Method For Solution Of Problems Research Articles

Acceleration of uncertainty propagation through Lagrange multipliers in partitioned stochastic method

A partitioned stochastic method (PSM) is proposed for the solution of static structural mechanics problems with uncertainties, whose solution vectors are the displacements for each partition and Lagrange multipliers along with the partition interfaces. The proposed partitioned stochastic method employs three stochastic basis selection steps: an arbitrary initial choice of displacement random bases, a set of conjugate bases for the Lagrange multipliers, and finally modification of the displacement bases affected by those of the Lagrange multipliers. The present PSM thus propagates the uncertainties instantly across partitioned substructures, resulting in an improved rate of convergence. Numerical experiments illustrate the proposed PSM outperforms a conventional partitioned solution method for structural mechanics problems.

A novel iterative integration regularization method for ill-posed inverse problems

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($$2^j$$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

A method for solving a boundary value problem in a multilayered area

A mathematical model of thermal process in an electrical machine was built as an example, presented as a three-layer cylinder where internal heat sources operate in one of the layers and heat is submitted to the other two by means of heat conduction. A method of solving the boundary-value problems for heat conduction equation in a complex area – a multi-layered cylinder with internal heat sources operating in one part of the layers and external ones in another part, is proposed. A method of problem solution in conditions of uncertainty of one of the boundary condition at the layers interface with conductive heat exchange between the layers is reviewed. The principle of method lies in the averaging of temperature distributions radially in the internal layers. As a result of transformations at the layers interface a boundary condition of the impedance-type conjugation appears. The analytical and numeric-analytical solutions of simplified problems were obtained.

REGULAR SOLUTION OF THE INVERSE PROBLEM WITH INTEGRAL CONDITION FOR A TIME-FRACTIONAL EQUATION

Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob\-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known. We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them. We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \;\;\;x\in \Bbb R^n\] where $u$ is the unknown solution of the Cauchy problem, $\eta_1$ and $\Phi_1$ are the given functions. Using the method of the Green's vector function, we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability. There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.

РЕАЛІЗАЦІЯ МОЖЛИВОСТЕЙ STEM-ОСВІТИ ЗАСОБАМИ ІНТЕГРАЦІЇ КРЕАТИВНИХ МЕТОДІВ НАВЧАННЯ

The article considers the problem of activation of the creative potential of students, their involvement in the searching activity using the integrated methods as the instrument to solve the contradictories between the requirements of innovative training at higher educational establishments and the practice of conventional methods application. The research argues the expediency of use of the opportunities of STEM education at technical higher educational establishments to train students to be able to make non-standard and adequate decision depending on the situation, to see the prospects and plan the strategies, etc. Creativity, cooperation and critical thinking are the key competences to get success in the 21st century, which secure development of the principal skills and provide knowledge not separately, but as an integration of courses into an unified educational system. The work outlines the main opportunities of STEM-education at technical higher educational establishments, particularly mastering tools of analysis and synthesis for specification of the problem and its solution; learning the mathematic apparatus for solution of the set professional tasks; readiness to integrate different aspects of knowledge and activities; ability to make situational integration of means, knowledge and actions in the process of problem solution; ability to creative integration in its dynamic development, focus on optimization of the professional activity, development of critical thinking as a principal feature of specialists; ability to set the problem correctly, etc. The article supplies examples of use of the cognitive teaching methods of propaedeutic block for the creative methods application. The research describes the opportunities to use such cognitive methods as empathy; semantic imagination; imaginary thinking; symbolic thinking; heuristic issues; comparison; heuristic observation; facts; investigation; constructing of notions, norms and theories; assumptions and trips into the future; forecasting, etc. The research considers peculiarities of the creative methods of training at technical higher educational establishments on the example of teacher’s proficiency in physics, which are based on the integrated approach by shaping of a logical chain of methods for solution of specific problems (methods of agglutination, synectics, analogy, inversion, excogitation, etc.). The conclusion is that the main task of introduction of the STEM-education at technical higher educational establishments expects appropriate pedagogical conditions to secure development of the creative potential of a personality, independent critical thinking, system of values, formation of the range of professional competences of the specialist.

