Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

The interaction of punches on the face of an elastic wedge

The interaction of two punches, which are elliptic in plan, on the face of an elastic wedge is investigated in a three-dimensional formulation for different types of boundary conditions on the other face. The wedge material is assumed to be incompressible. An asymptotic solution is obtained for punches which are relatively distant from one another and from the edge of the wedge. For the case when the punches are arranged relatively close to the edge of the wedge (or reach the edge, the contact area is unknown) the numerical method of boundary integral equations is used. The mutual effect of the punches is estimated by means of calculations. The asymptotic solution of the generalized Galin problem, concerning the effect of a concentrated force applied on the edge of the three-dimensional wedge on the contact pressure distribution under a circular punch relatively far from the edge, is obtained.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

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.     

The contact problem for a rectangle with stress-free side faces

The plane contact problem for an elastic rectangle into which two symmetrically positioned punches are impressed is considered. Homogeneous solutions are constructed that leave the side faces of the rectangle stress-free. When the modified boundary conditions using generalized orthogonality of the homogeneous solutions are satisfied, the problem reduces to a Friedholm integral equation of the first kind in the function describing the displacement of the surface of the rectangle outside the contact area. This function is sought in the form of the sum of a trigonometric series and a power function with a root singularity. The ill-posed infinite system of algebraic equations thereby obtained is regularized by introducing a small positive parameter (Ref. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978), and, after reduction, has a stable regularized solution. Since the matrix elements of the system are determined by a poorly converging number series, an effective method was developed for calculating the residues of the series. Formulae are found for the contact pressure distribution function and dimensionless indentation force. Since the first formula contains a third-order derivative of the functional series, when it is used, a numerical differentiation procedure is employed (Refs. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978; Danilina NI, Dubrovskaya NS, Kvasha OP et al. Numerical Methods. Textbook for Special Colleges. Moscow: Vysshaya Shkola; 1976). Examples of a calculation for a plane punch are given.     

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.     

The spatial contact problem for an elastic layer and wavy punch when there is friction and wear

The spatial (three-dimensional) problem of the wear of a wavy punch sliding over an elastic layer bonded to a rigid base, assuming there is complete contact between the punch and the layer, is considered. It is assumed that there is Coulomb friction and wear of the punch. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the harmonic functions in which are represented in the form of double Fourier integrals, after which the problem reduces to a linear system of differential equations. It is established that the harmonics constituting the shape of the punch and the contact pressure are shifted with respect to one another in time along the sliding line of the punch. The velocity of this shift depends on the longitudinal and transverse frequencies of the harmonic, that is, dispersion of the waves is observed.     

The solution of contact problems using boundary element method

The formulation of contact problems is extended to the case of moving punches and to the case when the state of the systems being investigated depends on the history of the change in the external actions. The quasi-static contact problem for a moving rigid rough punch and a single linearly deformable body is considered. A new iterational process is proposed for solving contact problems, taking friction in the contact area into account, and its convergence is proved. An algorithm of the solution, based on the boundary element method, is developed. Solutions of specific problems are given and analysed. Estimates of the difference of the solutions due to the difference in the impenetrability conditions and the difference in the steps of the loading parameter are obtained.     

A contact problem of the theory of consolidation for a strip

A plane mixed boundary-value problem of the linear theory of inertialess two-phase consolidation is considered [1]. A strip lying on a smooth undeformable foundation, impermeable to liquid, is under the pressure of a semi-infinite permeable. The material of the solid phase and the liquid are compressible. Using Laplace transformations with respect to time and the space coordinate, the problem is reduced to a Wiener-Hopf equation. The general features of the distribution of the roots of the characteristic equations, corresponding to different homogeneous conditions on the faces of the strip, are investigated. An effective solution is constructed in multiple integrals which converge exponentially with respect to all the variables. The temporal processes of the settling of the punch and the extrusion of the liquid are investigated.     

