupling method built on the Lagrangian framework. In their approach, the interface between the fluid and the deformable structure is represented as a line segment, allowing for an efficient analysis of fluid–structure interactions and simplifying the treatment of the deformable interface.

Klippel H et al. [11] proposed a hybrid method combining SPH and FEM, which is applied to the simulation of fluid flow and heat transfer in the casting process and can simulate the complex casting environment more accurately.

Fourey et al. [12] proposed an efficient SPH-FEM coupling strategy for fluid–structure interaction (FSI) problems. By verifying several benchmark cases, the advantages of this method in dealing with free-surface flow and complex structural deformation were proven.

Groenenboom et al. [13] studied the application of the SPH-FEM coupling method in fluid dynamics and fluid–structure interaction, especially in shock and explosion simulation, showing its potential in high dynamic problems.

Fourey et al. [14] proposed an improved SPH-FEM coupling method to deal with large deformation problems in material processing, showing its high efficiency and accuracy in simulating complex material behavior.

The present investigation developed a coupled SPH-FEM framework for simulating flow field evolution and thermal gradients during the casting filling processes, as conventional single-method approaches for flow–solidification coupling exhibit inherent constraints, including discontinuous interface discretization and suboptimal computational resource utilization. The novel physical field-adaptive coupling framework overcomes the rigidity of static partition models prevalent in current research, while experimental validation confirmed its enhanced accuracy in filling process simulation and thermal field prediction reliability. This validated framework establishes theoretical foundations for multiphase casting simulation while enabling a paradigm shift from single-physics modeling to intelligent multi-physics coupling systems.

2. Method

SPH method is a particle-based meshless numerical method proposed by Lucy and Gingold and Monaghan in 1977 [15,16]. Its basic idea is to discretize a continuous fluid or solid medium into a series of particles with physical properties such as mass and velocity. The FEM method is based on the structured mesh to discretize the continuous solution domain into a collection of finite elements, and the algebraic equations are established by interpolation approximation and the variational principle for each element.The process of SPH-FEM coupling algorithm is shown in Figure 1.

2.1. SPH Method

During casting filling processes, the liquid metal flow within mold cavities exhibits complex three-dimensional patterns accompanied by significant morphological variations. Free surface-involved computational fluid dynamics (CFD) constitutes the fundamental framework for filling process simulations. SPH discretizes physical field information through Lagrangian particles, eliminating grid-based connectivity requirements and inherently circumventing numerical inaccuracies caused by mesh distortion during large deformation events. In casting simulations, SPH’s particle-based formulation precisely resolves coupled field interactions encompassing fluid flow, thermal transport, and stress development [17].

In the SPH method, the particle approximation method is used to interpolate the field function and its spatial derivative. The approximate equation is as follows:

$$〈f(x_i)〉={\displaystyle \sum _j\frac{m_j}{{\rho }_j}}f(x_j)W_{ij}$$

(1)

$$〈\nabla ·f(x_i)〉=-{\displaystyle \sum _j\frac{m_j}{{\rho }_j}}f(x_j)\nabla W_{ij}$$

(2)

In the formula, $i$ is the central particle; $j$ is the particle in the support domain; $m_j$ is the mass of the $j$ th particle in the support domain; $\rho$ is density; N is the total number of particles in the support domain; $W_{ij}$ denotes the smooth kernel function of the influence of particles on particles; $W_{ij}=W\left(x_i-x_j,h\right)$.

The kernel function is used to describe the weight distribution of the interaction between particles, and the discrete particle attributes are interpolated into a continuous field. In this paper, the cubic spline kernel function is selected, and the expression is as follows:

$$W\left(r,h\right)=\alpha d\left\{\begin{array}{l}1-{\displaystyle \frac{3}{2}}{q}^2+q^3 \hspace{1em}\hspace{1em} 0\le \mathrm{q}\le 1 \\ {\displaystyle \frac{1}{4}}{\left(2-q\right)}^3\hspace{1em}\hspace{1em}\hspace{1em} 1\le \mathrm{q}\le 2\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{q}\ge 2\end{array}\right.$$

(3)

In the formula, $\alpha d$ is the normalized coefficient; $\alpha d$ takes ${\displaystyle \frac{10}{7\pi h^2}}$ when applied to two-dimensional problems and ${\displaystyle \frac{1}{\pi h^3}}$ when applied to three-dimensional problems; $q$ is the relative length, $q=r/h$.

