Solutions For Test Problems Research Articles

Using general-purpose integer programming software to generate bounded solutions for the multiple knapsack problem: a guide for or practitioners

An NP-Hard combinatorial optimization problem that has significant industrial applications is the Multiple Knapsack Problem. If approximate solution approaches are used to solve the Multiple Knapsack Problem there are no guarantees on solution quality and exact solution approaches can be intricate and challenging to implement. This article demonstrates the iterative use of general-purpose integer programming software (Gurobi) to generate solutions for test problems that are available in the literature. Using the software package Gurobi on a standard PC, we generate in a relatively straightforward manner solutions to these problems in an average of less than a minute that are guaranteed to be within 0.16% of the optimum. This algorithm, called the Simple Sequential Increasing Tolerance (SSIT) algorithm, iteratively increases tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. This approach is attractive to management for solving industrial problems because it is both cost and time effective and guarantees the quality of the generated solutions. Finally, comparing SSIT results for 480 large multiple knapsack problem instances to results using published multiple knapsack problem algorithms demonstrates that SSIT outperforms these specialized algorithms.

ПОЛУАНАЛИТИЧЕСКИЙ МЕТОД РЕШЕНИЯ УРАВНЕНИЙ ГАЗОВОЙ ДИНАМИКИ В ПЕРЕМЕННЫХ ЭЙЛЕРА

This paper presents a semi-analytical method for solving a system of equations of gas dynamics in Eulerian coordinates. Since only spatial derivatives are replaced by finite differences, the system of gas dynamic equations is reduced to a system of ordinary differential equations on a spatial grid. An approximate analytical solution of this system of differential equations for a small time-interval is used to describe the dynamics of a gas in the entire required time interval. Verification was carried out on one-dimensional test problems on the decay of an arbitrary discontinuity and the propagation of stationary shock waves of various intensities. To compare one-dimensional problems, the solution of test problems is given by the simple-to-implement basic particle-in-cell method. It is shown that the semianalytical method has high accuracy of calculations, and is also the most universal method for calculating applied problems.

Numerical solutions of differential equations having cubic nonlinearity using Boole collocation method

The aim of the study is to develop a numerical method for the solution of cubic nonlinear differential equations in which the numerical solution is based on Boole polynomials. That solution is in the form of the truncated series and gives approximate solution for nonlinear equations of cubic type. In this method, firstly, the matrix form of the serial solution is set and the nonlinear differential equation is converted into a matrix equation system. By adding the effect of both the conditions of the problem and the collocation points to this system of equations, we obtain the new system of equations. The coefficients of Boole-based serial solution are obtained from the solution of the resulting system of equations. The theoretical part is reinforced by considering three test problems. Numerical data for Boole solutions of test problems and absolute error functions are given in tables and figures.

The effect of negative temperatures on the retaining walls of deep pits

In countries with negative winter temperatures, pits and walls of underground structures are subjected to additional forces caused by frost heaving forces. In Russian construction practice, cases of the impact of such forces on the struts and anchors of construction walls are known, which led to the loss of stability of the struts and the failure of the anchors. Obviously, the design of retaining walls in winter should take into account the effect of additional forces caused by frost heaving and their evolution over time. The solution of test problems for an open pit shows that under certain conditions, the forces in the struts can exceed their values by an order of magnitude, determined when taking into account only the active pressure of the soil. Experimental studies were carried out to assess the processes of development of forces in the elements of the retaining walls in winter. The calculated assessment of the temperature fields and the stress-strain state in the excavation walls was performed using the Termoground program. The comparison of results of experimental and calculated data are shown in the article. The comparison of results of experimental and calculated data is shown in the article.

ЭМПИРИЧЕСКОЕ ИССЛЕДОВАНИЕ МОЩНОСТИ СТАТИСТИЧЕСКОГО КРИТЕРИЯ КОЛМОГОРОВА–СМИРНОВА В ЗАДАЧАХ ПРОВЕРКИ ГИПОТЕЗ О ЗАКОНЕ РАСПРЕДЕЛЕНИЯ

