Finite element and neural network approximations to measure forces using six-component wind tunnel balance

Nonlinear mathematical-programming-based design optimization can be an elegant method. However, the calculations required to generate the merit function, constraints, and their gradients, which are frequently required, can make the process computational intensive. The computational burden can be greatly reduced by using approximating analyzers derived from an original analyzer utilizing neural networks and linear regression methods. The experience gained from using both of these approximation methods in the design optimization of a high speed civil transport aircraft is the subject of this paper. The Langley Research Center's Flight Optimization System was selected for the aircraft analysis. This software was exercised to generate a set of training data with which a neural network and a regression method were trained, thereby producing the two approximating analyzers. The derived analyzers were coupled to the Lewis Research Center's CometBoards test bed to provide the optimization capability. With the combined software, both approximation methods were examined for use in aircraft design optimization, and both performed satisfactorily. The CPU time for solution of the problem, which had been measured in hours, was reduced to minutes with the neural network approximation and to seconds with the regression method. Instability encountered in the aircraft analysis software at certain design points was also eliminated. On the other hand, there were costs and difficulties associated with training the approximating analyzers. The CPU time required to generate the input-output pairs and to train the approximating analyzers was seven times that required for solution of the problem.

PurposeBuilding cost is an important part of construction projects, and its correct estimation has important guiding significance for the follow-up decision-making of construction units.Design/methodology/approachThis study focused on the application of back-propagation (BP) neural network in the estimation of building cost. First, the influencing factors of building cost were analyzed. Six factors were selected as input of the estimation model. Then, a BP neural network estimation model was established and trained by ten samples.FindingsAccording to the experimental results, it was found that the estimation model converged at about 85 times; compared with radial basis function (RBF), the estimation accuracy of the model was higher, and the average error was 5.54 per cent, showing a good reliability in cost estimation.Originality/valueThe results of this study provide a reliable basis for investment decision-making in the construction industry and also contribute to the further application of BP neural network in cost estimation.

SummaryThis paper studies the problem of stabilizing reference trajectories (also called as the trajectory tracking problem) for underactuated marine vehicles under predefined tracking error constraints. The boundary functions of the predefined constraints are asymmetric and time‐varying. The time‐varying boundary functions allow us to quantify prescribed performance of tracking errors on both transient and steady‐state stages. To overcome difficulties raised by underactuation and nonzero off‐diagonal terms in the system matrices, we develop a novel transverse function control approach to introduce an additional control input in backstepping procedure. This approach provides practical stabilization of any smooth reference trajectory, whether this trajectory is feasible or not. By practical stabilization, we mean that the tracking errors of vehicle position and orientation converge to a small neighborhood of zero. With the introduction of an error transformation function, we construct an inverse‐hyperbolic‐tangent‐like barrier Lyapunov function to show practical stability of the closed‐loop systems with prescribed transient and steady‐state performances. To deal with unmodeled dynamic uncertainties and external disturbances, we employ neural network (NN) approximators to estimate uncertain dynamics and present disturbance observers to estimate unknown disturbances. Subsequently, we develop adaptive control, based on NN approximators and disturbance estimates, that guarantees the prescribed performance of tracking errors during the transient stage of on‐line NN weight adaptations and disturbance estimates. Simulation results show the performance of the proposed tracking control.

A reactor integrated with separation process has been widely proposed and studied to enhance the production of high value added chemicals. An example of such a process is a pervaporative membrane reactor which is the combination of a reactor and a pervaporation part. The inclusive of a pervaporation can shift the chemical equilibrium of esterification by passing water out of reaction mixture and therefore increase the yield of a desired product. In addition, the temperature of the reactor is one of key factors to define the rate of reactions involved. In this work, a neural network inverse models based control (NIMC) has been designed to control an optimal temperature of a pervaporation membrane reactor. Furthermore, a neural network estimator (NNE) has been incorporated to the strategy as an estimator to estimate the amount of heat released of an esterification reaction. The performance of the designed controller has been evaluated in both nominal and plant/model mismatch cases, and compared with GMC coupled with the Kalman Filter. Simulation study has shown that the NIMC with the NNE can successfully control the reactor temperature at a desired set point in nominal cases and plant/model mismatches. The performance of the NIMC with the NNE is equivalent to that of the GMC coupled with the Kalman Filter. As a result, the NIMC with the NNE is applicable to control the pervaporative membrane reactor without first principle models of the reactor is available. This shows the advantage of the NIMC with NNE over the GMC with the Kalman filter.

