Weierstrass method for quaternionic polynomial root‐finding

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass‐like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.     

Object-oriented Design of Reusable Model Libraries of Hybrid Dynamic Systems – Part One: A Design Methodology

An object-oriented methodology for the design of reusable model libraries, dedicated to the composition of hybrid simulation models, is proposed. It takes into consideration all the phases of the library life cycle: design, programming, maintenance, modifications, extension, etc. The methodology distinguishes between the role of the library designers and the role of the library users. One of its main goals is guaranteeing that the users are able to use the libraries without having to know their internal details. In particular, users should not be confronted with numerical problems. The library designers should guarantee the numerical efficiency of the models based on the library predefined-models. In this respect, some of the numerical aspects of the model library design are discussed. The concept of library design rules is introduced as the cornerstone of the proposed methodology.     

Staggered discontinuous Galerkin methods for the incompressible Navier–Stokes equations: Spectral analysis and computational results

The goal of this paper is to create a fruitful bridge between the numerical methods for approximating PDEs in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear systems. Among the main objectives are the design of new, efficient iterative solvers and a rigorous analysis of their convergence speed. The link we have in mind is either the structure or the hidden structure that the involved coefficient matrices inherit, both from the continuous PDE and from the approximation scheme; in turn, the resulting structure is used for deducing spectral information, crucial for the conditioning and convergence analysis and for the design of more efficient solvers. As a specific problem, we consider the incompressible Navier–Stokes equations; as a numerical technique, we consider a novel family of high‐order, accurate discontinuous Galerkin methods on staggered meshes, and as tools, we use the theory of Toeplitz matrices generated by a function (in the most general block, the multilevel form) and the more recent theory of generalized locally Toeplitz matrix sequences. We arrive at a somehow complete picture of the spectral features of the underlying matrices, and this information is employed for giving a forecast of the convergence history of the conjugate gradient method, together with a discussion on new and more advanced techniques (involving preconditioning, multigrid, multi‐iterative solvers). Several numerical tests are provided and critically illustrated in order to show the validity and the potential of our analysis.     

Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus

University and older school students following scientific courses now use complex calculators with graphical, numerical and symbolic capabilities. In this context, the design of lessons for 11th grade pre-calculus students was a stimulating challenge.In the design of lessons, emphasising the role of mediation of calculators and the development of schemes of use in an 'instrumental genesis' was productive. Techniques, often discarded in teaching with technology, were viewed as a means to connect task to theories. Teaching techniques of use of a complex calculator in relation with 'traditional' techniques was considered to help students to develop instrumental and paper/pencil schemes, rich in mathematical meanings and to give sense to symbolic calculations as well as graphical and numerical approaches.The paper looks at tasks and techniques to help students to develop an appropriate instrumental genesis for algebra and functions, and to prepare for calculus. It then focuses on the potential of the calculator for connecting enactive representations and theoretical calculus. Finally, it looks at strategies to help students to experiment with symbolic concepts in calculus.This revised version was published online in September 2005 with corrections to the Cover Date.     

An efficient algorithm for regularization of Laplace transform inversion in real case

We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution.

In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix.     

The Design of Laminates as a Global Optimization Problem

The problem of formulating the design of laminated composite plates as a global optimization problem is discussed. In the paper, a general procedure, based upon the polar formalism for the use of tensor invariants as design variables is presented, along with a wide discussion of its peculiarities and of some numerical issues. An outlook about some open problems is also given, with a perspective on possible future issues and further developments.     

Analytical modeling of active structural acoustic control

Since the reduction of disturbing noise is highly important in industrial and civil engineering, much research and development has been done in this field over the past decades. The application of smart structures provides a way to reduce unwanted noise radiation. A smart structure is an integrated system consisting of the passive base structure attached with actuators and sensors. Piezoelectric ceramics are widely used as sensors and actuators, because they can easily be bonded on or imbedded into conventional structures. An often used concept for noise reduction is active structural acoustic control (ASAC). This concept means that the noise radiation will be reduced by minimizing or changing the vibrational behavior of a structure using additional actuator forces. The development and industrial use of smart structures for ASAC requires efficient and reliable simulation and design tools. Usually the design process of smart structures is based on numerical methods. In order to verify numerical techniques, the paper presents an analytical reference solution for an ASAC system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of the conjugate system method for optimal parametric design of stochastic dynamic systems for final position control

Efficient numerical algorithms are proposed for computer design of linear stochastic dynamic systems for final position control. The algorithms use the method of inverse-conjugate systems with dimension reduction of the space of optimized parameters.Sevastopol' Instrument-Building Institute. Translated from Dinamicheskie Sistemy, No. 10, pp. 51–57, 1992.     

