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Rheological parameters identification using in situ experimental data of a flat die extrusion
Abstract
Viscosity is an important characteristic of flow property's and process ability for polymeric materials. A flat die was developed by Maillefer-extrusion, to make rheological characterisations. In this paper, the rheological parameters of the melt are identified through optimisation by a response surface method. The objective is to minimize the differences between the measured pressure obtained in flat die and the pressures computed by one-dimensional finite difference code programme. An objective function is defined as the global relative error obtained through the differences between measured and computed pressures. This objective function is minimised by varying the rheological parameters. For this minimisation, two methods are used, i.e. the local response surface and the global response surface. The rheological parameters permit to calculate the viscosity. Then, we compare this calculated viscosity with an experimental viscosity measured on a capillary rheometer to validate our method.
... The figure 2 shows the boundary conditions, described by equations set (2). T reg is the regulation temperature of the die and p in is chosen such as imposing the experimental pressure difference between the die extremities. ...
... The objective of the method is to evaluate the rheology law taking in account the inhomogeneity of temperature inside the polymer. Such a method has been proposed by [2] and allows to estimate parameters of an Ostwald -de Waelle law (or power law) [3,4], which is adapted to high shears rates. ...
... This method is quite similar to the proposed one by Lebaal et al. [2]. The important difference holds on the discrete rheological model, which does not imply a specific form. ...
Simulation quality is determined by the knowledge of the parameters of the model. Yet the rheological models for polymer are often not very accurate, since the viscosity measurements are made under approximations as homogeneous temperature and empirical corrections as Bagley one. Furthermore rheological behaviors are often traduced by mathematical laws as the Cross or the Carreau-Yasuda ones, whose parameters are fitted from viscosity values, obtained with corrected experimental data, and not appropriate for each polymer. To correct these defaults, a table-like rheological model is proposed. This choice makes easier the estimation of model parameters, since each parameter has the same order of magnitude. As the mathematical shape of the model is not imposed, the estimation process is appropriate for each polymer. The proposed method consists in minimizing the quadratic norm of the difference between calculated variables and measured data. In this study an extrusion die is simulated, in order to provide us temperature along the extrusion channel, pressure and flow references. These data allow to characterize thermal transfers and flow phenomena, in which the viscosity is implied. Furthermore the different natures of data allow to estimate viscosity for a large range of shear rates. The estimated rheological model improves the agreement between measurements and simulation: for numerical cases, the error on the flow becomes less than 0.1% for non-Newtonian rheology. This method couples measurements and simulation, constitutes a very accurate mean of rheology determination, and allows to improve the prediction abilities of the model.
... Il existe plusieurs méthodes d'approximation plus au moins différentes, notamment la méthode diffuse [166] et la méthode krigeage [167][168][169], utilisées dans le contexte de la méthode des sur- ...
... Le deuxième terme, r T (x)α représente la correction du premier terme permettantà la courbe (ouà la surface) de passer par tous les points d'interpolation. Où ω est la fonction poids, gaussienne, donnée par la relation [167] : Où d i = n j=1 (x j − x j i ) 2 est la distance entre le point x et les points voisins x i , dans le domaine de rayon r w . c représente le paramètre de dilatation (dans cetteétude, c est supposéêtreégalà r w /4. ...
Les produits structuraux (poutres, poteaux…etc) en bois de hêtre reconstitué par collage, notamment le Bois Massif Reconstitué (BMR) présentent un fort potentiel tant au niveau de la performance mécanique que de la ressource disponible par comparaison aux essences résineuses. Par ailleurs, les normes européennes actuelles sont rédigées pour les essences résineuses et l’usage du bois de hêtre en structures est limité à la classe de service 1 et aux produits d’intérieur et d’ameublement. Le marché français montre une forte augmentation de l’importation de produits structuraux en bois reconstitué par collage, principalement d’Allemagne et d’Autriche. Au-delà du potentiel de résistance mécanique du hêtre, son utilisation comme bois d’œuvre contribuera positivement au bilan carbone pour l’environnement, par la réduction des transports des bois d’importation. Cependant, pour un usage structural, les produits en hêtre reconstitué par collage nécessitent de passer par une évaluation de leur performance, notamment la fiabilité des aboutages. Cette thèse présente une méthodologie expérimentale et numérique permettant l’analyse du comportement mécanique des poutres en BMR. La démarche a été appliquée avec succès dans le cadre de poutres constituées de 2 à 3 lamelles avec ou sans aboutages. Dans ce contexte, nous avons également réalisé une étude paramétrique pour étudier l’influence de plusieurs paramètres sur la résistance des aboutages. En fin, nous avons formulé le problème d’optimisation de la géométrie de l’enture multiple afin d’augmenter la résistance des aboutages et donc la résistance globale des poutres en BMR
... When applying RSM to a particular problem, two important issues have to be considered: the choice of the Design of Experiments (DoE) and the construction of accurate function approximations (Breitkopf et al., 2005;Lebaal et al., 2005;Naceur et al., 2006;Naceur et al., 2008;Lebaal et al., 2009a). The ultimate goal of this study is to develop a numerical optimisation strategy to obtain optimised clinching tool geometry leading to a greater strength of joints than that previously achieved (Oudjene et al., 2009). ...
... A weight function of Gaussian type with a circular support (Lebaal et al., 2005) is adopted for the kriging interpolation: discrete node x i to a sampling point x in the domain of support with radius r w , and c is the dilation parameter (for the computation, c was assumed to be equal to r w /4). ...
