Relevant bibliographies by topics / B-splines à plusieurs variables

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'B-splines à plusieurs variables'

Author: Grafiati
Published: 4 June 2021
Last updated: 11 February 2022

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Journal articles on the topic "B-splines à plusieurs variables"

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Simon, Thomas. "Produit Beta-Gamma et régularité du signe." Studia Scientiarum Mathematicarum Hungarica 51, no. 4 (2014): 429–53. http://dx.doi.org/10.1556/sscmath.51.2014.4.1280.

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On étudie la positivité complète du noyau de convolution multiplicatif T associé au produit de deux variables aléatoires indépendantes B(a, b) et Γ(c). Ce noyau T est complétement positif d’ordre infini si b ∈ ℕ* ou si d = a + b − c ∈ ℕ. Dans les autres cas la régularité du signe de T a toujours un ordre fini, qui est ici calculé. Plus précisément, pour tout n ≧ 1 on montre que T est complètement positif d’ordre n + 1 si et seulement si (d, b) est situé au dessus d’un certain escalier \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{E}_n$$ \end{document} dessiné dans le demi-plan supérieur. Cet escalier caractérise aussi la constance du signe de plusieurs déterminants associés á la fonction hypergéométrique confluente de seconde espèce.
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Vincent, Stephanie, Richard Planells, Catherine Defoort, et al. "Genetic polymorphisms and lipoprotein responses to diets." Proceedings of the Nutrition Society 61, no. 4 (2002): 427–34. http://dx.doi.org/10.1079/pns2002177.

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RésuméAlors que l'alimentation de l'homme a beaucoup évoluée depuis ses origines, le génome humain est resté très stable. Pourtant, de très nombreux gènes ont des polymorphismes connus. En fait, on considère maintenant que les principales pathologies humaines (maladies cardiovasculaires, diabète, obésité et cancers) résultent d'une interaction entre des facteurs de succeptibilité génétique et des facteurs de l'environement, dont l'alimentation. Dans le domaine du métabolisme des lipoprotéines et des maladies cardiovasculaires, des polymorphimes de plusieurs gènes ont été identifiés et associés aux niveaux des paramètres lipidiques ou aà des réponses variables aux régimes, comme pour les apoprotéines (apo) E, B, A-IV et C-III, le LDL récepteur, la protéine microsomiale de transport (MTP), la protéine de liason des acides gras (FABP), la protéine de transport des esters de cholésterol (CETP), la lipoprotéine lipase ou la lipase hepatique. Nous réalisons une étude d'intervention à Marseille dans le but d'étudier l'interaction
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Wright, John, Anne Lecours, Claude Duchesne, and Stéphane Sabourin. "Le profil psychosocial de couples consultant à une clinique de fertilité." Santé mentale au Québec 15, no. 2 (2006): 89–106. http://dx.doi.org/10.7202/031564ar.

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Résumé L'objectif de la présente étude est de tracer le profil psychosocial de couples consultant à une clinique de fertilité. Plus spécifiquement, il s'agit (a) de comparer le statut psychosocial de sujets consultant à une clinique de fertilité à celui de sujets normaux ou d'individus souffrant de problèmes psychologiques, (b) de déterminer s'il existe des différences de profil entre les hommes et les femmes consultant pour un problème de fertilité" et (c) d'évaluer s'il est possible de prédire la nature du profil psychosocial des sujets à partir de diverses caractéristiques sociodémographiques et médicales. L'échantillon se compose de 30 couples qui consultent un spécialiste de la clinique de fertilité d'un hôpital de la région de Montréal. L'analyse des résultats démontre que le profil psychosocial des sujets consultant à une clinique de fertilité se situe à mi-chemin entre celui de sujets normaux et celui d'individus souffrant de problèmes psychologiques. Le statut psychologique des hommes et des femmes consultant à une clinique de fertilité diffère sur trois variables: dépression, estime de soi et stress. Enfin, les analyses de régression multiple ont permis d'identifier plusieurs caractéristiques médicales reliées à l'ampleur des difficultés psychosociales rapportées par les sujets infertiles. Le profil psychosocial de couples consultant à une clinique de fertilité
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Sana, Madiha, and Muhammad Mustahsan. "Finite Element Approximation of Optimal Control Problem with Weighted Extended B-Splines." Mathematics 7, no. 5 (2019): 452. http://dx.doi.org/10.3390/math7050452.