To mitigate non-physical particle penetration during close-range interactions, the adopted SPH formulation incorporates the well-established Monaghan artificial viscosity model [18] and applies repulsive force between particles with too-small particle spacing to avoid excessive aggregation of particles. The specific expression is as follows:

$${\Pi }_{ij}=\left\{\begin{array}{l}{\displaystyle \frac{-a\Pi c_{ij}{\varphi }_{ij}+\beta \Pi {\varphi }_{ij}^2}{{\overline{\rho }}_{ij}}}\hspace{1em}\hspace{1em}{v}_{ij}\cdot x_{ij}<0 \\ \hspace{1em}\hspace{1em}\hspace{1em}0 \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{v}_{ij}\cdot x_{ij}\ge 0\end{array}\right.$$

(4)

In Equation (4), $a_{\prod }$ and ${\beta }_{\prod }$ are standard constants, generally of the value 0.1, but according to the actual situation, the value of the size will fluctuate; C represents the speed of sound. The main function of $\varphi$ is to not produce numerical divergence when the particle spacing is small. $\nu$ is the particle velocity, $x_{ij}=x_i-x_j$; ${\nu }_{ij}=v_i-v_j$.

2.2. SPH-FEM Coupling Algorithm

The SPH-FEM coupling algorithm provides a Lagrangian framework for resolving fluid–structure interaction (FSI) problems. This hybrid approach assigns SPH to fluid domain modeling and FEM to solid domain simulation [19,20]. Pioneered by Attaway [21], Johnson [22], and Beissel et al. [23], the algorithm was initially implemented for impact dynamics simulations. The underlying principle strategically deploys SPH in large deformation zones and FEM in regions with limited strain gradients. An adaptive conversion protocol automatically transforms distorted elements into particles when predefined distortion thresholds are exceeded. The original implementation employed a master–slave contact formulation to manage element–particle interactions. Subsequent algorithmic refinements have expanded SPH-FEM applications to encompass machining simulations and multi-physics coupling scenarios [24,25].

The SPH-FEM coupling framework synergistically integrates SPH’s large deformation capabilities in fluid domains with FEM’s precision in solid mechanics, establishing a novel computational framework for wall penetration analysis [26]. Conventional SPH implementations require element removal to maintain simulation continuity during extreme deformation events, inherently compromising mass conservation. The coupled framework preserves full element integrity, ensures strict mass conservation, resolves complex penetration physics, and enhances computational efficiency compared to standalone SPH [27].

Moreover, pure SPH formulations exhibit boundary penetration artifacts during solid contact scenarios, particularly near geometric discontinuities or high-pressure zones, due to incomplete interaction models. FEM’s structured mesh topology provides stabilized contact constraints that effectively mitigate SPH particle penetration during interfacial interactions. The hybrid algorithm demonstrates enhanced accuracy in predicting stress/strain distributions and fluid–structure interactions during penetration events, with particular advantages in multi-physics coupling scenarios [28].

The proposed coupling mechanism operates exclusively at fluid–solid interfaces and is governed by stress equilibrium and kinematic compatibility conditions. At the SPH particles far away from the coupling interface, the following is considered:

$$〈f\left(x_i\right)〉={\displaystyle \sum _{j=1}^N\frac{m_j}{{\rho }_j}}f\left(x_j\right)W_{ij}$$

(5)

The interfacial dynamics of SPH particles are governed by a coupled system comprising continuity, momentum, energy conservation, and Newtonian motion equations. This formulation enables rigorous analysis of energy transfer mechanisms driven by thermal gradients and particle kinematics during mold filling and resolves particle trajectories and phase transitions near the interface while systematically monitoring their dynamic responses under multi-physics conditions. This mathematical foundation facilitates precise modeling of interfacial particle behavior and enables in-depth investigation of coupled thermo-mechanical phenomena during casting solidification [29]. Equations (6)–(9) are the continuity equation, momentum equation, energy equation, and motion equation of SPH particles near the coupling interface, respectively:

$${\displaystyle \frac{d{\rho }_i}{dt}}={\displaystyle \sum _{j=1}^N{m}_j}\left({\vartheta }_i^{\beta }-{\vartheta }_j^{\beta }\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}+{\displaystyle \sum _{j=1}^{N_b}{m}_{bj}}\left({\vartheta }_i^{\beta }-{\vartheta }_{bj}^{\beta }\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}$$

(6)

$${\displaystyle \frac{d{\vartheta }_i^{\alpha }}{dt}}={\displaystyle \sum _{j=1}^N{m}_j}\left({\displaystyle \frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\sigma }_i^{\alpha \beta }}{{\rho }_j^2}}\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}+{\displaystyle \sum _{j=1}^{N_b}{m}_{bj}}\left({\displaystyle \frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\sigma }_{bi}^{\alpha \beta }}{{\rho }_{bj}^2}}\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}$$

(7)

$$\begin{array}{l}{\displaystyle \frac{d{e}_i}{dt}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^N{m}_j}\left({\vartheta }_j^{\beta }-{\vartheta }_i^{\beta }\right)\left({\displaystyle \frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\sigma }_i^{\alpha \beta }}{{\rho }_j^2}}\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}\\ +{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^{N_b}{m}_{bj}}\left({\vartheta }_{bj}^{\beta }-{\vartheta }_i^{\beta }\right)\left({\displaystyle \frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\sigma }_{bi}^{\alpha \beta }}{{\rho }_{bj}^2}}\right){\displaystyle \frac{\partial W_{ij}}{\partial x_i^{\beta }}}\end{array}$$

(8)

$${\displaystyle \frac{d{x}^{\alpha }}{dt}}={\vartheta }^{\alpha }$$

(9)

At the finite element, we consider the following:

$$〈f\left(x\right)〉={\displaystyle \sum _i{N}_i}\left(x\right)f\left(x_i\right)$$

(10)

In the formula, ${\rho }_i,{\vartheta }_i,e_i,m_i,{\sigma }_i$ represent the density, velocity, energy, mass, and stress tensor of particle $i$. $m_{bj},{\rho }_{bj},{\vartheta }_{bj},{\sigma }_{bj}$ represent the mass, density, velocity, and stress tensor of the background particle $j$. $N_i\left(x\right)$ denotes the finite element shape function.