The production of high-strength clinched joints is the ultimate goal of the manufacturing industry. The determination of optimum tool shapes in the clinch forming process is needed to achieve the required high strength of clinched joints. The design of the tools (punch and die) is crucial since the strength of the clinched joints is closely related to the tools geometry. To increase the strength of clinched joints, an optimisation procedure using the response surface methodology, based on an adaptive moving target zone, is presented. The cost function studied here is defined in terms of the maximum value of the tensile force computed during the simulation of the sheets separation. Limitations on the geometrical parameters due to feasibility issues are also taken into account. The kriging interpolation is used to provide an approximation to the optimisation problem and to build the response surfaces.
... Numerous approximation techniques, particularly the RSM [12][13][14][15] using a second-order polynomial approximation, have been developed for various optimization problems. Other researchers have focused on managing this RSM to achieve the global optimum, for example by employing a self-adaptive research space technique. ...
... Extrusion process parameters like the type of extruder, length of barrel, feed rate, speed, screw configuration; moisture content and so on can also be attributed to the system parameters, system parameter are a type of extruder, length of the barrel, feed rate, screw speed, moisture content, screw configuration etc. and momentum transfer parameters are; viscosity, torque, specific mechanical energy, extrusion pressure, and mass flow rate etc and thus to optimize and scale up the process appropriately (Kokini et al. 1992;Bouvier, 2001). The geometry of the extrusion die, as well as operating conditions such as temperature of control, flow rate, and melt theological parameters, affect extrusion defects (such as weld lines, a fairly uniform exit velocity distribution within the extrusion, and problems with stagnation zones) (Lebaal, et al. 2005). Four different die patterns and numerous operating requirements (including temperature and moisture level) were measured to evaluate a die design procedure that yielded a uniform velocity profile of extruded, the operating terms that influenced the velocity profile, the velocity deviations among the center and edge of extruded were more than 10% at the studied conditions, the die pattern was the main factor in realizing uniformity in velocity profile and thickness of extruded, when a concaveconcave die with proper curvature was used, while uniform speed and thickness of extruded were attained, with less than 5% speed deviations and thickness deviations between center and edge of extruded of mostly less than 10 % (Yeh and Hwang, 1995). ...
Die holes distribution is the most important factor of any feed pelleting system, as far as machine efficiency is concerned. This study was carried out with fish diet, where had been processed without steam addition (cold pelleting). The aim of the study was to explore the impact of fish feed extruder die surface holes distribution patterns on extruder productivity, total consumed power, specific energy requirements, pelleting efficiency, pellets bulk density and durability and economical evaluation. The optimum results were recorded with die that had holes distributed in the surface of die in star pattern, observed improvement in extruder productivity, pellets durability and pelleting efficiency by up to 10.75 and 15.64 %, 0.51 and 4.74 % , 0.53 and 1.63 %, respectively at constant moisture content of 36 % and screw speed of 96 rpm(12.7 m. min-1), comparing to triangular pattern and scattering pattern respectively, and decreased, bulk density, total consumed power, specific energy requirements and pelleting cost by up to 1.86 and 6.27 %, 14.98 and 47.34%, 12.2 and 6.2 % , 28.8 and 35.6%, respectively at constant moisture content of 36 % and screw speed of 96 rpm (12.7 m. min-1), compared to triangular pattern and scattering pattern respectively.
... Similarly, two systems of Equations (14) and (9) with the pressure gradient Equations (12) and (13) and flow rates from Equations (1) and (15) are solved by an iterative numerical scheme, such as the secant method. The convergence criterion is defined as follows: Figure 2a shows the algorithm used for calculations. ...
In the polymer sheet processing industry, the primary objective when designing a coat-hanger die is to achieve a uniform velocity distribution at the exit of the extrusion die outlet. This velocity distribution depends on the internal flow channels of the die, rheological parameters and extrusion process conditions. As a result, coat-hanger dies are often designed for each polymer based on its individual rheological data and other conditions. A multi-rheology method based on a flow network model and the Winter–Fritz equation is proposed and implemented for the calculation, design and optimization of flat sheeting polymer extrusion dies. This method provides a fast and accurate algorithm to obtain die design geometries with constant wall-shear rates and optimal outlet velocity distributions. The geometric design when complemented and validated with fluid flow simulations could be applied for multi-rheological fluid models such as the power-law, Carreau–Yasuda and Cross. This method is applied to sheet dies with both circular- and rectangular-shaped manifolds for several rheological fluids. The designed geometrical parameters are obtained, and the associated fluid simulations are performed to demonstrate its favorable applicability without being limited to only the power-law rheology. The two such designed dies exhibit 32.9 and 21.5 percent improvement in flow uniformity compared to the previous methods for dies with circular and rectangular manifolds, respectively.
... The moving least-square approximation is used in order to approximate the relationship between all parameters. The second order approximation is the most intensively used method in the literature [26][27][28][29][30][31]. The explicit relationship between real and approximate function, can be expressed as follows: ...
Compact heat exchangers for heat removal has become one of the most effective cooling techniques. In this paper, numerical scheme to simulate conjugated heat transfer using additively manufactured heat exchanger is presented. A lattice structure as porous medium is used as effective cooling techniques for heat removal. The objective of these lattices is to transmit heat from the hot part to the cold part while letting the system benefit from two phenomena, heat conduction and convection. An optimization procedure using response surface method is proposed for the presented compact heat exchanger to improve the efficiency. This method was introduced in order to reach the global optimum with a limited number of computer experiments. Two optimization variables are identified: edge thickness for the lattices and the inlet velocity. A constraint optimization problem is formulated to maximize the heat flux through the heat exchanger. Hence at the same time, limiting the increase of the pressure drop and the decrease of the exit temperature.
