Blast response of a lined cavity in a porous saturated soil

The paper proposes a comprehensive approach to simulate the blast response of a lined cavity in a porous soil. To calculate the soil–structure contact pressure, the coupled Godunov-variational-difference approach was developed. The lining is modeled by a Timoshenko elastic–plastic shell with kinematic linear hardening. To solve the problem in the lining domain, the variational-difference method is applied. The soil is modeled by the Lyakhov three-phase model that takes into account both bulk and shear elastic–plastic behavior, including the effect of soil pressure on the yield strength for the stress tensor deviator. The problem of blast wave propagation within the soil medium is solved by the Godunov method. The coupled approach to calculate the soil–lining contact pressure is based on the relationships on the shock and rarefaction waves with finite-difference equations of the shell motion using a simple iteration method. It allows the reduction of the contact problem to the self-similar symmetric Riemann problem. Solution of the problem of an explosion in a porous medium, and analysis of the soil–obstacle interaction under the blast action using the proposed method show good correspondence with available experimental results. Also, the plane problem of blast response of the circular cavity lined by a thin steel lining was solved. The effect of the gas volumetric content in the soil on the incident shock wave pressure as well as on the contact pressure and lining meridian strain was studied.     

Nonstationary torsion of a tapered shaft weakened by a spherical crack

We obtain an approximate effective solution of the problem of nonstationary stress concentration near a spherical crack located inside a tapered shaft whose face is subjected to the action of impact tangential stresses. To solve the problem, we propose to use an approach based on the discretization of the problem in time with the help of a difference scheme. By the method of integral transformations, the problem is reduced to an integral equation for the unknown jump of displacements on the crack. This equation is solved by the method of orthogonal polynomials. The relation for the evaluation of the stress intensity factors is obtained. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 1, pp. 49–55, January–February, 2008.     

Simultaneous determination for a space-dependent heat source and the initial data by the MFS

In this paper we propose a numerical algorithm based on the method of fundamental solutions for recovering a space-dependent heat source and the initial data simultaneously in an inverse heat conduction problem. The problem is transformed into a homogeneous backward-type inverse heat conduction problem and a Dirichlet boundary value problem for Poisson's equation. We use an improved method of fundamental solutions to solve the backward-type inverse heat conduction problem and apply the finite element method for solving the well-posed direct problem. The Tikhonov regularization method combined with the generalized cross validation rule for selecting a suitable regularization parameter is applied to obtain a stable regularized solution for the backward-type inverse heat conduction problem. Numerical experiments for four examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.     

An ablation problem in a finite medium with time‐dependent heat flux

Abstract

The objective of this work is to develop an approximate analytical solution for the transient ablation problem in a finite medium. The medium is subjected to time‐dependent boundary heat flux, i. e., q₀=at^P , and with this refined integral technique, the complicated nonlinear problem is reduced to an initial value problem, which is then solved by the Runge‐Kutta method. Results are more accurate than with the classical heat balance integral method and also indicate that the dependence of the solution on the assumed temperature profile is much weaker than is the case with the classical integral method.     

Oscillation of a floating body in a viscous fluid

The nonlinear viscous-flow problem associated with the heaving motion of a two-dimensional floating cylinder is considered. It is formulated as an initial-boundary-value problem in primitive variables and solved using a finite-difference method based on boundary-fitted coordinates. A fractional-step procedure is used to advance the solution in time. As a case study, results are obtained for a rectangular cylinder oscillating at a Reynolds number of 10³. The nonlinear viscous forces are compared with those of linear potential theory. An assessment on the importance of viscous and nonlinear effects is made. The solution technique is sufficiently robust that extensions to consider other single and coupled modes of motion are possible.     

