Convergence of a Modified SLP Algorithm for the Extended Linear Complementarity Problem

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｔｈｅｅｘｔｅｎｄｅｄｌｉｎｅａｒｃｏｍｐｌｅｍｅｎｔａｒｉｔｙｐｒｏｂｌｅｍ (ＸＬＣＰ)ｉｎｔｒｏｄｕｃｅｄｂｙＭａｎｇａｓａｒｉａｎａｎｄＰａｎｇ[1](ａｌｓｏｓｅｅＹｅ[2 ]) :Ｆｉｎｄａｖｅｃｔｏｒ(ｘ ,ｙ) ∈Ｒ2ｎｓｕｃｈｔｈａｔＭｘ-Ｎｙ∈Ｋ ,ｘＴｙ=0 ,　ｘ≥ 0 ,ｙ≥ 0 ,( 1 )ｗｈｅｒｅＭ ,Ｎ ∈Ｒｍ×ｎａｒｅｇｉｖｅｎ ,ａｎｄＫｉｓａｐｏｌｙｈｅｄｒａｌｓｅｔｉｎＲｍｄｅｆｉｎｅｄｂｙＫ :=ｕ∈Ｒｍ｜Ｇｕ≥ｇ ,　　Ｇ∈Ｒｌ×ｍ,ｇ∈Ｒｌ.Ｔｈｉｓｐｒｏ…     

On probabilistic norm of a linear operators and space of operators

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ［１］,ＸｉａｏｉｎｔｒｏｄｕｃｅｄｔｈｅｎｏｔｉｏｎｏｆｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｏｆａｌｉｎｅａｒｏｐｅｒａｔｏｒｏｎＭｅｎｇｅｒｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｅｄｓｐａｃｅ（ｂｒｉｅｆｙ,ＭｅｎｇｅｒＰＮ＿ｓｐａｃｅ）．Ａｎｄｂｙｕｓｉｎｇｔｈｅｎｏｔｉｏｎ,ｕｎｄｅｒｔｈｅｃｏｎｄｉｔｉｏｎｏｆｔ＿ｎｏｒｍΔｓａｔｉｓｆｙｉｎｇΔ（ｔ,ｔ）≥ｔ（ｔ∈（０,１］）,ｈｅｇａｖｅａｄｅｓｃｒｉｐｔｉｏｎｏｆｂｏｕｎｄｅｄｎｅｓｓｏｆｌｉｎｅａｒｏｐｅ…     

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY (Ⅱ)-LINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the forms similar to Biot’s equations for saturated porous media. The Darcy’s laws of unsaturated soil were proved. It is shown that Biot’s equations of saturated porous media are the simplification of the theory. All these illustrate that constructing constitutive relation of unsaturated soil on the base of mixture theory is rational.     

Robust H∝ filtering for discrete-time impulsive systems with uncertainty

This paper investigates robust filter design for linear discrete-time impulsive systems with uncertainty under H∞ performance. First, an impulsive linear filter and a robust H∞ filtering problem are introduced for a discrete-time impulsive systems. Then, a sufficient condition of asymptotical stability and H∞ performance for the filtering error systems are provided by the discrete-time Lyapunov function method. The filter gains can be obtained by solving a set of linear matrix inequalities （LMIs）. Finally, a numerical example is presented to show effectiveness of the obtained result.     

求解线性方程组的一种新方法

将线性方程组的一般系数矩阵转化为对称正定矩阵，从而把原线性方程组的求解问题转化为一个等价变分问题的极少值点寻优问题，借助对分寻优法进行求解。算例结果表明，本文方法不仅对于良态线性方程组的求解问题是有效的，而且对于病态线性方程组的求解问题同样是有效的。     

The representation problem of constrained linear elasticity

The title problem is given the following explicit solution: = _D S|D, where the elasticity tensor in the constrained case is the restriction to the constraint subspace D of a corresponding unconstrained elasticity tensor S, followed by composition with the orthogonal projection _D on D.     