3. Results and Discussion

The optimization of interdependent process parameters—including injection velocity, fluidity characteristics, and mold geometry—has emerged as a critical challenge in achieving defect-free castings through controlled mold filling. While empirical approaches face inherent limitations in temporal and financial efficiency, computational modeling techniques offer transformative potential for process optimization.

To validate the 3D casting filling model, an axisymmetric annular component was strategically selected for numerical verification. Leveraging geometric symmetry, the computational model was developed to analyze melt flow characteristics and free-surface dynamics. The coupled SPH-FEM framework was validated against experimental observations reported by Schmid et al. and bottom-filling simulations of symmetric plates enabled thermal field validation through experimental comparisons, confirming the coupling method’s accuracy in thermal modeling. This symmetry-based validation strategy enhances computational efficiency through dimensional reduction while demonstrating the method’s engineering viability for complex casting systems exhibiting axisymmetric flow behavior.

3.1. Numerical Parameter Selection of Casting Filling Process

Parametric optimization in casting filling simulations critically governs computational stability and solution accuracy. The inter-facial dynamics between molten metal and mold walls, coupled with bulk flow characteristics, directly determine flow behavior fidelity. Therefore, model selection must align with material constitutive properties and flow regimes to prevent numerical instability and penetration artifacts [30].

3.1.1. Contact Algorithm

The adopted coupling framework implements Johnson’s master–slave contact formulation [22], which enforces inter-facial velocity continuity and momentum conservation to iteratively resolve particle–element penetration, ensuring kinematic consistency and momentum conservation. The framework defines master surface segments along 2D continuum element boundaries, with segment endpoints designated as parent nodes. Fluid particles possess characteristic radii equivalent to half the initial inter particle spacing, maintained as constant throughout incompressible flow simulations per the initial configuration. Its algorithm consists of three parts, namely contact search, contact judgment, and contact processing.

The contact detection phase primarily detects potential contact pairs between slave particles and master surface facets within the computational domain. By traversing the nodes from the surface, the nodes that are close to the main surface and may have contact are found, mainly through the box test and penetration test.

The basic principle of box testing is to divide the entire model space into a series of regular three-dimensional boxes, which cover the areas where the master and slave objects may contact. Each box has a specific number and space range, forming a grid-like structure for quick positioning and search. If the subordinate node (fluid particle) may be in contact with the main facet, it must be located in a box composed of the main facet extension. The size $d_e$ of the expansion box must satisfy the inclusion of all fluid particles that may be in contact with the main section, expressed as follows:

$$d_e=V_{ref}\mathsf{\Delta }t$$

(11)

In the formula, $V_{ref}$ is the maximum relative velocity between the fluid particle and the main section, and $\mathsf{\Delta }t$ is the time step.

Precise contact condition assessment requires rigorous evaluation of particle-facet interactions to delineate interfacial contact regions. The geometric penetration analysis subsequently resolves contact states, eliminating non-physical penetration artifacts while ensuring algorithmic stability and convergence within the Lagrangian framework.

If the slave node (fluid particle) makes contact with or crosses the main segment line (or its extension) [31], it must satisfy Equation (12).

$$0\le {\delta }_{\mathrm{cro}}\le d_e$$

(12)

In the formula, ${\delta }_{cro}$ represents the distance of penetration between the fluid particle and the main section.

The contact resolution phase establishes interaction constraints between slave particles and corresponding master surface elements, employing an iterative force correction scheme to mitigate interfacial penetration [32].

Iterative velocity and positional updates between master/slave nodes typically require multiple cycles to achieve kinematic consistency at the particle–element interface. Single slave nodes interacting with isolated master segments can achieve convergence within a single iteration cycle. However, contact reinitialization occurs when slave nodes engage adjacent or previously contacted master segments, requiring dynamic constraint updates.

3.1.2. Boundary Treatment

The SPH-FEM coupling framework introduces virtual particles at finite element nodes within interfacial regions to bridge discrete particle–element interactions. These hybrid nodes inherit identical material properties (mass, velocity, and stress) from their parent finite elements while maintaining kinematic consistency with SPH particles. Passively embedded in the SPH neighbor search domain, these particles enhance interfacial force calculations without participating in SPH time integration. Their state variables evolve through finite element formulation updates governed by nodal solutions [33]. During SPH numerical integration, finite element nodes participate as ghost particles within kernel support domains. This formulation ensures consistent inclusion of all interfacial entities within the kernel radius in neighbor lists [34].

