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The numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.

The approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.

According to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.

The key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

The geometric parameters of the compressor blade have a noteworthy influence on compressor stability, which should be meticulously designed. However, machining inaccuracies cause the blade geometric parameters to deviate from the ideal design, and the geometric deviation exhibits high randomness. Therefore, the purpose of this study is to quantify the uncertainty and analyze the sensitivity of the impact of blade geometric deviation on compressor stability.

In this work, the influence of blade geometric deviation is analyzed based on a subsonic compressor rotor stage, and three-dimensional numerical simulations are used to compute samples with different geometric features. A method of combining Halton sequence and non-intrusive polynomial chaos is adopted to carry out uncertainty quantitative analysis. Sobol’ index and Spearman correlation coefficient are used to analysis the sensitivity and correlation between compressor stability and blade geometric deviation, respectively.

The results show that the compressor stability is most sensitive to the tip clearance deviation, whereas deviations in the leading edge radius, trailing edge radius and chord length have minimal impact on the compressor stability. And, the effects of various blade geometric deviations on the compressor stability are basically independent and linearly superimposed.

This work provided a new approach for uncertainty quantification in compressor stability analysis. The conclusions obtained in this work provide some reference value for the manufacturing and maintenance of rotor blades.

This paper aims to present an unconditionally energy-stable scheme for approximating the convective heat transfer equation.

The scheme stems from the generalized positive auxiliary variable (gPAV) idea and exploits a special treatment for the convection term. The original convection term is replaced by its linear approximation plus a correction term, which is under the control of an auxiliary variable. The scheme entails the computation of two temperature fields within each time step, and the linear algebraic system resulting from the discretization involves a coefficient matrix that is updated periodically. This auxiliary variable is given by a well-defined explicit formula that guarantees the positivity of its computed value.

Compared with the semi-implicit scheme and the gPAV-based scheme without the treatment on the convection term, the current scheme can provide an expanded accuracy range and achieve more accurate simulations at large (or fairly large) time step sizes. Extensive numerical experiments have been presented to demonstrate the accuracy and stability performance of the scheme developed herein.

This study shows the unconditional discrete energy stability property of the current scheme, irrespective of the time step sizes.

This study aims to develop an experimentally validated three-dimensional numerical model for predicting different flow patterns produced with a gas dynamic virtual nozzle (GDVN).

The physical model is posed in the mixture formulation and copes with the unsteady, incompressible, isothermal, Newtonian, low turbulent two-phase flow. The computational fluid dynamics numerical solution is based on the half-space finite volume discretisation. The geo-reconstruct volume-of-fluid scheme tracks the interphase boundary between the gas and the liquid. To ensure numerical stability in the transition regime and adequately account for turbulent behaviour, the k-ω shear stress transport turbulence model is used. The model is validated by comparison with the experimental measurements on a vertical, downward-positioned GDVN configuration. Three different combinations of air and water volumetric flow rates have been solved numerically in the range of Reynolds numbers for airflow 1,009–2,596 and water 61–133, respectively, at Weber numbers 1.2–6.2.

The half-space symmetry allows the numerical reconstruction of the dripping, jetting and indication of the whipping mode. The kinetic energy transfer from the gas to the liquid is analysed, and locations with locally increased gas kinetic energy are observed. The calculated jet shapes reasonably well match the experimentally obtained high-speed camera videos.

The model is used for the virtual studies of new GDVN nozzle designs and optimisation of their operation.

To the best of the authors’ knowledge, the developed model numerically reconstructs all three GDVN flow regimes for the first time.

A parabolic trough solar collector is an advanced concentrated solar power technology that significantly captures radiant energy. Solar power will help different sectors reach their energy needs in areas where traditional fuels are in use. This study aims to examine the sensitivity analysis for optimizing the heat transfer and entropy generation in the Jeffrey magnetohydrodynamic hybrid nanofluid flow under the influence of motile gyrotactic microorganisms with solar radiation in the parabolic trough solar collectors. The influences of viscous dissipation and Ohmic heating are also considered in this investigation.

Governing partial differential equations are derived via boundary layer assumptions and nondimensionalized with the help of suitable similarity transformations. The resulting higher-order coupled ordinary differential equations are numerically investigated using the Runga-Kutta fourth-order numerical approach with the shooting technique in the computational MATLAB tool.

The numerical outcomes of influential parameters are presented graphically for velocity, temperature, entropy generation, Bejan number, drag coefficient and Nusselt number. It is observed that escalating the values of melting heat parameter and the Prandl number enhances the Nusselt number, while reverse effect is observed with an enhancement in the magnetic field parameter and bioconvection Lewis number. Increasing the magnetic field and bioconvection diffusion parameter improves the entropy and Bejan number.

Nanotechnology has captured the interest of researchers due to its engrossing performance and wide range of applications in heat transfer and solar energy storage. There are numerous advantages of hybrid nanofluids over traditional heat transfer fluids. In addition, the upswing suspension of the motile gyrotactic microorganisms improves the hybrid nanofluid stability, enhancing the performance of the solar collector. The use of solar energy reduces the industry’s dependency on fossil fuels.

In the digital assembly system of large aircraft components (LAC), the docking trajectory of LAC is an important factor affecting the docking accuracy and stability of the LAC. The main content of docking trajectory planning is how to move the LAC from the initial posture and position to the target posture and position (TPP). This paper aims to propose a trajectory planning method of LAC based on measured data.