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two-fluid interfacial flows, having in mind possible interface topology changes (like merger or break-up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator-splitting for temporal discretization and the level-set method for interface representation. We show that the finite element implementation of the level-set approach brings some additional benefits as compared to the standard, finite difference level-set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second-order accuracy of the interface normal, curvature and mass conservation. The operator-splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal-order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.

In this paper an efficient algorithm is presented for the development of compact and passive macro‐models of electromagnetic devices through the systematic reduction of the order of discrete models for these devices obtained through the use of finite elements. The proposed methodology is founded on a new finite element formulation that casts Maxwell's curl equations in a state‐space form. Such state‐space representations are very compatible with a class of robust model order reduction techniques based on Krylov subspaces. However, the advantage of this compatibility appears to be hindered by the fact that the state matrix of the discretized Maxwellian system is of dimension almost twice that obtained from the finite element approximation of the electromagnetic vector wave equation. It is shown in this paper that the apparent penalty in both memory and computation efficiency can be avoided by a proper selection of the finite element expansion functions used for the discretization of the electromagnetic fields. More specifically, it is shown that the proper selection of expansion functions renders the state‐space form of the Maxwellian system equivalent to the discrete problem obtained from the approximation of the vector wave equation using tangentially continuous vector finite elements. This equivalence is then used to effect Krylov‐based model order reduction directly from the finite element approximation of the vector wave equation. In particular, a passive model order reduction algorithm is used for this purpose. The proposed reduced order macro‐modelling algorithm is demonstrated through its application to a variety of microwave passive components. Copyright © 2002 John Wiley & Sons, Ltd.

In theory, recent results in nonparametric regression show that neural network estimates are able to achieve good rates of convergence provided suitable assumptions on the structure of the regression function are imposed. However, these theoretical analyses cannot explain the practical success of neural networks since the theoretically studied estimates are defined by minimizing the empirical L_2 risk over a class of neural networks and in practice, solving this kind of minimization problem is not feasible. Consequently, the neural networks examined in theory cannot be implemented as they are defined. This means that neural network in applications differ from the ones that are analyzed theoretically. In this thesis we narrow the gap between theory and practice. We deal with neural network regression estimates for (p,C)-smooth regression functions m that satisfy a projection pursuit model. We construct three implementable neural network estimates and show that each of them achieve up to a logarithmic factor the optimal univariate rate of convergence. Firstly, for univariate regression functions with p contained in [-1/2,1] we construct a neural network estimate with one hidden layer where the weights are learned via gradient descent. The starting weights are randomly chosen from an interval independently of the data. The interval is large enough to guarantee that the estimate is close to a piecewise constant approximation. Secondly, for multivariate regression functions with p contained in (0,1] we construct a neural network estimate with one hidden layer where the weights are learned via gradient descent. The initial weights are chosen from specific intervals dependently on the data and the projection directions. This choice guarantees that the estimate is close to a piecewise constant approximation. The projection directions are repeatedly chosen randomly. Lastly, for multivariate regression functions with p>0 we construct a multilayer neural network estimate. The value of the inner weights are prescribed dependently on the projection directions by a new approximation result for a projection pursuit model by piecewise polynomials. The outer weights are chosen by solving a linear equation system. The projection directions are repeatedly chosen randomly. Since we are able to show a rate of convergence that is independent of the dimension of the data our second and third estimates are able to circumvent the curse of dimensionality.

