Vol 39, No 7 (2018)
- Year: 2018
- Articles: 20
- URL: https://journals.rcsi.science/1995-0802/issue/view/12616
Article
Modeling Groundwater Flow in Unconfined Conditions: Numerical Model and Solvers’ Efficiency
Abstract
A mathematical model for variably saturated flow in unconfined conditions is presented. The model is based on pseudo-unsaturated approach using Richards equation with piecewise linear dependencies between hydraulic head, water content and relative permeability. It is implemented in GeRa (Geomigration of Radionuclides) software package, which is designed for modeling groundwater flow and contaminant transport in porous media and uses finite volume methods on unstructured grids. We consider two nonlinear solvers for nonlinear equations arising from discretization of the Richards equation, namely Newton and Picard methods. A special method for correction of hydraulic head values within the iterations of nonlinear solvers is proposed. The developed numerical techniques are applied to two test cases: dam seepage and real-world groundwater flow problems.
Grid-Characteristic Numerical Method for Low-Velocity Impact Testing of Fiber-Metal Laminates
Abstract
The grid-characteristic numerical method (GCM) for hyperbolic equations systems is applied in many science fields—gas dynamics, hydrodynamics, plasma dynamics, etc. Its application for problems of dynamics of deformable solids is less popular, especially in comparison with finite elements methods. GCM shows good results and high performance for elastic wave problems in the approximation of small deformations—seismic survey and ultrasound non-destructive testing in medicine, aviation and railway industry. Low-velocity impacts (hail, dropped tool, bird strike, etc.) are one of the most dangerous load types for polymer composites. They cause barely visible impact damage (BVID) that can only be detected by a thorough ultrasound testing, but severely reduces the residual strength of the material, especially for a compression load along the surface. This testing increases the operating cost, and its necessity can be easily missed, which greatly reduces the reliability of polymer composites. Hybrid fiber-metal composites (GLARE, ARALL, titanium composite laminates) were developed to unify the advantageous properties of polymer composites and metal. The addition of a thin metal layer (1–2 mm) helps to reduce the impact vulnerability of polymer composites in case of a penetration or significant deformations of the material. The application of GCM for low-velocity impact problems can help to explain the damage pattern in fiber-metal composites in case of low-velocity strike, including delamination effects, by modelling elastic wave processes in the complex anisotropicmedium. This article contains the brief description of the GCM and numerical results that were obtained for model problems of a low-velocity impact on titanium composite laminates.
Investigation of Lagrange–Galerkin Method for an Obstacle Parabolic Problem
Abstract
The convergence and accuracy estimates are proved for Lagrange–Galerkin method, used for approximating the parabolic obstacle problem. The convergence analysis is based on the comparison of the solutions of Lagrange–Galerkin and backward Euler approximation schemes. First order in time step estimate for the difference of the solutions for above schemes in energy norm is proved under sufficiently weak requirements for the smoothness of the initial data. First order in time and space steps accuracy estimate for Lagrange–Galerkin method is derived in the case of discontinuous time derivative of the exact solution.
Bicubic Hermite Elements in a Domain with the Curved Boundary
Abstract
A family of bicubic Hermite elements is considered which involves a rectangular element and two triangular ones including a triangular element with a curved side. The triangular elements are used in combination with the rectangular ones only near the boundary of a domain and provide interelement continuity of an approximate solution. Special attention is paid to the triangular element with a curved side since it is nonconforming in the sense of the Dirichlet boundary condition. For the Poisson equation the convergence estimate in the energy norm is proved.
Some a Posteriori Error Bounds for Numerical Solutions of Plate in Bending Problems
Abstract
For the efficient error control of numerical solutions of the solid mechanics problems, the two requirements are important: an a posteriori error bound has sufficient accuracy and computation of the bound is cheap in respect to the arithmetic work. The first requirement can be formulated in a more specific form of consistency of an a posteriori bound, assuming that it is not improvable in the order and, at least, coincides in the order with the a priori error estimate. Several new a posteriori error bounds are presented, which improve accuracy and reduce the computational cost. Also for the first time a new consistent guaranteed a posteriori error bound is suggested.
Iterative Method for Solving Parabolic Linear-Quadratic Optimal Control Problem with Constraints on the Time Derivative of the State
Abstract
We consider a linear-quadratic optimal control problem of a system governed by parabolic equation with distributed in right-hand side control and control in Neumann boundary condition. Pointwise constraints for control functions and for time derivative of the state function are imposed. We construct a mesh approximation of this problem using two different approximations of the objective functional. Iterative solution methods are investigated for the constructed approximations of the optimal control problems. Numerical results confirm the effectiveness of the proposed methods.
Financial Bubbles Existence in the Cantor–Lippman Model for Continuous Time
Abstract
This article considers the problem of bubbles existence while using the pool of renewable investment projects. The formulation of the Cantor–Lippman model for continuous time is described in this paper. The result allows to classify pools of investment projects into the arbitration, and the ineffective and the standard is proved. The estimation of the yield is found for each of the classes. The classification of pools and their yield calculation are based on the functions of the upper envelope of the Laplace transform of the investment projects cash flow functions. It is shown that for the case of a standard pool the yield can be obtained by computing the minimal positive root of the upper envelope. Also, it is shown that for the case of a standard pool the roots different from the minimum one refer to bubble strategies requiring permanent reinvesting to support growth and are not able to result in liquid final state for investors.
Semi-Lagrangian Approximation of Conservation Laws in the Flow around a Wedge
Abstract
In the paper, the numerical modeling of a supersonic flow around a wedge by viscous heat-conducting gas is considered. A numerical algorithm is proposed for the initial boundary-value problem for the Navier–Stokes equations. These equations are modified and amplified by new boundary conditions to provide the conservation law for the full energy: kinetic and inner. Then the combination of the Lagrangian approximation for the transfer operators and the conforming finite element method for other terms provides an efficient algorithm. Particular attention has been given to the approximation providing the conservation laws for mass and full energy at discrete level. Test calculations have been performed for a wide range of Mach and Reynolds numbers.
