FEM and FVM Methods for Design and Manufacturing of Hierarchical Aerospace Composites: A Review

Abstract

The manufacturing of multiscale composite structures in aerospace engineering is governed by complex interactions among material heterogeneity, fluid rheology, and multiphysics phenomena—including thermal, chemical, electrical, and mechanical effects. These coupled processes introduce significant challenges during both processing and post-manufacturing stages, which are often difficult to resolve using traditional (experimental) trial-and-error approaches. This review explores the potential of advanced numerical methods and simulation frameworks to address these complexities. Emphasis is placed on the use of finite element and finite volume methods, along with their respective solution strategies and domain discretisation techniques, to solve the coupled governing equations involved in composite manufacturing processes. By integrating theory, computation, and physics-based understanding, these approaches enable predictive capability and design optimisation in the development of high-performance composite components for aerospace applications; many challenges though still remain in fabrication, design, and analysis.

1. Introduction

Numerical means continue to have potential for parameter predictions in a diverse range of engineering applications such as energy, marine, automotive, and aerospace. With applications in aerospace structures, the design and manufacturing of composites are vital when considering high strength–weight ratio, light weight, durability, shape flexibility, and chemical resistance [1]. Such composites consist of two or more materials combined together prior to the manufacturing phase. These materials, generally, are classified into two main parts—matrices and reinforcements. This review will examine numerical works on the use of thermoset polymers (e.g., epoxy, polyester) and synthetic/natural fibres (e.g., carbon) as composite matrices and reinforcements, respectively. The aforementioned combination has received much attention within aerospace industries for its pivotal role in substituting metal-based parts, cost-cutting, and continuously improving efficiency. The good news is that manufacturing processes, as well as the post-manufacturing tests of fibre-reinforced polymer composites (FRPCs), can now be modelled and simulated by a variety of accurate and reliable numerical techniques. The finite element method (FEM) and finite volume method (FVM) are some of the more widely used techniques that allow the transformation (conversion) of partial differential equations (PDEs), such as the Navier–Stokes equations (for flow), into an algebraic form that can be numerically solved throughout a discretised domain—see Figure 1. Other methods include the finite difference method (FDM) [2,3] and lattice Boltzmann method (LBM) [4,5,6]; however, the FDM is restricted to structured mesh (regular grids), andthe LBM is a memory-consuming algorithm and is limited to specific flow regimes. It should be noted that the aforementioned methods are applicable to flow problems. The FEM can also be used to simulate the behaviour of objects when subjected to stresses and strains (loads and boundary conditions); similarly, the so-called boundary element method (BEM)—analysing only boundary elements of the domain—is another promising numerical solution for structural analysis [7]. AI applications in composite manufacturing, particularly machine learning and deep learning, could help reduce analysis time and predict material properties [8,9]. The integration of machine learning with multiscale FEM and FVM frameworks is transforming how composite manufacturing processes—such as permeability prediction, curing, and residual stress analysis—are modelled. Karuppusamy et al. [9] highlight the use of supervised learning models, such as artificial neural networks and support vector machines, for predicting key composite properties based on fibre–matrix interactions and processing parameters. These models reduce the computational cost of traditional simulations and enhance insight into process–structure–property relationships. Similarly, Wu et al. [10] developed gated recurrent unit (GRU) surrogates embedded into FEM multiscale workflows to approximate meso-scale mechanical behaviour, achieving significant speed improvements for tasks like resin impregnation modelling. Furthermore, Yang et al. [11] demonstrated how convolutional neural networks (CNNs) trained on FEM-generated microstructure–response data can accurately predict stress fields, enabling defect-aware property mapping without repeated simulations. Collectively, these hybrid AI–physics models provide a scalable, efficient solution to modelling the complex, nonlinear, and path-dependent behaviour of composites across manufacturing scales. However, this still requires both the quantity and quality of data, which remain areas of active research and development [12]. Figure 2 illustrates the hierarchical architecture of fibre-reinforced composites, highlighting the transition from micro-scale fibre–matrix interactions to the macro-scale laminate structure. This inherent multiscale complexity introduces major challenges in modelling and simulation, particularly in capturing properties such as permeability, cure kinetics, and local stress distributions. A key issue is how to effectively transfer information across scales—for instance, employing representative volume elements (RVEs) at the micro-scale to estimate macroscopic permeability or using meso-scale woven models to inform laminate-level behaviour. In some cases, macro-scale process data may also be used to infer micro-scale conditions, enabling inverse estimation of internal fields. Addressing these scale-bridging challenges is essential for accurate and predictive modelling of composite manufacturing processes.