... The optimization process is based on the minimization of the welding energy and the improvement of the production efficiency by reducing the welding time and decreasing the HAZ. For this purpose, the CFD ANSYS 14.0 thermal model code is coupled with the Sequential Quadratic Programming gradient (SQP) optimization algorithm available in MATLAB code [35,36]. This combination allows the determination of the optimal parameters of the FSW process, i.e., the welding velocity u w , the tool rotational velocity ω, the tool shoulder radius R s and the forging force F. ...
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the author's institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights A B S T R A C T The present study deals with the improvement of the Friction Stir Welding process, through the prediction of the optimal operating conditions, necessary for welding typical Aluminum-Lithium alloy material AA2195-T8. An optimization strategy coupled with 3D transient heat transfer computation were used to improve the FSW process parameters such as welding velocity, tool rotation velocity, tool diameter and applied force. The optimization procedure is based on three criteria: the control of the maximum temperature during FSW; the minimization of the Heat Affected Zone (HAZ) length and finally the reduction of the total welding energy. The obtained optimal parameters have given an ideal welding temperature in the workpiece, thereby ensuring good welding quality, gain in energy consumption and decrease both the welding time and the HAZ length.
... Within this aim, many different optimisation algorithms have been developed (Levenberg, 1944;Marquardt, 1963;Nelder and Mead, 1965). Inverse methods have been used to determine food physical properties (Simpson and Cortes, 2004;Zueco et al., 2004;Mohamed, 2008;Monteau, 2008;Da Silva et al., 2009Fabbri et al., 2011Fabbri et al., , 2014, even if the studies concerning the determination of viscosity rheological are very few and not referred to non-Newtonian food materials (Lebaal et al. 2005;Guet et al., 2006;Fullana et al., 2007;Kalyon and Tang, 2007;Park et al., 2007;Bandulasena et al., 2007;Nascimento et al., 2010;Bandulasena et al., 2011). ...
The description of the rheological properties of food material plays an important role in food engineering. Particularly for the optimisation of pasta manufacturing process (extrusion) is needful to know the rheological properties of semolina dough. Unfortunately characterisation of non-Newtonian fluids, such as food doughs, requires a notable time effort, especially in terms of number of tests to be carried out. The present work proposes an alternative method, based on the combination of laboratory measurement, made with a simplified tool, with the inversion of a finite elements numerical model. To determine the rheological parameters, an objective function, defined as the distance between simulation and experimental data, was considered and the well-known Levenberg-Marqard optimisation algorithm was used. In order to verify the feasibility of the method, the rheological characterisation of the dough was carried also by a traditional procedure. Results shown that the difference between measurements of rheological parameters of the semolina dough made with traditional procedure and inverse methods are very small (maximum percentage error equal to 3.6%). This agreement supports the coherence of the inverse method that, in general, may be used to characterise many non-Newtonian materials.
... The method of response surface [21] consists in the Routine for change channel geometry ...
... Since we know the position of the nodes, we seek to determine the coordinates of all the nodes (Ni), which belong to both surfaces S1 and S2 (figure 2). For this, we use the diffuse approximation [14] with a linear interpolation. To change the geometry of the repartition channel we define the new surface (S1 k ), which corresponds to the new parameters of optimization A k , B k , and C k . ...
... The objective and constraint functions are implicit, compared to the optimization parameters and their evaluation requires a large amount of computing time using three dimensional analyses. In order to decrease the evaluation number of the objective and constraint functions we use in this work the response surface method [11,13] which consists in the construction of an approximate expression of the objective and constraint functions starting from a limited number of evaluations of the real function. In order to obtain a good approximation, we used a Kriging interpolation described in next section. ...
... These numerical technics are widely used in the literature. Identification based on Finite Element simulation loops are possible, however the simulation time can be consequent [11]. Kim and Choi [12] used inverse analysis to predict the deformation behavior and interfacial friction under hot working conditions. ...
The simulative prediction of material behaviour in forming processes necessitates a
precise determination of the material parameters. The present work focusses on the
modelling of the isostatic part of the flow stress using a flow curve with an analytical
suppression of the influence of friction and an adequate analytical law. The
experimental data are obtained from isothermal upsetting tests with various upsetting
ratios. The different ratios are based on a variation of the height of the sample,
remaining the diameter constant. For the proposed flow stress law five parameters are
identified. In order to decrease the number of function evaluations, a new reduction
model method based on both analytical and sequential quadratic programming (SQP)
algorithms is developed and applied to identify flow stress law parameters. A
comparison with traditional SQP algorithm is also done. A 3D finite element model is
built in order to simulate a side pressing test and an experimental validation is done. As
numerical results fit very well experimental data, the proposed model achieves a
precise prediction of the flow behaviour. The identification of the other parts of the
model (i.e. dependencies on strain-rate and temperature) are conducted in further
works.
... Since we know the nodes position, we seek to determine the coordinates of all the nodes (Ni), belonging to both surfaces SI and S2 (figure 2). For this, we use the diffuse approximation [14] with a linear interpolation. To change the geometry of the channel repartition we define the new surface (Sl k ), which corresponds to the new optimization parameters A k , B k , and C k . ...