Airfoil design optimization using the Navier-Stokes equations

A design optimization technique is presented which couples a computationally efficient Navier-Stokes code with a numerical optimization algorithm. The design method improves the aerodynamic performance of an airfoil subject to specified design objectives and constraints. Recent advances in computers and compputational fluid dynamics have permitted the use of the Navier-Stokes equations in the design procedure to include the nonlinear, rotational, viscous physics of transonic flows. Using numerical optimization guarantees that a better design will be produced even with strict design constraints. The method is demonstrated with several examples at transonic flow conditions.     

Optimal configuration of a decentralized, market-driven production/inventory system

Pull systems are inherently easier to implement on the shop-floor; however, they are quite difficult to plan and design for optimal operation, leaving little guidelines to system designers and practitioners. In this paper we use an effective and relatively fast numerical method to understand the optimal configuration of a multi-stage, multi-product, decentralized, market-driven production/inventory system that minimizes average inventory holding subject to a service level constraint through selection of various production and procurement control parameters. We have also conducted a number of numerical experiments to understand how the control policies respond to changes in the system parameters, such as the number of stages, system workload, demand arrival rates of products, and inventory holding costs.     

Automatic differentiation for the optimization of a ship propulsion and steering system: a proof of concept

We describe the optimization of the Voith-Schneider-Propeller (VSP) which is an industrial propulsion and steering system of a ship combined in one module. The goal is to optimize efficiency of the VSP with respect to different design variables. In order to determine the efficiency, we have to use numerical simulations for the complex flow around the VSP. Such computations are performed with standard (partly commercial) flow solvers. For the numerical optimization, one would like to use gradient-based methods which requires derivatives of the flow variables with respect to the design parameters. In this paper, we investigate if Automatic Differentiation (AD) offers a method to compute the required derivatives in the described framework. As a proof of concept, we realize AD for the 2D-code Caffa and the 3D-code Comet, for the simplified model of optimizing efficiency with respect to the angle of attack of one single blade (like an airfoil). We show that AD gives smooth derivatives, whereas finite differences show oscillations. This regularization effect is even more pronounced in the 3D-case. Numerical optimization by AD and Newton’s method shows almost optimal convergence rates.     

A non-standard finite difference method to solve a model of HIV–Malaria co-infection

In this paper, we design and analyse a non-standard finite difference numerical scheme for the numerical solution of the HIV–malaria co-infection model with a distributed delay representing the incubation period of malaria parasite in the mosquitoes vector. To come up with the efficient numerical method for the full co-infection model, we study a number of qualitative properties of sub-models and then use the information while designing the numerical methods for these sub-models. One of the salient features of these methods is that they preserve positivity of the solution which is very essential while studying epidemiological models. We also present numerical simulations to confirm the theoretical findings.     

Greedy sparse linear approximations of functionals from nodal data

In many numerical algorithms, integrals or derivatives of functions have to be approximated by linear combinations of function values at nodes. This ranges from numerical integration to meshless methods for solving partial differential equations. The approximations should use as few nodal values as possible and at the same time have a smallest possible error. For each fixed set of nodes and each fixed Hilbert space of functions with continuous point evaluation, e.g. a fixed Sobolev space, there is an error–optimal method available using the reproducing kernel of the space. But the choice of the nodes is usually left open. This paper shows how to select good nodes adaptively by a computationally cheap greedy method, keeping the error optimal in the above sense for each incremental step of the node selection. This is applied to interpolation, numerical integration, and numerical differentiation. The latter case is particularly important for the design of meshless methods with sparse generalized stiffness matrices. The greedy algorithm is described in detail, and numerical examples are provided. In contrast to the usual practice, the greedy method does not always use nearest neighbors for local approximations of function values and derivatives. Furthermore, it avoids multiple points from clusters and it is better conditioned than choosing nearest neighbors.     

Optimization of a cylindrical shell of reinforced plastic with a filler under dynamic loading

Conclusions 1. The numerical investigation of profiles of functions of the physical constraints carried out in this work allows us to assume that problems of optimal design of shells of reinforced plastics, strengthened by an elastic filler, for purposes of stability (static and dynamic) under axial compression, in the given formulation, are problems of convex programming. This guarantees uniqueness of their solution and allows us to use gradient methods for a numerical realization.2. Optimal shells of a composite material have a smaller mass than equivalent shells of high-strength metal alloys. The gain in the expenditure of the material is ensured not only as a result of higher specific characteristics of the composite, but basically as a result of optimizing the reinforcement structure of the pack of orthotropic layers of the shell.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 879–885, September–October, 1977.     