The SPH formulation integrates interfacial finite element nodes into particle neighbor lists, preventing kernel truncation artifacts at domain boundaries while maintaining integral continuity.However, numerical stability requires careful consideration of temporal synchronization between discrete and continuum domains. Asynchronous time integration between SPH particles and FEM nodes can induce interfacial oscillations through momentum transfer mismatches. When SPH particles require reduced timesteps per CFL constraints, unsynchronized updates of ghost particle states may compromise interfacial momentum conservation. FEM interfacial nodes receive displacement constraints from neighboring SPH particles, effectively enforcing hybrid boundary conditions. This strong coupling assumption enables bidirectional force transmission across the interface. Under extreme deformation or shear conditions compromising particle–node mapping, stabilization techniques enforce interfacial compatibility through augmented Lagrangian constraints. At the coupling interface, the SPH-FEM consolidation algorithm can simultaneously realize the influence of SPH particles on the finite element (SPH applies boundary conditions to FEM) and the influence of finite element on SPH particles (finite element nodes are added to the support domain of SPH particles). Two different calculation methods are used together in the same object to ensure that the continuity requirements of physical quantities at the coupling interface are met.

3.1.3. Time Integral

The coupled SPH-FEM framework employs explicit Lagrangian formulations with temporal discretization via the central difference method. This scheme maintains conditional stability while effectively resolving transient fluid–structure interaction phenomena. To maintain the stability of the calculation, the time step of both must meet the corresponding conditions, which is less than the critical time step [35].

$$\mathsf{\Delta }{t}_{FEM}=\lambda \mathsf{\Delta }{t}_{crFEM}$$

(13)

$$\mathsf{\Delta }{t}_{crFEM}=\min {\displaystyle \frac{l_e}{c_e}}$$

(14)

In the formula, $\lambda$ is a constant, generally $0.8\le \lambda \le 0.98$; $l_e$ is the characteristic length of the unit; $c_e$ is the sound speed of the unit.

For SPH, the time step must comply with the CFL condition (Courant–Friedrichs–Lewy condition) to ensure stability [36]:

$$\mathsf{\Delta }{t}_{SPH}<k_{CFL}\min \left({\displaystyle \frac{h}{c}}\right)$$

(15)

$k_{CFL}$ is 0.25; $h$ is the particle spacing, 0.001 m; $c=\sqrt{K/\rho }$; $K$ is the bulk modulus of the molten metal; ρ is the density. The calculation shows that $\mathsf{\Delta }{t}_{\max }=1.2\times {10}^{-6}s$.

In this paper, the same time step $\mathsf{\Delta }t$ is used for the SPH-FEM coupling algorithm.

$$\mathsf{\Delta }t=\min \left(\mathsf{\Delta }{t}_{SPH},\mathsf{\Delta }{t}_{FEM}\right)$$

(16)

3.1.4. Interparticle Distance

The initial interparticle distance optimization balances numerical accuracy and computational efficiency. Excessive resolution increases computational load, whereas insufficient resolution distorts free-surface tracking. Liu et al. [37] established optimal discretization criteria for metallic flows, recommending interparticle distances of 1–2% characteristic length for accuracy–stability equilibrium. For 0.1 m scale castings, 1 mm spacing provides optimal resolution while maintaining computational feasibility. Hu et al. [38] demonstrated 1 mm resolution achieves superior cost effectiveness for flow-front prediction through parametric filling simulations.

The numerical parameters for the filling process are provided in

Table 1. Numerical parameter table.

Process	Options
Dimension	Three-dimensional (3D)
Boundary	Dummy particle method
Time integration	Central difference
Viscosity	Artificial
Initial particle distance (Δx)	0.001 m
Time step	1 × 10−6
Smooth length	1.2Δx

.

3.2. Simulation Calculation of Ring Filling Process

Molten metal filling constitutes thermally driven transient flow phenomena, characterized by splashing and oxidation phenomena that induce internal casting defects. Water analog experiments have become a vital physical modeling technique for such investigations. Cleary et al. [39,40] successfully benchmarked SPH-based water filling simulations against Schmid et al.’s experimental data [41,42], validating SPH’s predictive capabilities. However, conventional SPH formulations exhibit particle penetration artifacts during weakly compressible flow simulations. To address these limitations, the proposed SPH-FEM coupling scheme implements interfacial stabilization mechanisms to maintain particle distribution uniformity while preventing non-physical penetration.

The ring part is a disk-symmetrical hole part, and its size structure diagram and finite element mesh diagram are shown in Figure 2. As a typical example, the liquid inlet is a regular vertical boundary during the whole calculation process of the annular part, and the cavity flow process is an arc boundary. The experimental mold is transparent to facilitate the intuitive observation of the entire liquid flow process.