The boundary-element method for the determination of a heat source dependent on one variable

This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions of the direct problem and a supplementary condition, called an overdetermination. In this problem, if the heat source is taken to be space-dependent only, then the overdetermination is the temperature measurement at a given single instant, whilst if the heat source is time-dependent only, then the overdetermination is the transient temperature measurement recorded by a single thermocouple installed in the interior of the heat conductor. These measurements ensure that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method with the discrepancy principle or the L-curve criterion for the choice of the regularization parameter. The boundary-element method (BEM) is developed for solving numerically the inverse problem and numerical results for some benchmark test examples are obtained and discussed     

London Penetration Depth Measurement Using a SQUID Microscope

A method for the solution of the problem of a superconductor in an external magnetic field is developed. The method allows to convert a three dimensional vector problem into a two dimensional surface problem. Using this method, the Meissner behaviour of thin film and bulk superconductors in an external magnetic field is considered. A scheme for measurement of the local London penetration depth using a SQUID microscope of appropriate design is proposed. It is shown that the signal of such a microscope is linear with the change of the London depth and it is possible to attain a resolution of 1 nm of the measurement.     

A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid Flow Over a Linearly Stretching Surface

Present contribution is concerned with the construction and application of a numerical method for the fluid flow problem over a linearly stretching surface with the modification of standard Gradient descent Algorithm to solve the resulted difference equation. The flow problem is constructed using continuity, and Navier Stoke equations and these PDEs are further converted into boundary value problem by applying suitable similarity transformations. A central finite difference method is proposed that gives third-order accuracy using three grid points. The stability conditions of the present proposed method using a Gauss-Seidel iterative procedure is found using VonNeumann stability criteria and order of the finite difference method is proved by applying the Taylor series on the discretised equation. The comparison of the presently modified optimisation algorithm with the Gauss-Seidel iterative method and standard Newton’s method in optimisation is also made. It can be concluded that the presently modified optimisation Algorithm takes a few iterations to converge with a small value of the parameter contained in it compared with the standard descent algorithm that may take millions of iterations to converge. The present modification of the steepest descent method converges faster than Gauss-Seidel method and standard steepest descent method, and it may also overcome the deficiency of singular hessian arise in Newton’s method for some of the cases that may arise in optimisation problem(s).     

Mathematical simulation of the dynamics of a dispersed phase in free gravitational convection of a viscous incompressible liquid in a square cavity

The problem of motion of monodisperse spherical particles in a heterogeneous medium in nonisothermal free convection of a carrying incompressible liquid in a square cavity with inhomogeneous distribution of temperature on the walls is considered. The problem is solved by the finite-difference method via joint solution of equations for the carrying phase in Euler variables and the equation for a disperse particle in Lagrange variables. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 1, pp. 81–89, January–February, 2008.     

Solution of a mixed boundary-value problem by the Bubnov-Galerkin method

A method is proposed of constructing a sequence of coordinate functions with the aid of R-functions [1–3], for the solution of a mixed boundary-value problem. The approximate solution to the given problem is qualitatively the same as that obtained by the relaxation method [6].Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 4, pp. 732–736, October, 1972.     

Energy-oriented bi-objective optimisation for a multi-module reconfigurable manufacturing system

This paper investigates a multi-module reconfigurable manufacturing system for multi-product manufacturing. The system consists of a rotary table and multiple machining modules (turrets and spindles). The production plan of the system is divided into the system design phase and the manufacturing phase, where the installation cost and the energy consumption cost correspond to the two phases, respectively. A mixed-integer programming model for a more general problem is presented. The objectives are to minimise the total cost and minimise the cycle time simultaneously. To solve the optimisation problem, the ε-constraint method is adopted to obtain the Pareto front for small size problems. Since the ε-constraint method is time consuming when problem size increases, we develop a multi-objective simulated annealing algorithm for practical size problems. To demonstrate the efficiency of the proposed algorithm, we compare it with a classic non-dominated sorting genetic algorithm. Experimental results demonstrate the efficiency of the multi-objective simulated annealing algorithm in terms of solution quality and computation time.     

Numerical identification for impedance coefficient by a MFS-based optimization method

In this paper, we propose a new numerical method to solve an inverse impedance problem for Laplace's equation. The Robin coefficient in the impedance boundary condition is recovered from Cauchy data on a part of boundary. A crucial step is to transform the problem into an optimization problem based on the MFS and Tikhonov regularization. Then the popular conjugate gradient method is used to solve the minimization problem. We compare several stopping rules in the iteration procedure and try to find an accurate and stable approximation. Numerical results for four examples in 2D and 3D cases will show the effectiveness of the proposed method.     

