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Optimal switching angles are investigated for minimizing accumulated numerical errors when the dual‐Euler method is used in the simulation of angular rotation.

First, round‐off errors are theoretically modeled with a simplified mathematical representation of rotation. Round‐off errors take critical roles in the vicinity of indefinite points because they cause major numerical inaccuracy in very large numerical values represented with limited binary numbers. Optimal switching angles of (±π/4, ±3π/4) are derived and numerically examined. With a more practical and severe rotational model, the switching angles are numerically tested.

In conclusion, switching pitch angles of (±π/4, ±3π/4) yield near minimum numerical errors in angular parameters of pitch, yaw, and roll if truncation errors are not dominant by using high‐order integration algorithms and small step sizes. It is also noticed that accumulated numerical errors increase dramatically if pitch and roll angles are switched beyond the optimal angles with a little margin.

Optimal switching angles in the dual‐Euler method are identified based on the truncation error analysis. The mechanism of accumulated numerical errors in the dual‐Euler method, which depends on switching angles, is also revealed.

An unconditionally positive definite finite difference scheme termed as UPFD has been derived to approximate a linear advection-diffusion-reaction equation which models exponential travelling waves and the coefficients of advection, diffusion and reactive terms have been chosen as one (Chen-Charpentier and Kojouharov, 2013). In this work, the author tests UPFD scheme under some other different regimes of advection, diffusion and reaction. The author considers the case when the coefficient of advection, diffusion and reaction are all equal to one and also cases under which advection or diffusion or reaction is more important. Some errors such as L₁ error, dispersion, dissipation errors and relative errors are tabulated. Moreover, the author compares some spectral properties of the method under different regimes. The author obtains the variation of the following quantities with respect to the phase angle: modulus of exact amplification factor, modulus of amplification factor of the scheme and relative phase error.

Difficulties can arise in stability analysis. It is important to have a full understanding of whether the conditions obtained for stability are sufficient, necessary or necessary and sufficient. The advection-diffusion-reaction is quite similar to the advection-diffusion equation, it has an extra reaction term and therefore obtaining stability of numerical methods discretizing advection-diffusion-reaction equation is not easy as is the case with numerical methods discretizing advection-diffusion equations. To avoid difficulty involved with obtaining region of stability, the author shall consider unconditionally stable finite difference schemes discretizing advection-diffusion-reaction equations.

The UPFD scheme is unconditionally stable but not unconditionally consistent. The scheme was tested on an advection-diffusion-reaction equation which models exponential travelling waves, and the author computed various errors such as L₁ error, dispersion and dissipation errors, relative errors under some different regimes of advection, diffusion and reaction. The scheme works best for very small values of k as k → 0 (for instance, k = 0.00025, 0.0005) and performs satisfactorily at other values of k such as 0.001 for two regimes; a = 1, D = 1, κ = 1 and a = 1, D = 1, κ = 5. When a = 5, D = 1, κ = 1, the scheme performs quite well at k = 0.00025 and satisfactorily at k = 0.0005 but is not efficient at larger values of k. For the diffusive case (a = 1, D = 5, κ = 1), the scheme does not perform well. In general, the author can conclude that the choice of k is very important, as it affects to a great extent the performance of the method.

The UPFD scheme is effective to solve advection-diffusion-reaction problems when advection or reactive regime is dominant and for the case, a = 1, D = 1, κ = 1, especially at low values of k. Moreover, the magnitude of the dispersion and dissipation errors using UPFD are of the same order for all the four regimes considered as seen from Tables 1 to 4. This indicates that if the author is to optimize the temporal step size at a given value of the spatial step size, the optimization function must consist of both the AFM and RPE. Some related work on optimization can be seen in Appadu (2013). Higher-order unconditionally stable schemes can be constructed for the regimes for which UPFD is not efficient enough for instance when advection and diffusion are dominant.