The numerical model adopts water density as 998 kg/m³, with dynamic viscosity (1.0 × 10⁻³ Pa s) characterizing shear flow behavior. The fluid enters the mold through a valve-controlled inlet at 18 m/s. Initial particle distributions feature uniformly spaced fluid particles at the mold inlet and boundary-aligned virtual particles along geometric contours. The calculation results of the SPH-FEM coupling method are compared with the experimental results of Schmid in the following figure.

As shown in Figure 3, the left figure is the experimental process diagram of the ring part, and the right figure is the calculation result diagram of the SPH-FEM coupling method. In the calculation result diagram of the coupling method, the red particles represent liquid water, and the blue represents the finite element mesh. Figure 3 illustrates three characteristic stages during the annular component’s water filling process. During initial filling (t = 8.82 ms), the fluid propagates directly toward the core through central flow channels. Core interaction triggers symmetrical flow bifurcation into two branches that subsequently impinge upon cavity walls. Wall engagement induces secondary hydrodynamic separation, generating four distinct flow paths along upper and lower cavity contours (Figure 3a), with each branch maintaining wall-adherent propagation. The results obtained from the SPH-FEM coupling method align well with the experimental data, whether it is the flow direction of water or the shape of the movement process. As the water filling continues (t = 11.76 ms), the two branches at the top converge. After the confluence, the two branches merge and flow in the negative direction of y. After the direct flow to the core and contact with the core, they are also divided into two symmetrical branches under their reaction. The two branches at the bottom continue to flow along the inner wall of the cavity and intersect with the fluid at the inlet. As shown in Figure 3b, the water filling results of the two cavities show that the flow direction of the water and the position of the cavity are highly consistent with the experiment. As the fluid continues to flow (t = 16.17 ms), the filling tributaries intersect to form the four cavities shown in Figure 3c. As the flow progresses, the cavity gradually decreases, and the filling process is eventually completed. The calculation results using the SPH-FEM coupling method are in close agreement with the experimental findings.

On the whole, the experimental results are basically consistent with the calculation results obtained by the SPH-FEM coupling method, which indicates that the validity and accuracy of the SPH-FEM coupling method in simulating the water filling process of ring parts are strongly verified. The symmetry-based validation strategy achieves computational economy through dimensional reduction while maintaining solution accuracy, and sustained particle distribution uniformity throughout filling stages prevents accuracy-compromising aggregation phenomena. This stability ensures robust physical quantity propagation and numerical convergence, enhancing simulation reliability. The demonstrated axisymmetric flow compatibility underscores the method’s engineering viability for complex casting systems, providing foundational insights for industrial implementation and multidisciplinary extensions.

3.3. Simulation Calculation of Filling Process of Bottom-Pouring Plate

Casting filling constitutes multi-physics phenomena involving coupled heat/mass transfer, momentum exchange, and phase transformations [43]. Predictive accuracy requires comprehensive consideration of interacting process variables, particularly thermal evolution modeling during filling stages for process fidelity. This methodology integrates hydrodynamic field characteristics [44] with multiphase coupling mechanisms. Initializing the thermal model with measured pouring temperatures establishes physically consistent boundary conditions. Subsequent thermal analysis demonstrates enhanced predictive capabilities in capturing transient thermal gradients during mold filling.

Mold filling processes inherently obey energy conservation principles, during which thermal energy exchange between molten metal and mold walls induces continuous melt-temperature attenuation. This thermal evolution constitutes transient convection-diffusion phenomena governed by coupled heat transfer mechanisms [45]. The heat exchange between the SPH particles and the FEM interface nodes is calculated by the contact algorithm. The thermal convection term $\rho c_{p}v\cdot \nabla T$ of the fluid particles is mapped to the equivalent heat flux density of the FEM boundary nodes. $q=k\nabla T\cdot n$, which is involved in the FEM integral as the Neumann boundary condition. The viscous stress (diffusion term) of SPH particles at the interface is transferred to the FEM node through the contact force, and the displacement response of the FEM is fed back to the SPH momentum equation through the velocity constraint to ensure the continuity of the interface velocity and the conservation of momentum. The equation expression is as follows:

$${\displaystyle \frac{\delta }{\delta x}}\left(\rho cT\right)+\nabla \cdot \left(\rho cUT\right)=\nabla \cdot \left(k\nabla T\right)+S_T$$

(17)

where c is the specific heat capacity; T is the thermodynamic temperature K; K is the thermal conductivity; ST is the heat source term.

The bottom-injected symmetric plate configuration employs a predefined particle distribution above the mold cavity, representing molten metal injection that progressively fills the cavity through controlled inflow dynamics. Convective heat transfer boundary conditions at mold walls capture liquid–solid phase interaction energetics. In order to better avoid the phenomenon of wall penetration during the filling process, the filling model of the bottom-filled symmetrical plate adopts the size of $100\ast 100\ast 10$ and is divided into finite element meshes, as shown in Figure 4.

