Comments on “Elasto-plastic solution for shallow tunnel in semi-infinite space”

This paper considers the approach proposed by Zou et al. for determining the plastic zone around a shallow circular tunnel in an elasto-plastic semi-infinite space that incorporates the gravitational effect based on the bipolar coordinate system. The paper aims to analyze the correctness of the analytic expressions for elastic and plastic stress, as well as the reasonableness of the procedure building for solving the elasto-plastic interface. Finally, a promising way of solving this problem more exactly is presented.     

Convex Semi-Infinite Parametric Programming: Uniform Convergence of the Optimal Value Functions of Discretized Problems

The continuity of the optimal value function of a parametric convex semi-infinite program is secured by a weak regularity condition that also implies the convergence of certain discretization methods for semi-infinite problems. Since each discretization level yields a parametric program, a sequence of optimal value functions occurs. The regularity condition implies that, with increasing refinement of the discretization, this sequence converges uniformly with respect to the parameter to the optimal value function corresponding to the original semi-infinite problem. Our result is applicable to the convergence analysis of numerical algorithms based on parametric programming, for example, rational approximation and computation of the eigenvalues of the Laplacian.     

Newton’s method on generalized Banach spaces

We present a weaker convergence analysis of Newton’s method than in Kantorovich and Akilov (1964), Meyer (1987), Potra and Ptak (1984), Rheinboldt (1978), Traub (1964) on a generalized Banach space setting to approximate a locally unique zero of an operator. This way we extend the applicability of Newton’s method. Moreover, we obtain under the same conditions in the semilocal case weaker sufficient convergence criteria; tighter error bounds on the distances involved and an at least as precise information on the location of the solution. In the local case we obtain a larger radius of convergence and higher error estimates on the distances involved. Numerical examples illustrate the theoretical results.     

Strain gradient plasticity solution for an internally pressurized thick-walled cylinder of an elastic linear-hardening material

An elastic-plastic solution is presented for an internally pressurized thick-walled plane strain cylinder of an elastic linear-hardening plastic material. The solution is derived in a closed form using a strain gradient plasticity theory. The inner radius of the cylinder enters the solution not only in non-dimensional forms but also with its own dimensional identity, which differs from that in classical plasticity based solutions and makes it possible to capture the size effect at the micron scale. The classical plasticity solution of the same problem is recovered as a special case of the current solution. To further illustrate the newly derived solution, formulas and numerical results for the plastic limit pressure are provided. These results reveal that the load-carrying capacity of the cylinder increases with decreasing inner radius at the micron scale. It is also seen that the macroscopic behavior of the pressurized cylinder can be well described by using classical plasticity based solutions.     

Strain gradient plasticity solution for an internally pressurized thick-walled cylinder of an elastic linear-hardening material

An elastic-plastic solution is presented for an internally pressurized thick-walled plane strain cylinder of an elastic linear-hardening plastic material. The solution is derived in a closed form using a strain gradient plasticity theory. The inner radius of the cylinder enters the solution not only in non-dimensional forms but also with its own dimensional identity, which differs from that in classical plasticity based solutions and makes it possible to capture the size effect at the micron scale. The classical plasticity solution of the same problem is recovered as a special case of the current solution. To further illustrate the newly derived solution, formulas and numerical results for the plastic limit pressure are provided. These results reveal that the load-carrying capacity of the cylinder increases with decreasing inner radius at the micron scale. It is also seen that the macroscopic behavior of the pressurized cylinder can be well described by using classical plasticity based solutions.     

Analytical stress and displacement around non-circular tunnels in semi-infinite ground

The construction of non-circular tunnels at shallow depths is becoming increasingly common to efficiently use excavation space. However, the stress concentration and ground movement are probably much greater around shallow non-circular tunnels than around deep ones; the greater stress and ground movement have significant negative effects on tunnel stability and the safety of nearby structures. This study provides a methodology for obtaining the analytical solutions of ground response induced by non-circular tunnel excavation. The surcharge loadings and gravity induced initial stresses, as well as the internal forces due to liners, are considered in the derivation.Using the Schwartz alternating method combined with complex variable theory, a series of two types of problems on simply connected domain, i.e., a non-circular hole in an infinite plane and a half-plane without holes, is in turn successfully solved. Convergent and accurate analytical solutions are finally achieved by superposing the solutions in all the alternating iterations. The analytical solutions exhibit a close agreement with the numerical results, except for the horizontal displacement along the line near the ground surface. Finally, a parametric study is performed for the elliptical/square tunnel subjected to infinite surcharge loadings.The proposed analytical solutions have great value for the conceptual understanding of the mechanical behavior of non-circular tunnels and for verifying the numerical models. Moreover, an alternative approach is provided for the preliminary designs of future shallow tunnels excavated in rock or medium/stiff clay.     

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump. In memory of Gene Golub.     

Farkas-type theorems for positively homogeneous systems in ordered topological vector spaces

In this paper, we present versions of the Farkas Lemma and the Gale Lemma for a semi-infinite system involving positively homogeneous functions in a topological vector space. In particular, we present two such versions for a semi-infinite system containing min-type functions. Our main theoretical tool is abstract convexity.     

Partial difference equation method for lattice path problems

Many problems concerning lattice paths, especially on the square lattice have been exactly solved. For a single path, many methods exist that allow exact calculation regardless of whether the path inhabits a strip, a semi-infinite space or infinite space, or perhaps interacts with the walls. It has been shown that a transfer matrix method using the Bethe Ansatz allows for the calculation of the partition function for many non-intersecting paths interacting with a wall. This problem can also be considered using the Gessel-Viennot methodology. In a concurrent development, two non-intersecting paths interacting with a wall have been examined in semi-infinite space using a set of partial difference equations.Here, we review thispartial difference equation method for the case of one path in a half plane. We then demonstrate that the answer for arbitrary numbers of non-intersecting paths interacting with a wall can be obtained using this method. One reason for doing this is its pedagogical value in showing its ease of use compared to the transfer matrix method. The solution is expressed in a new form as a constant term formula, which is readily evaluated. More importantly, it is the natural method that generalizes easily to many intersecting paths where there is inter-path interactions (e.g., osculating lattice paths). We discuss the relationship of the partial difference equation method to the transfer matrix method and their solution via a Bethe Ansatz.     

Generalized power series on a non-Archimedean field

Power series with rational exponents on the real numbers field and the Levi-Civita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the Levi-Civita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.     

DOMAIN DECOMPOSITION WITH NONMATCHING GRIDS FOR EXTERIOR TRANSMISSION PROBLEMS VIA FEM AND DTN MAPPING

In this paper, we are concerned with a non-overlapping domain decomposition method （DDM） for exterior transmission problems in the plane. Based on the natural boundary integral operator, we combine the DDM with a Dirichlet-to-Neumann （DtN） mapping and provide the numerical analysis with nonmatching grids. The weak continuity of the approximation solutions on the interface is imposed by a dual basis multiplier. We show that this multiplier space can generate optimal error estimate and obtain the corresponding rate of convergence. Finally, several numerical examples confirm the theoretical results.     

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.     

An accelerated central cutting plane algorithm for linear semi-infinite programming

An algorithm for linear semi-infinite programming is presented which accelerates the convergence of the central cutting plane algorithm first proposed in [4]. Compared with other algorithms, the algorithm in [4] has the advantage of being applicable under mild conditions and of providing feasible solutions. However its convergence has been shown to be rather slow in practical instances. The algorithm proposed in this paper introduces a simple acceleration scheme which gives faster convergence, as confirmed by several examples, as well as an interval of prefixed length containing the optimum value. It is also shown that the algorithm provides a solution of the dual problem and that it can be used for convex semi-infinite programming too.Mathematics Subject Classification (1991): 90C05, 90C34, 65K05, 90C51Acknowledgments. The author whishes to thank the three anonymous referees and an associate editor for many useful comments and valuable suggestions.     

