Time response of structure with interval and random parameters using a new hybrid uncertain analysis method

Practical structures often operate with some degree of uncertainties, and the uncertainties are often modelled as random parameters or interval parameters. For realistic predictions of the structures behaviour and performance, structure models should account for these uncertainties. This paper deals with time responses of engineering structures in the presence of random and/or interval uncertainties. Three uncertain structure models are introduced. The first one is random uncertain structure model with only random variables. The generalized polynomial chaos (PC) theory is applied to solve the random uncertain structure model. The second one is interval uncertain structure model with only interval variables. The Legendre metamodel (LM) method is presented to solve the interval uncertain structure model. The LM is based on Legendre polynomial expansion. The third one is hybrid uncertain structure model with both random and interval variables. The polynomial-chaos-Legendre-metamodel (PCLM) method is presented to solve the hybrid uncertain structure model. The PCLM is a combination of PC and LM. Three engineering examples are employed to demonstrate the effectiveness of the proposed methods. The uncertainties resulting from geometrical size, material properties or external loads are studied.     

A Lagrangean based branch-and-cut algorithm for global optimization of nonconvex mixed-integer nonlinear programs with decomposable structures

In this work we present a global optimization algorithm for solving a class of large-scale nonconvex optimization models that have a decomposable structure. Such models, which are very expensive to solve to global optimality, are frequently encountered in two-stage stochastic programming problems, engineering design, and also in planning and scheduling. A generic formulation and reformulation of the decomposable models is given. We propose a specialized deterministic branch-and-cut algorithm to solve these models to global optimality, wherein bounds on the global optimum are obtained by solving convex relaxations of these models with certain cuts added to them in order to tighten the relaxations. These cuts are based on the solutions of the sub-problems obtained by applying Lagrangean decomposition to the original nonconvex model. Numerical examples are presented to illustrate the effectiveness of the proposed method compared to available commercial global optimization solvers that are based on branch and bound methods.     

Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering

In recent years, it has been shown that strategies based on an interval-Newton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior. These strategies provide a mathematical and computational guarantee either that all solutions have been located in an equation solving problem or that the global optimum has been found in an optimization problem. The primary drawback to this approach is the potentially high computational cost. In this paper, we consider strategies for bounding the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method. Recent preconditioning techniques for this purpose are reviewed, and a new bounding approach based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to determine the desired bounds exactly (within round out), leading to significant overall improvements in computational efficiency. These techniques will be demonstrated using several global optimization problems, with focus on problems arising in chemical engineering, including parameter estimation and molecular modeling. These problems range in size from under ten variables to over two hundred, and are solved deterministically using the interval methodology.     

Reinforcing key combinatorial ideas in a computational setting: A case of encoding outcomes in computer programming

Counting problems are difficult for students to solve, and there is a perennial need to investigate ways to help students solve counting problems successfully. One promising avenue for students’ successful counting is for them to think judiciously about how they encode outcomes – that is, how they symbolize and represent the outcomes they are trying to count. We provide a detailed case study of two students as they encoded outcomes in their work on several related counting problems within a computational setting. We highlight the role that a computational environment may have played in this encoding activity. We illustrate ways in which by-hand work and computer programming worked together to facilitate the students’ successful encoding activity. This case demonstrates ways in which the activity of computation seemed to interact with by-hand work to facilitate sophisticated encoding of outcomes.     

Models for petroleum field exploitation

This paper presents some models for an early evaluation of a petroleum field. Based on crude assumptions about a reservoir, our models suggest decisions concerning platform capacity, drilling programme and production. We start out with a simple production planning model using linear programming. By mixed integer programming techniques the model is gradually extended. The most sophisticated version of the model can propose platform capacity, where and when wells should be drilled, and the production from the wells. The models are tested on numerical examples, and the results are discussed. From the experiments we conclude that the problems are very hard to solve, and that the size of problems that can be solved is limited by the computational burden. Finally we give some ideas for future work that may provide better solution methods.     

Solving mixed-integer robust optimization problems with interval uncertainty using Benders decomposition

Uncertainty and integer variables often exist together in economics and engineering design problems. The goal of robust optimization problems is to find an optimal solution that has acceptable sensitivity with respect to uncertain factors. Including integer variables with or without uncertainty can lead to formulations that are computationally expensive to solve. Previous approaches for robust optimization problems under interval uncertainty involve nested optimization or are not applicable to mixed-integer problems where the objective or constraint functions are neither quadratic, nor linear. The overall objective in this paper is to present an efficient robust optimization method that does not contain nested optimization and is applicable to mixed-integer problems with quasiconvex constraints (? type) and convex objective funtion. The proposed method is applied to a variety of numerical examples to test its applicability and numerical evidence is provided for convergence in general as well as some theoretical results for problems with linear constraints.     