The metal liquid is injected into the cavity from the right sprue and gradually filled from the bottom of the plate until the filling is completed. The filling process parameters are set as shown in

Table 2. Filling process parameter setting.

Process	Options
Material	ZL101
Pouring temperature	700 °C
Density	2372 kg/m3
Melting point	600 °C
Thermal conductivity	130 W/(m·K)

.

Figure 5 presents comparative analyses of plate casting filling: experimental observations (left), particle trajectory simulations (center), and SPH-FEM coupled 3D thermomechanical analysis (right). In the simulation result diagram, the temperature color strip is used for comparison.

Through the simulation and experimental observation of the filling process, it was found that at the initial stage of filling, when t = 1.2 s, the molten metal begins to flow from the spruce into the cavity, the flow and behavior at this stage shows obvious dynamic characteristics. Because the metal liquid collides with the cavity wall during the high-speed flow, the local energy is concentrated and released, and the metal liquid splashes first on the left side of the cavity. This kind of splashing phenomenon is expected in the high-speed casting process. When the molten metal flow rate exceeds the critical value [46], the conversion of kinetic energy to surface energy will cause droplet splashing, which is consistent with the experimental observation results in this paper and verifies the accuracy of the simulation. By further analyzing the temperature change, it can be found that during the preliminary stage of liquid metal injection into the runner, heat conduction begins to occur due to the significant temperature difference between the liquid metal and the mold. However, due to the short time of this stage, the heat conduction effect is not significant, so the temperature change is not obvious. The heat of the molten metal is mainly concentrated in the flow front, while the heat distribution of the mold is relatively uniform.

With the advancement of the filling process, when t = 3.83 s, the metal liquid is stably filled upward at the filling port. The flow behavior at this stage shows obvious directivity, and the liquid metal gradually fills the cavity under the combined action of gravity and inertial force. A comparison of the simulation results with the experimental data revealed that the filling states are essentially identical, and the simulation successfully reproduces the splash phenomenon of the molten metal; the occurrence of the splashing phenomenon is governed by both the flow velocity of the molten metal and the cavity geometry. During this phase, pronounced thermal gradients emerge with expanding influence zones. Conductive heat transfer from molten metal to mold walls initiates localized temperature elevation in proximal mold regions, and distinct thermal gradients characterize the system, with maximum temperatures concentrated at melt–mold interfaces that decay radially outward.

At t = 5.5 s, the molten metal initiates bilateral cavity filling, inducing secondary splashing upon wall impingement. This stage exhibits multidirectional flow characteristics with turbulent recirculation patterns. Recurrent wall collisions during cavity filling generate persistent splashing phenomena. Central upward flow generates a characteristic convex–concave–convex morphology through dynamic momentum redistribution. Morphological congruence with experimental observations validates the simulation’s predictive capacity. Concurrently, thermal gradients propagate radially with pronounced temperature elevation in mold–cavity interfacial zones, evidenced by isothermal band broadening. When the liquid metal is filled on both sides of the cavity, the contact area between the flow branch and the cavity wall increases significantly. According to the Fourier law $q=-k\nabla T$, the expansion of the contact area will directly increase the heat flux q between the mold and the liquid metal, resulting in a rapid increase in the temperature near the wall of the mold. The heat conduction effect of the molten metal is particularly significant at this stage, and the temperature distribution of the mold shows obvious non-uniformity.

During final filling stages (t = 7.76 s), quasi-steady flow conditions prevail while maintaining free-surface evolution. Hydrodynamic characteristics transition to laminar regimes with velocity attenuation and substantially diminished splashing. Numerical results revealed vertically stratified thermal profiles characterized by elevated basal temperatures. This thermal stratification directly correlates with upward melt propagation dynamics. Progressive bottom-to-top filling establishes thermal gradients through convective heat transfer dominance. Conduction-driven thermal equilibration emerges during terminal filling phases as convective effects subside.

The SPH-FEM coupled framework demonstrates excellent congruence with experimental observations in bottom-injection plate casting simulations, particularly in replicating three-dimensional flow patterns; this hybrid methodology leverages complementary particle–element interactions to capture complex filling dynamics. SPH excels in resolving free-surface flows and splashing phenomena, while FEM precisely models conductive heat transfer and mechanical responses in solid domains. This synergistic integration enables simultaneous tracking of transient thermal evolution and hydrodynamic behavior, achieving high fidelity to the actual casting processes. The methodology successfully resolves intricate flow characteristics, including vortex formation and momentum transfer dynamics. SPH-based analysis revealed detailed sprue-to-cavity flow trajectories and splashing patterns that align closely with experimental results, validating its capability in hydrodynamic modeling. Concurrently, FEM-based thermal monitoring captures spatial and temporal temperature variations in mold regions, providing critical insights for predicting solidification behavior and microstructural evolution. The coupled framework demonstrates robust performance in balancing computational accuracy with physical realism across all filling stages.