Polymer extrusion is one of the most important manufacturing methods used today. A flat die, is commonly used to extrude thin thermoplastics sheets. If the channel geometry in a flat die is not designed properly, the velocity at the die exit may be perturbed, which can affect the thickness across the width of the die. The ultimate goal of this work is to optimize the die channel geometry in a way that a uniform velocity distribution is obtained at the die exit. While optimizing the exit velocity distribution, we have coupled three-dimensional extrusion simulation software Rem3D®, with an automatic constraint optimization algorithm to control the maximum allowable pressure drop in the die; according to this constraint we can control the pressure in the die (decrease the pressure while minimizing the velocity dispersion across the die exit). For this purpose, we investigate the effect of the design variables in the objective and constraint function by using Taguchi method. In the second study we use the global response surface method with Kriging interpolation to optimize flat die geometry. Two optimization results are presented according to the imposed constraint on the pressure. The optimum is obtained with a very fast convergence (2 iterations). To respect the constraint while ensuring a homogeneous distribution of velocity, the results with a less severe constraint offers the best minimum.
... The response surface method (Myers and Montgomery, 2002;Lebaal et al., 2005) consists in the construction of an approximate expression of the objective function starting from a limited number of evaluations of the real function. To obtain a good approximation, we used a Kriging interpolation described in the next section. ...
A new optimisation methodology for the design of coat-hanger dies is presented. Two approaches are presented to optimise the velocities distribution across the die exit. In the first approach, we predict the optimal shape of a coat hanger die; in the second approach, to keep the same geometry and avoid design of a new die, we optimise the temperature of regulation in heterogeneous way. This method involves coupling a three-dimensional finite element simulation software and an optimisation strategy. For this optimisation, the Sequential Quadratic Programming algorithm and the global response surface method with Kriging interpolation are used.
... Gradient methods [7][8][9][10]12,14] require the computations of the gradients of the functions; the computation of gradients by finite difference is time consuming and depends on the perturbed parameters. For the above reasons we decided to chose a response surface method (RSM) [15]. ...
Balancing the distribution of flow through a die to achieve a uniform velocity distribution is the primary objective and one of the most difficult tasks of extrusion die design. If the manifold in a Coat-hanger die is not properly designed, the exit velocity distribution may be not uniform; this can affect the thickness across the width of the die. Yet, no procedure is known to optimize the coat hanger die with respect to an even velocity profile at the exit. While optimizing the exit velocity distribution, the constraint optimization algorithm used in this work enforced a limit on the maximum allowable pressure drop in the die; according to this constraint we can control the pressure in the die. The computational approach incorporates three-dimensional finite element simulations software Rem3D® and includes an optimization algorithm based on the global response surfaces with the Kriging interpolation and SQP algorithm within an adaptive strategy of the search space to allow the location of the global optimum with a fast convergence. The optimization results which represent the best die design are presented according to the imposed constraint on the pressure.
... Through analysis of various flow field characteristics such as pressure drop, velocity, shear rate, shear stress, temperature, residence time, the polymer extrusion process can provide information that would be impossible, or at least extremely difficult to physical measure. Compared with other numerical methods, finite element method (FEM) has been applied widely in the simulation of polymer extrusion processing [1][2][3]. Finite element method is generally divided into mixed FEM (e.g. general FEM software ANSYS) and penalty FEM [4,5]. ...
In this paper, penalty finite element method is applied to the simulation of three-dimensional incompressible viscous steady flow, and the program is coded for analysis of three-dimensional flow process of plastic melt in extrusion die. Slit die is typical in polymer extrusion process, and the relevant part is used widely. Because the flow law is different for various polymers with different rheological character even in the same die, the polymer flow in slit die is simulated for Newtonian fluid and power-law fluid by the program and ANSYS, respectively. Flow laws of two kinds of fluid are compared, and the result shows that the non-uniform velocity distribution in outlet is caused by the non-Newtonian effect of material. The penalty factor is important for the penalty finite element model, which decides the precision and efficiency to a great extent. The proper penalty value is decided by some factors, for example, the character of the whole equations and geometric model. Numerical solution with change of penalty value and viscosity is analyzed, and the result shows that the penalty value should be suited with viscosity in special range for penalty finite element model. The experiment has been developed, and the velocity distribution in outlet of slit die is measured. The result of finite element analysis is consistent with experimental result basically, and the reason of result divergence of the two ways is analyzed. The results show that the given model is suited for the polymer extrusion process, and the advice of the selection of penalty value is instructional for penalty element model.
... The objective and constraint functions are implicit, compared to the optimization parameters and their evaluation requires a large amount of computing time using three dimensional analyses. In order to decrease the evaluation number of the objective and constraint functions we use in this work the response surface method [11,13]. This method consists in the construction of an approximate expression of the objective and constraint functions starting from a limited number of evaluations of the real functions. ...
A new approach to the optimal design of the die wall temperature profile in polymer extrusion processes is presented. In this
approach, optimization of the design variables is conducted by a Response Surface Method (RSM) and the Sequential Quadratic
Programming (SQP) algorithm. Design of experiment (DoE) needed for the construction of the response surface is used to evaluate
the objective and the constraint functions on the basis of a finite element method (FEM). Two designs of experiments are used
and the performances of the optimization results are compared with respect to efficiency and ability to obtain a global optimum.
Typically, for extrusion die design, the objective function states that the average velocity across the die exit is uniform.
Constraints are used to limit the pressure drop in the die. For this purpose, we optimize the wall temperature profile of
a coat hanger die in a heterogeneous way, (i.e. the wall temperature may not be constant in the entire die). The melt temperature
enables us to locally control the viscosity, which influences the flows in the various zones. The effect of the design variables
in the objective and constraint functions is investigated using Taguchi method. The flow analysis results are then combined
with an automatic optimization algorithm to provide a new profile of the die wall temperature distributions.