Influence of measurement data metrics on the identification of Interval Fields for the representation of spatial variability in finite element models

Due to the exponential growth in computing power, numerical modelling techniques method have gained an increasing amount of interest for engineering and design applications. Nowadays, the deterministic finite element (FE) method, an efficient tool to accurately solve the Partial Differential Equations (PDE) that govern most real-world problems, has become an indispensable tool for an engineer in various design stages. A more recent trend herein is to use the ever increasing computing power incorporate uncertainty and variability, which is omnipresent is all real-live applications, into these FE models. Several advanced techniques for incorporating either variability between nominally identical parts or spatial variability within one part into the FE models, have been introduced in this context. For the representation of spatial variability on the parameters of an FE model in a possibilistic context, the theory of Interval Fields (IF) was proven to show promising results. Following this approach, variability in the input FE model is introduced as the superposition of base vectors, depicting the spatial ‘patterns’, which are scaled by interval factors, which represent the actual variability. Application of this concept, however, requires identification of the governing parameters of these interval fields, i. e. the base vectors and interval scalars. Recent work of the authors therefore was focussed on finding a solution to the inverse problem, where the spatial uncertainty on the output side of the model is known from measurement data, but the spatial variability on the input parameters is unknown. Based on an a priori knowledge on the constituting base vectors of the interval field, the simulated output of the IFFEM computation is compared to measured data, and the input parameters are iteratively adjusted in order to minimize the discrepancy between the variability in simulation and measurement data. This discrepancy is defined based on geometric properties of the convex sets of both measurement and simulation data. However, the robustness of this methodology with respect to the size of the measurement data set that is used for the identification, as yet remains unclear. This paper therefore is focussed on the investigation of this robustness, by performing the identification on different measurement sets, depicting the same variability in the dynamic response of a simple FE model, which contain a decreasing amount of measurement replica. It was found that accurate identification remains feasible, even under a limited amount of measurement replica, which is highly relevant in the context of a non-probabilistic representation of variability in the FE model parameters. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

Comparison of the Solution of the Riemann Problem for Inviscid/Viscous Flows

The aim of this paper is to present the influence of the method used to solve the convective fluxes on the transonic internal flow field. The governing equations are discretized using an upwind method based on different solutions of the Riemann problem, flux vector splitting (FVS) or flux difference splitting schemes (FDS). Turbulence effects are simulated by means of the low‐Reynolds‐number k – ϵ and the SST (Shear‐Stress‐Transport) turbulence models.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

Non-probabilistic Bayesian update method for model validation

Model validation is the principal strategy to evaluate the accuracy and reliability of computational simulations. A systematic model validation procedure including uncertainty quantification, model update and prediction is described based on a non-probabilistic interval model. The crucial technical challenge in model validation is limited data, thus the non-probabilistic interval model is adopted to describe uncertain parameters. To establish the model update formula, the concepts of the interval escape rate and interval coverage rate are first described. Then, not only can the possibility of failure be estimated but also the credibility of the possibility of failure based on the proposed model validation method. The data in the validation experiment are used to update the credibility of each interval model, while the data from the accreditation experiment are used to conduct a final check of the validated models. To demonstrate that the proposed method can be applied to model validation problems successfully, a validation benchmark, the static frame challenge problem, is implemented. In addition, a practical aviation structure engineering validation problem is described. The results of these two validation problems show the feasibility and effectiveness of the proposed model validation method. The theoretical framework proposed in this paper is also suitable for model validation of computational simulations in other research fields.     

Binary,unrelaxable and hidden constraints in blackbox optimization

This work proposes strategies to handle three types of constraints in the context of blackbox optimization: binary constraints that simply indicate if they are satisfied or not; unrelaxable constraints that are required to be satisfied to trust the output of the blackbox; hidden constraints that are not explicitly known by the user but are triggered unexpectedly. Using tools from classification theory, we build surrogate models of those constraints to guide the Mads algorithm. Numerical results are conducted on three engineering problems.     

The contribution of mathematical modelling to the practice of project management

This paper looks at the contribution that mathematical modellinghas made to project management over the past 50 years, and thecontribution it is currently making and can make in the future.Project Management started with well-defined foundations posingprecise, well-defined problems. In its growing phase, modellersplayed an essential role in taking the problems defined by theproject-management world and offering solutions, from the originalPERT, through resource allocation and levelling procedures,Monte Carlo simulation models, criticality analyses and so on.Since then, however, while the project management field itselfhas tried to establish its procedures, keeping to its philosophicalstance, much of the mathematical-modelling world has continuedalong its trajectory, producing ever more complex solutionsto ever more complex models, motivated by mathematical impressivenessrather than the need to solve real-world problems. This paperoutlines much of this work, some of which does find its wayinto project-network software but much of which languishes injournals. However, over the last decade or so, Operational Researchershave begun to build models of projects that are systemic anddynamic and explain many of the behaviours of projects thatconventional decomposition models do not; and at the same time,some of the Project Management world has started to realizethe limitations of its philosophical stance and started lookingto build new theory for modern, complex, dynamic projects. Asthese two trends come together, it is essential that modellersare at the forefront of building this new theory.     

Collaborative Verification-Driven Engineering of Hybrid Systems

Hybrid systems with both discrete and continuous dynamics are an important model for real-world cyber-physical systems. The key challenge is to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal verification to prove their correctness. Their combination forms the vision of verification-driven engineering. Often, hybrid systems are rather complex in that they require expertise from many domains (e. g., robotics, control systems, computer science, software engineering, and mechanical engineering). Moreover, despite the remarkable progress in automating formal verification of hybrid systems, the construction of proofs of complex systems often requires nontrivial human guidance, since hybrid systems verification tools solve undecidable problems. It is, thus, not uncommon for development and verification teams to consist of many players with diverse expertise. This paper introduces a verification-driven engineering toolset that extends our previous work on hybrid and arithmetic verification with tools for (1) graphical (UML) and textual modeling of hybrid systems, (2) exchanging and comparing models and proofs, and (3) managing verification tasks. This toolset makes it easier to tackle large-scale verification tasks.     

Integer linear programming models for grid-based light post location problem

Selecting optimal location is a key decision problem in business and engineering. This research focuses to develop mathematical models for a special type of location problems called grid-based location problems. It uses a real-world problem of placing lights in a park to minimize the amount of darkness and excess supply. The non-linear nature of the supply function (arising from the light physics) and heterogeneous demand distribution make this decision problem truly intractable to solve. We develop ILP models that are designed to provide the optimal solution for the light post problem: the total number of light posts, the location of each light post, and their capacities (i.e., brightness). Finally, the ILP models are implemented within a standard modeling language and solved with the CPLEX solver. Results show that the ILP models are quite efficient in solving moderately sized problems with a very small optimality gap.     

A network model for airline cabin crew scheduling

Airline crew scheduling problems have been traditionally formulated as set covering problems or set partitioning problems. When flight networks are extended, these problems become more complicated and thus more difficult to solve. From the current practices of a Taiwan airline, whose work rules are relatively simple compared to many airlines in other countries, we find that pure network models, in addition to traditional set covering (partitioning) problems, can be used to formulate their crew scheduling problems. In this paper, we introduce a pure network model that can both efficiently and effectively solve crew scheduling problems for a Taiwan airline using real constraints. To evaluate the model, we perform computational tests concerning the international line operations of a Taiwan airline.     

The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval

Based on two-dimensional tensor product B-spline wavelet on the interval (BSWI), a class of C0 type plate elements is constructed to solve plane elastomechanics and moderately thick plate problems. Instead of traditional polynomial interpolation, the scaling functions of two-dimensional tensor product BSWI are employed to form the shape functions and construct BSWI elements. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via the constructed transformation matrix in this paper. The method combines the versatility of the conventional finite element method (FEM) with the accuracy of B-spline functions approximation and various basis functions for structural analysis. Some numerical examples are studied to demonstrate the proposed method and the numerical results presented are in good agreement with the closed-form or traditional FEM solutions.     

Review and comparison of three methods for the solution of the car sequencing problem

The car sequencing problem is the ordering of the production of a list of vehicles which are of the same type, but which may have options or variations that require higher work content and longer operation times for at least one assembly workstation. A feasible production sequence is one that does not schedule vehicles with options in such a way that one or more workstations are overloaded. In variations of the problem, other constraints may apply. We describe and compare three approaches to the modeling and solution of this problem. The first uses integer programming to model and solve the problem. The second approaches the question as a constraint satisfaction problem (CSP). The third method proposes an adaptation of the Ant Colony Optimization for the car sequencing problem. Test-problems are drawn from CSPLib, a publicly available set of problems available through the Internet. We quote results drawn both from our own work and from other research. The literature review is not intended to be exhaustive but we have sought to include representative examples and the more recent work. Our conclusions bear on likely research avenues for the solution of problems of practical size and complexity. A new set of larger benchmark problems was generated and solved. These problems are available to other researchers who may wish to solve them using their own methods.     

Thermal effects in orthotropic porous elastic beams

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity.     

Interval linear systems: the state of the art

Summary  Linear systems represent the computational kernel of many models that describe problems arising in the field of social, economic as well as technical and scientific disciplines. Therefore, much effort has been devoted to the development of methods, algorithms and software for the solution of linear systems. Finite precision computer arithmetics makes rounding error analysis and perturbation theory a fundamental issue in this framework (Higham 1996). Indeed, Interval Arithmetics was firstly introduced to deal with the solution of problems with computers (Moore 1979, Rump 1983), since a floating point number actually corresponds to an interval of real numbers. On the other hand, in many applications data are affected by uncertainty (Jerrell 1995, Marino & Palumbo 2002), that is, they are only known to lie within certain intervals. Thus, bounding the solution set of interval linear systems plays a crucial role in many problems. In this work, we focus on the state of the art of theory and methods for bounding the solution set of interval linear systems. We start from basic properties and main results obtained in the last years, then we give an overview on existing methods.