Statistical tests that use A.N. Kolmogorov one-sample statistics and N.V. Smirnov two-sample statistics have been widely used for solving applied problems for almost a hundred years. Those criteria are present in many textbooks and implemented in computer software for data analysis. The purpose of the work is to empirically estimate the power of the Kolmogorov–Smirnov criterion on a set of test problems to check hypotheses about the distribution law, as well as to investigate the properties of estimates of the power of the criterion for various types of resampling. To obtain various estimates of the power of the Kolmogorov–Smirnov criterion in solving test problems, the classical bootstrap, nonparametric bootstrap, parametric bootstrap, bootstrap with a random term, and resampling without returns were used. Based on the results of multiple solution of test problems, medians of p-values, medians of estimates of the power of the Kolmogorov–Smirnov criterion were calculated, and medians of biases of estimates on test problems and medians of all pairwise differences of various estimates of the power of the criterion were found. The distribution laws for the considered Kolmogorov–Smirnov power estimates, for the biases of the power estimates, and for all pairwise differences of the power estimates were investigated. For test problems where real data were considered, calculating power estimates without resampling is impossible, but unbiased power estimates can be predicted as the half-sum of two other power estimates, one obtained with a nonparametric bootstrap and the other obtained with resampling without returns. Methods for restructuring sample data and software for estimating the power of a statistical test developed within the study are universal and can be used for analyzing the properties of power estimates of other one-sample statistical tests, as well as for developing the classical methodology for checking statistical hypotheses.

Nonlocal FEM Simulations of Ductile Damage with Regularized Crack Path Predictions

Constitutive equations of finite strain elasto-plasticity are coupled to continuum damage mechanics to simulate the initiation and propagation of cracks in ductile materials. The diffuse deterioration of material’s strength is modelled using porosity as a damage parameter. We show that the local damage model yields simulation results, pathologically dependent on the FEM mesh upon mesh refinement: (i) the structural behaviour becomes unrealistically brittle; (ii) the FEM mesh determines the crack path. To solve these problems, an integral-based approach to nonlocal damage is implemented. It enables physically sound results by introducing the linear size of the microstructure into the model’s formulation. To prevent the effect of unrealistic diffusion of damage, the averaging operator is applied not to the porosity, but to its dual parameter, material’s continuity. Solutions of test problems for crack propagation in a compact tension specimen are presented. The convergence of FEM solutions is demonstrated for different slopes of the mesh with fine and coarse discretizations.

Solution of the first basic physically nonlinear problem of elasticity theory for anisotropic bodies

The study looks upon the process of the physically non-linear deformation of transversely isotropic homogeneous continuous solid bodies made from composites where the reinforcing elements are far more rigid than the binder. The solution of the physically non-linear problems employs the simplified theory of plasticity as proposed by B.E. Pobedra. The study proposes an approach to the writing out of an explicit solution that builds on the small parameter method. Ilyushin plasticity functions (that fall within the generalized Hooke’s law) are assigned discrete small values, and the resulting equations are then decomposed into power series in small parameters. The values of such small parameters are the measures of the deflection of the non-linear vs. the linear medium. Such decomposition produces an analytical coordinate & small parameters function enabling an immediate solution of the non-linear problem using just one linear elasticity field, which, in turn, is also created exclusively in reliance on the method of boundary states. Below are the results of solutions of test problems featuring a transversely isotropic cylinder with homogeneous boundary conditions. High precision in this case is achieved as early on as the third iteration. In problems with non-trivial boundary conditions, maximal precision is achieved at the first iteration and heavily depends on small parameter values. Each of the problems presented provides a detailed convergence analysis and graphic illustrations of the results.

Solutions Of Boundary-Value Problems For A System Of Differential Equations Of The Fourth Order With The Method Of Finite Differences

The paper considers a system of fourth-order ordinary differential equations. The system of equations is written in vector form. Computational algorithms are presented using the finite difference method with an error of O(h2). The resulting algebraic equations are solved by the matrix sweep method. Exact and approximate solutions of test problems are given. And also the errors of the considered numerical method are estimated.

Solutions Of Boundary-Value Problems For A System Of Differential Equations Of The Fourth Order With The Method Of Finite Differences

The paper considers a system of fourth-order ordinary differential equations. The system of equations is written in vector form. Computational algorithms are presented using the finite difference method with an error of O(h2). The resulting algebraic equations are solved by the matrix sweep method. Exact and approximate solutions of test problems are given. And also the errors of the considered numerical method are estimated.