This review provides insights into numerical works based on FVM and FEM for flow analyses, and FEM for structural problems. Studies on coupled (twining) approaches—fluid structural interfaces (FSIs)—are also raised. Discretisation methods and calculation approaches for PDEs are thoroughly discussed.

2. FEM in Composite Manufacturing

2.1. FEM: Governing PDEs to Algebraic Form

In this subsection, an illustrative brief example is given for a unsteady incompressible Newtonian fluid (i.e., Stokes flow), usually the case for resin moulding of composites (impregnation of fibre preforms), to demonstrate the FEM numerical solution of a CFD (flow) problem [13,14,15,16]. This usually starts with partial differential equations (PDEs) of a governing equation (e.g., Stokes flow)—see Equations (1) and (2) below:

$${\displaystyle \frac{\partial \mathbf{u}}{dt}}=-\nabla p+\mu {\nabla }^2\mathbf{u}+\mathbf{f}\mathrm{in}{\mathrm{\Omega }}_f$$

(1)

$$\nabla ·\mathbf{u}=0\mathrm{in}{\mathrm{\Omega }}_f$$

(2)

where $\nabla p$ is the pressure gradient, $\mu {\nabla }^2\mathbf{u}$ is the viscous term, and $\mathbf{f}$ is the source term. Then, a weak/variational form is applied to integrate the PDEs over the fluid domain (${\mathrm{\Omega }}_f$) using test/basis functions ($\varphi$, $\psi$).

$${\int }_{{\mathrm{\Omega }}_f}{\displaystyle \frac{\partial \mathbf{u}}{dt}}·\varphi d{\mathrm{\Omega }}_f={\int }_{{\mathrm{\Omega }}_f}\left(-\nabla p+\mu {\nabla }^2\mathbf{u}\right)·\varphi d{\mathrm{\Omega }}_f+{\int }_{{\mathrm{\Omega }}_f}\mathbf{f}·\varphi d{\mathrm{\Omega }}_f$$

(3)

$${\int }_{{\mathrm{\Omega }}_f}\psi \nabla ·\mathbf{u}d{\mathrm{\Omega }}_f=0$$

(4)

This allows the integration of the PDEs to attain a saddle-point matrix (i.e., algebraic form/system).

$$\left[\begin{array}{cc}\mathbf{M}+∆t\mathbf{K}& ∆t\mathbf{G}\\ \mathbf{D}& 0\end{array}\right]\left[\begin{array}{c}{\mathbf{U}}^{n+1}\\ {\mathbf{P}}^{n+1}\end{array}\right]=\left[\begin{array}{c}\mathbf{F}\\ 0\end{array}\right]$$

(5)

where $\mathbf{M}$ and $\mathbf{K}$ are matrices, while $\mathbf{G}$ and $\mathbf{D}$ are gradient and divergence operators, respectively. $\mathbf{U}$ and $\mathbf{P}$ are vectors of unknown nodal velocity and pressure, and $\mathbf{F}$ is the body force or source term vector. $∆t$ is a time step size, while n and $(n+1)$ indicate the current time level and next/future time level, respectively.

