Optimization problems for periodic systems

Quadratic optimization algorithms for discrete-time linear periodic systems are discussed. Consideration is given to the conventional linear-quadratic problem statement where the optimal controller is found by solving the discrete algebraic Riccati equation and to input-feedback controller design problems. Considerable attention is focused on the synthesis of a reliable controller with a guaranteed stability margin. The following inverse problem is solved: given plant and controller matrices, find the weight matrices of the functional being optimized. The efficiency of the algorithms is demonstrated by way of examples     

Solution of some periodic elastoplastic problems

The article considers an elastoplastic problem for a plane weakened by an infinite number of round openings. It is assumed that the level of the stresses and the distance between the openings are such that the round openings are completely enveloped by the corresponding plastic zone; under these circumstances, the adjacent plastic regions do not coalesce. The article also considers the inverse elastoplastic problem under conditions of plane strain for an unbounded plane, weakened by a periodic series of openings. A number of communications have been devoted to periodic problems in the theory of elasticity and plasticity with an unknown boundary [1–8]. In distinction from [1–8], in which the method of perturbations was used, another method is used to solve periodic elastoplastic problems, making it possible to obtain a solution with any arbitrary relative dimensions of the region.     

奇异边界法模拟二维弹性力学问题

奇异边界法是一种新的边界型无网格数值离散方法.该方法使用基本解作为插值基函数,在继承传统边界型方法优点的同时,不需要费时费力的网格划分和奇异积分,数学简单,编程容易,是一个真正的无网格方法.为避免配置点与插值源点重合时带来的基本解源点奇异性,该方法提出了源点强度因子的概念,从而将边界型强格式方法的核心归结为求解源点强度因子.论文首次将该方法应用于求解平面弹性力学问题.数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度.     

Singular perturbation in bending problems for orthotropic plates

The paper deals with a number of singularly perturbed boundary-value problems and variational inequalities that arise in the theory of bending of orthotropic plates with strong anisotropy of elastic properties. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 195–201. September–October, 1999.     

Collisionless periodic solutions to some three-body problems

We provide sufficient conditions for the existence of periodic solutions to some three-body problems. Periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions.     

Inverse problems for scattering by periodic structures

Suppose the 3-dimensional space is filled with three materials having dielectric constants ? ₁ above S ₁={x ₂=f ₁(x ₁), x ₃ arbitrary}, ? ₂ below S ₂ = {x ₂ =f ₂(x ₁), x ₃ arbitrary} and ? _o in {f ₂(x ₁) <x ₂ 1(x ₁), x ₃ arbitrary} where f ₁ f ₂ are periodic functions. Suppose for a plane wave incident on S ₁ from above we can measure the reflected and transmitted waves of the corresponding time-harmonic solution of the Maxwell equations, say at x ₂=±b,b large. To what extent can we infer from these measurements the location of the pair (S ₁, S ₂ ? In this paper, we establish a local stability: If ( $\tilde S_1 ,\tilde S_2$ ) is another pair of periodic curves “close” to (S ₁, S₂), then, for any δ>0, if the measurements for the two pairs are δ-close, then $\tilde S_1$ and $\tilde S_2$ are 0(δ)-close to S ₁ and S ₂, respectively.     

Singular limit of equations for linear viscoelastic fluids with periodic boundary conditions

We consider a one-parameter family of problems, governing, for any fixed parameter, the motion of a linear viscoelastic fluid in a two-dimensional domain with periodic boundary conditions. The asymptotic behavior of each problem is analyzed, by proving the existence of the global attractor. Moreover, letting the parameter go to zero, since the memory effect disappears, we obtain a limiting problem, given by the Navier-Stokes equations. For any fixed parameter, we construct an exponential attractor. The resulting family is robust, meaning that these exponential attractors converge, in an appropriate sense, to an exponential attractor of the limiting problem.     

Singular boundary method for solving plane strain elastostatic problems

This study documents the first attempt to apply the singular boundary method (SBM), a novel boundary only collocation method, to two-dimensional (2D) elasticity problems. Unlike the method of fundamental solutions (MFS), the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest. Three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the MFS, boundary element method (BEM), and finite element method (FEM).     

Singular boundary method for 2D dynamic poroelastic problems

This paper extends a strong-form meshless boundary collocation method, named the singular boundary method (SBM), for the solution of dynamic poroelastic problems in the frequency domain, which is governed by Biot equations in the form of mixed displacement–pressure formulation. The solutions to problems are represented by using the fundamental solutions of the governing equations in the SBM formulations. To isolate the singularities of the fundamental solutions, the SBM uses the concept of the origin intensity factors to allow the source points to be placed on the physical boundary coinciding with collocation points, which avoids the auxiliary boundary issue of the method of fundamental solutions (MFS). Combining with the origin intensity factors of Laplace and plane strain elastostatic problems, this study derives the SBM formulations for poroelastic problems. Five examples for 2D poroelastic problems are examined to demonstrate the efficiency and accuracy of the present method. In particular, we test the SBM to the multiply connected domain problem, the multilayer problem and the poroelastic problem with corner stress singularities, which are all under varied ranges of frequencies.     

Boundary value problems for nonlinear second-order difference equations with impulse

In this paper, we consider a boundary value problem with impulse （BVPI）for nonlinear second-order difference equations. Existence and uniqueness of solutions of the considered BVPI are established.     

Singular finite elements for the sudden-expansion and the die-swell problems

The singular finite element method is used to solve the sudden-expansion and the die-swell problems in order to improve the accuracy of the solution in the vicinity of the singularity and to speed up the convergence. The method requires minor modifications to standard finite element schemes, and even coarse meshes give more accurate results than refined ordinary finite element meshes. Improved normal stress results for the sudden-expansion problem have been obtained for various Reynolds numbers up to 100 using the singular elements constructed for the creeping flow problem. In addition, the normal stresses at the walls appear to be insensitive to the singularity powers used in the construction of the singular basis functions. The die-swell problem is solved using the singular elements constructed for the stick–slip problem. The singular elements accelerate the convergence of the free surface dramatically.     

Singular integral equations in Hilbert space applied to crack problems

Crack problems are solved by application of a system of singular integral equations in Hilbert space. The method consists of replacing the singular integral equation by a system of linear algebraic equations for the values of the unknown function at specially chosen points within the range of integration. Obtained is a solution for a nonsymmetric cross-shaped crack in an infinite and isotropic solid subjected to a constant pressure, the symmetric configuration being a special case.     

Averaging method in problems of control over periodic systems

Explicit estimates are obtained for the closeness of trajectories of exact and averaged systems. We show that an optimal control of an averaged system is ε-optimal for an exact system both on asymptotically finite (of order 1/ε) time intervals and on infinite time intervals.     

Plane contact problems for a periodic two-layered elastic composite

Summary This paper deals with the two-dimensional static punch problems in the presence of friction for a periodically layered half-space. Within the framework of the homogenized model of linear elasticity with microlocal parameters [8–11] the exact solutions of the considered problems are obtained. The case of the indentation of the composite body by a rigid rectangular punch has been discussed in detail.

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Ebene Kontaktprobleme für ein periodisches elastisches Zweischichten-Komposit

Übersicht In dieser Arbeit wird das zweidimensionale statische Stempelproblem mit Reibung für einen periodisch geschichteten Halbraum betrachtet. Im Rahmen eines Homogenisierungsmodells der linearen Elastizitätstheorie mit mikropolaren Parametern werden die exakten Lösungen der betrachteten Probleme gewonnen. Im Detail wird das Eindringen eines rechteckigen starren Stempels behandelt.