An investigation of the impact and penetration of solids of revolution into soft earth

The numerical method proposed earlier in [1] is developed to solve problems of the impact and penetration of rigid and deformable bodies of revolution into soft soil, which are described by Grigoryan's model [2]. The effect of the surface Coulomb friction, the bulk compressibility and the shear strength of soft soil on the forces of resistance and contact pressures in the contact zone is analysed. The results of numerical solutions of problems in a coherent formulation are compared with analytical relations and experimental data on the determination of the forces and coefficients of resistance to the penetration of impactors of different shapes into soft soil.     

The non-linear deformation and stability of elliptical cylindrical shells under transverse bending

A finite-element formulation of the solution of problems of the stability of non-circular cylindrical shells taking into account their bending moments and the non-linearity of their precritical stress-strain state in proposed. Explicit expressions for the displacements of the elements of non-circular cylindrical shells as rigid bodies are derived by integration of the equations obtained by equating the components of the linear strains to zero. These expressions are used to construct shape functions of an effective quadrilateral finite element of the natural curvature. An effective algorithm is developed for investigating the non-linear deformation and stability of the shells. The stability of a cylindrical shell of elliptical cross-section under transverse bending is investigated. The influence of the ellipticity and non-linearity of the deformation on the shell's stability is determined. The results of the analysis are compared with experimental data.     

Solution of problems of oblique penetration of axisymmetric projectiles into soft soil based on local interaction models

A comparative analysis of the solutions of the three-dimensional problem of the oblique penetration of a rigid body into soft soil is carried out arsing interaction models based on one-dimensional solutions of the problem of the spherical cavity expansion. Both the well-known self-similar analytical solutions for an incompressible medium as well as the generalized solution for a compressible elastoplastic medium with separation of the shock wave which arises are considered. Use of the incompressible medium hypothesis, disregarding flow separation, in estimating the maximum values of the resistive forces leads to large errors. Taking account of compressibility enables the resistive forces to be refined appreciably and enables a satisfactory estimate of the deviation of the trajectories of bodies from the initial direction of motion to the obtained. In the proposed method of solving oblique penetration problems, a three-dimensional problem is reduced, on the basis of the plane sections hypothesis and disregarding peripheral mass and momentum flows, to the combined solution of a number of axisymmetric problems for each meridional section. It is shown that, with well-known local interaction models, this approach enables the reliability of the calculation of both the force and the kinematic characteristics of the penetration process to be increased considerably due to the fact that the dynamics of the free surface and cavitation effects of the covitating flow are taken into account.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

The momentum of waves and their effect on the motion of lumped objects along one-dimensional elastic systems

Taking the example of the small longitudinal oscillations of a rod, it is shown that, in order to answer the question concerning wave momentum and its action on an obstacle, the problem of the wave motion in the medium has to be solved in a non-linear formulation. The variational formulation of problems in the dynamics of one-dimensional elastic systems with moving clampings and loads is improved taking account of non-linear factors. The equations of motion and the natural boundary conditions are obtained. The small longitudinal-transverse oscillations of a string and the motion of a bead sliding along it are considered.     

On the game theory approach to problems of optimization of elastic bodies : PMM vol. 37, n6, 1973, pp. 1098–1108

Problems of optimization of elastic bodies are considered usually in deterministic formulation, and for their solution the methods of variational calculus and the theory of optimal control are applicable (c.f., e.g., [1] and [2–4]). In the present paper there are considered those cases when either the complete information concerning the applied loads is not available,or it is known that the structure may be subjected subsequently to various loads of a certain class. The formulation is given of the problem of the determination of the shape of the elastic body, optimal for a class of loads, and there is indicated a general scheme for its solution based on the “minimax” approach used in the theory of games. Problems of optimization of elastic beams are considered and as a result of their solution certain features of optimal shapes are exhibited.     

Study of a Numerical Approach for a Transient Flow Problem in Porous Media

In this work, we deal with the numerical study of the new approximation method proposed in [7] for a transient flow problem in porous media. The stationary problem, obtained from a time discretization of this transient problem, is considered as an optimal shape design formulation. We prove the existence of the solution of the discrete optimal shape problem obtained from finite element discretization. We study the convergence and give numerical results showing the efficiency of the proposed approach.     

Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

Spatial contact problems for rough elastic bodies under elastoplastic deformations of the unevenness

On the basis of /1, 2/, a model is constructed for the contact between a rigid stamp and a rough body taking elastoplastic deformations of the unevenness into account. The contact model for rough bodies with elastic deformations of the unevenness is a special case. A classical approach utilizing boundary integral equations is applied in the mathematical formulation of the contact problem. Under quite general assumptions (for instance, the multiconnectedness of the contact domain desired), the uniqueness and existence of the solution are investigated. A method is developed to determine the contact pressure, the closure of the bodies, and also the contact area which consists of two parts in the general case, a zone of elastoplastic deformation of the unevenness and a zone of their elastic deformation. The efficiency of the method is shown in examples of new contact problems. The solution is represented in a convenient form for analysing the influence of the roughness. This is of considerable value for material testing by a contact method. A fairly complete survey of research on contact problems for rough bodies can be found in /1–4/.     

A global optimization point of view to handle non-standard object packing problems

This work originates from research carried out in support to the cargo accommodation of space vehicles/modules. The goal of this activity is to maximize the loaded cargo, taking into account the given accommodation requirements. Items can often be modelled as parallelepipeds, but it is even more frequent that real-world issues make this approximation no longer acceptable. These aspects and the presence of additional overall conditions, such as balancing, give rise to very challenging non-standard packing problems, not only in the frame of space engineering, but also in different application areas. This article considers first the orthogonal packing of tetris-like items, within a convex domain and subsequently the packing of polygons with (continuous) rotations in a convex domain. The proposed approach is based on mixed integer linear/non-linear programming (MIP, MINLP), from a global optimization point of view. The tetris-like formulation is exploited to provide the MINLP solution process with an approximated starting solution.     

Variational principles and mathematical methods of solving boundary-value problems for piecewise uniform nonlinear media

We give a variational formulation and a scheme for numerical solution of boundary-value problems for piecewise-uniform nonlinear media taking account of frictional forces. We solve the problem of determining the stressed state of a thick three-layer cylinder of elastoplastic material with linear reinforcing under the action of internal pressure and an axially symmetric temperature field.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 64–67.     

An invariant variational problem of the laminar boundary layer

A variational problem on minimizing, by normal injection into a laminar boundary layer, the Newtonian drag of a blunt cylindrical body in a supersonic flow of an ideal gas is considered, taking into account the limitation on the power of the system to control the injection. Using the first integral obtained, the order of the conjugate system is reduced, which enables an effective algorithm to be constructed for finding the optimal control using the grid method. The results of a computational experiment are presented, according to which the gains in the values of the drag functional for the optimal controls obtained reach 65% compared with a uniform injection law.     

Computational Excavation Analysis of a Single Cutting Tool using a Hypoplastic Constitutive Model

We propose a computational model for the numerical analysis of the dynamic interaction between a single excavation tool and the surrounding soil. An incremental non-linear (hypoplastic) constitutive model is employed to capture the complex response of soft soils. Large displacements and deformations are handled by an Updated Lagrangian formulation, the particle finite element method (PFEM). (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Averaging of a finely laminated elastic medium with roughness or adhesion on the contact surfaces of the layers

The governing relations of a laminated elastic medium with non-ideal contact conditions in the interlayer boundaries are obtained by an asymptotic averaging method. The interaction of rough surfaces is described by a non-linear contact condition which simulates the local deformation of the microroughnesses using a certain penetration of the nominal surfaces of the elastic layers. The cohesive forces, caused by the thin adhesive layer, are described within the limits of the Frémond model which includes a differential equation characterizing the change in the cohesion function. A piecewise-linear approximation of the initial positive segment of the Lennard–Jones potential curve is proposed to describe of the adhesive forces between smooth dry surfaces. A comparison is made with the solution obtained within the limits of the Maugis–Dugdale model based on a piecewise-constant approximation. Solutions of the above problems are constructed taking account of the possible opening of interlayer boundaries.     