2.2. Review of FEM-Based Numerical Studies

The FEM has been and still is applied in composite manufacturing as a prediction and optimisation tool to characterise parameters like permeability, porosity, tortuosity, curing, etc. For instance, Tan et al. [17] used the FEM approach to predict resin flow progression in a dual-scale porous media during an RTM process. A PoreFlow, based on a Fortran modular package, was utilised to run the FEM numerical calculations. This was to predict two-scale permeability, macro and micro, in an effort to waive the use of fitting parameters as well as to optimise the infiltration of resin into the fibrous preforms, void formation, and reverse flow. The authors [17] validated FEM-based results with available experimental data, which manifested its capability to simulate liquid moulding of composites—see Figure 3. Likewise, a study by Simacek et al. [18] adopted an FEM approach using LIMS (Liquid Injection Moulding Simulation) software to account for capillary effects within the fibre tow scale for resin impregnation modelling. The authors [18] revealed that capillarity can slow intra-tow filling and, consequently, leave voids or dry spots at the micro-level. The study emphasised the model’s reliability in such advanced transport phenomena (capillary effect inclusion) via validation against experimental data. Oliveira et al. [19] applied the FEM approach, implemented in PAM-RTM software, to perform macro-scale flow simulations (i.e., overall permeability prediction). The study [19] validated the FEM-based PAM-RTM results of flow-front progression against experimental flow behaviour in the mould, manifesting its feasibility. Dealing with curing, a study by Cheung et al. [20] addressed such phenomena using the FEM implemented in commercial Apaqus software. This was used to couple transient heat conduction with resin cure reaction (an autocatalytic model) to capture the evolution of resin temperature and degree of cure during exothermic heat generation. The FEM-based simulations were validated against experimental data—literature benchmarks—for temperature distribution and cure degree profiles. Yet another work by Sandberg et al. [21] studied the resin flow, heat transfer, and cure kinetics of a thermosetting system (polyurethane resin) in the pultrusion process, as shown in Figure 4, using FEM-based COMSOL multiphysics 5.4 software. The authors [21] compared simulated temperature and degree-of-cure profiles with available measured data from pultrusion experiments, by which a good agreement was reached, validating the developed numerical model for use in composite manufacturing analysis.

Table 1. Selected numerical contributions evaluating FEM-based tools for modelling composite manufacturing processes.

References	Flow Model	Thermal Model	Cure Model	Numerical Method
Tan et al. [17]	✓	×	×	FEM/CV (PoreFlow)
Simacek et al. [18]	✓	×	×	FEM/CV (LIMS)
Oliveira et al. [19]	✓	×	×	FEM/CV (PAM-RTM)
Cheung et al. [20]	×	✓	✓	FEM (Apaqus)
Sandberg et al. [21]	✓	✓	✓	FEM (COMSOL)
Flow Model:	Navier–Stokes or Stokes equations are numerically solved (see Appendix A.1).
Thermal Model:	Energy or heat balance equations are numerically solved (see Appendix A.2).
Cure Model:	Species transport or chemical equations are numerically solved (see Appendix A.2).

highlights key FEM-based numerical studies focused on simulating composite manufacturing processes.

FEM-based tools such as Poreflow, LIMS, PAM-RTM, and COMSOL are widely used for modelling resin flow in fibrous porous media. Poreflow and LIMS excel in permeability-focused RTM simulations, PAM-RTM is valued in industry for robust process prediction, and COMSOL offers multiphysics flexibility for integrating flow, heat transfer, and cure kinetics. Ultimately, selecting the most suitable platform depends on the modelling objectives, whether the priority is process-specific simulation or multiphysics integration.

To sum up, the FEM is well-known for its strong multiphysics coupling, integrating flow, heat transfer, chemical reactions, and structural responses within a single solver framework. The literature shows accurate flow-front tracking using this method, via Level Set approach, which effectively captures the evolution of interfaces or moving fronts. Additionally, the FEM offers great adaptability to unstructured meshes and complex geometries. However, it remains computationally intensive when simulating transient (unsteady) flows. Further work is also needed to stabilise solutions in convection-dominated regimes and to reduce numerical oscillations (errors). Overall, the FEM is preferred when detailed coupling of structural, thermal, and chemical phenomena is required.

3. FVM in Composite Manufacturing

3.1. FVM: Governing PDEs to Algebraic Form

Like the FEM, the FVM starts with the strong PDE form of the momentum and continuity equations—Stokes Equations (1) and (2), as mentioned earlier in Section 2.1 [22,23,24,25]. Instead of multiplying the PDE by test functions (like in FEM) over the domain, the FVM integrates the PDE over each control volume (cell), and uses the divergence theorem (Gauss’s theorem) to covert the control volume to surface integrals—see Equations (6) and (7).

$${\int }_V{\displaystyle \frac{\partial \mathbf{u}}{dt}}dV=-{\int }_{\partial V}p·\mathbf{n}dS+\mu {\int }_{\partial V}\nabla \mathbf{u}·\mathbf{n}dS+{\int }_V\mathbf{f}dV$$