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In this research article, an optimal control problem (OCP) with boundary observations is approximated using finite element method (FEM) with weighted extended B-splines (WEB-splines) as basis functions. This type of OCP has a distinct aspect that the boundary observations are outward normal derivatives of state variables, which decrease the regularity of solution. A meshless FEM is proposed using WEB-splines, defined on the usual grid over the domain, R 2 . The weighted extended B-spline method (WEB method) absorbs the regularity problem as the degree of the B-splines is increased. Convergence analysis is also performed by some numerical examples.
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YEN, JOHN, REZA LANGARI, and LIANG WANG. "PRINCIPAL COMPONENTS, B-SPLINES, AND FUZZY SYSTEM REDUCTION." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 04, no. 06 (1996): 561–72. http://dx.doi.org/10.1142/s0218488596000317.

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This paper proposes to use the method of principal components to reduce the dimensionality of input space and the B-splines to represent the membership functions of input variables. A model reduction strategy, which is based on Johansen’s optimality theorem, also suggested. The utility of this approach is illustrated using a fuzzy system modeling example.
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Rahayu, Putri Indi, and Pardomuan Robinson Sihombing. "PENERAPAN REGRESI NONPARAMETRIK KERNEL DAN SPLINE DALAM MEMODELKAN RETURN ON ASSET (ROA) BANK SYARIAH DI INDONESIA." JURNAL MATEMATIKA MURNI DAN TERAPAN EPSILON 14, no. 2 (2021): 115. http://dx.doi.org/10.20527/epsilon.v14i2.2968.

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Sharia Bank Return On Assets (ROA) modeling in Indonesia in 2018 aims to analyze the relationship pattern of Retturn On Assets (ROA) with interest rates. The analysis that is often used for modeling is regression analysis. Regression analysis is divided into two, namely parametric and nonparametric. The most commonly used nonparametric regression methods are kernel and spline regression. In this study, the nonparametric regression used was kernel regression with the Nadaraya-Watson (NWE) estimator and Local Polynomial (LPE) estimator, while the spline regression was smoothing spline and B-splines. The fitting curve results show that the best model is the B-splines regression model with a degree of 3 and the number of knots 5. This is because the B-splines regression model has a smooth curve and more closely follows the distribution of data compared to other regression curves. The B-splines regression model has a determination coefficient of R ^ 2 of 74.92%,%, meaning that the amount of variation in the ROA variable described by the B-splines regression model is 74.92%, while the remaining 25.8% is explained by other variables not included in the model.
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Gu, Sheng Li, and Fu Ming Wang. "Evolutionary Structural Shape Optimization Based on Adaptive FEM and Boundary Representation of B-Spline." Advanced Materials Research 562-564 (August 2012): 1575–82. http://dx.doi.org/10.4028/www.scientific.net/amr.562-564.1575.

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This paper presents a structural shape optimization algorithm based on the evolutionary structural optimization (ESO) method in conjunction with element error estimate and adaptive FEM. B-splines are used to describe the boundary of the design domain; the shape of these B-splines is governed by a set of master nodes which can be taken as the design variables. The optimal shape of the design boundary with constant stress is achieved iteratively by the movement and update of the position of the master nodes based on nodal stress leveling. The result quality, in terms of accuracy and efficiency, is tested and discussed with an analytical solution.
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Zhang, Dai Yuan. "An Improved Algorithm Using B-Spline Weight Functions for Training Feedforward Neural Networks." Applied Mechanics and Materials 278-280 (January 2013): 1301–4. http://dx.doi.org/10.4028/www.scientific.net/amm.278-280.1301.

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An improved algorithm using B-splines as weight functions for training neural networks is proposed. There is no need for training neural networks or solving linear equations. The most important advantage is that we can get the forms of weight functions by the given patterns directly. Each of weight function is a one-variable function and takes one associated input point (input neuron) as its argument. The form of each weight function is a linear combination of some B-splines defined on the sets of given input variables (input knots or input patterns), whose coefficients are associated with the given output patterns. Some examples are presented to illustrate good performance of the new algorithm.
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Faezian, Ghassem, Ahmad Darabi, and Nader Sargolzaei. "Customized shape optimization of switched reluctance motor using B-Splines." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 40, no. 3 (2021): 402–13. http://dx.doi.org/10.1108/compel-07-2020-0250.