First, the posture and position error model of the wing is constructed according to the measured data of the measurement points (MPs) and the fork lug joints. Second, the particle swarm optimization algorithm based on the dynamic inertia factor is used to optimize the TPP of the wing. Third, to ensure the efficiency and stability of posture adjustment, the S-shaped curve is used as the motion trajectory of LAC, and the parameters of the trajectory are solved by the generalized multiplier method. Finally, a series of docking experiments are carried out.

During the process of posture adjustment, the motion of the numerical control locator (NCL) is stable, and the interaction force between the NCLs is always within a reasonable range. After the docking, the MPs are all within the tolerance range, and the coaxiality error of the fork lug hole is less than 0.2 mm.

In this paper, the measured data rather than the theoretical design model is used to solve the TPP, which improves the docking accuracy of LAC. Experiment results show that the proposed trajectory method can complete the LAC docking effectively and improve the docking accuracy.

This article aims to present a direct solution to handle linear constraints in finite element (FE) analysis without penalties or the Lagrange multipliers introduced.

First, the system of linear equations corresponding to the linear constraints is solved for the leading variables in terms of the free variables and the constants. Then, the reduced system of equilibrium equations with respect to the free variables is derived from the finite-dimensional virtual work equation. Finally, the algorithm is designed.

The proposed procedure is promising in three typical cases: (1) to enforce displacement constraints in any direction; (2) to implement local refinements by allowing hanging nodes from element subdivision and (3) to treat non-matching grids of distinct parts of the problem domain. The procedure is general and suitable for 3D non-linear analyses.

The algorithm is fitted only to the Galerkin-based numerical methods.

The proposed procedure does not need Lagrange multipliers or penalties. The tangential stiffness matrix of the reduced system of equilibrium equations reserves positive definiteness and symmetry. Besides, many contemporary Galerkin-based numerical methods need to tackle the enforcement of the essential conditions, whose weak forms reduce to linear constraints. As a result, the proposed procedure is quite promising.

This study aims to simulate the dendritic growth in Stokes flow by iteratively coupling a domain and boundary type meshless method.

A preconditioned phase-field model for dendritic solidification of a pure supercooled melt is solved by the strong-form space-time adaptive approach based on dynamic quadtree domain decomposition. The domain-type space discretisation relies on monomial augmented polyharmonic splines interpolation. The forward Euler scheme is used for time evolution. The boundary-type meshless method solves the Stokes flow around the dendrite based on the collocation of the moving and fixed flow boundaries with the regularised Stokes flow fundamental solution. Both approaches are iteratively coupled at the moving solid–liquid interface. The solution procedure ensures computationally efficient and accurate calculations. The novel approach is numerically implemented for a 2D case.

The solution procedure reflects the advantages of both meshless methods. Domain one is not sensitive to the dendrite orientation and boundary one reduces the dimensionality of the flow field solution. The procedure results agree well with the reference results obtained by the classical numerical methods. Directions for selecting the appropriate free parameters which yield the highest accuracy and computational efficiency are presented.

A combination of boundary- and domain-type meshless methods is used to simulate dendritic solidification with the influence of fluid flow efficiently.

The paper aims to expand the scope of application of the lattice Boltzmann method (LBM), especially in the field of aircraft engineering. The traditional LBM is usually applied to incompressible flows at a low Reynolds number, which is not sufficient to satisfy the needs of aircraft engineering. Devoted to tackling the defect, the paper proposes a developed LBM combining the subgrid model and the multiple relaxation time (MRT) approach. A multilayer adaptive Cartesian grid method to improve the computing efficiency of the traditional LBM is also employed.

The subgrid model and the multilayer adaptive Cartesian grid are introduced into MRT-LBM for simulations of incompressible flows at a high Reynolds number. Validated by several typical flow simulations, the numerical methods in this paper can efficiently study the flows under high Reynolds numbers.

Some numerical simulations for the lid-driven flow of cavity, flow around iced GLC305, LB606b and ONERA-M6 are completed. The paper presents the investigation results, indicating that the methods are accurate and effective for the separated flow after icing.

LBM is developed with the addition of the subgrid model and the MRT method. A numerical strategy is proposed using a multilayer adaptive Cartesian grid method and its treatment of boundary conditions. The paper refers to innovative algorithm developments and applications to the aircraft engineering, especially for iced wing simulations with flow separations.

In fire resistance tests (FRTs) of building materials, a crucial criterion to pass the test procedure is to avoid the leakage of the hot flue gases caused by gaps and cracks occurring due to the thermal exposure. The present study's aim is to calculate the deformation of a steel door, which is embedded within a wall made of bricks, and qualitatively determine the flue gas leakage.

A computational fluid dynamics/finite element method (CFD/FEM) coupling was introduced representing an intermediate approach between a one-way and a full two-way coupling methodology, leading to a simplified two-way coupling (STWC). In contrast to a full two way-coupling, the heat transfer through the steel door was simulated based on a one-way approach. Subsequently, the predicted temperatures at the door from the one-way simulation were used in the following CFD/FEM simulation, where the fluid flow inside and outside the furnace as well as the deformation of the door were calculated simultaneously.

The simulation showed large gaps and flue gas leakage above the door lock and at the upper edge of the door, which was in close accordance to the experiment. Furthermore, it was found that STWC predicted similar deformations compared to the one-way coupling.

Since two-way coupling approaches for fluid/structure interaction in fire research are computationally demanding, the number of studies is low. Only a few are dealing with the flue gas exit from rooms due to destruction of solid components. Thus, the present study is the first two-way approach dealing with flue gas leakage due to gap formation.