Superconvergent Patch Recovery for plate problems using statically admissible stress resultant fields

In this paper, we study an approach for recovery of an improved stress resultant field for plate bending problems, which then is used for a posteriori error estimation of the finite element solution. The new recovery procedure can be classified as Superconvergent Patch Recovery (SPR) enhanced with approximate satisfaction of interior equilibrium and natural boundary conditions. The interior equilibrium is satisfied a priori over each nodal patch by selecting polynomial basis functions that fulfil the point‐wise equilibrium equations. The natural boundary conditions are accounted for in a discrete least‐squares manner. The performance of the developed recovery procedure is illustrated by analysing two plate bending problems with known analytical solutions. Compared to the original SPR‐method, which usually underestimates the true error, the present approach gives a more conservative error estimate. Copyright © 1999 John Wiley & Sons, Ltd.     

An <Emphasis Type="Italic">hp</Emphasis>-adaptive finite element/boundary element coupling method for electromagnetic problems

We present an hp-version of the finite element / boundary element coupling method to solve the eddy current problem for the time-harmonic Maxwell’s equations. We use H(curl, Ω -conforming vector-valued polynomials to approximate the electric field in the conductor Ω and surface curls of continuous piecewise polynomials on the boundary Γ of Ω to approximate the twisted tangential trace of the magnetic field on Γ. We present both a priori and a posteriori error estimates together with a three-fold hp-adaptive algorithm to compute the fem/bem coupling solution with appropriate distributions of polynomial degrees on suitably refined meshes.     

Boundary condition for the equation of particle diffusion in nonuniform flow

The boundary condition on an absorbing surface for the equation of particle diffusion is obtained from the approximate solution of the Fokker-Planck equation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 55, No. 5, pp. 735–739, November, 1988.     

Heat and mass transfer in a binary laminar boundary layer with free convection on a vertical porous surface

An approximate analytical solution of heat and mass transfer in a binary laminar boundary layer with free convection on a vertical surface is presented. The numerical solution is compared with an approximate analytical solution obtained by another method.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 21, No. 1, pp. 19–28, July, 1971.     

Method of determining the failure parameters of structural members with edge cracks

Conclusions The theoretical-experimental method of determining the fracture parameters of structural members has been generalized for edge cracks on the basis of a solution for a half-plane with an edge crack.The error of determining the stress-intensity factors K_I and K_II does not exceed 2–3%. In many cases, the method makes it possible to determine, with the accuracy sufficient for practice, the stresses acting along the crack-line prior to crack appearance.The results show that the polynomials (7) can be used to approximate the load function. The number of terms, retained in the expansion (7), can be selected on the basis of the criterion (8). The approximation (9) is more suitable for solving the contact problems. Using this approximation, it is possible to have on average the error of determining the SIF and restore the loads applied to the panel.To apply the method, it is sufficient to measure the relative displacement of the edges in 1–3 points at a large distance from the tip.For the cracks extending to the curvilinear boundary the accuracy of restoration of the stresses by the proposed method depends on the b/R ratio, whereas there is no such dependence for a straight boundary.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 6, pp. 29–35, November–December, 1992.     

Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method

In this paper a singular integral equation method is applied to calculate the distribution of stress intensity factor along the crack front of a 3D rectangular crack. The stress field induced by a body force doublet in an infinite body is used as the fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r ^–3. In solving the integral equation, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function, which expresses stress singularity along the crack front in an infinite body. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary.     

有限元网格修正的自适应分析及其应用

本文在对有限元变量连续条件分析的基础上，将应力误差范数用于计算结果的误差估计，使非结构化网格生成系统与有限元计算有机地结合起来，并将网格单元修正的自适应分析应用于二维应力集中问题的研究，从而实现了有限元最佳化离散，提高了有限元数值求解的可靠性和近似程度。     

Error estimation using hypersingular integrals in boundary element methods for linear elasticity

A natural measure of the error in the boundary element method rests on the use of both the standard boundary integral equation (BIE) and the hypersingular BIE (HBIE). An approximate (numerical) solution can be obtained using either one of the BIEs. One expects that the residual, obtained when such an approximate solution is substituted to the other BIE is related to the error in the solution. The present work is developed for vector field problems of linear elasticity. In this context, suitable ‘hypersingular residuals’ are shown, under certain special circumstances, to be globally related to the error. Further, heuristic arguments are given for general mixed boundary value problems. The calculated residuals are used to compute element error indicators, and these error indicators are shown to compare well with actual errors in several numerical examples, for which exact errors are known. Conclusions are drawn and potential extensions of the present error estimation method are discussed.     

Error estimation based on Superconvergent Patch Recovery using statically admissible stress fields

In this paper we investigate an approach for a posteriori error estimation based on recovery of an improved stress field. The qualitative properties of the recovered stress field necessary to obtain a conservative error estimator, i.e. an upper bound on the true error, are given. A specific procedure for recovery of an improved stress field is then developed. The procedure can be classified as Superconvergent Patch Recovery (SPR) enhanced with approximate satisfaction of the interior equilibrium and the natural boundary conditions. Herein the interior equilibrium is satisfied a priori within each nodal patch. Compared to the original SPR-method, which usually underestimates the true error, the present approach gives a more conservative estimate. The performance of the developed error estimator is illustrated by investigating two plane strain problems with known closed-form solutions. © 1998 John Wiley & Sons, Ltd.     

