International Journal of Computational Methods

In this paper, we implement an efficient numerical technique which we call fractional Chebyshev finite difference method (FChFDM). The fractional derivatives are presented in terms of Caputo sense. The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a nonuniform finite difference scheme. The error bound for the fractional derivatives is introduced. We used the introduced technique to solve numerically the fractional-order delay BVPs. The application of the proposed method to introduced problem leads to algebraic systems which can be solved by an appropriate numerical method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

The atmospheric wind turbulence over a specified time period has a strong influence on the airplane performance characteristics. Forecasting of this unsteady aerodynamic phenomenon is complex one for designing the control systems to ensure the structural safety. A novel approach is developed to assess the influence of a gust structure on the aerodynamic coefficients of an airplane. The load factor enhancement because of the discrete gust is also quantified to ensure the safety margin. The Kussner’s function is used to determine the time varying increment of gust-dependent lift produced on an airplane wing entering a sharp-edged gust. A most general gust shape is assumed for the present study with quasi-steady approximation. The turbulent viscosity across the chord wise positions are quantified to calculate the velocity fluctuations because of eddies. Determining the gust influence on the fundamental lift and drag characteristics of a commercial airplane is focused in the present investigation. It is accomplished with the help of Wagner’s function in the time domain for the gust response-dependent lift. The outcome of the numerical simulation process is fully verified using the theoretical and experimental results. Solution convergence is attained for a range of input conditions and it shows that the proposed methodology is competent to assess the gust response for various airplane systems design.

Three-dimensional primitive equations (3DPE) become a reasonable approach in hydrodynamics in terms of computational costs when the length of the computational domain and/or computational time scales increases. However, given the simplified set of equations used in the analysis, results with 3DPE-based models are expected to be approximate and before attempting to reproduce complex natural flows they first need to be validated against more simple flows observed in laboratory settings. Here, the validity of Cartesian free-surface hydrodynamic models to reproduce three turbulent flows characteristic of river environments is tested: (1) the development of shallow mixing layers, (2) flow pass a lateral cavity and (3) flow in open channel with mild curvature. Errors between measured and modeled values were generally less than 10%, proving their validity to reproduce such turbulent flows and their potential for simulations in more complex natural environments, such it is the case of the confluence between the Ebro and Segre rivers into Ribarroja reservoir.

In this paper, tetrahedral background cells are used in nodal integration of radial point interpolation method (RPIM). The nodal integration is based on Taylor series terms and it is originally applied for the solutions of 2D problems in literature. Therefore, in this study, it is attempted that the tetrahedral integration cells are used in the solution of 3D elasto-static problems. The accuracy is seriously affected by order of Taylor series terms and it is investigated up to fifth order. A methodology is developed for prevention of negative volumes and calculation problems in subdivision of integration cells for each node. Three different case studies are solved with different support domain sizes and shape parameters. The best accuracy is achieved with fourth-order Taylor terms in nodal integration radial point interpolation method (NI-RPIM). ${\alpha }_c$-value of 3.00 and $q$ value of 1.03 in radial basis functions give good results in all cases.

In this paper, Golden Ratio Simulated Annealing (GRSA) for Protein Folding Problem (PFP) is presented. GRSA is similar to Multiquenching Annealing (MQA) and Threshold Temperature Simulated Annealing (TTSA) algorithms. In contrast to MQA and TTSA, GRSA uses several strategies in order to reduce the execution time for finding the best solution of PFP. Firstly, temperature parameters are tuned with an analytical-experimental approach; secondly, a heuristic technique for dividing the search space is implemented. GRSA has a special phase which detects the thermal equilibrium by a least squares method. In addition, a reheat strategy to escape from local optima is applied.