The SPH-FEM coupled framework demonstrates superior predictive capabilities in thermal modeling, with periodic boundary conditions in symmetric configurations enhancing computational efficiency through reduced iterative convergence requirements. The methodology enables real-time prediction of thermal gradients during mold filling, offering theoretical foundations for process optimization and metallurgical quality enhancement. Furthermore, the framework proactively identifies potential defect nucleation zones, enabling targeted process refinements.

3.4. Stress Field Simulation Analysis of Casting Filling Process

The thermal stress induced while the casting filling process is underway can lead to cracks and deformation in both the manufacturing and utilization of castings, significantly compromising their performance and functionality. Consequently, it is essential to numerically simulate the generation and evolution of thermal stress. Hot crack is a common defect that cannot be ignored. The formation of thermal stress is closely associated with the distribution of the temperature field, the concentration of thermal stress, and the flow behavior of molten metal [47]. Bottom-gated injection induces complex thermal gradients through differential heat transfer rates in plate geometries. Proximal regions experience rapid thermal transients from direct melt contact, whereas distal zones exhibit delayed thermal response. Resultant thermal expansion mismatches generate intrinsic residual stresses. Exceeding material-specific critical stress thresholds initiates surface/subsurface crack propagation. Crack-induced degradation mechanisms compromise tensile properties, fatigue life, and impact toughness, predisposing components to in-service failures [48]. These defects act as stress concentrators, potentiating progressive crack growth that undermines structural integrity during postprocessing or operational loading. Figure 6 shows the stress–temperature nephograms of t = 0.83 s, 2.5 s, and 7.5 s obtained by SPH-FEM coupling method. It can be seen from the nephogram that hot crack defects may occur at the right corner during the filling process.

Bottom-filled plate casting processes exhibit progressive melt advancement along geometric contours of the mold cavity, with corner regions susceptible to flow stagnation and incomplete filling caused by abrupt flow redirection. These flow anomalies induce localized thermal field heterogeneity and amplify solidification-induced thermal stresses. Microstructural heterogeneity in corner zones during aluminum alloy solidification creates preferential sites for crack nucleation, with subsequent stress-driven propagation culminating in hot tearing defects.

From the von Mises stress nephogram and isobaric diagram in Figure 6, it is evident that when t = 0.83 s, the metal liquid just flows into the corner of the cavity, and the flow direction will suddenly change due to the special geometry of the corner, resulting in the uneven distribution of the flow velocity of the metal liquid at the corner. On the outer side of the corner, the liquid metal directly impacts the mold wall, resulting in a large impact force, which in turn causes a higher stress. On the inner side of the corner, the metal liquid is prone to flow obstruction and accumulation, which makes the heat transfer inside the corner relatively slow, forming a large temperature difference with the surrounding area and resulting in a large thermal stress. The accumulation of thermal stress creates conditions for the subsequent generation of hot cracks. When t = 2.5 s, it can be clearly observed from the stress–temperature cloud diagram that the color at the corner is darker, and the stress value is higher, which indicates that there is stress concentration at the corner. During the solidification process, the metal at the corner cannot contract freely due to the surrounding constraints, resulting in additional shrinkage stress, which is superimposed with the impact stress, resulting in a significant increase in the stress level. At the same time, due to the geometric characteristics, the heat dissipation at the corner is faster, which solidifies before the surrounding area, and the subsequent liquid metal shrinks to produce a tensile effect, forming a thermal stress. Observing the pressure line diagram, it can be found that the pressure contour at the corner is densely distributed, and the pressure value changes drastically, indicating that the metal liquid flow is blocked, and the pressure accumulates. This high-pressure state not only affects the stress of the mold but also interferes with the flow stability of the molten metal, resulting in uneven temperature distribution, further aggravating thermal stress and creating mechanical conditions for the formation of hot cracks. With the passage of time to t = 7.5 s, the filling is close to completion, the flow rate of molten metal is reduced, and the stress at the corner is still relatively high. At this time, the solidification process of the liquid metal is further promoted. When the surrounding liquid metal continues to solidify and contract, it will have a greater constraint on the solidified corner part. The first solidified part at the corner is more seriously hindered during the contraction process, resulting in stress accumulation and further destroying the integrity of the metal structure at the corner and increasing the risk of hot cracks.

Figure 7 indicates that in the initial stage of filling (about 0–1 s), the metal liquid just flows into the corner of the mold. Because the metal liquid enters the cavity at a certain speed and pressure, the impact pressure will be generated at the corner. This impact pressure prompts the liquid metal to quickly transfer heat to the corner area, accelerating the rise in temperature. At the same time, the pressure makes the metal liquid and the mold wall more closely interact, which is conducive to the transfer of heat from the metal liquid to the mold, further promoting the temperature rise. The temperature rise rate is faster in the rapid heating stage (about 1–3 s). At this stage, the metal liquid continues to flow in, forming a certain pressure accumulation at the corner. The greater pressure makes the flow rate of the metal liquid relatively stable and faster and can continuously transport the high-temperature metal liquid to the corner, continuously providing heat for the region and ensuring the rapid rise in the temperature. The pressure affects the heat conduction inside the liquid metal, which makes the heat distribution more uniform at the corner and promotes the overall temperature rise. Due to the difference in the heating rate of different parts of the mold, a large temperature gradient is generated at the corner and the surrounding area. The metal expands in the region with high temperature, while the region with low temperature limits its expansion, resulting in thermal stress. When the thermal stress exceeds the strength limit of the metal at this temperature, cracks may be caused. In the heating slowdown stage (about 3–10 s), although the overall temperature change tends to be gentle, the thermal stress caused by the temperature gradient before still exists. If the heat dissipation rate of different parts of the mold is different or the thermal expansion coefficient is different due to the composition difference inside the molten metal, the residual thermal stress will be generated, which will result in a hidden danger for the generation of thermal cracks.

