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This paper aims to present an evolutionary algorithm (EA) to accelerate the convergence for the radiative transfer equation (RTE) numerical solution using high-order and high-resolution schemes by the relaxation coefficients optimization.

The objective function minimizes the residual value difference between iterations in each control volume until its difference is lower than the convergence criterion. The EA approach is evaluated in two configurations, a two-dimensional cavity with scattering media and absorbing media.

Experimental results show the capacity to obtain the numerical solution for both cases on all interpolation schemes tested by the EA approach. The EA approach reduces CPU time for the RTE numerical solution using SUPERBEE, SWEBY and MUSCL schemes until 97% and 135% in scattering and absorbing media cases, respectively. The relaxation coefficients optimized every two numerical solution iterations achieve a significant reduction of the CPU time compared to the deferred correction procedure with fixed relaxation coefficients.

The proposed EA approach for the RTE numerical solution effectively reduces the CPU time compared to the DC procedure with fixed relaxation coefficients.

This paper sets out to implement bounded high‐order (HO) resolution schemes in a hybrid finite volume/finite element method for the solution of the radiative transfer equation.

The hybrid finite volume/finite element method had formerly been developed using the step scheme, which is only first‐order accurate, for the spatial discretization. Here, several bounded HO resolution schemes, namely the MINMOD, CLAM, MUSCL and SMART schemes, formulated using the normalized variable diagram, were implemented using the deferred correction procedure.

The results obtained reveal an interaction between spatial and angular discretization errors, and show that the HO resolution schemes yield improved accuracy over the step scheme if the angular discretization error is small.

Although the HO resolution schemes reduce the spatial discretization error, they do not influence the angular discretization error. Therefore, the global error is only reduced if the angular discretization error is also small.

The use of HO resolution schemes is only effective if the angular refinement yields low‐angular discretization errors. Moreover, spatial and angular refinement should be carried out simultaneously.

The paper extends a methodology formerly developed in computational fluid dynamics, and aimed at the improvement of the solution accuracy, to the hybrid finite volume/finite element method for the solution of the radiative transfer equation.

The purpose of this study is to propose a novel relaxation modified factor to accelerate the numerical solution of the radiative transfer equation (RTE) with several high-resolution total variation diminishing schemes. The methodology proposed is denoted as the X-factor method.

The X-factor method was compared with the technique deferred-correction (DC) for the calculations of a two-dimensional cavity with absorting-emiting-scatteting gray media using the discrete ordinates method. Four parameters were considered to evaluate: the absorption coefficient, the emissivity of boundary surface, the scattering albedo and under-relaxation factor.

The results showed the central processing unit (CPU) time of X-factor method was lower than DC. The reductions of CPU time with the X-factor method were observed from 0.6 to 75.4%.

The superiority of the X-factor method over DC was showed with the reduction of CPU time of the numerical solution of RTE for evaluated cases.

The paper aims to use the finite volume method widely used in computational fluid dynamics to avoid the serious remeshing and mesh distortion during aluminium profile extrusion processes simulation when using the finite element method. Block-structured grids are used to fit the complex domain of the extrusion. A finite volume method (FVM) model for aluminium extrusion numerical simulation using non-orthogonal structured grids was established.

The influences of the elements ' nonorthogonality on the governing equations discretization of the metal flow in aluminium extrusion processes were fully considered to ensure the simulation accuracy. Volume-of-fluid (VOF) scheme was used to catch the free surface of the unsteady flow. Rigid slip boundary condition was applied on non-orthogonal grids.

This paper involved a simulation of a typical aluminium extrusion process by the FVM scheme. By comparing the simulation by the FVM model established in this paper with the ones simulated by the finite element method (FEM) software Deform-3D and the corresponding experiments, the correctness and efficiency of the FVM model for aluminium alloy profile extrusion processes in this paper was proved.

This paper uses the FVM widely used in CFD to calculate the aluminium profile extrusion processes avoiding the remeshing and mesh distortion during aluminium profile extrusion processes simulation when using the finite element method. Block-structured grids with the advantage of simple data structure, small storage and high numerical efficiency are used to fit the complex domain of the extrusion.

The purpose of this paper is to develop a finite volume approach for the simulation of three-dimensional two-phase (polymer melt and air) flow in plastic injection molding which is capable of robustly handling the mesh non-orthogonality and the discontinuities in fluid properties.

The presented numerical method is based on a cell-centered unstructured finite volume discretization with a volume-of-fluid technique for interface capturing. The over-relaxed approach is adopted to handle the non-orthogonality involved in the discretization of the face normal derivatives to enhance the robustness of the solutions on non-orthogonal meshes. A novel interpolation method for the face pressure is derived to address the numerical stability issues resulting from the density and viscosity discontinuities at the melt–air interface. Various test cases are conducted to evaluate the proposed method.

The presented method was shown to be satisfactorily accurate by comparing simulations with analytical and experimental results. Besides, the effectiveness of the proposed face pressure interpolation method was verified by numerical examples of a two-phase flow problem with various density and viscosity ratios. The proposed method was also successfully applied to the simulation of a practical filling case.

The proposed finite volume approach is more tolerant of non-orthogonal meshes and the discontinuities in fluid properties for two-phase flow simulation; therefore, it is valuable for engineers in engineering computations.

To present the detailed implementation processes of the IDEAL algorithm for two-dimensional compressible flows based on Delaunay triangular mesh, and compare the performance of the SIMPLE and IDEAL algorithms for solving compressible problems. What’s more, the implementation processes of Delaunay mesh generation and derivation of the pressure correction equation are also introduced.