The purpose of this study is to propose a precise and standardized strategy for numerically simulating vehicle aerodynamics.

Error sources in computational fluid dynamics were analyzed. Additionally, controllable experiential and discretization errors, which significantly influence the calculated results, are expounded upon. Considering the airflow mechanism around a vehicle, the computational efficiency and accuracy of each solution strategy were compared and analyzed through numerous computational cases. Finally, the most suitable numerical strategy, including the turbulence model, simplified vehicle model, calculation domain, boundary conditions, grids and discretization scheme, was identified. Two simplified vehicle models were introduced, and relevant wind tunnel tests were performed to validate the selected strategy.

Errors in vehicle computational aerodynamics mainly stem from the unreasonable simplification of the vehicle model, calculation domain, definite solution conditions, grid strategy and discretization schemes. Using the proposed standardized numerical strategy, the simulated steady and transient aerodynamic characteristics agreed well with the experimental results.

Building upon the modified Low-Reynolds Number k-e model and Scale Adaptive Simulation model, to the best of the authors’ knowledge, a precise and standardized numerical simulation strategy for vehicle aerodynamics is proposed for the first time, which can be integrated into vehicle research and design.

To propose novel predictor‐corrector time‐integration algorithms for pseudo‐dynamic testing.

The novel predictor‐corrector time‐integration algorithms are based on both the implicit and the explicit version of the generalized‐α method. In the non‐linear unforced case second‐order accuracy, stability in energy, energy decay in the high‐frequency range as well as asymptotic annihilation are distinctive properties of the generalized‐α scheme; while in the non‐linear forced case they are the limited error near the resonance in terms of frequency location and intensity of the resonant peak. The implicit generalized‐α algorithm has been implemented in a predictor‐one corrector form giving rise to the implicit IPC‐ρ_∞ method, able to avoid iterative corrections which are expensive from an experimental standpoint and load oscillations of numerical origin. Moreover, the scheme embodies a secant stiffness formula able to approximate closely the actual stiffness of a structure. Also an explicit algorithm has been implemented, the EPC‐ρ_b method, endowed with user‐controlled dissipation properties. The resulting schemes have been tested experimentally both on a two‐ and on a six‐degrees‐of‐freedom system, exploiting substructuring techniques.

The analytical findings and the tests have indicated that the proposed numerical strategies enhance the performance of the pseudo‐dynamic test (PDT) method even in an environment characterized by considerable experimental errors. Moreover, the schemes have been tested numerically on strongly non‐linear multiple‐degrees‐of‐freedom systems reproduced with the Bouc‐Wen hysteretic model, showing that the proposed algorithms reap the benefits of the parent generalized‐α methods.

Further developments envisaged for this study are the application of the IPC‐ρ_∞ method and of EPC‐ρ_b scheme to partitioned procedures for high‐speed pseudo‐dynamic testing with substructuring.

The implicit IPC‐ρ_∞ and the explicit EPC‐ρ_b methods allow a user to have defined dissipation which reduces the effects of experimental error in the PDT without needing onerous iterations.

The paper proposes novel time‐integration algorithms for pseudo‐dynamic testing. Thanks to a predictor‐corrector form of the generalized‐α method, the proposed schemes maintain a high computational efficiency and accuracy.

The purpose of this paper is to evaluate some numerical integration strategies used in generalized (G)/extended finite element method (XFEM) to solve linear elastic fracture mechanics problems. A range of parameters are here analyzed, evidencing how the numerical integration error and the computational efficiency are improved when particularities from these examples are properly considered.

Numerical integration strategies were implemented in an existing computational environment that provides a finite element method and G/XFEM tools. The main parameters of the analysis are considered and the performance using such strategies is compared with standard integration results.

Known numerical integration strategies suitable for fracture mechanics analysis are studied and implemented. Results from different crack configurations are presented and discussed, highlighting the necessity of alternative integration techniques for problems with singularities and/or discontinuities.

