Numerical Solution Of Problem Research Articles

On Inverse Coefficient Heat-Conduction Problems on Reconstruction of Nonlinear Components of the Thermal-Conductivity Tensor of Anisotropic Bodies

The authors are the first to present a closed procedure for numerical solution of inverse coefficient problems of heat conduction in anisotropic materials used as heat-shielding ones in rocket and space equipment. The reconstructed components of the thermal-conductivity tensor depend on temperature (are nonlinear). The procedure includes the formation of experimental data, the implicit gradient-descent method, the economical absolutely stable method of numerical solution of parabolic problems containing mixed derivatives, the parametric identification, construction, and numerical solution of the problem for elements of sensitivity matrices, the development of a quadratic residual functional and regularizing functionals, and also the development of algorithms and software systems. The implicit gradient-descent method permits expanding the quadratic functional in a Taylor series with retention of the linear terms for the increments of the sought functions. This substantially improves the exactness and stability of solution of the inverse problems. Software systems are developed with account taken of the errors in experimental data and disregarding them. On the basis of a priori assumptions of the qualitative behavior of the functional dependences of the components of the thermal-conductivity tensor on temperature, regularizing functionals are constructed by means of which one can reconstruct the components of the thermal-conductivity tensor with an error no higher than the error of the experimental data. Results of the numerical solution of the inverse coefficient problems on reconstruction of nonlinear components of the thermal-conductivity tensor have been obtained and are discussed.

Global–Local ROM for the solution of parabolic problems with highly concentrated moving sources

Problems characterised by steep moving gradients are challenging for any numerical technique and even more for the successful formulation of Reduced Order Models (ROMs). The aim of this work is to study the numerical solution of problems with steep moving gradients, by placing the focus on parabolic problems with highly concentrated moving sources. More specifically, a Global–Local scheme well-suited for reduction methods is formulated. With this Global–Localscheme, the local nature of the steep moving gradients is exploited by modelling the neighbourhood of the heat source with a moving local domain. This domain is coupled to the global domain without requiring any re-mesh and preserving the meshes of both domains during the whole simulation. Then, a ROM based on the Proper Orthogonal Decomposition (POD) technique is developed for the moving local domain. The proposed technique establishes a valid approach for tackling the non-separability of the space and time dimensions of these problems. In order to assess the numerical performance of the proposed numerical techniques, several evaluation tests are performed.

On the Existence of a Progressive Variational Vademecum based on the Proper Generalized Decomposition for a Class of Elliptic Parameterized Problems

In this study, we present the mathematical analysis needed to explain the convergence of a progressive variational vademecum based on the proper generalized decomposition (PGD). The PGD is a novel technique that was developed recently for solving problems with high dimensions, and it also provides new approaches for obtaining the solutions of elliptic and parabolic problems via the abstract separation of variables method. This new scenario requires a mathematical framework in order to justify its application to the solution of numerical problems and the PGD can help in the change to this paradigm. The main aim of this study is to provide a mathematical environment for defining the notion of progressive variational vademecum. We prove the convergence of this iterative procedure and we also provide the first order optimality conditions in order to construct the numerical approximations of the parameterized solutions. In particular, we illustrate this methodology based on a robot path planning problem. This is one of the common tasks when designing the trajectory or path of a mobile robot. The construction of a progressive variational vademecum provides a novel methodology for computing all the possible paths from any start and goal positions derived from a harmonic potential field in a predefined map.

Численное решение задач теории упругости методом конечных элементов

Численное решение задач теории упругости методом конечных элементов

Homogeneous–heterogeneous reactions in peristaltic flow of Prandtl fluid with thermal radiation

This attempt is made to look at the influence of homogeneous–heterogeneous reactions on MHD peristaltic flow of Prandtl fluid. The configuration of channel is considered as inclined and asymmetric. Mathematical modelling is developed through consideration of Joule heating, viscous dissipation and thermal radiation. Heat and mass transfer phenomenons are addressed simultaneously. Numerical solutions of problem are attained after employing long wavelength and low Reynolds number assumptions. The results for velocity, temperature, concentration and heat transfer rate have been emphasized through graphical illustration. Physical interpretation for sundry parameters of interest is carried out in detail. Results indicate that concentration decreases during both homogeneous and heterogenous reactions where thermal radiation lessens fluid temperature and rate of heat transfer.

Numerical simulation of the two-phase fluid filtration in heterogeneous media

Under study is numerical solution of the problems of two-phase filtration. The formulation of the problem is given in terms of velocity, pressure and saturation. To approximate the velocity and pressure, the mixed finite elementmethod is used. The flux schemes are applied for discretization of the convection term in the saturation equation. We present the results of numerical solution of a model problem for heterogeneous media.