Estimation of the numerical efficiency of global optimization methods

Currently, test problems are used to test the effectiveness of new global optimization methods. In this article, we analyze test global optimization problems to test the numerical efficiency of methods for their solution. At present, about 200 test problems of unconditional optimization and more than 1000 problems of conditional optimization have been developed. We can find these test problems on the Internet. However, most of these test problems are not informative for testing the effectiveness of global optimization methods. The solution of test problems of conditional optimization, as a rule, has trivial solutions. This allows the parameters of the algorithms to be tuned before these solutions are obtained. In test problems of conditional optimization, the accuracy of the fulfillment of constraints is important. Often, small errors in the constraints lead to a significant change in the value of an objective function. Construction of a new package of test problems to test the numerical efficiency of global optimization methods and compare the exact quadratic regularization method with existing methods.The author suggests limiting oneself to test problems of unconstrained optimization with unknown solutions. A package of test problems of unconstrained optimization is pro-posed, which includes known test problems with unknown solutions and modifications of some test problems proposed by the author. We also propose to include in this package J. Nie polynomial functions with unknown solutions. This package of test problems will simplify the verification of the numerical effectiveness of methods. The more effective methods will be those that provide the best solutions. The paper compares existing global optimization methods with the exact quadratic regularization method proposed by the author. This method has shown the best results in solving most of the test problems. This paper presents some of the results of the author's numerical experiments. In particular, the best solutions were obtained for test problems with unknown solutions. This method allows solving multimodal problems of large dimensions and only a local search program is required for its implementation.

Smoothed particle hydrodynamics simulation of biphasic soft tissue and its medical applications.

Modeling the coupled fluid and elastic mechanics of blood perfused soft tissues is important for medical applications. In particular, the current study aims to capture the effect of tissue swelling and the transport of blood through damaged tissue under bleeding or hemorrhaging conditions. The soft tissue is considered a dynamic poro-hyperelastic material with blood-filled voids. A biphasic formulation-effectively, a generalization of Darcy's law-is utilized, treating the phases as occupying fractions of the same volume. A Stokes-like friction force and a pressure that penalizes deviations from volume fractions summing to unity serve as the interaction force between solid and liquid phases. The resulting equations for both phases are discretized with the method of smoothed particle hydrodynamics (SPH). The solver is validated separately on each phase and demonstrates good agreement with exact solutions in test problems. Simulations of oozing, hysteresis, swelling, drying and shrinkage, and tissue fracturing and hemorrhage are shown in the paper. Graphical Abstract In the paper, a new methodology for the numerical simulation of the full dynamic response of blood-perfused soft tissues was developed.

Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

The least-squares collocation method in the mechanics of deformable solids

The paper is devoted to the application of the least-squares collocation method for solving the two- and one-dimensional problems in the mechanics of deformable solids. Calculation results of the deflections of isotropic and orthotropic elastic plates within the framework of various theories are presented. This paper shows that approximate solutions obtained by the hp-version of the least-squares collocation method converge with a high order and agree with analytical solutions of test problems with a high degree of accuracy. Numerical and mathematical modeling of three-point bending of composite beams was carried out taking into account nonlinear stress-strain relationships, multi-modulus behavior, and incipient fracture. It was found that the simulation results are in good agreement with the results of mechanical tests of three-point bending of composite beams.

New versions of the least-squares collocation method for solving differential and integral equations

This paper describes new versions of the least-squares collocation method for solving differential and integral equations. A p-version of the method has been proposed and implemented to solve nonlinear systems of partial differential equations. The stationary Navier-Stokes equations are used as an example. A hp-version of the method has been implemented for the numerical solution of the Fredholm integral equations of the second kind in the one-and two-dimensional cases. This paper shows that approximate solutions obtained by various versions of the least-squares collocation method converge with a high order and agree with analytical solutions of test problems with a high degree of accuracy.

Solution of Test Problems with Grey Wolf Optimization Algorithm and Comparison with Particle Swarm Optimization

In this study, Grey Wolf Optimization (GWO), which is a new method with swarm intelligence is compared with another metaheuristic optimization method, Particle Swarm Optimization (PSO), using optimization benchmark functions. Simulation studies on test functions are presented as a table by obtaining mean, standard deviation, best and worst values. In addition, the effects of population and iteration number change on the GWO algorithm are presented in separate tables. The GWO algorithm has establish a good balance between exploration and exploitation. Simulation studies have shown that GWO has better convergence performance and optimization accuracy.