Efficient preconditioning techniques for finite‐element quadratic discretization arising from linearized incompressible Navier–Stokes equations

We develop an efficient preconditioning techniques for the solution of large linearized stationary and non‐stationary incompressible Navier–Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non‐stationary case. The time discretization procedure uses the Gear scheme and the second‐order Taylor–Hood element P₂?P₁ is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r?(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P₁?P₂ hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Bounds and Error Estimates in Structural Eigenvalue Problems∗

The fundamental eigenvalue P₁ of a linear eigenvalue problem represents the smallest positive value of P for which the quadratic functional U(u) — PW(u) is not positive definite. If the norm of all admissible functions u is chosen to be W, as is usually the case, then the customary variational formulation leads to upper bounds if approximate modes u are selected.

It is shown in the current study that the choice of an appropriate norm D may reduce an otherwise intractable problem to a simpler one whose eigenvalue X becomes a function of an assumed value of P in such a way that P₁ is bounded between P and λ. Moreover, since λ(P) = P₁, the gap between the bounds can be brought within arbitrarily small limits.

For the purpose of demonstration the method is applied to the buckling and the vibration problem of a stepwise prismatic column, with an obvious iteration procedure leading to extremely rapid convergence. Other potential areas of application are also discussed.     

Solving frictional contact problems by two aggregate-function-based algorithms

Three dimensional frictional contact problems are formulated as linear complementarity problems based on the parametric variational principle. Two aggregate-function-based algorithms for solving complementarity problems are proposed. One is called the self-adjusting interior point algorithm, the other is called the aggregate function smoothing algorithm. Numerical experiment shows the efficiency of the proposed two algorithms. The project supported by the National Natural Science Foundation of China(10225212, 50178016, 10302007), the National Key Basic Research Special Foundation and the Ministry of Education of China The English text was polished by Ron Marshall.     

Analysis and control of a class of uncertain linear periodic discrete-time systems

Feedback control problems for linear periodic systems （LPSs） with interval- type parameter uncertainties are studied in the discrete-time domain. First, the stability analysis and stabilization problems are addressed. Conditions based on the linear matrices inequality （LMI） for the asymptotical stability and state feedback stabilization, respec-tively, are given. Problems of L2-gain analysis and control synthesis are studied. For the L2-gain analysis problem, we obtain an LMI-based condition such that the autonomous uncertain LPS is asymptotically stable and has an L2-gain smaller than a positive scalar γ. For the control synthesis problem, we derive an LMI-based condition to build a state feedback controller ensuring that the closed-loop system is asymptotically stable and has an L2-gain smaller than the positive scalar γ. All the conditions are necessary and sufficient.     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

Lower bound limit analysis of three-dimensional elastoplastic structures by boundary element method

IntroductionThelimitanalysisofstructuresisoneofthemostpracticalandusefulbranchesinplasticity .Ithasimportantapplicationbackgroundforproblemssuchasthedeterminationofloadcarryingcapacityandplasticformingofmetal.Thepurposeofthelimitanalysisofstructuresistoprovidereliabletheoreticalbasesforengineeringdesignandsafetyassessment.Asasimplifiedmethodforelastoplasticproblems,limitanalysisneednotrequirethehistoryofloadandcancomputethelimitloadsdirectlyinsteadofelastoplasticincrementalcomputationwhichisus…     

Krylov subspace projection method and its application on oil reservoir simulation

ＩｎｔｒｏｄｕｃｔｉｏｎＭａｎｙｐｒｏｂｌｅｍｓｏｆｅｎｇｉｎｅｅｒｉｎｇｉｎｔｅｒｅｓｔ,ｉｎｃｌｕｄｉｎｇｐｒｏｂｌｅｍｓｏｆｏｉｌｒｅｓｅｒｖｏｉｒｓｉｍｕｌａｔｉｏｎ,ｌｅａｄｔｏｔｈｅｔａｓｋｏｆｓｏｌｖｉｎｇａｓｙｓｔｅｍｏｆｌｉｎｅａｒｅｑｕａｔｉｏｎｓＡｘ=ｆ,(1)ｗｈｅｒｅＡｉｓａｎｎ×ｎ,ｒｅａｌ,ｌａｒｇｅｓｐａｒｓｅ,ｎｏｎ＿ＳＰＤｍａｔｒｉｘ.Ｔｈｅｄｉｒｅｃｔｄｅｃｏｍｐｏｓｉｔｉｏｎｍｅｔｈｏｄｉｓｍｏｓｔｅｆｆｉｃｉｅｎｔｓｏｌｖｅｒｗｈｅｎｎｉｓｓｍａｌｌ.Ｈｏｗｅｖｅｒ…     