Hermite wavelet method for the numerical solution of nonlinear singular initial value problems

Hermite wavelet method for the numerical solution of nonlinear singular initial value problems

A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Abstract The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.

Direct methods for finding optimal solution of assignment problems are not always dependable

Direct methods for finding optimal solution of assignment problems are not always dependable

A Numerical Method for the Solution of Two Inverse Problems in the Mathematical Model of Redox Sorption

For a mathematical model with internal-diffusion kinetics and an oxidation-reduction reaction, we consider two inverse problems of determining the sorption isotherm or the redox reaction rate constant from the output dynamic curve. A numerical method for the solution of these inverse problems is proposed, computational results are reported, and its potential is investigated.

Analytical solutions to contact problem with fractional derivatives in the sense of Caputo

The current study extends the applications of the variational iteration method for the analytical solution of fractional contact problems. The problem involves Caputo sense while calculating the derivative of fractional order, we apply the Penalty function technique to transform it into a system of fractional boundary value problems coupled with a known obstacle. The variational iteration method is employed to find the series solution of fractional boundary value problem. For different values of fractional parameters, residual errors of solutions are plotted to make sure the convergence and accuracy of the solution. The reasonably accurate results show that one of the highly effective and stable methods for the solution of fractional boundary value problem is the method of variational iteration.

Efficient Solution Method Based on Inverse Dynamics of Optimal Control Problems for Fixed-Based Rigid-Body Systems

We propose an efficient solution method of finite horizon optimal control problems (FHOCPs) for fixed-based rigid-body systems based on inverse dynamics. Our method can reduce the computational cost compared with the conventional FHOCP based on forward dynamics. We reformulate the FHOCP for the rigid-body systems by utilizing the generalized acceleration as the decision variables and inverse dynamics as the equality constraint. We derive the necessary conditions of the optimal control, namely, the optimality conditions, and formulate a two-point boundary-value problem that can be solved efficiently by using the recursive Newton Euler algorithm (RNEA) and the partial derivatives of RNEA. The results of the several numerical experiments on nonlinear model predictive control using the proposed formulation demonstrate the effectiveness of our approach.

General Euler-Poisson-Darboux Equation and Hyperbolic B-Potentials

In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz B-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler-Poisson-Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

Special issue on advanced solution methods for modeling complex electromagnetic problems

Special issue on advanced solution methods for modeling complex electromagnetic problems

Celanese is the ICIS Company of the Year

Off-site construction (OSC) is an innovative and sustainable construction method. One of its critical success factors is the proper management of its supply chain (SC). Recently, significant research attention has been focused on the modelling of OSC-SC to improve its performance and sustainability. However, the literature still lacks a comprehensive review of the modelling studies on off-site construction -supply chain management (OSC-SCM). Therefore, this research contributes by providing a comprehensive and up-to-date mapping and clustering of 309 journal articles on the modelling of OSC-SC to identify its trends and gaps, and hence, highlight future research opportunities. To achieve these objectives, a mixed review method, consisting of scientometric and systematic reviews, is used. The scientometric review identifies the most prolific journals, researchers, co-occurrence network of keywords, and their citation bursts. On the other hand, the systematic review classifies the articles based on OSC type and supply chain (SC) stage. At each SC stage, the articles are further classified based on problem settings and solution methods. The results show that production problems have received the most attention (i.e., by 28% of the included articles), followed by on-site construction (19%), design (13%), and logistics (7%) problems. However, researchers are more inclined to address problems at multiple SC stages (33%) due to the interrelationship between these stages. Besides, the systematic analysis shows that OSC-SC problems have been solved frequently by a variety of solution methods such as optimization (25%), simulation (13%) and building information modelling (BIM) (9.5%). However, researchers tend more to integrate multiple solution methods (35%) to address the complexities of OSC-SC problems. Finally, the included studies are classified based on the three sustainability dimensions. The results show that economic, environmental, and social sustainability dimensions have been considered in previous studies by 72%, 24%, and 4%, respectively. Therefore, the study identifies research gaps at each SC stage of each OSC type to incentivize future studies to consider more environmental and social sustainability factors in OSC-SC models. Since modular integrated construction (MiC) has the highest prefabrication level, this study provides future research directions for sustainable supply chain management (SCM) in MiC. The present research is an important reference guide that helps researchers and practitioners to understand different problem settings and their solution methods in OSC-SCM.

Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi‐Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi‐Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation. Also, a vector‐matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method.

Solution method for ill-conditioned problems based on a new improved fruit fly optimization algorithm

Abstract Based on deeply analysis for optimization process of basic fruit fly optimization algorithm (FOA), a new improved FOA (IFOA) method is proposed, which modifies random search direction, increases the adjustment coefficient of search radius, and establishes a multi-sub-population solution mechanism. The proposed method can process the nonlinear objective function that has non-zero and non-negative extreme points. In the paper, IFOA method is applied to ill-conditioned problem solution in the field of surveying data processing. Application of the proposed method on two practical examples show that solution accuracy of IFOA is superior to that of three well-known intelligent optimization algorithms and two existing improved FOA methods, and it is also better than truncated singular value decomposition method and ridge estimation method. In addition, compared with intelligent search method represented by particle swarm optimization algorithm, The IFOA method has the advantages of less parameter settings, simple optimization process and easy program implementation. So, IFOA method is feasible, effective and practical in solving ill-conditioned problems.

Discrete-Continual Finite Element Method for Semianalytical Analysis of Plates on Two-Parameter Elastic Foundation. Part 2: Discrete-Continual Formulation of the Problem, Numerical Samples

This paper is devoted to application of discrete-continual finite element method (DCFEM) for semianalytical solution of problems of analysis of plates on two-parameter elastic foundation. Corresponding resultant multipoint boundary problem for system of ordinary differential equations of the first order with piecewise-constant coefficients is presented. Numerous samples of problems of structural mechanics have been considered. These samples allow demonstration of some advantages of application of the suggested approach to semianalytical analysis of plates on elastic foundation.

GMsFEM on unstructured grids for single-phase flow in fractured porous media

In this work, we consider an unstructured Generalized Multiscale Finite Element Method (GMsFEM) for solution of the filtration problem in a fractured media. The basic idea is that coarse grid blocks are formed as sets of fine grid triangular cells and, thus, can be of an almost arbitrary polygonal shape. In this approach, when constructing the fine grid, the faces of coarse blocks are not considered. Discrete fractured model is used to describe a flows in a fractured media.

A penalization scheme for the numerical solution of optimal control problems with control complementarity constraints

AbstractWe suggest a simple penalty method for the numerical solution of optimal control problems with control complementarity constraints which is based on the famous Fischer–Burmeister function. The distinct advantage of this approach is the resulting smoothness of the penalty term and the unconstrainedness of the associated penalized surrogate problems. A numerical example from elliptic PDE control is presented as well.

Adaptive Large Neighborhood Search for Multitrip Vehicle Routing with Time Windows

We consider a multitrip vehicle routing problem with time windows (MTVRPTW), in which each vehicle can perform several trips during its working shift. This problem is especially relevant in the context of city logistics. Heuristic solution methods for multitrip vehicle routing problems often separate routing and assignment phases to create trips and then assign them to the available vehicles. We show that this approach is outperformed by an integrated solution method in the presence of time windows. We use an automatic configuration tool to obtain efficient and contextualized implementations of our solution methods. We provide suitable instances for the MTVRPTW as well as an instance generator. Also, we discuss the relevance of two objective functions: the total duration and the total travel time. When minimizing the travel time, large increases in waiting time are incurred, which is not realistic in practice.