The turbulent boundary layer in the initial segment of axisymmetric channels when flow swirling occurs at the inlet

We present an approximate solution for the problem of a turbulent boundary layer in an incompressible liquid in the case of flow swirling at the inlet.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 1, pp. 95–102, July, 1969.     

Two-model iteration method for the solution of an inverse boundary heat-exchange problem

A method is proposed for restoration of the boundary condition, with an iterative correction of the initial data used in this method, involving the utilization of both exact and approximate heat-transfer models.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 2, pp. 313–319, February, 1989.     

Error in the numerical solution to the heat conduction problem with a nonlinear boundary condition

Concerning the heat conduction problem with a nonlinear boundary condition, as in the case of a boiling process, the effect of the time interval in the numerical solution scheme on the error of the solution is analyzed here.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.22, No. 5, pp. 894–898, May, 1972.     

The nonsteady temperature field of a solid under the nonsymmetric and time-variable conditions of heat transfer

A method is proposed for the approximate solution of nonsteady heat-conduction problems under the nonsymmetric and time-variable boundary conditions of the third kind, applicable to bodies of arbitrary shape.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 1, pp. 118–126, July, 1969.     

Solution of a boundary-value problem for a nonlinear differential equation by the method of successive approximations

A method is proposed for obtaining an approximate solution of a boundary value problem for the ordinary nonlinear heat-transfer equation in plane Poiseuille flow when the viscosity varies exponentially with the temperature.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 26. No. 3, pp. 539–543, March, 1974.     

“Averaging” method in problems of convective mass and heat transfer

An approximate method is proposed for the solution of nonlinear boundary problems of convective heat and mass transfer; the method is based on the procedure of averaging equations or boundary conditions.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 2, pp. 205–215, August, 1984.     

Residual Correction Procedure with Bernstein Polynomials for Solving Important Systems of Ordinary Differential Equations

One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems. Systems of ordinary differential equations like systems of second-order boundary value problems (BVPs), Brusselator system and stiff system are significant in science and engineering. One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations. Bernstein polynomials method with residual correction procedure is used to treat those challenges. The aim of this paper is to present a technique to approximate solutions of such differential equations in optimal way. In it, we introduce a method called residual correction procedure, to correct some previous approximate solutions for such systems. We study the error analysis of our given method. We first introduce a new result to approximate the absolute solution by using the residual correction procedure. Second, we introduce a new result to get appropriate bound for the absolute error. The collocation method is used and the collocation points can be found by applying Chebyshev roots. Both techniques are explained briefly with illustrative examples to demonstrate the applicability, efficiency and accuracy of the techniques. By using a small number of Bernstein polynomials and correction procedure we achieve some significant results. We present some examples to show the efficiency of our method by comparing the solution of such problems obtained by our method with the solution obtained by Runge-Kutta method, continuous genetic algorithm, rational homotopy perturbation method and adomian decomposition method.     

Tangential-displacement effects in the wedge indentation of an elastic half-space – an integral-equation approach

The idea of considering tangential-displacement effects in a classical elastostatic contact problem is explored in this paper. The problem involves the static frictionless indentation of a linearly elastic half-plane by a rigid wedge, and its present formulation implies that the tangential surface displacements are not negligible and should thus be coupled with the normal surface displacements in imposing the contact zone boundary conditions. L.M. Brock introduced this idea some years ago in treating self-similar elastodynamic contact problems, and his studies indicated that such a formulation strongly influences the contact-stress behavior at half-plane points making contact with geometrical discontinuities of the indentor. The present work again demonstrates, by studying an even more classical problem, that the aforementioned considerations eliminate contact-stress singularities and therefore yield a more natural solution behavior. In particular, the familiar wedge-apex logarithmic stress-singularity encountered within the standard formulation of the problem (i.e. by avoiding the tangential displacement in the contact boundary condition) disappears within the proposed formulation. The contact stress beneath the wedge apex takes now a finite value depending on the wedge inclination angle and the material constants. By utilizing pertinent integral relations for the displacement/stress field in the half-plane, an unusual mixed boundary-value problem results whose solution is obtained through integral equations.     

Application of a differential-difference method to the solution of one-dimensional nonstationary heat conduction problems with a moving boundary

A differential-difference method is applied to obtain an approximate solution of one-dimensional nonstationary heat conduction problems with a moving boundary in rectangular and cylindrical systems of coordinates. Recursion formulas are obtained for the determination of successive values of the unknown functions.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 4, pp. 719–724, October, 1969.     

Approximate method for determining nonstationary one-dimensional temperature fields

An approximate method for solution of the heat-conduction equation is considered; it can be used to reduce a boundary-value problem for a partial-differential equation to a Cauchy problem for a system of ordinary differential equations. A generalization to a problem with unknown boundary is given.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.17, No. 2, pp. 354–358, August, 1969.     

Mathematical modeling of the process of plane polymer film formation

The approximate solution is considered of a nonlinear nostationary problem of heat conduction with conjugate boundary conditions.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 58, No. 5, pp. 862–866, May, 1990.