This study presents a variety of fracture mechanics examples solved by G/XFEM in which the use of standard numerical integration with Gauss quadratures results in loss of precision. It is discussed the behaviour of subdivision of elements and mapping of integration points strategies for a range of meshes and cracks geometries, also featuring distorted elements and how they affect strain energy and stress intensity factors evaluation for both strategies.

The purpose of this paper is to illustrate how the numerical solution of the Burgers' equation is obtained using the methods of cubic B‐spline collocation and quadratic B‐spline Galerkin over the geometrically graded mesh.

The spatial domain is partitioned into geometrically graded mesh. The finite element methods are constructed within the Galerkin and collocation methods using an expansion of the quadratic and cubic B‐splines as an approximate function, respectively, over this mesh.

When the higher errors are observed at near boundaries for shock‐like and travelling wave solutions of the Burgers' equation, accuracy of the defined methods increase by using finer mesh at near this boundary.

Over the geometrically graded mesh definitions of the quadratic B‐spline Galerkin and cubic B‐spline collocation are given.

The purpose of this paper is to present outcomes of efforts made over the last 20 years to extend the applicability of the Richardson extrapolation procedure to numerical predictions of multidimensional, steady and unsteady, fluid flow and heat transfer phenomena in regular and irregular calculation domains.

Pattern-preserving grid-refinement strategies are proposed for mathematically rigorous generalizations of the Richardson extrapolation procedure for numerical predictions of steady fluid flow and heat transfer, using finite volume methods and structured multidimensional Cartesian grids; and control-volume finite element methods and unstructured two-dimensional planar grids, consisting of three-node triangular elements. Mathematically sound extrapolation procedures are also proposed for numerical solutions of unsteady and boundary-layer-type problems. The applicability of such procedures to numerical solutions of problems with curved boundaries and internal interfaces, and also those based on unstructured grids of general quadrilateral, tetrahedral, or hexahedral elements, is discussed.

Applications to three demonstration problems, with discretizations in the asymptotic regime, showed the following: the apparent orders of accuracy were the same as those of the numerical methods used; and the extrapolated results, measures of error, and a grid convergence index, could be obtained in a smooth and non-oscillatory manner.

Strict or approximate pattern-preserving grid-refinement strategies are used to propose generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. Such extrapolation procedures play an indispensable role in the verification and validation techniques that are employed to assess the accuracy of numerical predictions which are used for designing, optimizing, virtual prototyping, and certification of thermofluid systems.

Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and uniform temperature or heat flux, assuming that the rest of the cavity is completely adiabatic. The paper used suitable numerical methods to follow the interface temporal evolution with a good accuracy. The purpose of this paper is to show how the evolution of the latent energy absorbed to melt the PCM depends on the temperature imposed on the lower wall of the cavity.

The problem is written with non-homogeneous boundary conditions. Momentum and energy equations are numerically solved in space by a spectral collocation method especially oriented to this situation. A Crank-Nicolson scheme permits the resolution in time.

The results clearly show the evolution of multicellular regime during the process of fusion and the kinetics of phase change depends on the boundary condition imposed on the bottom cell wall. Thus the charge and discharge processes in energy storage cells can be controlled by varying the temperature in the cell PCM. Substantial modifications of the thermal convective heat and mass transfer are highlighted during the transient regime. This model is particularly suitable to follow with a good accuracy the evolution of the solid/liquid interface in the process of storage/release energy.

The time-dependent physical properties that induce non-linear coupled unsteady terms in Navier-Stokes and energy equations are not taken into account in the present model. The present model is actually extended to these coupled situations. This problem requires smoother geometries. One can try to palliate this disadvantage by constructing smoother approximations of non-smooth geometries. The augmentation of polynomials developments orders increases strongly the computing time. When the external heat flux or temperature imposed at the PCM is much greater than the temperature of the PCM fusion, one must choose carefully some data to assume the algorithms convergence.

Among the areas where this work can be used, are: buildings where the PCM are used in insulation and passive cooling; thermal energy storage, the PCM stores energy by changing phase, solid to liquid (fusion); cooling and transport of foodstuffs or pharmaceutical or medical sensitive products, the PCM is used in the food industry, pharmaceutical and medical, to minimize temperature variations of food, drug or sensitive materials; and the textile industry, PCM materials in the textile industry are used in microcapsules placed inside textile fibres. The PCM intervene to regulate heat transfer between the body and the outside.

The paper's originality is reflected in the precision of its results, due to the use of a high-accuracy numerical approximation based on collocation spectral methods, and the choice of Chebyshev polynomials basis in both axial and radial directions.

The purpose is to reduce round-off errors in numerical simulations. In the numerical simulation, different kinds of errors may be created during analysis. Round-off error is one of the sources of errors. In numerical analysis, sometimes handling numerical errors is challenging. However, by applying appropriate algorithms, these errors are manageable and can be reduced. In this study, five novel topological algorithms were proposed in setting up a structural flexibility matrix, and five different examples were used in applying the proposed algorithms. In doing so round-off errors were reduced remarkably.

Five new algorithms were proposed in order to optimize the conditioning of structural matrices. Along with decreasing the size and duration of analyses, minimizing analytical errors is a critical factor in the optimal computer analysis of skeletal structures. Appropriate matrices with a greater number of zeros (sparse), a well structure and a well condition are advantageous for this objective. As a result, a problem of optimization with various goals will be addressed. This study seeks to minimize analytical errors such as rounding errors in skeletal structural flexibility matrixes via the use of more consistent and appropriate mathematical methods. These errors become more pronounced in particular designs with ill-suited flexibility matrixes; structures with varying stiffness are a frequent example of this. Due to the usage of weak elements, the flexibility matrix has a large number of non-diagonal terms, resulting in analytical errors. In numerical analysis, the ill-condition of a matrix may be resolved by moving or substituting rows; this study examined the definition and execution of these modifications prior to creating the flexibility matrix. Simple topological and algebraic features have been mostly utilized in this study to find fundamental cycle bases with particular characteristics. In conclusion, appropriately conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.

(1) Five new algorithms were proposed in order to optimize the conditioning of structural flexibility matrices. (2) A JAVA programming language was written for all five algorithms and a friendly GUI software tool is developed to visualize sub-optimal cycle bases. (3) Topological and algebraic features of the structures were utilized in this study.

This is a multi-objective optimization problem which means that sparsity and well conditioning of a matrix cannot be optimized simultaneously. In conclusion, well-conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.

Engineers always finding mathematical modeling of real-world problems and make them as simple as possible. In doing so, lots of errors will be created and these errors could cause the mathematical models useless. Applying decent algorithms could make the mathematical model as precise as possible.

Errors in numerical simulations should reduce due to the fact that they are toxic for real-world applications and problems.

This is an original research. This paper proposes five novel topological mathematical algorithms in order to optimize the structural flexibility matrix.

The purpose of the method is to develop a numerical method for the solution of nonlinear partial differential equations.

A new numerical approach based on Barycentric Rational interpolation has been used to solve partial differential equations.

A numerical technique based on barycentric rational interpolation has been developed to investigate numerical simulation of the Burgers’ and Fisher’s equations. Barycentric interpolation is basically a variant of well-known Lagrange polynomial interpolation which is very fast and stable. Using semi-discretization for unknown variable and its derivatives in spatial direction by barycentric rational interpolation, we get a system of ordinary differential equations. This system of ordinary differential equation’s has been solved by applying SSP-RK43 method. To check the efficiency of the method, computed numerical results have been compared with those obtained by existing methods. Barycentric method is able to capture solution behavior at small values of kinematic viscosity for Burgers’ equation.

To the best of the authors’ knowledge, the method is developed for the first time and validity is checked by stability and error analysis.