KeywordsOptimization-DoE-Response surface method-FEM
... Trois résultats d'optimisations seront présentés [LEB05a], [LEB05b]. D'abord en utilisant l'approximation diffuse, puis l'interpolation Krigeage avec deux plans d'expérience différents (plan composite et plan enrichi). ...
Au cours de cette étude, différents aspects d'optimisation ont été abordés. L'objectif de nos travaux de recherches est d'homogénéiser la répartition des vitesses à la sortie des filières, par la maîtrise et l'optimisation des paramètres géométrique et opératoires. Dans cette étude l'optimisation a été effectuée en utilisant un logiciel commercial Rem3D®, basé sur la méthode des éléments finis.
Une procédure d'optimisation, basée sur la méthode de surface de réponse, a été proposée. Celle-ci nous a permis de résoudre un problème d'optimisation implicite dont l'évaluation des fonctions est très coûteuse en temps de calcul. Pour cela, toutes les fonctions sont écrites sous une forme explicite en utilisant soit l'approximation diffuse ou l'interpolation Krigeage. Compte tenu de la présence des contraintes non linéaires, un algorithme itératif de type SQP, a été utilisé. Pour localiser l'optimum global avec précision et à moindre coût, une procédure d'échantillonnage auto adaptatif de l'espace de recherche a été appliquée et plusieurs stratégies permettant de réactualiser les approximations et le point initial ont été adoptées.
Dans la première étape l'objectif était d'identifier le comportement rhéologique d'une matière plastique en production. La comparaison avec des mesures en rhéométrie capillaire nous a permis de vérifier la pertinence des paramètres rhéologique obtenus par optimisation.
Les résultats de trois autres applications mettent en évidence l'intérêt de l'optimisation des paramètres géométriques et opératoires du procédé d'extrusion.
Une filière optimisée numériquement pour une gamme différente de polymère a été réalisée et une comparaison expérimentale a permis de valider toute la procédure de simulation et d'optimisation mise en place. Les résultats expérimentaux et de simulations montrent une bonne homogénéisation de la répartition des vitesses à la sortie de la filière optimale pour une gamme très large de débits et pour différents polymères.
The present study is focused mainly on the comparative analysis of mechanical properties and microstructure of nitrogen stainless steel using friction stir welding (FSW). Temperature max out is the major factor for quality welding and for this reason water run is used for cooling while welding nitrogen stainless steel and also air stream cooling is used for comparison of strength in the welding. It is found that by using water run cooling in the welding area, the heat affected zone shows superior density in dislocation and nugget zone shows a grain size which is in better-quality when compared to air stream cooling in the joints. In addition to the above tests immersion corrosion test is conducted to analyze how much wear the tool undergone during the welding process and it has observed better corrosion resistance for water run cooling
The new method presented in this paper falls into the category of sampling methods and model management in the optimization process of surrogate related methods. This method was introduced in order to reach the global optimum with a limited number of computer experiments. During these developments, the Particle Swarm Optimization (PSO) was used as a smart sampling tool to construct the metamodel. These methods with their stochastic nature can also overcome the problems of local minima.
In order to improve the efficiency and accuracy of the metamodel (Kriging), a knowledge database with smart sampling methods has been integrated into the optimization model management, to avoid unnecessary finite elements calculations and enrich the collection (sampling) in each optimization iteration. This method makes it possible to reduce the sampling size and at the same time increases the accuracy of the metamodel.
For validation of the developed method, different benchmark functions were chosen in terms of features and has successfully then minimized. Finally, a practical engineering optimization problem for polymer extrusion was implemented with suggested Kriging Swarm Optimization algorithm (KSO). In this procedure, the Finite Element Analysis (FEA) was combined for simulation procedures to resolve non-isothermal non-Newtonian flow. Polymer extrusion results were applied for gathering information from design space samples and Kriging.
L'objectif de nos travaux de recherches est d'homogénéiser la répartition des vitesses à la sortie des filières, par la maîtrise et l'optimisation des paramètres géométrique et opératoires. Une procédure d'optimisation, basée sur la méthode de surface de réponse, a été proposée. Toutes les fonctions sont écrites sous une forme explicite en utilisant soit l'approximation diffuse ou l'interpolation Krigeage. Compte tenu de la présence des contraintes non linéaires, un algorithme de type SQP, a été utilisé. Pour localiser l'optimum global avec précision, une procédure d'échantillonnage auto adaptatif de l'espace de recherche a été adoptée. Les résultats d'optimisation mettent en évidence l'intérêt de l'optimisation des paramètres géométriques et opératoires du procédé d'extrusion. Une filière optimisée numériquement pour une gamme différente de polymère a été réalisée et une comparaison expérimentale a permis de valider toute la procédure de simulation et d'optimisation mise en place
The knowledge of fluid food rheological properties plays an important role in process engineering. Unfortunately, characterization of non-Newtonian fluids requires a notable effort in terms of time and resources. As a consequence, the aim of this research was to set up an method, based on the combination of the inversion of a simple finite element model and a laboratory measurement, carried on with a simplified tool. Although the method has a more general applicability, for illustrative purpose, it is here mainly shown with reference to power-law rheological model and two different materials (tara gum mixture and dough of water and flour). To measure the rheological parameters, the square of distance between simulated and experimental data was considered as an objective function, and some well-known optimization algorithms were tested. In order to verify the feasibility of the method, the experimental rheological characterization of the considered materials were carried out using a rotational rheometer. The calculated rheological parameter values were comparable with those obtained by traditional procedure (mean percentage error 6.47 ± 5.57). The most efficient optimization algorithm, in terms of iterations number, computational speed and minimum of the objective function, was the Levenberg–Marquardt one, but even other tested algorithms drove to similar final results (maximum difference of 18% between the optimized k and n values). The results demonstrated also that the precision of the calculated rheological parameters does not depend on the initial parameter values.
Balancing the distribution of flow through a die to achieve a uniform velocity distribution is the primary objective and one of the most difficult tasks of extrusion die design. If the extrusion die is not properly designed, the exit velocity distribution may be not uniform; this can affect the thickness across thewidth of the die. The objective of our research is to ensures a homogeneous exit velocity distribution on the outlet side of the die, through the control and the optimization of the geometrical parameters and operating conditions. An optimization procedure, based on the response surface method, was proposed. All the functions are written in an explicit form by using the either diffuse approximation or the Kriging interpolation. Due to the presence of the nonlinear constraints, an iterative algorithm of type SQP, was used. To find the global optimum with precision and at lower cost an auto-adaptive research space is adopted.
In the first stage the objective was to identify the rheological behaviour of a plastic using in situ experimental data. The comparison with measurements in capillary rheometry enabled us to verify the rheological parameters obtained by optimization.
The results of three other applications shows the interest of the optimization of the geometrical and operating conditions of the extrusion process.
A coat-hanger melt distributor is optimized numerically to ensures a homogeneous exit velocity distribution that will best accommodate for a different range of materials and multiple operating conditions. The results of numerical simulation are then validated by comparison with experimental measurements.
The numerical optimization algorithm presented in this work shows its suitability and robustness as a tool for extrusion die design. It proves its capability of predicting optimal die geometry with uniform velocity distribution for a range of polymers, at satisfactory computing times for 3D Finite Element flow analysis.
Rheology at high shear rate for both polymers, semi-crystalline HDPE and amorphous PMMA was studied. Classical rheometry tools, plane-plane dynamic mode rheometer and capillary rheometer, were used in extreme conditions (with channels diameters for capillary rheometry up to 0.3 mm). However, analyses of these results were made following conventional assumptions neglecting instabilities and physical phenomena involved in these flows.We then developed a mathematical model of a capillary flow in to reflect the importance of different physical phenomena that can occur in extreme flows, namely heating, pressure dependency of viscosity, compressibility and the wall slip. The results of the developed model were compared with experimental results.We helped develop a press microinjection original and we tested it with instrumented plate mold with thickness up to 0.2 mm. We have shown that it is possible to make quality parts which are proven by well reproducible pressure, speed and temperature measurements. We used the experimental rheological data in filling phase modeling with the calculation software Rem3D. Correlations between experimental measurements and calculations were carried out by comparing the pressure in the filling system and the cavity.
During a stretch-blow-moulding cycle, the heat conditioning of the preform is a critical step. We propose an optimisation method allowing to automatically compute the best set of process parameters for the infrared oven. Firstly, we couple the Nelder-Mead optimisation algorithm with finite-element simulations of the blowing step performed using ABAQUS®. The goal is to calculate the optimal temperature distribution inside the preform, in order to provide a bottle with a uniform thickness. Secondly, we couple a sequential quadratic programming method with an in-lab software devoted to the 3D simulation of the infrared heating. This method aims to calculate the best set of parameters for the infrared oven. Numerical results of the optimisation have been qualitatively validated by comparison with experimental measurements performed using a semi-industrial blowing machine and a simple-design bottle. The radiative properties of PET are measured using an infrared spectrometer. These properties are used to calculate the spectral radiative absorption of PET. Heating simulations are validated with temperature measurements performed with an infrared camera. We also have developed a sensor in order to measure the thermal contact resistance between the polymer and the mould. In addition, the air flow rate was measured inside the preform, and applied as an input parameter. Then, the air pressure is automatically computed using a thermodynamic model. Experimental measurements, performed on an in-lab blowing machine, are used to validate the blowing kinematics and the thickness distribution of the bottle predicted by the simulations.
Milling a cobalt-based refractory material has been very little studied being proved to be very expensive due to rapid toolwear. The aim of this work was cost optimisation of cutting conditions in slot milling with carbide tools. It used a Kriging interpolation technique of search space that avoids the local optima and decreases the function evaluation toward the global minimum. The objective and constraint functions taken into account were tool cost and surface roughness. Experimental results showed that optimal cutting conditions still led to an important abrasion of the tool, but minimised the adhesive wear at lower cutting speeds.
This paper shows a complete approach to solve a given problem, from the experimentation to the optimization of different cutting parameters. To solve an industrial problem of slotting CoCr29Ni10W7, a Cobalt-based refractory material, we have implemented a design of experiment to determine the effect of cutting parameters on tool life, surface roughness, and cutting forces. After theses trials, an optimization approach has been implemented to find the lowest manufacturing cost while respecting the roughness constraints and cutting force limitation constraints. The optimization approach is based on the Response Surface Method using the Sequential Quadratic programming algorithm and Kriging interpolation for a constrained problem.
This paper shows a complete approach to solve a given problem, from the experimentation to the optimization of different cutting parameters. In response to an industrial problem of slotting FSX 414, a Cobalt-based refractory material, we have implemented a design of experiment to determine the most influent parameters on the tool life, the surface roughness and the cutting forces. After theses trials, an optimization approach has been implemented to find the lowest manufacturing cost while respecting the roughness constraints and cutting force limitation constraints. The optimization approach is based on the Response Surface Method (RSM) using the Sequential Quadratic programming algorithm (SQP) for a constrained problem. To avoid a local optimum and to obtain an accurate solution at low cost, an efficient strategy, which allows improving the RSM accuracy in the vicinity of the global optimum, is presented. With these models and these trials, we could apply and compare our optimization methods in order to get the lowest cost for the best quality, i.e. a satisfying surface roughness and limited cutting forces.
In this work, we aimed to develop an optimum design for a spider die used for the extrusion of high-density polyethylene tubes. For this purpose, a computational-fluid-dynamics-based model using the generalized Newtonian approach was employed to investigate the pressure drop, flow, and temperature uniformity in the die. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011
In the study described in this article, we aimed to apply multiparametric optimization to the processing conditions in a spider die used for the extrusion of polymers, in this case, high-density polyethylene tubes. Product quality is affected by the homogeneity of the flow speed and the temperature at the die exit. Inhomogeneity causes shape distortions and material weak spots (weld lines), and its major cause in spider dies is the discontinuity and distortion of the flow caused by spider legs, the ties by which the inner mandrel of the die is secured to its external casing. The Nelder-Mead nonlinear optimization technique was applied to the numerical model to pinpoint the processing conditions, namely, the inlet pressure, inlet temperature of the melt, temperature of the die walls, and temperature of the spider legs. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012 Copyright © 2012 Wiley Periodicals, Inc.
THE coat-hanger melt distributor is a device commonly used in the wire coating process. Its task is to distribute the melt around the conductor uniformly. It is quite common that materials and flow rates differ from what had been specified during the design procedure. This may lead to bad performance with materials of very different rheological properties from the design material. In this article, we present an optimal design approach to avoid this loss of performances. This approach involves coupling a three-dimensional finite element simulation software with an optimization strategy based on a response surface method. The objective is to determine a coat-hanger melt distributor geometry that ensures a homogeneous exit velocity distribution that will best accommodate for a different range of materials. A coat-hanger melt distributor with a manifold of constant width is designed, and a set of flow distribution measurements is established for two different materials. The results of numerical simulation are then validated by comparison with experimental measurements. The effect of material change is also investigated. POLYM. ENG. SCI., 2009. © 2008 Society of Plastics Engineers
Refractory titanium alloys are often used in aerospace industry for applications requiring high temperature strength and high mechanical resistance. However, the main problems encountered when machining this kind of material are the low material removal rate and the short cutting tool life. Due to their hard nature, an effective manufacturing strategy is required by industrials to optimize cutting parameters and improve machining efficiency. In this work, a new technique of optimization for dry machining refractory titanium alloys has been developed. The optimisation procedure is a new strategy based on both response surface method with “Kriging interpolation” and sequential quadratic programming algorithm. In comparison with previous models taken from the literature, two functions have been formulated in the machining problem to describe the non linear and complex relationship between variables and responses. The first function called “objective function” represents the volume of material removal produced during the cutting tool life and which has to be maximised. The second function called “constraint function” represents the surface roughness and has to be taken below a critical value. Thanks to this new approach, a robust optimization algorithm has been carried out to optimize cutting conditions taking into account productivity and quality.
A modified power law, the so-called Carreau formula, for the viscosity of pseudoplastic fluids in simple shear flow is considered. The appropriateness of introducing this modification is demonstrated by comparative calculations of some simple types of flow, namely the pressure flow through a slit die and the superimposed pressure and drag flow between two parallel plates. These calculations yield substantially different results according to whether the flow curve of the fluid is approximated by the usual power law or by the Carreau formula, which can be more closely matched to the rheological properties.
Subsequently, analytic expressions for the above-mentioned flow types are derived on the basis of the Carreau formula enabling the flow to be computed using a pocket calculator. Moreover, a simplified but still very accurate formula for the throughput characteristic (or equivalently the apparent flow curve) of a slit die and a circular capillary is presented.
The results obtained here are applicable to the design of polymer processing machines and dies as well as to the interpretation of viscosity measurements made with capillaries.
Die design is a daily concern in the manufacturing of extruded profiles. There are very few available Computer Aided Design (CAD) tools for profile die designers and considerable time and resources must be devoted to trial and error design procedures. This paper examines the applicability of the flow analysis network (FAN) to the design of profile extrusion dies. The FAN method is found to provide useful design information at a fraction of the computing cost required for 2.5 and 3D finite element methods. Experimental flow distribution obtained by extruding PVC compounds and polyethylene through simple profile dies are compared to model predictions. The numerical and experimental results provide evidence about the significant contribution of transverse flow to the flow distribution at die exit.
The application of quasi-Newton methods is widespread in numerical optimization. Independently of the application, the techniques used to update the BFGS matrices seem to play an important role in the performance of the overall method. In this paper, we address precisely this issue. We compare two implementations of the limited memory BFGS method for large-scale unconstrained problems. They differ in the updating technique and the choice of initial matrix. L-BFGS performs continuous updating, whereas SNOPT uses a restarted limited memory strategy. Our study shows that continuous updating techniques are more effective, particularly for large problems.
It is of great importance in designing extruders to predict the screw characteristics, that is, the throughput in terms ofthe screw speed and the pressure gradient along the down-channel direction. The screw characteristics depend on the extruder geometry, the operating conditions, and material properties of non-Newto nian fluids. This paper suggests a simple approach to determining the screw characteristics for a three-dimensional flow in a channel with a finite aspect ratio (ratio of a width to a depth, W/H) by introducing a Total Shape Factor (F-t) to correct a two-dimensional flow analysis for a channel with an infinite aspect ratio. In the present study, the Total Shape Factor (F-t) was defined as a ratio of a net flow rate obtained by the three-dimensional analysis to that by the two-dimensional analysis. In the proposed approach, the quantity, partial derivative F-t/partial derivative(H/W), which turns out to be almost constant, offers important information for understanding the effects of the flights. Therefore, partial derivative F-t/partial derivative(H/W) is extensively reported in this paper in terms of several dimensionless parameters. This simple approach with such a database will be very useful for extruder designers to predict screw characteristics.