On numerical optimization theory of infinite kernel learning

In Machine Learning algorithms, one of the crucial issues is the representation of the data. As the given data source become heterogeneous and the data are large-scale, multiple kernel methods help to classify “nonlinear data”. Nevertheless, the finite combinations of kernels are limited up to a finite choice. In order to overcome this discrepancy, a novel method of  “infinite”  kernel combinations is proposed with the help of infinite and semi-infinite programming regarding all elements in kernel space. Looking at all infinitesimally fine convex combinations of the kernels from the infinite kernel set, the margin is maximized subject to an infinite number of constraints with a compact index set and an additional (Riemann–Stieltjes) integral constraint due to the combinations. After a parametrization in the space of probability measures, it becomes semi-infinite. We adapt well-known numerical methods to our infinite kernel learning model and analyze the existence of solutions and convergence for the given algorithms. We implement our new algorithm called “infinite” kernel learning (IKL) on heterogenous data sets by using exchange method and conceptual reduction method, which are well known numerical techniques from solve semi-infinite programming. The results show that our IKL approach improves the classifaction accuracy efficiently on heterogeneous data compared to classical one-kernel approaches.     

Mathematical model for crack arrest of an infinite plate weakened by a finite and two semi-infinite cracks

The problem of an unbounded plate weakened by three quasi-static coplanar and collinear straight cracks: two semi-infinite cracks and a finite crack situated symmetrically between two semi-infinite cracks, is investigated. The plate is subjected to uniform unidirectional in-plane tension at infinite boundary. Developed plastic zones are arrested by distributing cohesive yield point stress over their rims. The solution is obtained using complex variable technique. Closed form analytic expressions are derived for load bearing capacity and crack-tip-opening displacement (CTOD). A case study is presented for CTOD and load bearing capacity versus crack length, plastic zone length and inter-crack distance etc. Results are presented graphically and analyzed.     

On solving convex quadratic semi-infinite programming probelms

In this paper we study a from of convex quadratic semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. An entropic path-following algorithum is introduced with a convergence proof. Some practical implementations and numerical experiments are also included     

一个锥模型的自适应信赖域算法及其收敛性

本文研究了无约束最优化问题的基于锥模型的自适应信赖域算法.利用理论分析得到一个新的自适应信赖域半径.算法在每步迭代中以变化的速率、当前迭代点的信息以及水平向量信息调节信赖域半径的大小.从理论上证明了新算法的全局收敛性和Q-二阶收敛性.用数值试验验证了新算法的有效性.推广了已有的自适应信赖域算法的可行性和有效性.     

Generalized semi-infinite programming: numerical aspects

Generalized semi-infinite optimization problems (GSIP) are considered. It is investigated how the numerical methods for standard semi-infinite programming (SIP) can be extended to GSIP. Newton methods can be extended immediately. For discretization methods the situation is more complicated. These difficulties are discussed and convergence results for a discretization- and an exchange method are derived under fairly general assumptions on GSIP. The question is answered under which conditions GSIP represents a convex problem.     

A general finite plasticity model with a smooth elastic-plastic transition and isotropic hardening. A finite element formulation and numerical verification

A number of approaches for finite deformation elastoplasticity with different classes of kinematic decomposition have been published in the literature (e.g. additive split of the Lagrange strain, multiplicative split of the deformation gradient, additive split of the rate of deformation, etc.). In the present work, a general theoretical framework for modeling a smooth elastic inelastic transition for large deformations of rate independent elastic-plastic and rate dependent elastic-viscoplastic materials has been proposed. It is well known that in classical rate independent elastic-plastic models the transition from the elastic regime to the plastic regime is rather sharp, while in the present model this transition is smooth and both rate independent and rate dependent models are characterized by overstress. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.