(6)

$${\int }_{\partial V}\mathbf{u}·\mathbf{n}dS=0$$

(7)

Thereby, this leads to an algebraic form (see Equations (8) and (9)) that is appropriate for numerical calculations—typically solved using iterative schemes like SIMPLE (Semi-Implicit Method for Pressure-Linked Equations).

$$\mathbf{A}{\mathbf{u}}^{n+1}+\mathbf{B}{p}^{n+1}=\mathbf{b}$$

(8)

$$\mathbf{C}{\mathbf{u}}^{n+1}=0$$

(9)

where $\mathbf{A}$ is a matrix that combines transient and viscous terms, $\mathbf{B}$ and $\mathbf{C}$ stand for the pressure-gradient operator and divergence operator, respectively, and $\mathbf{b}$ here includes previous time step velocity and body forces.

3.2. Review of FVM-Based Numerical Studies

The use of the FVM may be preferred to the FEM for CFD problems due to its simplicity on arbitrary meshes (e.g., tetrahedrons) and no requirement for basis or test functions—direct flux calculations. A work by Grössing et al. [26], based on the FVM, uses OpenFoam (i.e., open-source CFD) to characterise resin flow advancement in dual-scale porous medium. The FVM-based OpenFoam capability was attested in capturing unsaturated permeabilities as it agreed well with the flow-front experimental data—see Figure 5. A study by Alotaibi et al. [27] utilised ANSYS Fluent 19.2 (FVM solver) to calculate Stokes–Brinkman equations (micro–meso-scale simulation) with consideration of the yarn curvature effect on resin flow. The source term, provided in the momentum Equation, was employed for micro-permeability (intra-tow pores) while the Stokes equation (remaining terms) was applied for inter-tow pores. The numerical analysis [27] promoted the reliability of the FVM used in Fluent, and the ability to adopt a more customised flow problem—curvature/waviness impact. Wei et al. [28] adopted the FVM using the Moldex3D tool to describe the behaviour of resin flow in fibre preforms. The authors [28] concluded that the FVM-based flow-front simulations indicated consistency with the data from infusion experiments, and thus an estimation of the permeability/porosity ratio was obtained as a means to stabilise flow front in the mould. With respect to the curing of polymer composites, the work by Alotaibi et al. [29] managed to couple filling and curing simulations, using FVM-based Fluent, by incorporating customised codes into the commercial code for heat generation, viscosity evolution, and degree of cure. The coupled numerical approach was verified with the literature, and also demonstrated its capability to accurately simulate resin flows under varying process conditions (e.g., ability to predict early cure, notably during the fill stage, as can be seen in Figure 6), employing high-order upwinding interpolation schemes to enhance stability and convergence. A similar study by Yang et al. [30] used FVM-based Fluent to optimise the cure cycle (during cure stage) with the aim of achieving uniformity in cure, besides avoiding residual stresses. The numerical method was eventually validated with temperature development profiles, see Figure 7, in a resin transfer moulding (RTM) experiment. Wittemann et al. [31] analysed curing behaviour by integrating viscosity and cure models into a CFD code using OpenFoam—FVM solver. The study [31] modelled thermosets in reinforced reactive injection moulding (RRIM), whereby the FVM-based calculations matched well with experimental pressure measurements over the filling stage, showing the method’s high potential. An overview of representative FVM-based modelling efforts applied to composite manufacturing is presented in

Table 2. Selected numerical contributions evaluating FVM-based tools for modelling composite manufacturing processes.

References	Flow Model	Thermal Model	Cure Model	Numerical Method
Grössing et al. [26]	✓	×	×	FVM/VOF (OpenFoam)
Alotaibi et al. [27]	✓	×	×	FVM/VOF (Fluent)
Wei et al. [28]	✓	×	×	FVM/VOF (Moldex3D)
Alotaibi et al. [29]	✓	✓	✓	FVM/VOF (Fluent)
Yang et al. [30]	✓	✓	✓	FVM/VOF (Fluent)
Wittemann et al. [31]	×	✓	✓	FVM (OpenFoam)
Flow Model:	Navier–Stokes or Stokes equations are numerically solved (see Appendix A.1).
Thermal Model:	Energy or heat balance equations are numerically solved (see Appendix A.2).
Cure Model:	Species transport or chemical equations are numerically solved (see Appendix A.2).