The heat and mass transfer in an unsteady boundary layer flow of an incompressible, laminar, natural convective third-grade fluid is studied. The flow is taken over a semi-infinite vertical porous plate with the temperature-dependent fluid properties by taking into account the effect of viscous dissipation and variable suction. The partial differential equations governing the problem are reduced into the nonlinear, coupled, nondimensional ordinary differential equations with the help of suitable similarity transformations. The Galerkin Finite Element Method is implemented to solve this acquired system. The effects of various significant parameters such as Grashof number, Prandtl number, Eckert number, Solutal Grashof number and Schmidt number on dimensionless velocity, temperature and concentration profiles are presented graphically. The Nusselt number is found to be depressed whereas the Sherwood number is observed to be enhanced with the increasing values of Grashof number and Prandtl number. The study has important applications in chemical process industries such as filtration in food industry, production of drinking water and recovering salts from solutions.

A modified smoothed particle hydrodynamics (SPH) method is applied to simulate the dynamic tensile failure of rock-like heterogeneous materials. The material heterogeneities are represented by different mechanical properties with a Weibull distribution at different particles over the specimen. An elasto-plastic-damage model is adopted to simulate material failure. Numerical simulations are performed for three specimens with different degrees of heterogeneity under quasi-static and dynamic loading conditions, respectively. The effects of strength criterion coefficient $b$ on the predicted tensile strength are also discussed. Results show that the material heterogeneity has significant effect on the dynamic tensile failure, especially at a higher strain rate loading condition. At a lower strain rate, the failure mode of heterogeneous material is very much similar to that of homogeneous materials. At a higher strain rate, plenty of micro cracks occur forming a failure band in the tension zones. Moreover, the value of strength criterion coefficient $b$ has a considerable influence on the tensile strength under a higher loading rate.

Symplectic numerical methods for optimal control problems with irregular interpolation schemes are developed and the comparisons between irregular interpolation schemes and equidistant scheme are made in this paper. The irregular interpolation points, which are the collocation points usually adopted by pseudospectral (PS) methods, such as Legendre–Gauss, Legendre–Gauss–Radau, Legendre–Gauss–Lobatto and Chebyshev–Gauss–Lobatto points, are taken into consideration in this study. The symplectic numerical method with irregular points is proposed firstly. Then, several examples with different complexities highlight the differences in performance between different kinds of interpolation schemes. The numerical results show that the convergence of the present symplectic numerical methods can be obtained by increasing the number of sub-intervals or the number of interpolation points. Moreover, the comparison results show that the convergence of the symplectic numerical methods are generally independent on the type of interpolation points and the computational efficiency is not sensitive to the choice of interpolation points in general. Thus, the symplectic numerical methods with different interpolation schemes have obvious difference with the PS methods.

In this paper, two compact implicit finite difference methods are developed and analyzed for solving the one-dimensional time fractional diffusion equation. The temporal derivative is approximated by using Grünwald–Letnikov formula. Compact finite difference approximation is used for the second-order derivative in space. The local truncation errors are discussed. The stability analysis and the convergence of the proposed methods are investigated by means of Fourier series method. A comparison between the results of these methods and the exact solution is made. Numerical tests are given to verify the feasibility and accuracy of the methods.

In simulations of multiphase fluid flow using the phase-field method (PFM), the wetting boundary condition is required for off-grid objects. In this study, we propose an improved implementation of the wetting boundary condition for off-grid objects to reduce anisotropic errors arising from use of a rectangular grid. Our implementation of the phase-field wetting boundary condition conforms to the immersed-boundary formulation of solid–fluid interfaces; therefore, we call the immersed-boundary phase-field implementation (IB-PFI). We performed simulations with and without IB-PFI for (a) droplets adhering to circular objects and (b) capillary flow in a parallel-plate channel. In simulations without IB-PFI, anisotropic errors were induced by off-grid objects, and the results deviated from theoretical predictions. In contrast, simulations with IB-PFI suppressed the anisotropic errors and agreed with the theoretical predictions. Thus, IB-PFI extends the applicability of the PFM to simulations of multiphase fluid flows under numerous geometric conditions.