Optimal operation of multireservoir power systems

The optimal monthly operating policy of a multireservoir hydroelectric power system is a stochastic nonlinear problem. This paper presents a new method for determining the optimal monthly operating policy of a power system ofn reservoirs in series on a river taking into account the stochasticity of the river flows. Functional optimization techniques and minimum-norm formulation have been used to find the optimal release policy of the system. Results for a numerical example composed of four reservoirs are presented.This work was supported by the National Research Council of Canada, Grant No. A4146.     

Optimization of axisymmetric membrane shells

Problems of the joint optimization of the shape and distribution along the meridian of the thickness of membrane shells of revolution under the action of axisymmetric loads are considered, taking account of the constraints concerning the strength of the shell and the volume of its cavity. General formulations of problems of the optimal design of shells of revolution are given and the optimal shape of a shell and the corresponding thickness distribution are investigated. Results of the exact solution of problems of the optimal design of closed shells of revolution when there is an internal pressure are presented. The simultaneous introduction of two control functions, describing the shape of the shell and the distribution of its thickness, not only ensures a substantial reduction in the mass of a shell but also leads to significant mathematical simplifications, which enable the solution of the optimization problem being considered to be obtained in an analytical form.     

Stochastic shape sensitivity in powder metallurgy processing

Stochastic shape sensitivity in forming process of powder metallurgy materials is analyzed. For this purpose the rigid-poroplastic material model has been assumed. The theoretical formulation for stochastic shape sensitivity is described which presents probabilistic distributions taking into account random initial and boundary conditions. The control volume approach is discussed. Stochastic finite element equations for rigid – poroplastic materials are solved for the first two probabilistic moments. Numerical simulations were performed to illustrate shape sensitivity problems in the process of compression of rigid-poroplastic cylinder. The differences in deterministic and stochastic sensitivities are presented. The results derived can be used for the subsequent quantitative stochastic shape design as well as stochastic shape optimization.     

Influence of the saturation–suction relationship in the formulation of non-saturated soil consolidation models

The main scope of this paper is to present a fully coupled numerical model for isothermal soil consolidation analysis based on a combination of different stress states. Being originally a non-symmetric problem, it may be straightforward reduced to a symmetric one, and general guidelines for the conditions in which this reduction may be carried out, are addressed. Non-linear saturation–suction and permeability-suction functions were incorporated into a Galerkin approach of the non-saturated soil consolidation problem, which was solved using the finite element method.In order to validate the model, various examples, for which previous solutions are known, were solved. The use of either a strongly non-linear and non-symmetric formulation or a simple symmetric formulation with accurate prediction in deformation and pore-pressures is extremely dependent on the soil characteristic curves and their derivatives and this aspect is taken into account in the present mathematical approach. The emergent coupling effects may be easily uncoupled in the computer model by merely recasting some coefficients of the discrete equation system.     

Multiobjective Stochastic Control in Fluid Dynamics via Game Theory Approach: Application to the Periodic Burgers Equation

The purpose of the present work is to implement well-known statistical decision and game theory strategies into multiobjective stochastic control problems of fluid dynamics. Such goal is first justified by the fact that deterministic (either singleobjective or multiobjective) control problems that are obtained without taking into account the uncertainty of the model are usually unreliable. Second, in most real-world problems, several goals must be satisfied simultaneously to obtain an optimal solution and, as a consequence, a multiobjective control approach is more appropriate. Therefore, we develop a multiobjective stochastic control algorithm for general fluid dynamics applications, based on the Bayes decision, adjoint formulation and the Nash equilibrium strategies. The algorithm is exemplified by the multiobjective stochastic control of a periodic Burgers equation.     

A Benders decomposition algorithm for a multi-area,multi-stage integrated resource planning in power systems

The aim of this paper is to propose an integrated model for resource planning in power systems by taking into account both supply and demand sides options simultaneously. At supply-side, investment in generation capacity and transmission lines is considered. Demand side management (DSM) technologies are also incorporated to correct the shape of the load duration curve in terms of peak clipping and load shifting programmes. A mixed integer non-linear programming model is developed to find the optimal location and timing of electricity generation/transmission as well as DSM options. To solve the resulting complex model, nonlinearity caused by transmission loss terms are first eliminated using the piecewise linearization technique. Then, a Benders decomposition (BD) algorithm is developed to solve the linearized model. The performance of the proposed BD algorithm is validated via applying it to the 6-bus Garver test system and a modified 21-bus IEEE reliability test system.