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Purpose This study aims to design the rotor geometry of switched reluctance motor (SRM) in a completely flexible way. In the proposed method, there is no default geometry for the rotor. The initial geometry of the rotor can start from a circle or any other shape and depending on the required performance takes the final shape during the optimal design. In this way, the best performance, possible with geometric design, can be achieved. Design/methodology/approach The rotor boundary of a 4/2 SRM is defined by a few B-splines. Some control points are located around the rotor and changing their locations causes customized changes in the rotor boundary. Locations of these points are defined as design variables. A 2-D finite element analysis using MATLAB/PDE is applied to the SRM model and sensitivity analysis is used to optimization design by means of minimizing of objective function. Findings The proposed method has many more capabilities for matching different objective functions. For the suggested objective function, while the conventional rotor torque profile difference with the desired torque profile reaches 40%, this difference for B-spline rotor is about 17%. Experimental results from a prototype motor have a close agreement with analysis results. Originality/value The B-splines have been used to design machines and electromagnetic devices. However, this method is used for the first time in design of the whole rotor of a SRM.
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Lara-Ramirez, Jose Edgar, Carlos Hugo Garcia-Capulin, Maria de Jesus Estudillo-Ayala, Juan Gabriel Avina-Cervantes, Raul Enrique Sanchez-Yanez, and Horacio Rostro-Gonzalez. "Parallel Hierarchical Genetic Algorithm for Scattered Data Fitting through B-Splines." Applied Sciences 9, no. 11 (2019): 2336. http://dx.doi.org/10.3390/app9112336.

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Curve fitting to unorganized data points is a very challenging problem that arises in a wide variety of scientific and engineering applications. Given a set of scattered and noisy data points, the goal is to construct a curve that corresponds to the best estimate of the unknown underlying relationship between two variables. Although many papers have addressed the problem, this remains very challenging. In this paper we propose to solve the curve fitting problem to noisy scattered data using a parallel hierarchical genetic algorithm and B-splines. We use a novel hierarchical structure to represent both the model structure and the model parameters. The best B-spline model is searched using bi-objective fitness function. As a result, our method determines the number and locations of the knots, and the B-spline coefficients simultaneously and automatically. In addition, to accelerate the estimation of B-spline parameters the algorithm is implemented with two levels of parallelism, taking advantages of the new hardware platforms. Finally, to validate our approach, we fitted curves from scattered noisy points and results were compared through numerical simulations with several methods, which are widely used in fitting tasks. Results show a better performance on the reference methods.
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Dissertations / Theses on the topic "B-splines à plusieurs variables"

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Gormaz, Arancibia Raul. "Floraisons polynomiales : applications à l'étude des B-splines à plusieurs variables." Phd thesis, Grenoble 1, 1993. http://tel.archives-ouvertes.fr/tel-00342512.

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Les courbes de Bezier et les courbes splines ont trouve un cadre de présentation simple et naturel avec la notion de floraison d'une fonction polynomiale, telle qu'elle a été présentée dans les travaux de Lyle Ramshaw (1987). Notre but a consiste a étendre cette présentation au cas des surfaces et aussi des variétés de dimension supérieure. Les splines simpliciales sont une généralisation naturelle des b-splines au cas de plusieurs variables. Nous présentons leurs principales propriétés ainsi qu'une définition de différences divisées pour fonctions de plusieurs variables. Un algorithme d'évaluation d'une spline simpliciale est propose et teste. Floraisons et splines simpliciales sont les éléments essentiels d'un nouveau schéma de b-splines introduit par Dahmen, Micchelli et Seidel (1992). Ce schéma est étudié et ses principales propriétés sont présentées. Une grande similarité avec l'étude des courbes est retrouvée
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Gormaz, Arancibia Raul Laurent Pierre Jean. "Floraisons polynomiales applications à l'étude des B-splines à plusieurs variables /." S.l. : Université Grenoble 1, 2008. http://tel.archives-ouvertes.fr/tel-00342512.

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BOUHAMIDI, ABERRAHMAN. "Interpolation et approximation par des fonctions splines radiales a plusieurs variables." Nantes, 1992. http://www.theses.fr/1992NANT2019.