Shape optimization of a body in the Oseen flow

This article is concerned with a numerical simulation of shape optimization of the Oseen flow around a solid body. The shape gradient for shape optimization problem in a viscous incompressible flow is computed by the velocity method. The flow is governed by the Oseen equations with mixed boundary conditions containing the pressure. The structure of continuous shape gradient of the cost functional is derived by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. A gradient type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Hypersonic shapes of maximum drag for a given heat-transfer rate

The problem of determining the shape of a blunt, axisymmetric body which maximizes the drag for a given heat-transfer rate and diameter is considered. For blunt bodies, pressure drag predominates and is estimated from modified Newtonian flow considerations. With regard to heat transfer, it is assumed that the body is operating in the range of hypersonic speeds where the radiative heating rate can be neglected with respect to the convective heating rate. The latter is estimated from the boundary-layer analysis due to Lees. Optimum power-law shapes as well as variational shapes are determined and are shown to yield almost identical results. For low-to-moderate values of the convective heat-transfer parameter, the optimum shape is very flat and is approximately a one-half power-law shape; in this range, spherical segments are approximately one-half power-law shapes and, hence, are nearly maximum drag shapes. There exists a maximum value of the convective heat-transfer parameter for which maximum drag shapes exist, and the corresponding optimum shape is a cone, or a power-law shape of exponent unity. This limiting shape is shown to be that which maximizes the convective heat-transfer rate for a given diameter.This research was supported in part by NASA-Manned Spacecraft Center under Contract No. NAS-6963. The authors are indebted to Dr. John J. Bertin for helpful discussions and suggestions concerning heattransfer aspects of this paper.     

Shape and topology optimization of contact problems by level set methods

This paper deals with the numerical solution of a topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is considered. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. In the paper shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization. Level set based numerical algorithm for the solution of the shape optimization problem is proposed. Level set method is used to describe the position of the boundary of the body and its evolution on a fixed mesh. This evolution is governed by Hamilton – Jacobi equation. The speed vector field driving the propagation of the boundary of the body is given by the shape derivative of a cost functional with respect to the free boundary. Numerical examples are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hypersonic airfoils of maximum lift-to-drag ratio

The problem of determining the slender, hypersonic airfoil shape which produces the maximum lift-to-drag ratio for a given profile area, chord, and free-stream conditions is considered. For the estimation of the lift and the drag, the pressure distribution on a surface which sees the flow is approximated by the tangent-wedge relation. On the other hand, for surfaces which do not see the flow, the Prandtl-Meyer relation is used. Finally, base drag is neglected, while the skin-friction coefficient is assumed to be a constant, average value. The method used to determine the optimum upper and lower surfaces is the calculus of variations. Depending on the value of the governing parameter, the optimum airfoil shapes are found to be of three types. For low values of the governing parameter, the optimum shape is a flat plate at an angle of attack followed by slightly concave upper and lower surfaces. The next type of solution has a finite thickness over the entire chord with the upper surface inclined so that the flow is an expansion. Finally, for the last type of solution, the upper surface begins with a portion which sees the flow and is followed by an inclined portion similar to that above. For all of these solutions, the lower surface sees the flow. Results are presented for the optimum dimensionless airfoil shape, its dimensions, and the maximum lift-to-drag ratio. To calculate an actual airfoil shape requires an iteration procedure due to the assumption on the skin-friction coefficient. However, simple results can be obtained by assuming an approximate value for the skin-friction coefficient.This research was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR Grant No. 69-1744.     

Design optimization and forming methods for toroidal composite shells

Based on the example of a toroidal membrane, a model for calculating the winding trajectory and the shape of a shell billet and its transformation into given surface elements, as well as for calculating the shape of the membrane under an internal pressure loading, is developed. The problem of choosing optimum design variables and manufacturing parameters of the membrane is also investigated. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 2, pp. 147–164, March–April, 2006.     