Programming completely in C++.

Five compressible examples are used to test the SIMPLE and IDEAL algorithms, and the comparison with measurement data shows good agreement. The IDEAL algorithm has much better performance in both convergence rate and stability over the SIMPLE algorithm.

The detail solution procedure of implementing the IDEAL algorithm for compressible flows based on Delaunay triangular mesh is presented in this work, seemingly first in the literature.

This paper aims to develop a numerical method for analysing the time‐dependent conjugate heat and fluid flows inside and around single bubbles rising in a hot liquid.

The procedure combines the moving mesh method for flows in time‐dependent geometries and the zoned calculation algorithm for conjugate viscous flows. A moving axisymmetric boundary‐fitted mesh is used to track the deformable gas‐liquid interface, while conjugate flows in both gas and liquid sides are calculated by a two‐block zoned method. The interfacial stresses are employed to calculate the velocity value and to decide the time‐dependent bubble shape simultaneously. Governing equations for the rising velocity and acceleration of the bubble are derived according to the forces acting on the bubble.

A calculating procedure for time‐dependent conjugate heat and fluid flows inside and around a rising single bubble has been developed. The algorithm has been verified, and can be employed for further analysing heat, mass and momentum transfer phenomena and their relevant mechanisms.

The paper developed a method to obtain high fidelity results for the heat and fluid flow details in the vicinity of a time‐dependent moderately deformable rising bubble; the physically zero‐thickness of a gas‐liquid interface is guaranteed. The governing equations for the time‐dependent rising velocity and acceleration are derived.

The purpose of this paper is to develop a coupled approach to solve the pressure–velocity-coupled problem efficiently in the three-dimensional injection molding simulation.

A fully coupled pressure–velocity algorithm is developed to solve the coupled problem, by treating the pressure gradient term implicitly. And, the Schur complement preconditioned FGMRES is applied to decompose the resulting coupled pressure–velocity equation into pressure and velocity subsystems. Then, BoomerAMG is adopted to solve the pressure subsystem, and block Jacobi preconditioned FGMRES is applied to the velocity subsystem.

According to the several experiments, the fully coupled pressure–velocity algorithm was demonstrated to have faster convergence than the traditional SIMPLE algorithm, and the calculating time was reduced by up to 70 per cent. And, the Schur complement preconditioned FGMRES worked more efficiently than block Gauss–Seidel preconditioned FGMRES, block-selective AMG and AMG with block ILU(0) smoother and could take at least 47.4 per cent less time. The proposed solver had good scalability for different-size problems, including various cases with different numbers of elements. It also kept good speedup and efficiency in parallel performance.

A coupled solver has been proposed to effectively solve the coupled problem in the three-dimensional injection molding simulation, which is more robust and efficient than existing methods.

This paper aims to focus on the temporal and spatial fourth‐order finite volume discretization of the incompressible form of the Navier‐Stokes equations on structured uniform grids. The main purpose of the paper is to assess the accuracy enhancement with the inclusion of a high‐order reconstruction of the point‐wise velocity field on a fourth‐order accurate numerical scheme for the solution of the unsteady incompressible Navier‐Stokes equations.

The present finite volume method uses a fractional time‐step for decoupling velocity and pressure. A Runge‐Kutta integration scheme is implemented for integrating the momentum equation along with a polynomial interpolation and Simpson formula for space‐integration. The formulation is based on step‐by‐step de‐averaging process applied to the velocity field.

The reconstruction of the point‐wise velocity field on a higher‐order basis is essential to obtain solutions that effectively stand for a fourth‐order approximation of the point‐wise one. Results are provided for the Taylor vortex decay problem and for co‐ and counter‐rotating vortices to assess the increase in accuracy promoted by the inclusion of the high‐order de‐averaging procedure.

High‐order reconstruction of the point‐wise velocity field should be considered in high‐order finite volume methods for the solution of the unsteady incompressible form of the Navier‐Stokes equations on structured grids.

The inclusion of a high‐order reconstruction of the point‐wise velocity field is a simple and effective method of enhancing the accuracy of a finite volume code for the computational fluid dynamics analysis.

The paper develops an improved version of a fourth‐order accurate finite volume projection method with the inclusion of a high‐order reconstruction step.

The purpose of this paper is to numerically investigate steady, laminar natural and mixed convection heat transfer in a two-dimensional cavity by using a finite volume method with a fourth-order approximation of convective terms, with and without the presence of nanoparticles. Highly accurate benchmark results are also provided.

A finite volume method on a non-uniform staggered grid is used for the solution of two-dimensional momentum and energy conservation equations. Diffusion terms, in the momentum and energy equations, are approximated using second-order central differences, whereas a non-uniform four-point fourth-order interpolation (FPFOI) scheme is developed for the convective terms. Coupled mass and momentum conservation equations are solved iteratively using a semi-implicit method for pressure-linked equation method.

For the case of natural convection problem at high-Rayleigh numbers, grid density must be sufficiently high in order to obtain grid-independent results and capture reality of the physics. Heat transfer enhancement for natural convection is observed up to a certain value of the nanoparticle volume fraction. After that value, heat transfer deterioration is found with increasing nanoparticle volume fraction.

Developed a non-uniform FPFOI scheme. Highly accurate benchmark results for the heat transfer of Al₂O₃-water nanofluid in a cavity are provided.