Performance predictions in solid oxide fuel cells

The results of numerical calculations performed for planar solid oxide fuel cells are presented. Two different approaches are developed: (i) A detail numerical method and (ii) a presumed flow method. In the first approach, a commercial computational fluid dynamics code is employed, and user-defined-functions are developed to account for electro-chemical considerations. In the second approach, where the momentum equations do not require to be solved, an in-house code is developed and used to perform calculations. In both cases the following coupled physicochemical phenomena are modelled; heat and mass transfer, electrochemistry and electric potential. The polarisation curve is generally accepted as an important performance measure of the fuel cell. Performance predictions for this characteristic made by the two different approaches are compared. Results show voltage losses due activation, Ohmic resistance, and mass transfer in a typical solid oxide fuel cell, over a range of current density values. The results for the detailed numerical method are discussed in some detail with regard to the influence of different parameters on the overall performance of the device.     

A constitutive model for continuous fiber reinforced polymer plies

In the present paper a constitutive model is reviewed which can be used to predict the non-linear behavior of continuous fiber reinforced laminates with polymeric matrix materials. The constitutive model considers stiffness degradation and plastic strain accumulation at the length scale of the individual plies (laminae). These effects are modeled via two different phenomenological approaches, however, their interaction is considered when the constitutive equations are solved by an implicit integration scheme. To demonstrate the predictive capabilities of the individual model parts, examples are given where the above mentioned effects are decoupled. This way, their impact on the laminate's response can be studied independently. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control applied on an HIV‐1 within‐host model

The treatment of human immunodeficiency virus (HIV) remains a major challenge, even if significant progress has been made in infection treatment by ‘drug cocktails’. Nowadays, research trend is to minimize the number of pills taken when treating infection. In this paper, an HIV‐1 within host model where healthy cells follow a simple logistic growth is considered. Basic reproduction number of the model is calculated using next generation matrix method, steady states are derived; their local, as well as global stability, is discussed using the Routh–Hurwitz criteria, Lyapunov functions and the Lozinskii measure approach. The optimal control policy is formulated and solved as an optimal control problem. Numerical simulations are performed to compare several cases, representing a treatment by Interleukin2 alone, classical treatment by multitherapy drugs alone, then both treatments at the same time. Objective functionals aim to (i) minimize infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood. Copyright © 2015 John Wiley & Sons, Ltd.     

A second-order reduced multiscale approach for non-linear axisymmetric structures with periodic configurations

An effective second-order reduced multiscale (SORM) approach is proposed for axisymmetric inelastic heterogeneous structures with periodic configurations. The axisymmetric structures studied in this work are periodical in radial and axial directions and homogeneous in circumferential directions. At first, the high-order linear and non-linear local solutions at microscale are gotten by solving distinct multiscale auxiliary functions. Further, the homogenized parameters are introduced, and the related non-linear homogenization equations defined on global domain are given. The significant feature of the presented approaches are (i) high-order homogenization solutions that do not require high-order continuities of the coarse-scale solutions, (ii) a novel reduced-model form based on transformation field analysis (TFA) to analyze non-linear local cell problem with less computing time compared with direct numerical simulations and (iii) a new SORM algorithm derived for simulating the axisymmetric inelastic structures. Finally, by some representative examples, the effectiveness and accuracy of the presented algorithm are confirmed.     

Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization

Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL ^T factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraints.     

Recursive formulation of the relativistic law of superposition of multiple collinear velocities

We study the recursive formulation of the law of superposition of multiple collinear velocities. We start with the non-linear equation, transform it into two linear coupled difference equations with variable cofficients, and then decouple these latter equations. The coupled difference equations are solved by three different, but interrelated, methods: (i) via the graph theoretic discrete path approach, (ii) by using the general closed form solution of two coupled first order difference equations with variable coefficients, and (iii) in terms of the symmetric functions via the pochhammers of 2 × 2 non-autonomous matrices. The solutions of the decoupled equations are factorial polynomials.     

Optimal homotopy analysis method for nonlinear differential equations in the boundary layer

Optimal homotopy analysis method is a powerful tool for nonlinear differential equations. In this method, the convergence of the series solutions is controlled by one or more parameters which can be determined by minimizing a certain function. There are several approaches to determine the optimal values of these parameters, which can be divided into two categories, i.e. global optimization approach and step-by-step optimization approach. In the global optimization approach, all the parameters are optimized simultaneously at the last order of approximation. However, this process leads to a system of coupled, nonlinear algebraic equations in multiple variables which are very difficult to solve. In the step-by-step approach, the optimal values of these parameters are determined sequentially, that is, they are determined one by one at different orders of approximation. In this way, the computational efficiency is significantly improved, especially when high order of approximation is needed. In this paper, we provide extensive examples arising in similarity and non-similarity boundary layer theory to investigate the performance of these approaches. The results reveal that with the step-by-step approach, convergent solutions of high order of approximation can be obtained within much less CPU time, compared with the global approach and the traditional HAM.     