An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

The Navier–Stokes–Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flows. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicit–explicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths.

Influence of the Parameters of Discretization on the Accuracy of Numerical Solution of the Three-Dimensional Problem of Hydrogen Diffusion

We propose an algorithm and programs for the solution of the three-dimensional problem of hydrogen diffusion. Some numerical solutions of this problem and the problem of heat conduction in the three dimensional statement are verified by analyzing test problems whose analytic solutions are known. The dependence of the relative error on the density of elements of the grid and time intervals is constructed by using the finite-element method.

Numerical Simulation of the Nonlinear Deformation and Progressive Destruction of Composite Cylindrical Shells at Nonaxisymmetric Explosive Actions

On the basis of the applied theory of shells, a resolving system of equations is formulated and a method for the numerical solution of problems of nonlinear nonaxisymmetric deformation and fracture of composite cylindrical shells at explosive loadings of different intensity is developed. A model of progressive destruction of a composite shell based on the degradation of stiffness characteristics of individual layers in a multilayer package is elaborated, which depends on the fracture mode of the binder and fiber. The reliability of the technique considered is proved by comparing calculation results with known experimental data. Results of an analysis of the effect of nonaxisymmetric arrangement of an explosive charge on the fracture behavior of composite cylindrical shells with different reinforcement structures are presented.

Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space.

This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain 'Wood frequencies' at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function-that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) This paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

Solving eigenvalues problems for Helmholtz equation by point-source method

A method of problem solution of the eigenvalues and eigenfunctions for the Helmholtz equation in the domains with arbitrary configuration is worked out. In developing the approach of the numerical solution of problems, the point-source method (PSM) is used. The proposed method is based on the analysis of the condition number of the PSM system or error of the problem numerical solution. The concept of “eigenvalues criterion” is introduced. The research result is a developed effective method - an algorithm for solving problems of eigenvalues and eigenfunctions for the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problem eigenvalue, the condition number of the PSM system and the error of the numerical solution rise sharply. Therefore, the dependence of the condition number of the PSM system or the error of the problem numerical solution can be calculated from the Helmholtz parameter. Then, according to the position of the maximum of the obtained dependences, the eigenvalues of the Helmholtz equation in a given domain are found. It allows searching the eigenvalues. After finding the eigenvalues, it is possible to proceed to the determination of the eigenfunctions. At that, if the eigenvalue appears degenerate, that is some eigenfunctions correspond to it, then it is possible to find all the eigenfunctions taking into account the symmetry of the solution domain. The two-dimensional and three-dimensional test problems are solved. Upon the results obtained, the conclusion about the efficiency of the proposed method is made.

Generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions

Purpose – 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. Design/methodology/approach – 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. Findings – 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. Originality/value – 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.

Determination of the Frequencies and Modes of Natural Oscillations of Liquids in Composed Vessels

UDC 517.9 We develop a variation method for the solution of the spectral problems for free oscillations of liquids in axially symmetric vessels of complex geometry. The problem is formulated in terms of the conjugation method. We obtain a generalized functional for which the conjugation conditions on the adjacent parts of the introduced subdomains are natural boundary-value conditions. We use the Trefts method for the reduction of the original problem to a problem of solution of an algebraic problem of low dimension. The results of calculations confirm the efficiency of the proposed method. The solution of numerous problems of the mechanics of continua for composed domains can often be significantly simplified if we split the domain into separate subdomains by introducing the corresponding conjugation conditions on the interfaces. In the construction of approximate solutions of these problems in the indicated statement, we encounter certain difficulties caused by the necessity of satisfying the conjugation conditions. The problem of linear natural oscillations of a perfect incompressible liquid in an immobile vessel in terms of the potential of shiftsˆ.x;y;z/ takes the form [1]

Wetting transitions on patterned surfaces with diffuse interaction potentials embedded in a Young-Laplace formulation.

The Minimum Energy Paths (MEPs) of wetting transitions on pillared surfaces are computed with the Young-Laplace equation, augmented with a pressure term that accounts for liquid-solid interactions. The interactions are smoothed over a short range from the solid phase, therefore facilitating the numerical solution of problems concerning wetting on complex surface patterns. The patterns may include abrupt geometric features, e.g., arrays of rectangular pillars, where the application of the unmodified Young-Laplace is not practical. The MEPs are obtained by coupling the augmented Young-Laplace with the modified string method from which the energy barriers of wetting transitions are eventually extracted. We demonstrate the method on a wetting transition that is associated with the breakdown of superhydrophobic behavior, i.e., the transition from the Cassie-Baxter state to the Wenzel state, taking place on a superhydrophobic pillared surface. The computed energy barriers quantify the resistance of the system to these transitions and therefore, they can be used to evaluate superhydrophobic performance or provide guidelines for optimal pattern design.

Numerical solution of a semi-linear inverse parabolic problem via HPM

In this work, we consider the problem of finding an unknown time-dependent function in a semi-linear parabolic equation with given initial, boundary and energy over-specified conditions. The homotopy perturbation method, a powerful technique, is applied to obtain the numerical solution of this problem. Using this method, a rapid convergent sequence can be constructed which tends to the exact solution of the problem. Some examples are presented to illustrate the strength of the method.

Numerical solution of a semi-linear inverse parabolic problem via HPM

In this work, we consider the problem of finding an unknown time-dependent function in a semi-linear parabolic equation with given initial, boundary and energy over-specified conditions. The homotopy perturbation method, a powerful technique, is applied to obtain the numerical solution of this problem. Using this method, a rapid convergent sequence can be constructed which tends to the exact solution of the problem. Some examples are presented to illustrate the strength of the method.

Humidity sensor based on microfiber resonator with reduced graphene oxide

A simple relative humidity (RH) sensor based on a tapered single mode fiber loop resonator (MLR) is proposed and experimentally verified. The refractive index of the microfiber in the MLR was modified by environmental humidity, and its emission wavelength changed with RH level. Reduced graphene oxide (rGO) material was used for this device due to its inherent characteristics of high water permeability and high sensitivity whereby long response and recovery times are reduced. Moreover, humidity sensors have been widely used in different fields and are used in the solutions of numerous problems. The performance of tapered fiber was measured with and without rGO and at fiber diameter of 4.04 μm. The spectrum was observed to shift with the wavelength as relative humidity increased, and measured sensitivity with rGO was 0.0537 nm/%RH and without rGO was 0.0316 nm/%RH in the range of 30 to 50% relative humidity.

Effective conductivity of matrix composites and foam materials by self-consistent field method using commercial FEA software

The paper is dedicated to the numerical determination of the physical fields in matrix composite materials and foams and their effective properties. The main attention is paid to the comparative analysis of using the Lagrangian Finite Elements Analysis (FEA) and the class of Gaussian approximation used by the author earlier for the numerical solution of the problem. It is intended in the paper to show that the commercial FEA software is an effective practical solver for the calculation of the physical field in the RVE of composite materials and foams and accurate enough for determination of the effective conductive properties of composites. In the paper, results of commercial FEA calculations are compared with results obtained earlier by other methods and experimental data. The results and analysis presented in the paper may be interested for the wide cycle of researchers dealing with numerical solution of the homogenization problem and the paper contains some new informative material.

Stochastic Domain Decomposition for Time Dependent Adaptive Mesh Generation

The efficient generation of meshes is an important component in the numerical solution of problems in physics and engineering. Of interest are situations where global mesh quality and a tight coupling to the solution of the physical partial differential equation (PDE) is important. We consider parabolic PDE mesh generation and present a method for the construction of adaptive meshes in two spatial dimensions using stochastic domain decomposition that is suitable for an implementation in a multi– or many–core environment. Methods for mesh generation on periodic domains are also provided. The mesh generator is coupled to a time dependent physical PDE and the system is evolved using an alternating solution procedure. The method uses the stochastic representation of the exact solution of a parabolic linear mesh generator to find the location of an adaptive mesh along the (artificial) subdomain interfaces. The deterministic evaluation of the mesh over each subdomain can then be obtained completely independently using the probabilistically computed solutions as boundary conditions. The parallel performance of this general stochastic domain decomposition approach has previously been shown. We demonstrate the approach numerically for the mesh generation context and compare the mesh obtained with the corresponding single domain mesh using a representative mesh quality measure.

Mathematical modelling of the spread of contamination during fires in forests exposed to radioactive contamination

The paper suggested in the context of the general mathematical model of forest fires [1] gives a new mathematical setting and method of numerical solution of a problem of a radioactive spread above the forest region. Numerical solution of problems of radioactive smoke spread during crown fire in exemplified heat energy release in the forest fire front was found. Heat energy release in the forest fire front was found to cause further radioactive particles spread by the action of wind. In the absence of wind, radioactive smoke particles deposit again on the underlying surface after a time. As a wind velocity increases, these particles are transferred in the ground layer over distances proportional to a wind velocity.