A Two-Level Procedure for the Global Optimum Design of Composite Modular Structures—Application to the Design of an Aircraft Wing

This work concerns a two-level procedure for the global optimum design of composite modular structures. The case-study considered is the least weight design of a stiffened wing-box for an aircraft structure. The method is based on the use of the polar formalism and on a genetic algorithm. In the first level of the procedure, the optimal structure is designed as composed by a single equivalent layer, while a laminate realizing the optimal structure is found in the second level. The method is able to automatically find the optimal number of modules; no simplifying assumptions are used and it can be easily generalized to other problems. The work is divided into two parts: the theoretical formulation in the first part, the genetic procedure and some numerical examples in this second one.     

A wastewater treatment problem: study of the numerical convergence

We deal with the numerical approximation of a problem related to the management of sewage disposal and the design of wastewater treatment systems. We use a characteristics finite element method for the discretization of the state system. The main difficulty is due to the existence of Radon measures in the right-hand side of the system. We give convergence results and we present numerical experiences for a realistic problem.     

Leader–Follower Dynamic Game of New Product Diffusion

The objective of this paper is to study optimal pricing strategies in a duopoly, under an asymmetric information structure, where the appropriate solution concept is the feedback Stackelberg equilibrium. In order to take into account effects such as imitation (e.g., word of mouth) and saturation, the demand (state equation) is assumed to depend on past cumulative sales, market potential, and both players' prices. We assume also that the unit production cost decreases with cumulative production (learning effects). Each player maximizes his total discounted profit over the planning horizon.The problem is formulated as a two-player discrete-time finite-horizon game. Existence results are first obtained under rather mild conditions. Since the solution of this problem is intractable by analytical methods, we use a numerical approach. Thus, we design a numerical algorithm for the computation of feedback Stackelberg equilibria and use it to obtain strategies in various representative cases. The numerical results presented are intented to give some insights into the optimal pricing strategies in the context of an asymmetrical feedback information structure.     

An Outer Approximations Approach to Reliability-Based Optimal Design of Structures

We present a new formulation of the problem of minimizing the initial cost of a structure subject to a minimum reliability requirement, expressed in terms of the so-called design points of the first-order reliability theory, i.e., points on limit-state surfaces that are nearest to the origin in a transformed standard normal space, as well as other deterministic constraints. Our formulation makes it possible to use outer approximations algorithms for the solution of such optimal design problems, eliminating some of the major objections associated with treating them as bilevel optimization problems. A numerical example is presented that illustrates the reliability and efficiency of the algorithm.     

Computational simulation of fluid and dilute particulate flows on spiral concentrators

Spiral separators are used globally in the fine coal and heavy mineral processing industries as gravity-concentration devices. Consisting of an open trough that spirals vertically downwards in helix configuration about a central axis, a slurry mix of particles and water is fed to the top of the concentrator. Particles are then separated radially on the basis of density and size as they gravitate downwards. To enhance performance, the geometric design has evolved historically by experimental trial-and-error investigations to develop a prototype suited to the given industrial application. This approach has proved somewhat prohibitive for design purposes however, and researchers have accordingly turned to numerical techniques in an attempt to develop a fully predictive and reliable model for use in the design process. Towards this end, the present paper uses Computational Fluid Dynamics (CFD) analysis to simulate fluid and dilute particulate flows on one operational spiral unit. The free-surface Volume-of-Fluid (VOF) algorithm, isotropic RNG k–ε turbulence model and Lagrangian method have been used for this purpose. Satisfactory predictions have been obtained with respect to a collaborative experimental program, and the model forms the basis for future examination of the two-way fluid-particle coupling processes and inter-particle effects.     

Reduced quasi-Newton method for simultaneous design and optimization

We consider the task of design optimization where the constraint is a state equation that can only be solved by a typically rather slowly converging fixed point solver. This process can be augmented by a corresponding adjoint solver and based on the resulting approximate reduced derivatives also an optimization iteration which actually changes the design. To coordinate the three iterative processes, we use an exact penalty function of doubly augmented Lagrangian type. The main issue here is how to derive a design space preconditioner for the approximated reduced gradient which ensures a consistent reduction of the employed penalty function as well as significant design corrections. Some numerical experiments for an alternating approach where any combination and sequencing of steps are used to improve feasibility and optimality done on a variant of the Bratu problem are presented.