The contact problem for an elastic strip and a wavy punch when there is friction and wear

The plane problem of the mutual wear of a wavy punch and an elastic strip, bonded to an undeformable foundation under the condition of complete contact between the punch and the strip is considered. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the two harmonic functions in which are represented in the form of Fourier integrals after which the problem reduces to a non-linear system of differential equations. In the case of a small degree of wear of the strip, this system becomes linear and admits of a solution in explicit form. The harmonics, constituting the profile of the punch and the contact pressure, move along the strip with respect to one another and are shifted in time. Conditions are obtained that ensure the hermetic nature of the contact between the wavy punch and the strip when there is friction and wear.     

The periodic contact problem of the plane theory of elasticity. Taking friction,wear and adhesion into account

A solution of the plane problem of the contact interaction of a periodic system of convex punches with an elastic half-plane is given for two forms of boundary conditions: 1) sliding of the punches when there is friction and wear, and 2) the indentation of the punches when there is adhesion. The problem is reduced to a canonical singular integral equation on the arc of a circle in the complex plane. The solution of this equation is expressed in terms of simple algebraic functions of a complex variable, which considerably simplifies its analysis. Asymptotic expressions are obtained for the solution of the problem in the case when the size of the contact area is small compared with the distance between the punches.     

Shape optimization in contact problems of the theory of elasticity with incomplete external loading data

The contact interaction without friction of an absolutely rigid punch with an elastic half-space is considered. The external loads on the elastic medium are not fixed in advance, but a set containing all the admissible forms of applied forces is assumed to be specified. Using a guaranteed (minimax) approach, problems of optimizing the shape of the punch from the condition that its mass is a minimum are formulated. Inequality-type constraints, imposed on the total force and moments applied to the punch from the elastic-medium side, are assumed. Using Betti's reciprocal theorem and calculating the “worst” case for different types of constraints, the corresponding forces are determined and the optimum shape of the punch is obtained in analytical form.     

Robust optimal ship hulls based on Michell's wave resistance

We seek the hull of a ship with a given volume which minimizes the water resistance with uncertainties on the cruising speed. The water resistance is based on Michell's wave resistance functional, and the speed is a random variable whose probability distribution is known. We first handle the case where the support of the hull is given and then we also optimize this support for a given area. In each case, an optimal hull is shown to exist. The numerical simulations are costly so we adapt to our problem Newton's method for shape optimization. The Dirichlet energy is used as a test case. The numerical results for the robust optimal hull are compared with the case where the cruising speed is known.     

On the optimization of linear,time-variant,multivariable control systems using the contraction mapping principle

The problem of optimizing a linear, time-variant, multivariable control system with quadratic cost functional is first phrased in the context of functional analysis. It is shown that the optimum control is given implicitly as the solution of a matrix integral equation. After the fixed-point contraction mapping theorem is invoked, an iterative method for solving this equation is developed and conditions for its convergence to a unique optimum are derived. Techniques for transforming the system operator are discussed so that, when convergence of the original sequence of iterations cannot be assured, that of the transformed system can. However, it is shown specifically that there is always a finite optimization interval for which the procedure may be used. Bounds are also given for the errors, in the sense of norms, between the control aftern iterations and its ultimate value and between the cost functional aftern iterations and its ultimate value. These bounds are used to decide when to terminate the sequence. Solutions of the iterative scheme using a hybrid computer in parallel and serial modes are discussed and the delays inherent in both methods calculated. It is concluded that the method can be used to track an optimum control system, which drifts from optimum because of parameter variations, with little delay and particularly when the optimization interval is extrapolated only a little into the future. Comparison of the proposed scheme with the steepest-descent approach developed by Balakrishnan shows that the present scheme requires one-third of the computations per step and, therefore, may converge more quickly.The author is indebted to the Principal and the Governors of Sunderland Polytechnic for the facilities placed at his disposal and permission to publish this work.     

The application of adjoint method for shape optimization in Stokes–Oseen flow

This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes–Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient‐type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd.