Optimal Shape Design for Metal Forming Problems by the Finite Element Method

In this work, a method for calculation of the optimal shapes of axisymmetrical converging dies by the finite element method is presented. The shape optimization problem considered in this paper is to find the best shape of the die such that the flow rate will be uniform at the die exit.The optimization problem is to minimize an objective function by varying a part of boundary (ie: the shape of die) subject to constraints imposed by the metal forming problem. In this method, the B-spline functions allow us to determine the shape of the die, using its control points as design variables. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic modelling of die melt temperature profile in polymer extrusion: Effects of process settings,screw geometry and material

Extrusion is one of the major methods for processing polymeric materials and the thermal homogeneity of the process output is a major concern for manufacture of high quality extruded products. Therefore, accurate process thermal monitoring and control are important for product quality control. However, most industrial extruders use single point thermocouples for the temperature monitoring/control although their measurements are highly affected by the barrel metal wall temperature. Currently, no industrially established thermal profile measurement technique is available. Furthermore, it has been shown that the melt temperature changes considerably with the die radial position and hence point/bulk measurements are not sufficient for monitoring and control of the temperature across the melt flow. The majority of process thermal control methods are based on linear models which are not capable of dealing with process nonlinearities. In this work, the die melt temperature profile of a single screw extruder was monitored by a thermocouple mesh technique. The data obtained was used to develop a novel approach of modelling the extruder die melt temperature profile under dynamic conditions (i.e. for predicting the die melt temperature profile in real-time). These newly proposed models were in good agreement with the measured unseen data. They were then used to explore the effects of process settings, material and screw geometry on the die melt temperature profile. The results showed that the process thermal homogeneity was affected in a complex manner by changing the process settings, screw geometry and material.     

Thermoelastic contact during pressing of a heated elliptical cross-section die into a transversely isotropic piezoceramic half space

The three-dimensional thermoelastic contact as a nonplanar elliptical cross-section die is pressed into a transversely isotropic half space is examined under conditions such that the die has a previously specified temperature while the boundary of the half space outside the die is kept at zero temperature. Contact between the die and the half space is assumed to be ideal. The stressed state in the elastic medium is characterized by potential functions which are constructed explicity. Expressions are given for determining the translational displacement of points on the base of the stamp as well as the major half axis and eccentricity of the contact area. Some possible forms of boundary conditions for the electrical variables at the contact area are examined. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 37–53, 1999.     

FEM analysis for residual stress prediction in hot rolled steel strip during the run-out table cooling

With the increasing demand of higher quality hot rolled strips, flatness defects occurred on the strips during the cooling process on the run-out table have received significant attention and should be considered in the online shape control model. Non-uniform temperature distribution and cooling across the strip width are the main reasons why the strip becomes unflatten after cooling process although the strip is rolled flat at the finishing mill. A thermal, microstructural and mechanical coupling analysis model for predicting flatness change of steel strip during the run-out table cooling process was established using ABAQUS Finite Element Software. In this model, Esaka phase transformation kinetics model was employed to calculate the phase transformation, and coupled with temperature calculation by means of the user subroutine program HETVAL. An elasto-plasticity constitutive model of the material, in which conventional elastic and plastic strains, thermal strain, phase transformation strain and transformation induced plastic strain were taken into account, was derived and realized using the user subroutine program UMAT. The conclusion that the flatness of the steel strip will develop to edge wave defect under the functions of the different thermal and microstructural behaviors across strip width direction during the run-out table cooling procedure was acquired through the analysis results of this model. The calculation results of this analysis model agree with the actual measurements and observation, therefore this model has a high accuracy. To better control the flatness quality of hot rolled steel strip, the shape compensation control strategy of slight center wave rolling is proposed based on the analysis result. This control strategy has been verified by actual measurements, and applied in actual production.     

Comparing Optimization Algorithms for Shape Optimization of Extrusion Dies

The classical approach to extrusion die design relies heavily on the experience of the die designer; Especially the designer's ability to create an initial die design from a product design, the designer's constructional knowledge and performance during the running-in trials. Furthermore, the relative unpredictability of the running-in trials combined with the additional resource usage introduce uncertainties and delays in the time-to-market of a given product. To lower these delays and resource usage, extrusion die design can benefit greatly from numerical shape optimization. In this application, however, plastics melts pose a difficult obstacle, due to their rather unintuitive and nonlinear behavior. These properties complicate the numerical optimization process, which mimics running-in trials and relies on a minimal number of optimization iterations. As part of the Cluster of Excellence Integrative Production Technologies for High-Wage Countries at the RWTH Aachen University, an effort is made to shorten the manual running-in process by the means of numerical shape optimization. Using an in-house numerical shape optimization framework, a set of optimization algorithms, consisting of global, derivative-free and gradient-based optimizers, are evaluated with respect to the best die quality and a minimal number of optimization iterations. This evaluation is an important step on the way to include more computationally intensive material models into the optimization framework and identify the best possible optimization strategy for the numerical design of extrusion dies. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic channel control and pricing of a single perishable product on multiple distribution channels

This paper studies dynamic channel control and pricing of a single perishable product distributed through multiple channels with the objective of maximizing the total expected profit over a finite horizon. We consider two types of commissions, namely proportional and fixed commissions, on the third-party channels and utilize stylized linear functions to characterize dependent demand flows from different channels. We show that, the magnitude of the opportunity cost of capacity uniquely determines the optimal channel control, at any given inventory level and periods to go. Consequently, we are able to derive the optimal price offered on each channel as a function of the opportunity cost of capacity in closed form. This significantly reduces the computational complexity of the stochastic dynamic program when parameters are constant with time. When channels are independent, we provide a necessary and sufficient condition for the optimality of a nested channel control policy by commission rates. The same condition is also sufficient for the optimality of the nested channel control policy in a distribution system with two dependent channels. We then characterize the structural properties of the optimal pricing and channel control policies. Finally, we explore the impact of the substitution effect on the channel control through numerical studies and gain managerial insights.     

Complete efficiency and interdependencies between objective functions in vector optimization

Summary Vector optimization problems in linear spaces with respect to general domination sets are investigated including corollaries to Pareto-optimality and weak efficiency. The results contain equivalences between vector optimization problems, interdependencies between objective functions and domination sets, statements about the influence of perturbed objective functions on the decision maker's preferences and thus on the domination set, comparisons of efficiency with respect to polyhedral cones with Pareto-optimality, changes in the objective functions which restrict, extend or do not alter the set of Pareto-optima, possibilities for the use of domination sets immediately in the decision space. Conditions for complete efficiency are given.

---

Zusammenfassung Untersucht werden Vektoroptimierungsprobleme in linearen Räumen bezüglich allgemeiner Dominanzmengen einschließlich Folgerungen für Pareto-Optimalität und schwache Effizienz. Die Ergebnisse enthalten Äquivalenzen zwischen Vektoroptimierungsproblemen, Wechselwirkungen zwischen Zielfunktionen und Dominanzmengen, Aussagen über den Einfluß gestörter Zielfunktionen auf die Präferenzen des Entscheidungsträgers und somit auf die Dominanzmenge, Vergleiche von Effizienz bezüglich polyedrischer Kegel mit Pareto-Optimalität, Änderungen in den Zielfunktionen, die die Menge der Pareto-Optima einschränken, erweitern oder nicht beeinflussen, Möglichkeiten für die Nutzung von Dominanzmengen unmittelbar im Entscheidungsraum. Bedingungen für vollständige Effizienz werden gegeben.

     

Resource Allocation with Wobbly Functions

We consider resource allocation with separable objective functions defined over subranges of the integers. While it is well known that (the maximisation version of) this problem can be solved efficiently if the objective functions are concave, the general problem of resource allocation with functions that are not necessarily concave is difficult.In this article, we focus on a large class of problem instances, with objective functions that are close to a concave function or some other smooth function, but with small irregularities in their shape. It is described that these properties are important in many practical situations.The irregularities make it hard or impossible to use known, efficient resource allocation techniques. We show that, for this class of functions the optimal solution can be computed efficiently. We support our claims by experimental evidence. Our experiments show that our algorithm in hard and practically relevant cases runs up to 40–60 times faster than the standard method.     