Thermal stress concentrations arising from heterogeneous temperature fields represent a critical failure mechanism in mold systems. Material optimization through composite coatings improves thermal management, effectively mitigating stress magnitudes through enhanced heat dissipation. Numerical simulations establish theoretical frameworks for informed material selection and topology-optimized designs. Engineered thermal gradients during mold filling critically determine final casting integrity and process stability. Controlled thermal gradients promote directional solidification while regulating interdendritic feeding behavior, informing advanced cooling system configurations. Gradient cooling strategies achieve thermal equilibrium to enhance melt feeding efficiency during solidification. Active gradient control via pouring parameter modulation and mold preheating enables tailored mechanical property profiles, balancing wear resistance with fracture toughness for enhanced performance in multiaxial loading scenarios.

Future investigations will pursue multi-objective optimization through physics-informed machine learning frameworks that integrate multi-physics coupling datasets, synergistically enhancing mold longevity, solidification integrity, and filling dynamics. Building upon these foundations, implementation of physics-aware digital twins enables adaptive process control via real-time sensor fusion, optimizing thermal–fluid parameters (e.g., cooling flux and injection velocity) and enabling paradigm shifts from empirical to predictive process governance. This roadmap accelerates sustainable foundry operations through cyber–physical system integration, driving simultaneous advancements in production throughput and energy-efficient intelligent manufacturing.

4. Conclusions

This investigation develops a SPH-FEM coupled methodology for multi-physics simulations of mold filling processes, demonstrating robust predictive capabilities in resolving coupled thermofluidic phenomena. Through rigorous theoretical framework development and experimental validation, the key findings can be summarized as follows:

(1) The SPH-FEM coupled framework implements a dynamic fluid–structure interaction mechanism through synergistic integration of meshless Lagrangian particle dynamics and finite element formulations. By maintaining interfacial field quantity continuity, this approach achieves enhanced topological tracking precision for free-surface flows while optimizing computational efficiency in multi-physics computations. While SPH remains intrinsically constrained in resolving solid-phase thermal conduction and stress relaxation phenomena, and FEM exhibits inherent limitations in modeling extreme free-surface deformations, this complementary synergy establishes an innovative cross-scale framework for high-fidelity mold filling simulations;

(2) Focusing on bottom-gated symmetric plate filling processes, the methodology achieves high-fidelity reproduction of flow-front kinematics and thermal gradient evolution patterns. The framework demonstrates predictive capability in detecting localized stress concentrations and hot-crack risk mapping, validating its engineering efficacy in multi-physics environments. Benchmarking against conventional thermoelastic models and finite volume approaches reveals superior robustness in resolving dynamic flow-front behavior and thermomechanical interactions, establishing an effective platform for defect root-cause analysis;

(3) A symmetry-informed adaptive resource allocation framework enables concurrent optimization of flow-front particle resolution and solid-phase mesh density, effectively balancing computational accuracy with thermal–fluid interaction efficiency. Conventional uniform discretization approaches incur significant computational overhead through redundant nodal allocations. The proposed adaptive allocation mechanism maintains interfacial continuity while eliminating redundant computations through physics-aware domain partitioning, establishing a practical framework for full-process casting simulations.

While the SPH-FEM coupled framework demonstrates superior multi-physics modeling capabilities, its computational scalability remains constrained for industrial-scale casting simulations due to substantial resource requirements. The concurrent resolution of SPH particle interactions and FEM grid coupling demands extensive parallel computing infrastructure, currently precluding real-time industrial deployment. With the continuous advancement of computer technology and the deepening understanding of the casting process, the SPH-FEM coupling method is expected to be further extended to multi-material and multi-scale complex casting simulation, such as the development of an adaptive particle–grid dynamic conversion mechanism to improve the simulation efficiency of complex geometric features, the establishment of a melt flow–heat transfer–solidification–stress full-process coupling model to enhance multi-physical field prediction capabilities, and the combination of experimental data and machine learning algorithms to optimize process design. Fundamental investigations into heterogeneous interfacial heat transfer and quantitative sustainability assessments will accelerate adoption in advanced manufacturing domains, including aerospace component fabrication and automotive mega-casting processes, driving eco-efficient foundry innovations.