Numerical investigation of supersonic flow for axisymmetric cones

An existing curvilinear finite-volume code with robust shock-capturing scheme was modified to allow for simulations of supersonic flow for axisymmetric cone geometries. It is shown how for an axisymmetric coordinate system the convective and viscous flux derivatives in the circumferential direction reduce to a y-momentum equation source term. The advantage of this approach over an axisymmetric code is that the governing equations and the discretization do not need to be changed. This paper provides a detailed derivation of the axisymmetric source terms from the full Navier-Stokes equations. Results are shown for a sharp and a blunt cone for approach flow Mach numbers of M = 3.5 and M = 7.99.     

Use of fundamental solutions in the collocation method in axisymmetric elastostatics

A method referred to as the fundamental collocation method is applied to problems of axisymmetric elastostatics. In the method the governing field equations are satisfied exactly using fundamental solutions corresponding to concentrated forces while the boundary conditions are satisfied approximately using an overdeterminate collocation technique. Numerical results are given for two stress concentration problems. The paper is concluded by a critical discussion of the merits of the method.     

Numerical solution of a turning point problem in elasticity

Summary The axially symmetric equilibria of a stress-free thin elastic spherical cap are studied in dependence of a “thickness” parameter. The attention is focused on the turning points of the corresponding bifurcation diagram. The numerical method is based on an approximation of the osculating parabola in the turning point. The rate of convergence is related to the Fibonacci numbers.     

Comparison of elasticity theory and shell theory solutions

The axisymmetric bending of cylinder-cone shell junction has been analysed by the finite element method using three dimensional elasticity theory. Numerical results have been worked out for three shells having cylinder diameter to thickness ratios of 41, 21 and 13. These results have been compared with the results obtained by the classical thin shell theory of Sanders. The validity of the assumptions made in the classical thin shell theory is discussed. The nature of singularity at the re-entrant corner has been studied.     

Numerical prediction of discontinuous central bursting in axisymmetric forward extrusion by continuum damage mechanics

The prediction of the central burst defects in axisymmetric cold extrusion is analyzed numerically by using 2D finite element analysis (FEA) accounting for the ductile damage effect. The coupling between the ductile damage and the thermoelastoplastic constitutive equations is formulated in the framework of the thermodynamics of irreversible processes together with the continuum damage mechanics (CDM) theory. A simple isotropic ductile damage model is fully coupled with thermoelastoplastic constitutive equations of Prandtl-Reuss type including non-linear isotropic hardening and thermal effects. A modified ductile damage criterion based on linear combination of the stress tensor invariants is proposed in order to predict the occurrence of micro-crack initiation as a discontinuous central bursts along the bar axis. The implicit integration scheme of the fully coupled constitutive equations and the iterative resolution scheme to solve the associated thermomechanical equilibrium problem are presented. A three fields (velocity, hydrostatic pressure and temperature) variational formulation is used to solve the resulting algebraic system. The effects of various process parameters, namely, the diameter reduction ratio, the die semi-angle, the friction coefficient and the material ductility, …, on the central bursts occurrence are discussed. The quantitative effects of ductile damage on the extrusion parameters are studied and qualitative comparison with some available experimental data are given.     

规整化人脸图像超分辨率重建的数值解法

提出了基于特征子空间规整化的人脸图像超分辨率重建（SRR）算法并给出了三种数值解法。在仿射变换运动模型下,将图像的四邻域插值方法拓展为图像的梯度场估计问题,推导出了待求高分辨率（HR）图像关于运动参数的雅科比矩阵;并根据对SRR代价函数的全微分和偏微分展开,将非线性的SRR问题转换为线性问题迭代求解,讨论了三种运动参数与HR图像的联合迭代估计算法。给出了SRR规整化参数的自适应计算方法以实现自动SRR。仿真结果证实：采用的人脸子空间规整化方法优于传统规整化方法（拉普拉斯、全变差）,尤其在低信噪比时可以获得良好的人脸图像SRR效果。     

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.     

Similarity solutions for axisymmetric plane radial power law fluid flows through a porous medium

This paper investigates the unsteady flow of non-Newtonian fluids of power low behavior through a porous medium in a plane radial geometry. The equation governing the flow is a nonlinear parabolic partial differential equation with a source term whose solution satisfies certain fixed and moving boundary conditions. The attention is focused on the finding of similarity solution when the fixed boundary condition and the source term satisfy certain restrictions. In this case similarity transformations are determined and the resulting ordinary differential equations are deduced. For shear thinning fluids the existence of a pressure disturbance front moving with finite velocity is shown and expression for its location as a function of time is determined. The solutions in closed form have been given for certain particular cases where the resulting differential equations can be analytically solved. A numerical procedure has also been presented.     

