Approximate solution methods in mechanics analysis of a class of nonlinear seepage problems by a finite-element method

We analyze the nonlinear boundary-value problem of seepage under a subsurface hydrotechnical construction over an inclined rectilinear aquifer. The method of inverse boundary-value problems is applied, using the velocity hodograph plane in which the original problem is reduced to a linear problem. The linear problem is solved in the general case using the finite-element method. A computer program realizing the proposed algorithms has been developed. We have used this program to run a series of numerical experiments, reaching certain conclusions about the behavior of the main seepage characteristics.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 75–80, 1985.     

Approximate solution to the diffusion equation and its application to seepage-related problems

Effective dam safety procedures require pore pressure measurements to be interpreted as soon as possible after readings have been taken. Direct interpretations based on partial differential equations are not appropriate. Here, we took a different approach which consisted in applying the impulse response function concept to a dam site. Consequently, two models were designed so that their parameters matched the temporal moments of Green’s function of the associated parabolic problem. These models were compared with closed-form solutions. The first model was the exponential decay model, which gave good results, but did not account for high loading harmonics. The second model, based on the fundamental solution of a parabolic problem, was surprisingly accurate. As an example problem we considered monitoring data obtained from a zoned earth dam. The results show that the main aspects of the processes reflected in most cell recordings can be described in a linear framework, and that they are accounted for by both models.     

A class of simple exponential B-splines and their application to numerical solution to singular perturbation problems

Summary We shall consider an application of simple exponential splines to the numerical solution of singular perturbation problem. The computational effort involved in our collocation method is less than that required for the other methods of exponential type.     

Parameter extension for combined hybrid finite element methods and application to plate bending problems

Based on a weighted average of the modified Hellinger-Reissner principle and its dual, the combined hybrid finite element (CHFE) method was originally proposed with a combination parameter limited in the interval (0, 1). In actual computation this parameter plays an important role in adjusting the energy error of discretization models. In this paper, a novel expression of the combined hybrid variational form is used to show the relationship between the resultant method and some Galerkin/least-squares stabilized finite scheme for plate bending problems. The choice of combination parameter is then extended to (−∞, 0) ? (0, 1). Existence, uniqueness and convergence of the solution of discrete schemes are proved, and the advantage of the parameter extension in computation is discussed. As an application, improvement of Adini’s rectangular element by the CHFE approach is performed.     

Approximate analytical solution for supercritical flow in rectangular curved channels

In this study, we present an asymptotical mathematical model and an analytical solution for a supercritical flow in curved rectangular open channels. An original approach is proposed for solving the free-surface configuration and features of the flow in the presence of cross shock waves. The two-dimensional steady depth-averaged shallow water equations are transformed into an equivalent one-dimensional (1D) unsteady flow problem and a first order approximation is then obtained using small perturbation theory. Furthermore, the 1D asymptotic model is solved analytically by Laplace integral transformation and the two-dimensional flow field solution is reconstructed according to the translating planes. The free-surface profile along the outer chute wall and downstream channel was compared with the available experimental data, and the results indicated the satisfactory agreement of the maximum flow depth, peak positions, and wavelength. The proposed approach provides accurate predictions of the flow features and it facilitates the safe design of curved channel transitions.     

Approximate solution of a class of singular control problems

A class of singular control problems involving amplitude constraints on the controls is examined. IfL is the space of control functionsU, the control constraint setS can be identified with the unit ball inL . Now, for anyn (1, ), an analogous problem may be set up withL ⁿ forU and the unit ball inL ⁿ forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn is investigated.This research was supported by the Science Research Council of Great Britain and the Commonwealth Fund (Harkness Fellowship).     

Approximate analytical solution of fourth order boundary value problems

In this paper, the decomposition method is applied to boundary-value problems of fourth order for ordinary differential equations. AMS subject classification 65L10, 65L20, 34B15Waleed Al-Hayani: He was a professor in the Department of Mathematics, College of Science, Mosul University, Mosul, Iraq.     

Variational methods for constructing approximate solutions to some continuum mechanics problems

In order to find approximate solutions to some continuum mechanics problems that admit variational statements, we use an approach that is based on restricting the class of functions in which we seek an extremal for the action functional. We demonstrate the method by some examples for the problem of forced oscillations of a nonlinear elastic membrane (in particular, a string), the problem of a fluid flow through a porous obstacle, and the problem of stationary waves on the surface of a heavy fluid.     