A sequential partial linearization algorithm for the symmetric eigenvalue complementarity problem

In this paper, we introduce a Sequential Partial Linearization (SPL) algorithm for finding a solution of the symmetric Eigenvalue Complementarity Problem (EiCP). The algorithm can also be used for the computation of a stationary point of a standard fractional quadratic program. A first version of the SPL algorithm employs a line search technique and possesses global convergence to a solution of the EiCP under a simple condition related to the minimum eigenvalue of one of the matrices of the problem. Furthermore, it is shown that this condition is verified for a simpler version of the SPL algorithm that does not require a line search technique. The main computational effort of the SPL algorithm is the solution of a strictly convex standard quadratic problem, which is efficiently solved by a finitely convergent block principal pivoting algorithm. Numerical results of the solution of test problems from different sources indicate that the SPL algorithm is in general efficient for the solution of the symmetric EiCP in terms of the number of iterations, accuracy of the solution and total computational effort.

The approach to optimization of finite-difference schemes for the advective stage of finite-difference-based lattice Boltzmann method

The approach to optimization of finite-difference (FD) schemes for the linear advection equation (LAE) is proposed. The FD schemes dependent on the scalar dimensionless parameter are considered. The parameter is included in the expression, which approximates the term with spatial derivatives. The approach is based on the considering of the dispersive and dissipative characteristics of the schemes as the functions of the parameter. For the proper choice of the parameter, these functions are minimized. The approach is applied to the optimization of two-step schemes with an asymmetric approximation of time derivative and with various approximations of the spatial term. The cases of schemes from first to fourth approximation orders are considered. The optimal values of the parameter are obtained. Schemes with the optimal values are applied to the solution of test problems with smooth and discontinuous initial conditions. Also, schemes are used in the FD-based lattice Boltzmann method (LBM) for modeling of the compressible gas flow. The obtained numerical results demonstrate the convergence of the schemes and decaying of the numerical dispersion.

The least squares collocation method for the biharmonic equation in irregular and multiply-connected domains

This paper reports new h-and p-versions of the least squares collocation method of high-order accuracy proposed and implemented for solving boundary value problems for the biharmonic equation in irregular and multiply-connected domains. This paper shows that approximate solutions obtained by the least squares collocation method converge with high order and agree with analytical solutions of test problems with high degree of accuracy. There has been a comparison made for the results achieved in this study and results of other authors who used finite difference and spectral methods.

Non-linear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutions

In Cauchy problems with blow-up solutions there exists a singular point whose position is unknown a priori (for this reason, the application of standard fixed-step numerical methods for solving such problems can lead to significant errors). In this paper, we describe a method for numerical integration of blow-up problems for non-linear systems of coupled ordinary differential equations of the first order (xm)t′=fm(t,x1,…,xn), m=1,…,n, based on the introduction a new non-local independent variable ξ, which is related to the original variables t and x1,…,xn by the equation ξt′=g(t,x1,…,xn,ξ). With a suitable choice of the regularizing function g, the proposed method leads to equivalent problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore, the transformed problems admit the use of standard numerical methods with a fixed stepsize in ξ. Several test problems are formulated for systems of ordinary differential equations that have monotonic and non-monotonic blow-up solutions, which are expressed in elementary functions. Comparison of exact and numerical solutions of test problems showed the high efficiency of numerical methods based on non-local transformations of a special kind. The qualitative features of numerical integration of blow-up problems for single ODEs of higher orders with the use of non-local transformations are described. The efficiency of various regularizing functions is compared. It is shown that non-local transformations in combination with the method of lines can be successfully used to integrate initial–boundary value problems, described by non-linear parabolic and hyperbolic PDEs, that have blow-up solutions. We consider test problems (admitting exact solutions) for nonlinear partial differential equations such as equations of the heat-conduction type and Klein–Gordon type equations, in which the blowing-up occurs both in an isolated point of space x=x∗, and on the entire range of variation of the space variable 0≤x≤1. The results of numerical integration of test problems, obtained when approximating PDEs by systems with a different number of coupled ODEs, are compared with exact solutions.

Calculation of thermal stress-strain state of soil base under the building on permafrost soil

Article presents problem statement and solution for joint calculation of temperature regime and stress-strain state of permafrost soils under building footing, with consideration of thermal impact from on-surface and buried high-temperature utilities. Based on a series of optimization tasks, a design model for stress-strain state has been developed, solutions of test problems have been obtained and analyzed to determine the degree of influence of temperature stresses and relation of the deformation characteristics of frozen soil in the negative temperature spectrum on the stress distribution and settlement values of frozen soil base under the building.