Synthesis of shape and topology of multi-material structures with a phase-field method

In this paper, we present a phase-field method to the problem of shape and topology synthesis of structures with three materials. A single phase model is developed based on the classical phase-transition theory in the fields of mechanics and material sciences. The multi-material synthesis is formulated as a continuous optimization problem within a fixed reference domain. As a single parameter, the phase-field model represents regions made of any of the three distinct material phases and the interface between the regions. The Van der Waals–Cahn-Hilliard theory is applied to define a dynamic process of phase transition. The Γ-convergence theory is used for an approximate numerical solution to this free-discontinuity problem without any explicit tracking of the interface. Within this variational framework, we show that the phase-transition theory leads to a well-posed problem formulation with the effects of “domain regularization” and “region segmentation” incorporated naturally. The proposed phase-field method is illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods. It is further suggested that such a phase-field approach may represent a promising alternative to the widely-used homogenization models for the design of heterogeneous materials and solids, with a possible extension to a general model of multiple material phases.     

Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Thermodynamics of flows of a polymerizing,nonlinearly viscoplastic fluid having a free surface

The finite elements method is used as a basis for proposing a numerical algorithm for the solution of the problem of the flow of a polymerizing viscoplastic fluid with a free surface. The effect of the main parameters of the problem on the character of the hydrodynamic process is explained.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 59, No. 5, pp. 764–771, November, 1990.     

Axisymmetric oscillations of a cupola-shaped shell

A spectral boundary problem on axisymmetric eigenoscillations of a cupola-shaped shell is considered with emphasis on small shell thickness. The problem deals with a singularly perturbed system of ordinary differential equations. The paper examines analytical properties of the solution and, based on that, constructs an appropriate functional basis for Ritz’ method. Employing this basis provides fast convergence in the C ³-metrics.     

A plane dynamic axisymmetric problem for a hollow cylinder

A dynamic problem for an elastic hollow cylinder whose surfaces are subjected to the action of arbitrary forces is solved by a new method based on the use of finite differences only with respect to time [12]. The numerical analyses of the time dependences of stress concentration are carried out for hollow cylinders of different thickness under impact loading. The proposed method is tested by solving a problem of circular hole in an infinite body subjected to the action of impact loads over the entire surface. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 1, pp. 7–13, January–February, 2008.     

An analytical method for thermal stresses of a functionally graded material cylindrical shell under a thermal shock

In this paper, the thermal stresses of a thin functionally graded material (FGM) cylindrical shell subjected to a thermal shock are studied. An analytical method is developed. The studied problem for an FGM cylindrical shell is reduced to a plane problem. A perturbation method is used to solve the thermal diffusion equation for FGMs with general thermal properties. Then, the transient thermal stresses are obtained. The results show that the thermal shock is much easier to result in failure than the steady thermal loading. The present method can also be used to solve the crack problem of an FGM cylindrical shell with general thermal properties.     

On a robust and scalable linear elasticity solver based on a saddle point formulation

This paper presents a newly developed iterative algorithm for solving problems of linear isotropic elasticity discretized by means of mixed finite elements. It continues work started in References 1–5. The proposed method uses a pressure Schur complement approach to solve a saddle‐point system arising in the mixed formulation. As an inner solver for the displacement field variables it uses an extension to the robust black‐box multilevel procedure suggested in Reference 4. The proposed method works on a hierarchical sequence of finite element meshes to solve the problem with an arithmetic cost, nearly proportional to the dimension of the arising algebraic system. The coarsest mesh in the above sequence of meshes can consist of almost arbitrary triangular patches, which allows in practice to capture the solution even using a moderate number of successive refinement steps. The rate of convergence of the algorithm is bounded uniformly with respect to the problem coefficients, namely the Young's modulus E and the Poisson ratio ν. This makes it possible to apply the method for a broad class of engineering problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Solving an eigenvalue problem with a periodic domain using radial basis functions

A meshless method based on radial basis functions (RBFs) is proposed for solving an eigenvalue problem with a periodic domain. We compare Wendland's and Wu's compactly supported RBFs. The computational experiments examine the accuracy of the method as a result of variation in the number and layout of the original points and in addition, the shape factor. The results obtained from the method are in good agreement with the analytical solutions of the problem.