.

Among the main FVM-based tools used to model resin flow in fibrous porous media, OpenFOAM, ANSYS Fluent, and Moldex3D are the most commonly applied. OpenFOAM is popular in research because it is open-source and allows users to fully customise the simulation to include important physical effects like Darcy’s law. ANSYS Fluent is a commercial tool known for its reliable results and flexibility through user-defined functions, which help model complex behaviours such as resin curing and variable viscosity. Moldex3D is often used in industry, especially for thermoplastic composites, because it is easy to use and includes built-in features for mould filling and porous flow. Overall, OpenFOAM and Fluent are preferred for more detailed and accurate simulations, while Moldex3D is chosen when ease of use and process integration are more important.

In summary, the finite volume method (FVM) is well-suited for large-scale computational fluid dynamics (CFD) problems, offering strong parallel scalability and computational efficiency. However, it has limitations in structural integration, as the finite element method (FEM) is still required to model the mechanical response. Additionally, incorporating cure kinetics or micro-scale permeability effects often requires customisation of the governing PDEs, typically through user-defined functions (UDFs). Overall, this numerical approach is generally preferred when simulation scalability is of great importance.

4. Multiphysics Coupling: Fluid, Thermo-Chemical, and Structural Domains

4.1. Coupling Hypothesis

A detailed linkage equation via coupling or the so-called digital twinning (interaction) between fluid (liquid resin) and structure (fibre tows) is given within this Section by an illustrative numerical approach for residual stresses [33,34,35,36]—see Figure 8 for more insight.

These residual stresses are usually the remaining internal stresses after applying a force/load. This load can be induced by flow, heat (e.g., thermal expansion), or reaction (e.g., cure shrinkage). In this manner, Von Mises theory is used by the structure side (explained within this Section), while the fluid and/or thermo-chemical side is governed by unsteady incompressible Newtonian Stokes equations along with energy and species transport equations—the methodology is outlined in greater detail in Appendix A.

Here, $d_s$ is a structural displacement vector, ${\sigma }_s$ is the stress tensor, and ${\mathrm{\Omega }}_s$ is the solid domain. Pressure p and velocity $u_f$ are computed from the fluid side to provide traction ($t_{fsi}$) at the interface (${\mathrm{\Gamma }}_{fsi}$), likewise for temperature $T_f$—see Equations (10) and (11). This traction will act as an external force (load) on a body. Based on this premise, displacements (deformation) $d_s$ can be caused at the solid (structure) side, and from that deformation, strains and stresses can be evaluated. Following the results of the stress tensor (${\sigma }_s$) in Equation (12), Von Mises stress (${\sigma }_{vm}$), from Equation (13), is calculated to map residual stresses throughout the structure body, in particular fibre tows.

$${\sigma }_s·{\mathbf{n}}_s={\sigma }_f·{\mathbf{n}}_f={\mathbf{t}}_{fsi}\mathrm{on}{\mathrm{\Gamma }}_{fsi}$$

(10)

$$T_f=T_s\to -k_f\nabla T_f·n_f=-k_s\nabla T_s·n_s\mathrm{on}{\mathrm{\Gamma }}_{fsi}$$

(11)

$${\rho }_s{\displaystyle \frac{{\partial }^2{d}_s}{\partial t^2}}=\nabla ·{\sigma }_s\mathrm{in}{\mathrm{\Omega }}_s$$

(12)

$${\sigma }_{vm}=\sqrt{\frac{1}{2}[{({\sigma }_{xx}-{\sigma }_{yy})}^2+{({\sigma }_{yy}-{\sigma }_{zz})}^2+{({\sigma }_{zz}-{\sigma }_{xx})}^2+6({\tau }_{xy}^2+{\tau }_{yz}^2+{\tau }_{zx}^2)]}$$

(13)