Finite Element Approximation of the Minimal Eigenvalue of a Nonlinear Eigenvalue Problem
Abstract
The problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radiofrequency discharge at reduced pressures. A necessary and sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem is established. The original differential eigenvalue problem is approximated by the finite element method on a uniform grid. The convergence of approximate eigenvalue and approximate positive eigenfunction to exact ones is proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter.
Efficient Branching Programs for Quantum Hash Functions Generated by Small-Biased Sets
Abstract
In the paper we consider quantum (δ, ϵ)-hash functions in so called phase form (phase quantum (δ, ϵ)-hash function). It is known that ϵ-biased sets generate phase quantum (δ, ϵ)- hash function. We show that the construction is invertible, that is, phase quantum (δ, ϵ)-hash function defines ϵ-biased sets. Next, we present an efficient (in the sense of time and qubits needed) Branching program construction for phase quantum (δ, ϵ)-hash function.
On Block Sensitivity and Fractional Block Sensitivity
Abstract
We investigate the relation between the block sensitivity bs(f) and fractional block sensitivity fbs(f) complexity measures of Boolean functions. While it is known that fbs(f) = O(bs(f)2), the best known separation achieves \({\rm{fbs}}\left( f \right) = \left( {{{\left( {3\sqrt 2 } \right)}^{ - 1}} + o\left( 1 \right)} \right){\rm{bs}}{\left( f \right)^{3/2}}\). We improve the constant factor and show a family of functions that give fbs(f) = (6−1/2 − o(1)) bs(f)3/2.
On the Weight Lifting Property for Localizations of Triangulated Categories
Abstract
As we proved earlier, for any triangulated category \(\underline C \) endowed with a weight structure w and a triangulated subcategory \(\underline D \) of \(\underline C \) (strongly) generated by cones of a set of morphism S in the heart \(\underline {Hw} \) of w there exists a weight structure w' on the Verdier quotient \(\underline {C'} = \underline C /\underline D \) such that the localization functor \(\underline C \to \underline {C'} \) is weight-exact (i.e., “respects weights”). The goal of this paper is to find conditions ensuring that for any object of \(\underline {C'} \) of non-negative (resp. non-positive) weights there exists its preimage in \(\underline C \) satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that that these weight lifting properties are fulfilled whenever the set S satisfies the corresponding (left or right) Ore conditions. Moreover, if \(\underline D \) is generated by objects of \(\underline {Hw} \) then any object of \(\underline {Hw'} \) lifts to \(\underline {Hw} \). We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper.
Polynomial Greatest Common Divisor as a Solution of System of Linear Equations
Abstract
In this article we present a new algebraic approach to the greatest common divisor (GCD) computation of two polynomials based on Bezout’s identity. This approach is based on the solution of system of linear equations. Also we introduce the dmod operation for polynomials. This operation on polynomials f, g is used to reduce the degree of the larger polynomial f in a finite field Fp. This operation saves GCD(f, g). Also we present some ideas how to reduce spurious factors that arise at the procedure.
Symmetric Blind Information Reconciliation and Hash-function-based Verification for Quantum Key Distribution
Abstract
We consider an information reconciliation protocol for quantum key distribution (QKD). In order to correct down the error rate, we suggest a method, which is based on symmetric blind information reconciliation for the low-density parity-check (LDPC) codes. We develop a subsequent verification protocol with the use of ϵ-universal hash functions, which allows verifying the identity between the keys with a certain probability.
New Size Hierarchies for Two Way Automata
Abstract
We introduce a new type of nonuniform two-way automaton that can use a different transition function for each tape square. We also enhance this model by allowing to shuffle the given input at the beginning of the computation. Then we present some hierarchy and incomparability results on the number of states for the types of deterministic, nondeterministic, and bounded-error probabilistic models. For this purpose, we provide some lower bounds for all three models based on the numbers of subfunctions and we define two witness functions.
The Error Probability of the Miller–Rabin Primality Test
Abstract
In our paper we give theoretical and practical estimations of the error probability in the well-known Miller–Rabin probabilistic primality test. We show that a theoretical probability of error 0.25 for a single round of the test is very overestimated and, in fact, error is diminishing with the growth of length of numbers involved by a rate limited with ln n/\(\sqrt n \).
On the Probability of Finding Marked Connected Components Using Quantum Walks
Abstract
Finding a marked vertex in a graph can be a complicated task when using quantum walks. Recent results show that for two or more adjacent marked vertices search by quantum walk with Grover’s coin may have no speed-up over classical exhaustive search. In this paper, we analyze the probability of finding a marked vertex for a set of connected components of marked vertices. We prove two upper bounds on the probability of finding a marked vertex and sketch further research directions.
On Calculation of Monomial Automorphisms of Linear Cyclic Codes
Abstract
A description of the monomial automorphisms group of an arbitrary linear cyclic code in term of polynomials is presented. This allows us to reduce a task of code’s monomial automorphisms calculation to a task of solving some system of equations (in general, nonlinear) over a finite field. The results are illustrated with examples of calculating the full monomial automorphisms groups for two codes.
Attacking Quantum Hashing. Protocols and Their Cryptanalysis
Abstract
Quantumhash functions are similar to classical (cryptographic) hash functions and their security is guaranteed by physical laws. However, security of a primitive does not automatically mean that protocols based on this primitive are secure. We propose protocols based on quantum hash function and assess their security using Holevo entropy and recently introduced notion of quantum information cost.