Approximate solution of boundary layer problems with injection or suction

Zusammenfassung Der grundlegende linearisierte Ansatz zur Näherungslösung von Grenzschichtproblemen wird auf neue Anwendungen erweitert, um die Behandlung von Problemen mit Massenaustausch durch die Oberfläche zu ermöglichen und die Beschränkung auf die Prandtl-Zahl 1 zu mildern. Das klassische Problem der konstanten, d. h. nicht ähnlichen Geschwindigkeitsverteilung an der Oberfläche wird behandelt, und die Ergebnisse werden mit der exakten numerischen Lösung verglichen. Es ergibt sich eine ausgezeichnete Übereinstimmung. Sodann wird eine Lösung des schwierigen und praktisch wichtigen Problems einer Stufenfunktion für die Oberflächengeschwindigkeit gegeben. Die freie Konvektion über einer vertikalen Platte mit konstanter Temperatur und Oberflächengeschwindigkeit wird erstmalig gelöst. Schliesslich wird das der Schlitzeinspritzung entsprechende Anfangswertproblem unter Milderung der bisherigen AnnahmeP _r =1 behandelt.

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This work was supported by the National Science Foundation, Grant No. GK-310,     

Approximate method for the numerical solution of singular perturbation problems

We present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is divided into inner and outer region problems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem. In turn, the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm. The proposed method is iterative on the terminal point. Some numerical experiments have been included to demonstrate its applicability.     

Numerical solution of the plate bending and free vibrations problems

The problem of the bending and free vibrations of a clamped and edge-supported plate is considered. The proposed algorithm is the algorithm described in /1/, made specific for the case of the biharmonic equation. It does not have saturation /2/, i.e., its accuracy will be the higher, the smoother the solution. The program is constructed in such a manner that if the plate boundary is sufficiently smooth and given parametrically, then several of the first eigenvalues can be calculated and the bending problem can be solved. An illustration is presented of the eigenfrequency computation for an edge-supported plate whose boundary (an epitrochoid) has a curvature of the order of 10³ at twelve points (the curvatures enter explicitly in the appropriate boundary condition). The first eigenfrequencies are calculated with 7–8 places after the decimal point. The solution is obtained because of the accurate method of discretization and the study of the structure of the appropriate finitedimensional problem. This would permit execution of computations with a large number of points (up to 1230). A comparison is given with the results of computations of other authors for a circle and an ellipse /3–5/.     

Numerical resolvent methods for constrained problems in mechanics

Resolvent methods are presented for generating systematically iterative numerical algorithms for constrained problems in mechanics. The abstract framework corresponds to a general mixed finite element subdifferential model, with dual and primal evolution versions, which is shown to apply to problems of fluid dynamics, transport phenomena and solid mechanics, among others. In this manner, Uzawa’s type methods and penalization-duality schemes, as well as macro-hybrid formulations, are generalized to non necessarily potential nonlinear mechanical problems.     

An analytical solution for bending of transversely isotropic thick rectangular plates with variable thickness

In this study, the bending solution of simply supported transversely isotropic thick rectangular plates with thickness variations is provided using displacement potential functions. To achieve this purpose, governing partial differential equations in terms of displacements are obtained as the quadratic and fourth order. Then, the governing equations are solved using the separation of variables method satisfying exact boundary conditions. The advantage of the purposed method is that there is no limitation on the thickness of the plate or the way the plate thickness is being varied. No simplifying assumption in the analysis process leads to the applicability and reliability of the present method to plates with any arbitrarily chosen thickness. In order to confirm the accuracy of the proposed solution, the obtained results are compared with existing published analytical works for thin variable thickness and thick constant thickness plate. Also, due to the lack of analytical research on thick plates with variable thickness, the obtained results are verified using the finite element method which shows excellent agreement. The results show that the maximum displacement of the plates with variable thickness is moved from the center toward the thinner plate edge. In addition, results exhibit the profound effects of both thickness and aspect ratio on stress distribution along the thickness of the plate. Results also show that varying thickness has not a profound impact on bending and twisting moments in transversely isotropic plates. Five different materials consist of four transversely isotropic and one isotropic, as a special case, are considered in this paper, which it is shown that the material properties have a more considerable impact on higher thickness plate.     

Formulation and solution of boundary value problems of polymer mechanics

Certain aspects of the formation of boundary value problems of polymer mechanics are stressed and effective methods of solution are discussed. Recent solutions of boundary value problems of polymer mechanics are reviewed.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 54–62, January–February, 1969.     

Discrete mechanics and its application to the solution of the n-body problem

A theory of discrete mechanics is developed based on the results of D. Greenspan. Discrete dynamical equations in an inertial frame, in a coordinate system related to some material point, and in a rotating frame are given and the consistency, stability, and convergence of the methods are studied and some numerical examples presented.