Duality and admissible transformations in combinatorial optimization

For special combinatorial optimization problems different kind of objective functions are of practical interest. In this paper we investigate the applicability ofBurkard's duality concept for algebraic objective functions. Thereafter we discuss the concept of admissible transformations, a more combinatorial motivated way of problem solving which enables us to treat different objective functions in one approach. We demonstrate the applicability of this approach to special combinatorial optimization problems.

---

Zusammenfassung Bei der Behandlung spezieller kombinatorischer Optimierungsaufgaben sind verschiedene Zielfunktionen von praktischer Bedeutung. In dieser Arbeit untersuchen wir zunächst die Anwendbarkeit einer Dualitätstheorie für algebraische Zielfunktionen, die vonBurkard vorgeschlagen wurde. Danach diskutieren wir das Konzept der zulässigen Transformationen, einen mehr kombinatorisch motivierten Zugang, der es erlaubt verschiedene relevante Zielfunktionen in einem Ansatz zu behandeln. Wir demonstrieren die Anwendbarkeit dieses Konzeptes bei speziellen kombinatorischen Optimierungsproblemen.

     

Optimal Control of Continuous Casting by Nondifferentiable Multiobjective Optimization

A new version of an interactive NIMBUS method for nondifferentiable multiobjective optimization is described. It is based on the reference point idea and the classification of the objective functions. The original problem is transformed into a single objective form according to the classification information. NIMBUS has been designed especially to be able to handle complicated real-life problems in a user-friendly way.The NIMBUS method is used for solving an optimal control problem related to the continuous casting of steel. The main goal is to minimize the defects in the final product. Conflicting objective functions are constructed according to certain metallurgical criteria and some technological constraints. Due to the phase changes during the cooling process there exist discontinuities in the derivative of the temperature distribution. Thus, the problem is nondifferentiable.Like many real-life problems, the casting model is large and complicated and numerically demanding. NIMBUS provides an efficient way of handling the difficulties and, at the same time, aids the user in finding a satisficing solution. In the end, some numerical experiments are reported and compared with earlier results.     

Optimal control of the end-temperature in a semi-infinite rod

The temperature distribution in a semi-infinite rod is controlled by heat input at one end. The objective is to keep the temperature at this end of the rod close to a given value, without excessive heat input. This distributed parameter optimal control problem is reformulated as a calculus of variations problem for the optimal end-temperature. An explicit solution is derived, and its general properties are examined. Two example cases are provided to illustrate the results.

---

Zusammenfassung Die Wärmeverteilung in einem einseitig unendlichen Stab wird durch Wärmezufuhr an einem Ende gesteuert. Man versucht, die Temperatur an diesem Ende ohne übermässige Wärmezufuhr möglichst nahe bei einem vorgegebenen Wert zu halten. Das optimale Steuerungsproblem des Systems, dessen mathematisches Modell durch eine partielle Differentialgleichung gegeben ist, wird in ein Variationsrechnungsproblem für die optimale Endtemperatur transformiert. Eine explizite Lösung wird hergeleitet, ihre allgemeinen Eigenschaften werden untersucht und an zwei Beispielen illustriert.

     

Stretched plates weakened by inner curvilinear holes

Complex variable methods are used to obtain exact and closed expressions for Gaursat's functions for the stretched infinite plate weakened by one inner hole which is free from stresses. The plates considered are conformally mapped on the area of the right half-plane. The interesting cases of an infinite plate weakened by a crescent-like hole or by a cut having the shape of a circular arc, are included as special cases.

---

Zusammenfassung Mit Hilfe komplexer Methoden werden exakte und geschlossene Ausdrücke für die Gaursat-Funktionen einer gezogenen Scheibe erhalten, welche durch ein spannungsfreies Loch geschwächt ist. Die betrachteten Scheiben werden konform auf die rechte Halbebene abgebildet. Als Sonderfälle sind neumondförmige Löcher und Schlitze in Form von Kreisbogen in den Resultaten mitenthalten.

     

A modified Artificial Bee Colony algorithm for structural damage identification under varying temperature based on a novel objective function

This paper presents a modified Artificial Bee Colony algorithm for structural damage identification. Meanwhile, the effect of temperature variation is considered and the change of temperature will lead to the alteration of Young's modulus of material. A novel objective function is proposed as the combinations of the partial mode shape curvature data, alterations of natural frequencies, and a sparse penalty term. Such an objective is found to be sensitive to structural damage while not sensitive to environmental effects. On the other hand, To render the standard Artificial Bee Colony algorithm more powerful and robustness, two local search strategies are introduced into the employed and onlooker bee phase of the Artificial Bee Colony algorithm, respectively. Two numerical examples and a laboratory verification are employed to verify the efficiency and advantage of the proposed algorithm. The final results show that the present algorithm could yield more satisfactory identification results compared with other state-of-the-art evolutionary algorithms, even high-level noise and temperature variation are considered; and the proposed novel objective function is more sensitive to structural damages, compared with the traditional mode-shape-based objective function.     

Optimal PID controller of robots under stochastic uncertainty

In practice often it is not possible to specify exact model parameters. Hence, precomputed controller based on some parameter estimates can produce bad results. In this presentation the aim is to combine classical PID control theory and stochastic optimisation methods in order to obtain robust optimal feedback control. The method works with cost functions being minimized and takes into account stochastic parameter varations. After Taylor expansion to calculate expected cost functions and a few transformations an approximate deterministic substitute PID control problem follows. Here, retaining only linear terms, approximation of expectations and variances of the expected cost functions can be calculated explicitly. By means of splines, numerical approximations of the objective function and the differential equations are obtained then. Using stochastic optimization methods, random parameter variations are incorporated into the optimal control process. Hence, robust optimal feedback controls are obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An extension of the Bézier model

In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein-Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.     

Hierarchic control

Distributed control is applied to a system modelled by a parabolic evolution equation. One considers situations where there are two cost (objective) functions. One possible way is to cut the control into 2 parts, one being thought of as “the leader” and the other one as “the follower”. This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper. Dedicated to the memory of Professor K G Ramanathan     

Deformations of elastic membranes—Effect of different constitutive relations

Results for the nonlinear axisymmetric deformations of flat circular membranes and of spherical cap membranes subjected to a uniform pressure are presented. The emphasis of the study is on the effect of two different constitutive relations on the nature of the deformation when the membrane deforms into an approximate hemispherical shape.

---

Zusammenfassung Ergebnisse für nichtlineare axialsymmetrische Deformationen bei ebenen Kreismembranen und sphärischen Haubenmembranen unter gleichförmigem Druck werden besprochen. Besondere Beachtung findet die Auswirkung zweier verschiedener Stoffbeziehungen auf die Natur der Verformung, wenn sich die Membran in die ungefähre Gestalt einer Halbkugel deformiert.

     

Scattering phenomena and their application in optical anemometry

Theoretical and experimental investigations into the physics of optical anemometry are reported. The theoretical investigations involve considerations of light scattering by small moving particles suspended in a flowing fluid. The analysis yields general expressions for the light intensity at a point on the photodetector if several scattering beams penetrate the measuring control volume in different directions. The derivations also lead to the time dependence of the light intensity caused by the movement of the scattering particles relative to the light source and the photodetector. The relationship between the frequency of the photodetector signal and the instantaneous velocity of the scattering particles is derived on the basis of phase changes of the scattered light waves and is subsequently related to previously published derivations.The azimuthal distributions of signal quality and signal strength are evaluated based on results of computation of scattered light intensity distributions. These yield a dependence of local signal quality and signal strength on the particle size and its optical properties. In addition, a heuristic explanation of this dependence is presented that relates the signal quality influence by the particle size to the apparent fringe spacing in the measuring control volume. The need for finite apertures in optical anemometry is apparent and their influence on optical anemometer signals is discussed.The experimental programme included velocity measurements of rotating discs to demonstrate the correctness of the theoretical derivations. The decay of signal quality with signal mismatch from many particles moving at different velocities is demonstrated and the need for careful control of the measuring region is pointed out.

---

Zusammenfassung Es wird über theoretische und experimentelle Grundlagen der Physik optischer Anemometrie berichtet. Die theoretischen Betrachtungen befassen sich mit Untersuchungen der Lichtstreuung durch kleine Teilchen, die in einem strömenden Medium suspendiert sind. Die analytischen Untersuchungen resultieren in allgemeinen Ausdrücken für die Lichtintensität an einem Punkt des Photodetektors, für den Fall, dass mehrere Streustrahlen das Messvolumen in verschiedenen Richtungen durchdringen. Die Ableitungen führen ferner zu der Zeitabhängigkeit der Lichtintensität, die durch die Bewegung der Streuteilchen relativ zur Lichtquelle und dem Photodetektor verursacht wird. Die Beziehung zwischen der Frequenz des Photodetektorsignals und der Momentangeschwindigkeit der Streuteilchen ist abgeleitet. Sie basiert auf Phasenverschiebungen der gestreuten Lichtwellen und wird nachträglich anderen bereits publizierten Ableitungen gegenübergestellt.Basierend auf den Berechnungen der Intensitätsverteilungen des Streulichtes wird die räumliche Verteilung der Signalqualität und der Signalstärke berechnet. Dies führt zu einer Abhängigkeit der lokalen Signalqualität und Signalstärke von der Teilchengrösse und den optischen Eigenschaften des Teilchenmaterials. Zusätzlich wird eine heuristische Erklärung dieser Abhängigkeit gegeben, die den Einfluss der Signalqualität durch die Teilchengrösse durch den Interferenzstreifenabstand im Messvolumen zu erklären versucht. Die Notwendigkeit endlicher Aperturen in der optischen Anemometrie ist augenscheinlich und ihr Einfluss auf optische Anemometersignale wird diskutiert.Der experimentelle Teil umfasst Geschwindigkeitsmessungen rotierender Scheiben, um die Korrektheit der theoretischen Ableitungen und Berechnungen zu demonstrieren. Der Abfall der Signalqualität durch Auftreten von Signalen verschiedener Frequenzen, die von Teilchen mit unterschiedlichen Geschwindigkeiten herrühren, wird gezeigt, und es wird auf die Notwendigkeit einer sorgfältigen Kontrolle des Messvolumens hingewiesen.

     

Numerical experiments on optimal shape parameters for radial basis functions

A numerical investigation on a technique for choosing an optimal shape parameter is proposed. Radial basis functions (RBFs) and their derivatives are used as interpolants in the asymmetric collocation radial basis method, for solving systems of partial differential equations. The shape parameter c in RBFs plays a major role in obtaining high quality solutions for boundary value problems. As c is a user defined value, inexperienced users may compromise the quality of the solution, often a problem of this meshless method. Here we propose a statistical technique to choose the shape parameter in radial basis functions. We use a cross‐validation technique suggested by Rippa 6 for interpolation problems to find a cost function Cost(c) that ideally has the same behavior as an error function. If that is the case, the parameter c that minimizes the cost function will be an optimal shape parameter, in the sense that it minimizes the error function. The form of the cost and error functions are analized for several examples, and for most cases the two functions have a similar behavior. The technique produced very accurate results, even with a small number of points and irregular grids. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Convergence of the Simulated Annealing Algorithm for Continuous Global Optimization

A class of simulated annealing algorithms for continuous global optimization is considered in this paper. The global convergence property is analyzed with respect to the objective value sequence and the minimum objective value sequence induced by simulated annealing algorithms. The convergence analysis provides the appropriate conditions on both the generation probability density function and the temperature updating function. Different forms of temperature updating functions are obtained with respect to different kinds of generation probability density functions, leading to different types of simulated annealing algorithms which all guarantee the convergence to the global optimum.