The dual quadratic programming algorithm of Goldfarb and Idnani is implemented as a solver for a sequential quadratic programming algorithm. Initially the algorithm is briefly described. As the algorithm requires the inverse of the Cholesky factor of the Hessian matrix at each iteration a procedure is presented to directly obtain a matrix that multiplied by its transpose gives the BFGS update of the Hessian. A procedure is then presented to triangularise the updated factor using two series of Givens rotations. In order to increase efficiency a ‘warm start’ strategy is proposed whereby the choice of constraints to enter the active set is based on information of previous SQP iterations. Finally two examples are given to demonstrate the efficiency and robustness of the implementation.
Viscosity, including extensional and shear viscosities, is an important characteristic of flow property and processability for polymeric materials. The flow pattern in a die extrusion of polymer melts is mainly shear flow. The viscosity and the factors affecting it in a steady shear flow of polymer melts has been analyzed in the present paper. On the basis of this analysis, the characteristic of shear viscosity was discussed. The flow properties for several thermoplastic resins including high-density polyethylene (HDPE), low-density polyethylene (LDPE), polypropylene (PP) and high-impact polystyrene (HIPS) were investigated by using a capillary rheometer under relevant test conditions, and a shear viscosity model with four parameters proposed previously was preliminarily verified. The results showed that the predictions of shear viscosity were consistent with the measured data from the experimental of the samples.
In this paper, we adopt the first-order shear deformation theory in the moving least squares differential quadrature (MLSDQ) procedure for predicting the free vibration behavior of moderately thick symmetrically laminated composite plates. The transverse deflection and two rotations of the laminate are independently approximated with the moving least squares (MLS) approximation. The weighting coefficients used in the MLSDQ approximation are obtained through the fast computation of the MLS shape functions and their partial derivatives. The natural frequencies of vibration are computed for various laminated plates and compared with the available published results. Through numerical experiments, the capability and efficiency of the MLSDQ method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined. Effects of the size of support, order of completeness of the basis functions and node irregularity on the numerical accuracy are also investigated.
Shape Memory Alloys (SMA) exhibit remarkable mechanical properties, which permit to contemplate a large number of potential applications as actuators, connectors as well as passive or active damping devices. Although a large body of existing patents is available on SMA materials, only a limited number of industrial applications have been implemented up to now. This is probably due to a series of factors, including the absence of recognized mechanical standards, the relatively high cost of raw material, the high temperature dependence of SMA mechanical properties, the sensitivity to heat treatments and finally, the difficulty to model correctly hysteretic material laws. All this is about to change, thanks to recent research advances in modelling and to the appearance of new and less expensive alloy families. The purpose of this paper is to describe the computer aided design tools developed at École Polytechnique de Montréal to model the mechanical behavior of shape memory materials and to calculate the mechanical response of shape memory devices. A new kind of material law for hysteretic materials can be generated by combining a new phenomenological model based on dual kriging with a micromechanical model. The interface between the material law and finite elements can be improved by using, instead of a Von Mises transformation criterion, a Prager type criterion that is well suited for shape memory alloys. Finally, two examples of industrial applications, a Belleville washer for electrical contacts and a medical stent for non-invasive surgery, demonstrate the usefulness of this approach in the design and optimization of shape memory devices.
A technique for easily treating essential boundary conditions for approximations which are not interpolants, such as moving least-square approximations in the element-free Galerkin method, is presented. The technique employs a string of elements along the essential boundaries and combines the finite element shape functions with the approximation. In the resulting approximation, essential boundary conditions can be treated exactly as in finite elements. It is shown that the resulting approximation can exactly reproduce linear polynomials so that it satisfies the patch test. Numerical studies show that the method retains the high rate of convergence associated with moving least-square approximations.
Global convergence results are established for unconstrained optimization algorithms that utilize a nonmonotone line search procedure. This procedure allows the user to specify a flexible forcing function and includes the nonmonotone Armijo rule, the nonmonotone Goldstein rule, and the nonmonotone Wolfe rule as special cases.
Plans d'Expé Pour Surfaces de Ré, Industries Techniques
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Filì ere Rhéomé ThéFilì ere Plate, Internal Re-port of Nokia Maillefer
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D. Schlä,Filì ere Rhéomé ThéFilì ere Plate, Internal Re-port of Nokia Maillefer, 1997.
Filì ere Rhéométriques ThéorieFilì ere Plate
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D. Schläfli,Filì ere Rhéométriques ThéorieFilì ere Plate, Internal Report of Nokia Maillefer, 1997.
User's Manuals, Version 2-2, Transvalor
REM3D, User's Manuals, Version 2-2, Transvalor, 2004.
Plans d'Expériences Pour Surfaces de Réponse, Industries Techniques
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J. Goupy, Plans d'Expériences Pour Surfaces de Réponse, Industries
Techniques, Paris, 1999.
Filière Rhéométriques Théorie/Filière Plate, Internal Report of Nokia Maillefer
- D Schläfli
D. Schläfli, Filière Rhéométriques Théorie/Filière Plate, Internal Report of Nokia Maillefer, 1997.