Large Discrete Systems Research Articles

APPLICATION OF THE THEORY OF PETRI NETS IN THE DEVELOPMENT OF SIMULATION MODELS OF BUSINESS PROCESSES BASED ON THE IDEF3 METHODOLOGY

In this study, we propose a model of an artificial neural network used as a specialized superstructure over a genetic algorithm, which allows influencing the process of finding solutions directly during the synthesis of solutions. Such a combination of methods will allow controlling the trajectory of the population in the solution space, which is especially important when working with big data processing technology, when stopping the solution search process due to the attenuation of the evolutionary procedure or finding the population in a local extremum requires stopping the genetic algorithm, performing additional adjustment of operators and restarting, the use of such an approach is ineffective, especially when working with big data and labor-intensive calculations. This article proposes a model of an artificial neural network that allows recognizing the state of the population of the genetic algorithm and making a decision to change the operating parameters of the genetic algorithm operators. The proposed model allows recognizing the processes of attenuation of the evolutionary procedure when solving the problem of structural and parametric synthesis of large discrete systems and determining measures of influence on the operating parameters of the genetic algorithm. This model recognizes the state of the population with an accuracy of more than 95%, which allows to significantly reduce the time for finding solutions in problems of applying a genetic algorithm to work with big data.

Destructive Ability Analysis of Genetic Algorithm Operator When Solving Business Process Problem of Structural and Parametric Synthesis

The work analyzes the destructive ability of operators of a classical genetic algorithm when solving the problem of controlling the trajectory of a population during the search for solutions. In accordance with the hypothesis put forward, when solving the problem of structural-parametric synthesis of large discrete systems with a given behavior, it is advisable to change the settings of the functioning of the genetic algorithm operators to ensure: better convergence of the genetic algorithm, avoid attenuation, signal the need to restart the solution synthesis procedure, and also facilitate the derivation of the population from local extrema. It is proposed to use an artificial neural network as a control element (control add-on), which should implement control of the process of synthesis of simulation models of business processes based on a given behavior (the ability of a simulation model to convert a given input vector into a reference output), which is especially important when working with large systems. In accordance with the same hypothesis, an increase in the destructive ability of the crossing over and mutation operators allows the population to be dispersed across the solution space, which is advisable when attenuation occurs and is in a local extremum, and a decrease in the destructive ability contributes to a more thorough search for solutions in a certain area of the solution space. The paper provides examples of the work of operators and the behavior of the population when synthesizing simulation models of business processes based on the theory of Petri nets. To model the operation of a genetic algorithm and an artificial neural network, it is proposed to use the theory of Petri nets, which simplifies the process of managing the synthesis procedure and allows the use of parallel programming tools with distributed GPGPU computing on CUDA technology. As part of this study, an analysis of the behavior of the population when changing the operating settings of operators was carried out, which confirmed the hypothesis put forward.

Mathematical apparatus of artificial neural networks for genetic algorithm controlling under structural parametric synthesis of large discrete systems

The article considers the appliance possibility of mathematical apparatus of artificial neural networks for controlling processes of intellectual synthesis of large discrete systems models with a specified behavior. The analysis of the influence of different methods and its classification has carried out, which allows defining the moving trajectory of population in a space of decisions: diffusion to search new extrema and compact study the field in order to increase the speed of population convergence. To solve the problem, using the neural network approach, the two graphical presentations of population conditions have been done: the first, the value of fitness functions of each individual and, the second, some individuals with the same value of adaptation functions. The article defines the structure of the neural network in the form of the network without feed-back and number of neurons in the hidden layer. The training of neural networks has been done and the examples of management decision making based on graphical presentation of the population condition have been provided. The unification of evolutionary procedures by one mathematical apparatus of the Petri net was proposed, which allows using the technology of parallel calculation under programmed realization. Using parallel calculation, combined with proposed models and methods should increase the processing speed of decision-making support systems based on genetic algorithms.

DG-GMsFEM for Problems in Perforated Domains with Non-Homogeneous Boundary Conditions

Problems in perforated media are complex and require high resolution grid construction to capture complex irregular perforation boundaries leading to the large discrete system of equations. In this paper, we develop a multiscale model reduction technique based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions on perforations. This method implies division of the perforated domain into several non-overlapping subdomains constructing local multiscale basis functions for each. We use two types of multiscale basis functions, which are constructed by imposing suitable non-homogeneous boundary conditions on subdomain boundary and perforation boundary. The construction of these basis functions contains two steps: (1) snapshot space construction and (2) solution of local spectral problems for dimension reduction in the snapshot space. The presented method is used to solve different model problems: elliptic, parabolic, elastic, and thermoelastic equations with non-homogeneous boundary conditions on perforations. The concepts for coarse grid construction and definition of the local domains are presented and investigated numerically. Numerical results for two test cases with homogeneous and non-homogeneous boundary conditions are included, as well. For the case with homogeneous boundary conditions on perforations, results are shown using only local basis functions with non-homogeneous boundary condition on subdomain boundary and homogeneous boundary condition on perforation boundary. Both types of basis functions are needed in order to obtain accurate solutions, and they are shown for problems with non-homogeneous boundary conditions on perforations. The numerical results show that the proposed method provides good results with a significant reduction of the system size.

Multiscale dimension reduction for flow and transport problems in thin domain with reactive boundaries

In this paper, we consider flow and transport problems in thin domains. Modeling problems in thin domains occur in many applications, where thin domains lead to some type of reduced models. A typical example is one dimensional reduced-order model for flows in pipe-like geometries (e.g., blood vessels). In many reduced-order models, the model equations are described apriori by some analytical approaches. In this paper, we propose the use of multiscale methods, which are alternative to reduced-order models and can represent reduced-dimension modeling by using fewer basis functions (e.g., the use of one basis function corresponds to one dimensional approximation).The mathematical model considered in the paper is described by a system of equations for velocity, pressure, and concentration, where the flow is described by the Stokes equations, and the transport is described by an unsteady convection-diffusion equation with non-homogeneous boundary conditions on walls (reactive boundaries). We start with the finite element approximation of the problem on unstructured grids and use it as a reference solution for two and three-dimensional model problems. Fine grid approximation resolves complex geometries on the grid level and leads to a large discrete system of equations that is computationally expensive to solve. To reduce the size of the discrete systems, we develop a multiscale model reduction technique, where we construct local multiscale basis functions to generate a lower-dimensional model on a coarse grid. The proposed multiscale model reduction is based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsGEM). In DG-GMsFEM for flow problems, we start with constructing the snapshot space for each interface between coarse grid cells to capture possible flows. For the reduction of the snapshot space size, we perform a dimension reduction via a solution of the local spectral problem and use eigenvectors corresponding to the smallest eigenvalues as multiscale basis functions for the approximation on the coarse grid. For the transport problem, we construct multiscale basis functions for each interface between coarse grid cells and present additional basis functions to capture non-homogeneous boundary conditions on walls. Finally, we will present some numerical simulations for three test geometries for two and three-dimensional problems to demonstrate the method's performance.

Model of an Artificial Neural Network for Solving the Problem of Controlling a Genetic Algorithm Using the Mathematical Apparatus of the Theory of Petri Nets

The aim of the study was to increase the speed, quantity and quality of solutions in intelligent systems aimed at solving the problem of structural–parametric synthesis of models of large discrete systems with a given behavior. As a hypothesis, it was assumed that the adapted model of an artificial neural network is able to control changes in the parameters of the functioning of the operators of the genetic algorithm directly in the process of solving the problem of intelligent structural–parametric synthesis of models of large discrete systems. To solve the problem of managing the process of intelligent search for solutions based on a genetic algorithm, an artificial neural network, which is used as an add-in, must dynamically change the “destructive” ability of operators based on data about the current and/or historical state of the population. In the article, the theory of Petri nets is used as a single mathematical device capable of modeling the work of evolutionary procedures. This mathematical tool is able to simulate the operation of a genetic algorithm adapted to solving the problem of structural–parametric synthesis of models of large discrete systems with a given behavior; simulate the operation and training of an artificial neural network; combine the genetic algorithm with a control add-in based on an artificial neural network to prevent attenuation and premature convergence; simulate the process of recognizing the state of the population; and simulate the operation of the models obtained as a result of the synthesis. As an example of the functioning of the proposed approach, the article presents the results of a computational experiment, which considers the problem of structural–parametric synthesis of computer technology based on the developed models of the element base-RS, D and T triggers that are capable of processing a given input vector into the required (reference) output. In the software implementation of the proposed approach, calculations on the CPU and CPU+GPGPU technologies were used.

A Deep Neural Network Model of Particle Thermal Radiation in Packed Bed

Prediction of particle radiative heat transfer flux is an important task in the large discrete granular systems, such as pebble bed in power plants and industrial fluidized beds. For particle motion and packing, discrete element method (DEM) now is widely accepted as the excellent Lagrangian approach. For thermal radiation, traditional methods focus on calculating the obstructed view factor directly by numerical algorithms. The major challenge for the simulation is that the method is proven to be time-consuming and not feasible to be applied in the practical cases. In this work, we propose an analytical model to calculate macroscopic effective conductivity from particle packing structures Then, we develop a deep neural network (DNN) model used as a predictor of the complex view factor function. The DNN model is trained by a large dataset and the computational speed is greatly improved with good accuracy. It is feasible to perform real-time simulation with DNN model for radiative heat transfer in large pebble bed. The trained model also can be coupled with DEM and used to analyze efficiently the directional radiative conductivity, anisotropic factor and wall effect of the particle thermal radiation.

Strong Stubborn Set Pruning for Star-Topology Decoupled State Space Search

Analyzing reachability in large discrete transition systems is an important sub-problem in several areas of AI, and of CS in general. State space search is a basic method for conducting such an analysis. A wealth of techniques have been proposed to reduce the search space without affecting the existence of (optimal) solution paths. In particular, strong stubborn set (SSS) pruning is a prominent such method, analyzing action dependencies to prune commutative parts of the search space. We herein show how to apply this idea to star-topology decoupled state space search, a recent search reformulation method invented in the context of classical AI planning.
 Star-topology decoupled state space search, short decoupled search, addresses planning tasks where a single center component interacts with several leaf components. The search exploits a form of conditional independence arising in this setting: given a fixed path p of transitions by the center, the possible leaf moves compliant with p are independent across the leaves. Decoupled search thus searches over center paths only, maintaining the compliant paths for each leaf separately. This avoids the enumeration of combined states across leaves.
 Just like standard search, decoupled search is adversely affected by commutative parts of its search space. The adaptation of strong stubborn set pruning is challenging due to the more complex structure of the search space, and the resulting ways in which action dependencies may affect the search. We spell out how to address this challenge, designing optimality-preserving decoupled strong stubborn set (DSSS) pruning methods. We introduce a design for star topologies in full generality, as well as simpler design variants for the practically relevant fork and inverted fork special cases. We show that there are cases where DSSS pruning is exponentially more effective than both, decoupled search and SSS pruning, exhibiting true synergy where the whole is more than the sum of its parts. Empirically, DSSS pruning reliably inherits the best of its components, and sometimes outperforms both.

Constrained energy minimization based upscaling for coupled flow and mechanics

In this paper, our aim is to present (1) an embedded fracture model (EFM) for coupled flow and mechanics problem based on the dual continuum approach on the fine grid and (2) an upscaled model for the resulting fine grid equations. The mathematical model is described by the coupled system of equation for displacement, fracture and matrix pressures. For a fine grid approximation, we use the finite volume method for flow problem and finite element method for mechanics. Due to the complexity of fractures, solutions have a variety of scales, and fine grid approximation results in a large discrete system. Our second focus in the construction of the upscaled coarse grid poroelasticity model for fractured media. Our upscaled approach is based on the nonlocal multicontinuum (NLMC) upscaling for coupled flow and mechanics problem, which involves computations of local basis functions via an energy minimization principle. This concept allows a systematic upscaling for processes in the fractured porous media, and provides an effective coarse scale model whose degrees of freedoms have physical meaning. We obtain a fast and accurate solver for the poroelasticity problem on a coarse grid and, at the same time, derive a novel upscaled model. We present numerical results for the two dimensional model problem.

Efficient stabilization and acceleration of numerical simulation of fluid flows by residual recombination

The study of the stability of a dynamical system described by a set of partial differential equations (PDEs) requires the computation of unstable states as the control parameter exceeds its critical threshold. Unfortunately, the discretization of the governing equations, especially for fluid dynamic applications, often leads to very large discrete systems. As a consequence, matrix based methods, like for example the Newton–Raphson algorithm coupled with a direct inversion of the Jacobian matrix, lead to computational costs too large in terms of both memory and execution time.We present a novel iterative algorithm, inspired by Krylov-subspace methods, which is able to compute unstable steady states and/or accelerate the convergence to stable configurations. Our new algorithm is based on the minimization of the residual norm at each iteration step with a projection basis updated at each iteration rather than at periodic restarts like in the classical GMRES method. The algorithm is able to stabilize any dynamical system without increasing the computational time of the original numerical procedure used to solve the governing equations. Moreover, it can be easily inserted into a pre-existing relaxation (integration) procedure with a call to a single black-box subroutine.The procedure is discussed for problems of different sizes, ranging from a small two-dimensional system to a large three-dimensional problem involving the Navier–Stokes equations. We show that the proposed algorithm is able to improve the convergence of existing iterative schemes. In particular, the procedure is applied to the subcritical flow inside a lid-driven cavity. We also discuss the application of Boostconv to compute the unstable steady flow past a fixed circular cylinder (2D) and boundary-layer flow over a hemispherical roughness element (3D) for supercritical values of the Reynolds number. We show that Boostconv can be used effectively with any spatial discretization, be it a finite-difference, finite-volume, finite-element or spectral method.

Large Discrete Systems Evolutionary Synthesis Procedure

Large Discrete Systems Evolutionary Synthesis Procedure

A Comprehensive Survey of Flexible Manufacturing System Scheduling Using Petri Nets

A Flexible Manufacturing System (FMS) is an integrated, computer-controlled system of machines, automated handling systems, and storage systems that can be used to simultaneously manufacture a variety of jobs. FMSs can be characterized as asynchronous, concurrent, distributed and parallel systems in which multiple operations share multiple resources so that the performance criteria are optimized. Petri nets (PNs) have recently become a promising approach for modeling FMSs. PNs are formal graphical modeling tool that can be efficiently utilized as a process analysis and modeling tool, because it shows graphically and dynamically to simulate a process in an integrated manner. It is a mathematical modeling technique that is useful for modeling concurrent, asynchronous, distributed, parallel, nondeterministic, and stochastic systems. Unreasonably the dispatching resources/jobs to machine in FMS may result in a deadlock situation and the situation is studied thoroughly and avoided through PN techniques. From the design and analysis point of view, the uses of nets have many advantages in modeling, performance evaluation, qualitative analysis and code generation. Scheduling a manufacturing system is usually a Non-Polynomial hard problem. This means that only heuristic algorithms can be used to provide near-optimal schedule when it is merged with PN. The merging of PNs with knowledge based heuristic techniques seems to be very promising to deal with large complex discrete event dynamic systems. This paper presents a comprehensive survey of FMS that combines PNs with other methods.

STABILITY AND CHAOS IN REACTION SYSTEMS

Reaction systems are an abstract model of biochemical reactions in the living cell within a framework of finite (though often large) discrete dynamical systems. In this setting, this paper provides an analytical and experimental study of stability. The notion of stability is defined in terms of the way in which small perturbations to the initial state of a system are likely to change the system's eventual behavior. At the stable end of the spectrum, there is likely to be no change; but at the unstable end, small perturbations take the system into a state that is probabilistically the same as a randomly selected state, similar to chaotic behavior in continuous dynamical systems.

Fourier mode analysis of multigrid methods for partial differential equations with random coefficients

Partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models. Various approaches exist for computing the stochastic characteristics of the solution of such a differential equation. In this paper, we consider the spectral expansion approach. This method transforms the continuous model into a large discrete algebraic system. We study the convergence properties of iterative methods for solving this discretized system. We consider one-level and multi-level methods. The classical Fourier mode analysis technique is extended towards the stochastic case. This is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem. We show how the convergence properties depend on the particulars of the algorithm, on the discretization parameters and on the stochastic characteristics of the model. Numerical results are added to illustrate some of our theoretical findings.

Amplitudes for magnon scattering by vortices in two-dimensional weakly easy-plane ferromagnets

We study magnon modes in the presence of a vortex in a circular easy-plane ferromagnet. The problem of vortex-magnon scattering is investigated for partial modes with different values of the azimuthal quantum number m over a wide range of wave numbers. The analysis was done by combining analytical and numerical calculations in the continuum limit with numerical diagonalization of adequately large discrete systems. The general laws governing vortex-magnon interactions are established. We give simple physical explanations of the scattering results: the splitting of doublets for the modes with opposite signs of $m$, which takes place for the long wavelength limit, is an analogue of the Zeeman splitting in the effective magnetic field of the vortex. A singular behavior for the scattering amplitude, $\sigma_m\propto k$, takes place as $k$ diverges; it corresponds to the generalized Levinson theorem and can be explained by the singular behavior of the effective magnetic field at the origin.

MODAL MODIFICATION OF VIBRATING SYSTEMS: SOME PROBLEMS AND THEIR SOLUTIONS

Experimental modal testing data form an incomplete set of natural frequencies and mode shapes. The incompleteness of modal representation is inherent also to continuous systems that are represented by discrete models. Furthermore, current state-of-the-art numerical methods for determining the eigenvalues of large discrete systems can extract only some eigenvalues and eigenvectors associated either with low frequencies or the high-frequency range. Therefore, when predicting the effect of structural modification on the spectrum of a modified structure an error called a ‘truncation error’ exists in the calculations.The paper focuses on the prediction of the error in the spectrum of a modified structure resulting from the incomplete representation of the model. An optimal solution in a Rayleigh–Ritz sense is derived and the truncation error is bounded. Then the inverse problem of determining the structural modification that is needed to assign the spectrum is considered. A mathematical formulation characterising a continuous family of solutions is first given. Then a procedure for determining physical realisable solutions is developed. Finally, it is shown that if both left and right eigenvectors of the system are known then it is possible to solve the inverse structural modification problem without truncation error. The extraction of left eigenvectors from experimental data, however, can be an ill-conditioned problem. Thus, the modification could be sensitive to noise in the measured data, and the truncation error alone might not be the dominant mechanism of error.

When Is a One-Dimensional Lattice Small?

Dynamical systems may be either discrete or continuous, though it is often convenient to blur this distinction in order to formulate a continuum model of a large discrete system. For example, in systems in which macroscopic behavior alone is of interest, a continuum approach is justified because of the vast number of molecules present. But just how many molecules are necessary for a system to be accorded a continuum description? Due to recent advances in interrogating, fabricating, and controlling small collections of molecules [1], it now seems appropriate to consider the manner in which discrete physical systems acquire a continuum description. Such a study is not limited, of course, to molecular systems. As a preliminary question, we may ask, “How can the terms ‘small’ and ‘large’ be made precise?” To formulate this question in a physical context, we consider a model dynamical problem originally investigated by Fermi, Pasta, and Ulam (FPU) [2]. The model consists of a one-dimensional chain of point masses that interact with their nearest neighbors via nonlinear forces. FPU were surprised to find that even in the presence of nonlinearity the dynamics of the system was almost periodic, with a period known as the recurrence time. Further investigations of the lattice dynamics led to the discovery that the recurrence time has a power-law dependence on system parameters [3] (see also [4,5]). This can be explained in the continuum limit by soliton solutions of the Korteweg-de Vries (KdV) equation [6–9]. In this model, derived by letting the number of lattice points tend to infinity while keeping the total length of the lattice fixed [10], solitons flow around the lattice independently; the recurrences are due to the periodic return of these solitons to the initial configuration. The well-established scaling of recurrence time in the continuum limit makes the FPU lattice an ideal system to study the transition from a discrete lattice to a continuum system. We focus here on the behavior of the lattice as it passes through the twilight between “small” and “large” by examining the FPU recurrences as the system size changes. For large systems, when the dynamics is well modeled by the continuum KdV equation, we find that the recurrence can be characterized by a single parameter that measures the strength of the nonlinearity of the system. As the system size shrinks, however, a second parameter is required: the size of the system itself. The transition between these two regimes is well-defined and occurs at a critical value of the system size that appears to coincide with the onset of an instability in the dynamics. We are therefore able to formulate a precise criterion for where the transition between “small” and “large” lattices occurs. The FPU model consists of a one-dimensional lattice of point masses coupled by a linear force augmented by a quadratic nonlinearity and is described by the Hamiltonian

Integration of the human operator into responsive discrete production management systems

Abstract The need for production systems that can react or respond to dynamic changes is continuously increasing because of the reduction of product life cycle time and the rise of competition. To improve responsiveness, we show that integrating the intelligence of the human operator into the system helps to face complexity. However, little work has been done about the optimization of such integration considering production system constraints (real time decision making, observability, etc.) and human operator constraints (mental workload, trust in management system, self-confidence, etc.). This paper aims at discussing ways to take account of the cognitive abilities of the human operator and offers some advice on how to take accurately into account the integration of the human operator by the proposal of a set of global specifications. To illustrate how it is possible to contribute to the optimized design of a system based upon such specifications, we propose the concept of “distributed production management system”. We first address the specific interface issue. A classical example of “advanced display” designed for continuous systems, that is, the Ecological Interface Design (EID) approach, is applied to discrete production systems. We show that such an approach is coherent with parts of the introduced specifications but can be adapted to large complex and discrete systems with difficulty. To solve this issue, we propose the development of a distributed DSS where each local DSS integrates an advanced display and manages a set of production resources in cooperation with an operator, which reduces the global complexity.

On the Minimal Control and Minimal Observation of Large Scale Discrete Event Dynamical Systems

Abstract In the framework of supervisory control of discrete event dynamical systems(DEDS) [1] , given a target language K, if all events in Σ can be controlled and observed, then K is both controllable and observable, and thus the closed loop system L(S/G)=K. In addition to that, there may exist many event subset Σ c of Σ and event subset Σ o of Σ, which also guarantee K is both controllable and observable. It is of interest to find the Σ c and Σ o , of which each has the minimal elements. In this paper, some necessary and sufficient conditions are given for the existence of the minimal observation Σ o m (the minimal control Σ c m always exists), and some efficient algorithms to compute the minimal control and the minimal observation are provided. Furthermore, the similar results holds when Σ is partly controllable and observable.

Ritz method for dynamic analysis of large discrete linear systems with non‐proportional damping

AbstractReal and complex Ritz vector bases for dynamic analysis of large linear systems with non‐proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.