4.2. Review of Numerical Studies on Multiphysics Coupling

In fluid–structure interaction (FSI) simulations, the FEM is mainly used in the solid domain, while the FVM or FEM may be used in the fluid domain. Such linkage (transient fluid to structural response) is already discussed within Section 4.1. Reviews on structural response to loads caused by resin flow or cure are discussed within this Section. Yuan et al. [38] used Abaqus coupled with UMATHT and USEFLD (user subroutines) to link thermal/chemical effects during the curing process of composite materials. This was to predict macro-scale simulations of temperature, and degree of cure at the part level, as a means to use the aforementioned results for calculating micro-stresses (residual stresses) at fibre level. The proposed twinning method (thoroughly FEM-based) was validated with temperature data obtained from the literature, as illustrated in Figure 9, suggesting that micro-stresses occur at both the fibre and polymer (matrix) sides due to thermal expansion and curing shrinkage. A study by Alotaibi et al. [37] performed fluid–solid coupling simulations using FVM-FEM linkage—Fluent+ANSYS-Mechanical—to characterise residual stresses of fibre structures at the meso-micro-level. The fluid–solid coupling included effects of flow-induced load/deformation, thermally induced expansion, and cure-induced shrinkage during the two stages: filling and curing. The developed model had the potential to twin the FVM-based convection–reaction–diffusion flow (resin) problem with the FEM-based structural (carbon fibre) response for the interpretation of deformation, and residual stresses or strains in fibre preforms. Goncalves et al. [39] investigated the development of micro-residual stresses in carbon/epoxy polymer composites during the curing process using numerical simulation. The authors [39], similar to [38], employed the commercial FEM software Abaqus with user subroutines to account for thermal, chemical, and mechanical interactions, but for a micro-scale representative volume element (RVE). The numerical method was effective in capturing micro-stresses during cure stage, revealing that higher micro-mechanisms occurred at thinner fibre gaps due to chemical shrinkage or thermal expansion. Dewangan et al. [40] studied how porosity in fibrous porous materials could affect the development of residual stresses during curing, particularly under autoclave conditions. This FEM-based work [40], utilising Abaqus–Fortran, successfully integrated energy and cure equations with structural analyses, indicating that residual stress increases as porosity decreases—see Figure 10. A study by Kim et al. [41] evaluated thermoset resin cure-induced deformation in plain woven fabrics, coupling FEM Abaqus 2022 with MATLAB R2022a to calculate cure kinetics (by MATLAB) and deformation response (by Abaqus). The accuracy and reliability of the proposed modelling approach was confirmed via comparison with experimental deformation measurements. As findings, the work stressed that larger yarn spacing could provoke distortions in woven fabrics, and such a model would be a worthy option for predicting such defects (like warping). A selection of numerical studies that evaluated multiphysics coupling for modelling composite manufacturing processes is summarized in

Table 3. Selected numerical contributions evaluating flow and/or thermo-chemical-induced loads for structural analysis.

References	Flow Model	Thermo-Chemical Model	Structural Model	Numerical Method
Yuan et al. [38]	×	✓	✓	FEM (Abaqus)
Alotaibi et al. [37]	✓	✓	✓	FVM + FEM (Fluent + ANSYS Mechanical)
Goncalves et al. [39]	×	✓	✓	FEM (Abaqus)
Dewangan et al. [40]	×	✓	✓	FEM (Abaqus + Fortran)
Kim et al. [41]	×	✓	✓	FEM (Abaqus + MATLAB)
Flow Model:	Navier–Stokes or Stokes equations are numerically solved (see Appendix A.1).
Thermo-chemical Model:	Both energy and chemical equations are numerically solved (see Appendix A.2).
Structural Model:	Deformations or residual stresses/strains are numerically solved (see Section 4.1).

.

5. Conclusions

This review has presented a comparative analysis of the finite element (FEM) and finite volume (FVM) methods in the context of composite manufacturing, with emphasis on their mathematical formulation and application in modelling multiscale, multiphysics phenomena. The discussion of fluid–structure interaction (FSI) further highlights the importance of coupled modelling in capturing the dynamic behaviour between fluid and solid domains during processing. By integrating theoretical insights with numerical strategies, this work underscores the potential of simulation-driven approaches to enhance process understanding, prediction, and optimization in the manufacturing of advanced composite structures. Continued development of efficient and robust computational frameworks remains essential for future advancements in this field.