Modelling of a bioprocess was developed by combining three neural network estimators (NNEs). The bioprocess was for fed-batch culture of L-lysine production. The NNEs were NNE1, NNE2, and NNE3. NNE1 was applied to estimate the consumed sugar concentration, L-lysine concentration, and cell optical density. NNE2 was utilized to determine the off-gas concentration of O2 and CO2 while NNE3 was employed to predict the agitation speed. These NNEs possessed their own distinct functions and could be used independently. Combination of the NNEs based on neural network gave an easy and significant way for modelling a bioprocess.

The workshop Nonstandard Finite Element Methods , organized by Susanne C. Brenner (Baton Rouge), Carsten Carstensen (Berlin) and Peter Monk (Newark) was held August 10 – 16, 2008. This meeting was well attended with over 40 participants with broad geographic representation. Although Courant is often credited with the discovery of finite elements [2, 3], the first practical use of such methods in engineering seems to date back to the work of Argyris and Clough et. al. (apparently Clough coined the name “finite elements”) in the 1950s [1, 4]. By the 1970s the finite element method was firmly established in engineering practice and the basic theory of conforming elements for elliptic problems was well understood [2]. Engineers and mathematicians have since expanded the use of finite elements to a wide variety of new applications and have improved the theoretical underpinnings of the method. In addition, efficient computational algorithms have been developed, such as the multigrid method, that allow for the application of finite elements to sophisticated three dimensional problems. Within current research, we can distinguish classical or standard finite element methods, and newer nonstandard finite elements The former include conforming elements applied to “standard” Galerkin discretizations of problems. Usually accuracy is obtained by mesh refinement. As the range of applications has increased, and to alleviate perceived shortcomings in standard conforming elements, numerical analysts and engineers have sought more general finite element methods based, for example, on mixed formulations, and non-conforming or even discontinuous Galerkin schemes. These methods may be superior to standard finite elements in particular applications because of enhancements to stability, robustness or conservation properties. We refer to them as non-standard finite element methods. This Oberwolfach workshop was devoted to non-standard finite element methods and their analyses. Inevitability such a snapshot of the subject is biased, but we have tried to include non-standard methods in the broadest sense including mixed, discontinuous Galerkin, generalized, partition of unity and mortar FEM methods. All these schemes have in common that stability and convergence is not obvious and requires mathematical analysis. This is even more true for developing fast solvers, a posteriori error estimation and adaptive mesh design. Through the week there were 29 presentations on various non-standard topics. These provided an overview of current research directions, new developments and open problems.

The article is devoted to the development of machine learning methods for classes of technical problems, including determining the properties of materials. According to the authors, the neural network approximation algorithm is able to take into account the behavior of materials in various experimental conditions. The article provides illustrative examples of how a neural network with a single hidden layer can approximate a function of several variables with a given accuracy. As part of the study, a number of experimental measurements were made. The structure of the neural network and its main components are described.

Approximation of functions and their derivatives: A neural network implementation with applications

Chapter 11 - Finite Element Approximation with Splines

Discontinuous finite-element approximations for the analysis of shock waves in nonlinearly elastic materials

This work is concerned with the mathematical and numerical analyses of the Navier–Stokes equations driven by threshold power law slip boundary condition. The tangential shear belongs to the sub-differential of a power law function, giving rise to quasi-variational inequalities. We construct a unique weak solution and formulate a mixed finite element approximation to the system using conforming finite element spaces. The unique solvability of the finite element approximation is investigated and an abstract convergence result is established. We derive a-priori error estimate by providing a bound on the error between the weak solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We introduce a Lagrange multiplier and propose a Uzawa-type iterative scheme to solve the nonlinear problem resulting from the finite element approximation. Finally, numerical results are provided to validate the theoretical findings.

Lower Bound Finite Element Limit Analysis of Geo-Structures with Non-Associated Flow Rule