Numerical solutions of fourth order variational inequalities

We use variational inequality theory along with finite difference technique to obtain an approximation for the solution of a class of obstacle problem in elasticity, like those describing the equilibrium configuration of an elastic stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the obstacle problems.     

Numerical solutions of the regularized long-wave equation

The combined approach of quasilinearization and invariant imbedding is used for computing solutions of the nonlinear regularized long-wave (RLW) equation. The accuracy and efficiency of the scheme is tested by obtaining a solitary wave solution of the equation. In another example the development of an undular bore is discussed. The results are in good agreement with the available results.     

Numerical solutions of a nonlinear reaction-diffusion system

This paper is concerned with finite difference solutions of a coupled system of nonlinear reaction-diffusion equations. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. The existence and uniqueness of a non-negative finite difference solution and three monotone iterative algorithms for the computation of the solutions are given. It is shown that the time-dependent problem has a unique non-negative solution, whereas the steady-state problem may have multiple non-negative solutions depending on the parameters in the problem. The different non-negative steady-state solutions can be computed from the monotone iterative algorithms by choosing different initial iterations. Also discussed is the asymptotic behaviour of the time-dependent solution in relation to the steady-state solutions. The asymptotic behaviour result gives some conditions ensuring the convergence of the time-dependent solution to a positive or semitrivial non-negative steady-state solution. Numerical results are given to demonstrate the theoretical analysis results.     

Returns to scale and scale elasticity in the presence of weight restrictions and alternative solutions

In this paper some new results about two important topics in performance analysis, including scale elasticity (SE) and returns to scale (RTS), in the presence of weight restrictions and alternative solutions are proved. Since SE and RTS help managers to make decisions about the expansion or contraction of the operation of decision making units under assessment, the established results can be useful from both theoretical and applied points of view. The provided implications are devoted to some mathematical characterizations and properties of SE and RTS as well as their relationships.     

Numerical simulations of the flow and sound fields of a heated axisymmetric pulsating jet

The flow and sound fields of a heated axisymmetric pulsating jet have been investigated by direct numerical solution of the compressible Navier-Stokes equations in cylindrical coordinates using highly accurate numerical methods. Effects of pulsating frequency and amplitude on the flow structure and sound generation have been examined. The results show that the organized unsteadiness associated with the periodic pulsation leads to a variety of vortical structures in the pulsating flow field. The pulsating frequency and amplitude strongly affect the vortical flow structures and the radiated sound fields. At the higher pulsating frequency, the vortices in the pulsating jet become smaller and the radiated sound field has smaller wavelength. The pressure fluctuation amplitude of the sound field is inversely proportional to the distance from the sound source. Both the larger pulsating frequency and larger pulsating amplitude lead to stronger sound emissions from the pulsating jet. It has been found that the pressure fluctuation amplitude of the sound field is approximately proportional to the pulsating amplitude and frequency.     

Numerical solutions to the Witsenhausen counterexample byapproximating networks

Approximate solutions to the Witsenhausen counterexample (1968) are derived by constraining the unknown control functions to take on fixed structures containing “free” parameters to be optimized. Such structures are given by “nonlinear approximating networks”, i.e., linear combinations of parametrized basis functions that benefit by density properties in normed linear spaces. This reduces the original functional problem to a nonlinear programming one which is solved via stochastic approximation. The method yields lower values of the costs than the ones achieved so far in the literature, and, most of all, provides rather a complete overview of the shapes of the optimal control functions when the two parameters that characterize the Witsenhausen counterexample vary. One-hidden-layer neural networks are chosen as approximating networks     

Numerical stabilization of recurrence relations with vanishing solutions

Homogeneous recurrence relations exhibit a highly numerical unstable behaviour in step-by-step evaluation of succesive terms. It is pointed out that this is a result of the presence of vanishing solutions, which are always added to initial values for the recursion scheme, due to finite machine accuracy. Stabilization of the recursion is shown to be identical with resolving these vanishing contributions with sufficient accuracy. To this end, explicit analytical expressions for these solutions, as products of continued fractions, are given. Application of these vanishing solutions enables us to construct the self-consistent, numerical stable general solution of the recursion relation.     

Numerical solutions for constrained time-delayed optimal control problems

In this paper, time-delayed optimal control problems governed by delayed differential equation are solved. Two different techniques based on integration and differentiation matrices are considered. The time-delayed term of the problem has been approximated by Chebyshev interpolating polynomials. On this basis, the optimal control problem can be solved as a mathematical programming problem. The example illustrates the robustness, accuracy and efficiency of the proposed numerical techniques.     

Numerical verification of solutions for elasto-plastic torsion problems

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for the elasto-plastic torsion problems governed by the variational inequality. Based upon the finite element approximations and the explicit a priori error estimates for a simple problem, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. This paper is an extension of the previous paper [1] in which we mainly dealt with the obstacle problems, but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem. A numerical example is illustrated.