Set Containment Characterization

Characterization of the containment of a polyhedral set in a closed halfspace, a key factor in generating knowledge-based support vector machine classifiers [7], is extended to the following: (i) containment of one polyhedral set in another; (ii) containment of a polyhedral set in a reverse-convex set defined by convex quadratic constraints; (iii) Containment of a general closed convex set, defined by convex constraints, in a reverse-convex set defined by convex nonlinear constraints. The first two characterizations can be determined in polynomial time by solving m linear programs for (i) and m convex quadratic programs for (ii), where m is the number of constraints defining the containing set. In (iii), m convex programs need to be solved in order to verify the characterization, where again m is the number of constraints defining the containing set. All polyhedral sets, like the knowledge sets of support vector machine classifiers, are characterized by the intersection of a finite number of closed halfspaces.     

An exemplified semi-analytical approach to the transient sensitivity of nonlinear systems

The internal logic of a model formulated as coupled differential equations is severely challenged in cases where the value of a poorly determined parameter greatly affects the solution of the model. Thus, the sensitivity to parameter changes is essential in the verification stage of the modelling process. Since a sensitivity analysis points out where the influence of flaws in present knowledge has important consequences, a sensitivity analysis will also act as a guideline for future research.In the literature known to the author, the approach to sensitivity has been either by parameter changes or local linearization. However, neither of these approaches will indicate the role played by parameter interactions in sensitivity unless a prohibitively large number of simulations are carried out.The present paper shows a known analytical expression for the sensitivity to parameter changes in nonlinear systems. A simplified version of the expression is solved for a model of the biological processes in activated sludge with 10 state-variables and 31 parameters leading to the presentation of simulated sensitivity of sensitivity, i.e. to an estimate of parameter interactions in sensitivity. The result of the approach is demonstrated through a comparison of actual and predicted changes in solution. Finally, a proposal of the extent of a complete sensitivity analysis is given.     

Derivation and analytical investigation of three direct boundary integral equations for the fundamental biharmonic problem

We derive and investigate three families of direct boundary integral equations for the solution of the plane, fundamental biharmonic boundary value problem. These three families are fairly general so that they, as special cases, encompass various known and applied equations as demonstrated by giving many references to the literature. We investigate the families by analytical means for a circular boundary curve where the radius is a parameter. We find for all three combinations of equations (i) that the solution of the equations is non-unique for one or more critical radius/radii, and (ii) that this lack of uniqueness can always be removed by combining the integral equations with a suitable combination of one or more supplementary condition(s). We conjecture how the results obtained can, or cannot, be generalized to other boundary curves through the concept logarithmic capacity. A few published general results about uniqueness are compared with our findings.     

Pseudospectral method for transient viscoelastic flow in an axisymmetric channel

A new pseudospectral method for simulating transient viscoelastic flows is presented. The governing equations are a system of seven first-order equations of mixed type. The essential features of the method are (i) all seven independent flow variables are represented on a common Chebyshev-Gauss-Lobatto grid; (ii) the pressure is treated in such a way as to give a globally divergence-free velocity field, i.e., the divergence of the velocity field vanishes globally within the region, and (iii) different time scales pertaining within the hyperbolic constitutive equations are treated using the splitting technique of LeVeque and Yee originally proposed in a finite-difference context. The method is applied to transient axisymmetric flow of an Oldroyd B fluid in a channel formulated in two ways: (I) as an initial boundary-value problem, and (II) as a body-force problem. © 1993 John Wiley & Sons, Inc.     

Hiemenz flow and heat transfer of a third grade fluid

The laminar flow and heat transfer of an incompressible, third grade, electrically conducting fluid impinging normal to a plane in the presence of a uniform magnetic field is investigated. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). By means of the similarity transformation, the governing non-linear partial differential equations are reduced to a system of non-linear ordinary differential equations and are solved by a second-order numerical technique. Effects of various non-Newtonian fluid parameters, magnetic parameter, Prandtl number on the velocity and temperature fields have been investigated in detail and shown graphically. It is found that the velocity gradient at the wall decreases as the third grade fluid parameter increases.     

Numerical and analytical simulation of peristaltic flow of a Jeffrey-six constant fluid

This paper describes the Peristaltic flow of a Jeffrey-six constant fluid in an endoscope. The two-dimensional equation of Jeffrey-six constant fluid is simplified by making the assumptions of long wave length and low Reynolds number. The reduced momentum equations are solved with three methods, namely (i) Perturbation method, (ii) Homotopy analysis method, and (iii) shooting method. The comparison of the three solutions shows a very good agreement between the three results. The expressions for pressure rise and frictional forces per wave length have been also computed numerically. Finally, the pressure rise, frictional forces are plotted for different parameters of interest.     