NON-INTERIOR SMOOTHING ALGORITHM FOR FRICTIONAL CONTACT PROBLEMS

A new algorithm for solving the three-dimensional elastic contact problem with friction is presented. The algorithm is a non-interior smoothing algorithm based on an NCP-function. The parametric variational principle and parametric quadratic programming method were applied to the analysis of three-dimensional frictional contact problem. The solution of the contact problem was finally reduced to a linear complementarity problem, which was reformulated as a system of nonsmooth equations via an NCP-function. A smoothing approximation to the nonsmooth equations was given by the aggregate function. A Newton method was used to solve the resulting smoothing nonlinear equations. The algorithm presented is easy to understand and implement. The reliability and efficiency of this algorithm are demonstrated both by the numerical experiments of LCP in mathematical way and the examples of contact problems in mechanics.     

The axisymmetric mixed boundary-value problem of the vertical vibration of a rigid foundation on saturated layered soil subgrade

Based on the theory of elastic wave propagation in saturated soil subgrade established by the author of this paper, the axisymmetric vertical vibration of a rigid circular foundation resting on partially saturated soil subgrade which is composed of a dry elastic layer and a saturated substratum is studied. The analysis relied on the use of integral transform techniques and a pair of dual integral equations governing the vertical vibration of the rigid foundation is listed under the consideration of mixed boundary-value condition. The results are reduced to the case for saturated half-space. The set of dual integral equations are reduced to a Fredholm integral equation of the second kind and solved by numerical procedures. Numerical examples are given at the end of the paper and plots of the dynamic compliance coefficient C_v versus the dimensionless frequency a₀ are presented. Foundation item: the National Natural Science Foundation of China (59579018) Biography: CHEN Sheng-li (1974-)     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅰ)—BASIC PROPERTIES AND PRELIMINARY RESULTS

The controllability for switched linear system with time-delay in controls was first investigated. The whole work contains three parts. This is the first part, including problem formulation and some preliminaries. Firstly, the mathematical model of switched linear systems with time-delay in control functions was presented. Secondly, the concept of column space, cyclic invariant subspace and generalized cyclic invariant subspace were introduced. And some basic properties, such as separation lemma, were presented. Finally, a basic lemma was given to reveal the relation between the solution set of a centain integral equations and the generalized cyclic invariant subspace. This lemma will play an important role in the determination of controllability. All these definitions and lemmas are necessary research tools for controllability analysis. Contributed by YE Qing-kai Foundation items: the National Natural Science Foundation of China (69925307, 60274001); the National Key Basic Reasearch and Development Program (2002CB312200); the Postdoctoral Program Foundation of China Biography: XIE Guang-ming (1972∼), Doctor (E-mail: xiegming@mech.pku.edu.cn)     

Qualitative Behavior of Local Minimizers of Singular Perturbed Variational Problems

We consider a non-convex variational problem (P) and the corresponding singular perturbed problem (P _ε ). The qualitative behavior of stable critical points of (P _ε ) depending on ε and a lower order term is discussed and we prove compactness of a sequence of stable critical points as ε ↘ 0. Moreover we show whether this limit is the global minimizer of (P). Furthermore uniform convergence is considered as well as the convergence rate depending on ε.      

The site model theory and the standard linear solid

Summary The site model theory (SMT) is shown to lead to the same deformation behaviour as that displayed by the standard linear solid (SLS), group I, for all loading conditions. If a second deformation mechanism (inter-molecular slip) is introduced the result is the same as that obtained with the standard linear solid, group II, and models the behaviour of a polymer melt near to the solidification temperature.

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Zusammenfassung Es wird gezeigt, daß ein einfaches Platzwechsel-Modell (site model theory) bei allen Belastungsbedingungen das gleiche Deformationsverhalten voraussagt wie der lineare Drei-Parameter-Festkörper (standard linear solid, group I). Wenn ein weiterer Deformationsmechanismus (zwischenmolekulare Gleitung) eingeführt wird, entspricht das Verhalten dagegen demjenigen einer linearen Drei-Parameter-Flüssigkeit (standard linear solid, group II), welche das Verhalten einer Polymerschmelze in der Nähe der Schmelztemperatur beschreibt.