Active constraints,indefinite quadratic test problems,and complexity

The observation that at leasts constraints are active when the Hessian of the Lagrangian hass negative eigenvalues at a local minimizer is used to obtain two results: (i) a class of nearly concave quadratic minimization problem can be solved in polynomial time; (ii) a class of indefinite quadratic test problems can be constructed with a specified number of positive and negative eigenvalues and with a known global minimizer.The authors thank the reviewers for their constructive comments. The first author was supported by the National Science Foundation Grant DMS-85-20926 and by the Air Force Office of Scientific Research Grant AFOSR-ISSA-86-0091.     

A new adaptive trust-region method for system of nonlinear equations

This study presents a new trust-region procedure to solve a system of nonlinear equations in several variables. The proposed approach combines an effective adaptive trust-region radius with a nonmonotone strategy, because it is believed that this combination can improve the efficiency and robustness of the trust-region framework. Indeed, it decreases the computational cost of the algorithm by decreasing the required number of subproblems to be solved. The global and the quadratic convergence of the proposed approach is proved without any nondegeneracy assumption of the exact Jacobian. Preliminary numerical results indicate the promising behavior of the new procedure to solve systems of nonlinear equations.     

Natural convection flow from a horizontal circular cylinder with uniform heat flux in presence of heat generation

Natural convection boundary layer laminar flow from a horizontal circular cylinder with uniform heat flux in presence of heat generation has been investigated. The governing boundary layer equations are transformed into a non-dimensional form and the resulting non-linear systems of partial differential equations, which are solved numerically by two distinct methods namely: (i) implicit finite difference method together with the Keller-box scheme and (ii) perturbation solution technique. The results of the surface shear stress in terms of local skin-friction and the rate of heat transfer in terms of local Nusselt number, velocity distribution, velocity vectors, temperature distribution as well as streamlines, isotherms and isolines of pressure are shown by graphically for a selection of parameter set consisting of heat generation parameter.     

An interior-point trust-funnel algorithm for nonlinear optimization

We present an interior-point trust-funnel algorithm for solving large-scale nonlinear optimization problems. The method is based on an approach proposed by Gould and Toint (Math Prog 122(1):155–196, 2010) that focused on solving equality constrained problems. Our method is similar in that it achieves global convergence guarantees by combining a trust-region methodology with a funnel mechanism, but has the additional capability of being able to solve problems with both equality and inequality constraints. The prominent features of our algorithm are that (i) the subproblems that define each search direction may be solved with matrix-free methods so that derivative matrices need not be formed or factorized so long as matrix-vector products with them can be performed; (ii) the subproblems may be solved approximately in all iterations; (iii) in certain situations, the computed search directions represent inexact sequential quadratic optimization steps, which may be desirable for fast local convergence; (iv) criticality measures for feasibility and optimality aid in determining whether only a subset of computations need to be performed during a given iteration; and (v) no merit function or filter is needed to ensure global convergence.     

Symmetry analysis for uniaxial compression of a hypoplastic granular material

A variety of modelling approaches currently exist to describe and predict the diverse behaviours of granular materials. One of the more sophisticated theories is hypoplasticity, which is a stress-rate theory of rational continuum mechanics with a constitutive law expressed in a single tensorial equation. In this paper, a particular version of hypoplasticity, due to Wu [2], is employed to describe a class of one-dimensional granular deformations. By combining the constitutive law with the conservation laws of continuum mechanics, a system of four nonlinear partial differential equations is derived for the axial and lateral stress, the velocity and the void ratio. Under certain restrictions, three of the governing equations may be combined to yield ordinary differential equations, whose solutions can be calculated exactly. Several new analytical results are obtained which are applicable to oedometer testing. In general this approach is not possible, and analytic progress is sought via Lie symmetry analysis. A complete set or “optimal system” of group-invariant solutions is identified using the Olver method, which involves the adjoint representation of the symmetry group on its Lie algebra. Each element in the optimal system is governed by a system of nonlinear ordinary differential equations which in general must be solved numerically. Solutions previously considered in the literature are noted, and their relation to our optimal system identified. Two illustrative examples are examined and the variation of various functions occuring in the physical variables is shown graphically.     

Coradiant sets and $$\varepsilon $$-efficiency in multiobjective optimization

In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine constraints to investigate problems where these conditions are not met. Based on the combinatorial solution approach for mp-QP problems featuring bounded and continuous parameters, we show that (i) for unbounded parameters, it is possible to obtain the multi-parametric solution if there exists one realization of the parameters for which the optimization problem can be solved and (ii) for binary parameters, we present the equivalent mixed-integer formulations for the application of the combinatorial algorithm. These advances are combined into a new, generalized version of the combinatorial algorithm for mp-QP problems, which enables the solution of problems featuring both unbounded and binary parameters. This novel approach is applied to mixed-integer bilevel optimization problems and the parametric solution of the dual of a convex problem.     

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.