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a = ₁₂ ⁰ + ₂₁ ⁰ , see eq. [1] - b =N ₁ ⁰ ₁₂ ⁰ (V ₁₂ +V ₂₁), see eq. [1] - c = 2N _s ⁰ V _s see eq. [6] - k Boltzmann constant - t time - E,E ₁,E ₂ spring constants, see figures 1 and 3 - E _u unrelaxed modulus - N ₁ ⁰ site 1 equilibrium population in the unstressed state - N _s number of units available for slip - N(t) decrease in site 1 population - N _s (t) net number of slip jumps in the stressaided direction - T temperature (K) - V _i,j activation volume for jumps in directioni j - V _s activation volume for the slip process - strain - strain rate - incremental change in strain per unit change in site population - µ,µ ₁,µ ₂ dashpot constants, see figures 1 and 3 - applied stress - ₀ initial applied stress, (stress relaxation) =(t) (creep) - incremental change in stress per unit change in site population - ⁰ jump rate for slip in the unstressed state - _i,j ⁰ jump rate in the directioni j in the unstressed state With 3 figures and 3 tables     

Construction of high-order accuracy implicit residual smoothing schemes

ＩｎｔｒｏｄｕｃｔｉｏｎＧｅｎｅｒａｌｉｍｐｌｉｃｉｔｍｅｔｈｏｄｓ (ＦｕＤ .Ｘ .[1],Ｈｉｒｓｃｈ[2 ])ｃａｎｂｅｗｒｉｔｔｅｎａｓＩｍｐｌｉｃｉｔＰａｒｔ =ＥｘｐｌｉｃｉｔＰａｒｔ . (1 )ＩｔｈａｓｂｅｅｎｐｒｏｐｏｓｅｄｂｙＭａｃＣｏｒｍａｃｋ[3]ｔｈａｔｍｏｄｅｒｎｉｍｐｌｉｃｉｔｍｅｔｈｏｄｓｃａｎｂｅｗｒｉｔｔｅｎａｓＮｕｍｅｒｉｃａｌＰａｒｔδｉＵ =ＰｈｙｓｉｃａｌＰａｒｔ . (2 )　　Ｔｈｅｐｈｙｓｉｃａｌｐａｒｔｒｅｆｌｅｃｔｓｔｈｅｃｈａｎｇｅｒｕｌｅｏｆｐｈｙｓｉｃａｌｐａｒａｍｅ…     

Linear viscoelasticity,simple and planar melt extension of linear polybutadienes with bimodal molar mass distributions

Shear oscillations, simple and planar elongations have been performed with anionically polymerized polybutadienes (PB) and their blends at room temperature. The PB components were of different molar mass averages and of narrow molar mass distributions; the blends had bimodal molar mass distributions and are represented by the weight ratio w of the high molecular component. The crossover G() = G() obtained from oscillatory measurements shows correlations with molecular parameters. For the zero shear viscosity the well-known relation ₀ M _w ^3.4 is found. The recoverable equilibrium shear compliance J _e ⁰ is nearly the same for the components; for the blends it strongly depends on w with a pronounced maximum at small w. In elongation outside the linear region strain hardening is found; its magnitude depends on M _w of the components, the composition w of the blend, the mode of elongation (simple or planar), and the elongational strain rate. The hardening revealed in the increase of the elongational viscosity above the linear viscoelastic limit increases as a function of w up to a maximum similar to J _e ⁰ such that, for both properties, the molecular processes may be the same. The elongational viscosity µ₂ (from the lateral stress in planar elongation) is above the linear viscoelastic limit for bimodal and below this limit for conventional broad molar mass distributions. In general, it can be stated that with a more narrow molar mass distribution of linear polymers the elongational behavior of the melts comes closer to the linear viscoelastic limit.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.Extended version of a paper presented at the Annual Conf. German Soc. of Rheology, Berlin, May 13–15, 1991.