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Nous presentons une generalisation des fonctions splines plaques minces introduites par duchon, pour l'interpolation de lagrange. Utilisant des operateurs differentiels plus generaux, et les noyaux reproduisants associes, il est possible de construire des splines minimisant des normes correspondant par exemple a des splines sous tension, ou bien permettant une interpolation non plus seulement de lagrange, mais de type hermite-birkhoff a plusieurs variables. Comme autre exemple de telles fonctions radiales, on peut montrer que les multiquadriques de hardy sont un cas particulier de splines de duchon. Nous donnons une autre vue des courbes splines sous tension, que nous considerons comme un cas particulier des fonctions splines radiales associees a un noyau reproduisant dans un espace de hilbert dont l'operateur differentiel associe depend d'un parametre de tension. Des essais numeriques permettent de montrer comment ces diverses splines peuvent etre utilisees pour modeler des formes, deformer des surfaces plaques minces existantes, et en particulier mettre en evidence l'effet de la tension. Enfin, nous presentons une etude sur le calcul d'energie de minimisation, et nous donnons un algorithme recursif permettant de calculer cette energie en fonction des coefficients de bernstein-bezier
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Wiesenfarth, Manuel. "Estimation and Inference in Special Nonparametric Models with Applications to Topics in Development Economics." Doctoral thesis, 2012. http://hdl.handle.net/11858/00-1735-0000-000D-F09E-B.

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Book chapters on the topic "B-splines à plusieurs variables"

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Kokainis, Martins, and Svetlana Asmuss. "Higher Degree F-transforms Based on B-splines of Two Variables." In Information Processing and Management of Uncertainty in Knowledge-Based Systems. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-40596-4_54.

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Conference papers on the topic "B-splines à plusieurs variables"

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Wang, Mingming, and Xiaoping Qian. "Efficient Filtering in Topology Optimization via B-Splines." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-34712.

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This paper presents a B-spline based approach for topology optimization of three-dimensional (3D) problems where the density representation is based on B-splines. Compared with the usual density filter in topology optimization, the new B-spline based density representation approach is advantageous in both memory usage and CPU time. This is achieved through the use of tensor-product form of B-splines. As such, the storage of the filtered density variables is linear with respect to the effective filter size instead of the cubic order as in the usual density filter. Numerical examples of 3D topology optimization of minimal compliance and heat conduction problems are demonstrated. We further reveal that our B-spline based density representation resolves the bottleneck challenge in multiple density per element optimization scheme where the storage of filtering weights had been prohibitively expensive.
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Wessels, Francois J. L., G. Venter, and T. W. Von Backström. "An Efficient Scheme for Describing Airfoils Using Non-Uniform Rational B-Splines." In ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/gt2012-69468.

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In the well developed field of airfoil design there are a number of established schemes for describing airfoils, each with its own limitations and suitability to a particular application. Although less parsimonious, B-splines are often used to find solutions unconstrained by the characteristics of these schemes. They have proven useful to improve on existing designs and are useful in a numerical optimization process. As part of a numerical design project of a small wind turbine, a parametric scheme using Non-Uniform Rational B-Splines (NURBS) offering intuitive control over airfoil shape and unambiguous definitions for chord length and angle of attack is needed. These features are achieved at the lowest number of variables possible to ensure numerical optimization efficiency. Two previously uninvestigated NURBS arrangements are defined and investigated. The quality of the resulting airfoil reproductions are geometrically and aero-dynamically investigated and based on the results, the use of a thickness-camber scheme is motivated.
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Turner, Cameron J., and Richard H. Crawford. "Adapting Non-Uniform Rational B-Spline Fitting Approaches to Metamodeling." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85544.

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Modern engineering design requires the exploration of complex nonlinear design spaces that relate large numbers of design variables and performance criteria. Effective exploration and decision making requires the availability of efficient representations of high dimensional design spaces. Techniques from the field of metamodeling, which builds models of models, are often used to represent complex design relationships. We have developed an alternative, spline-based metamodeling technique that is amenable to adaptive representation of high dimensional spaces and facilitates optimization-based exploration of such spaces. This paper describes our technique for fitting Non-Uniform Rational B-splines (NURBs) to points in a design space generated from simulations or experiments. The algorithm is illustrated with examples from metamodeling and nonlinear optimization literature. The results indicate that our method achieves an average global correlation of more than 99% and an average maximum local RMS error of less than 3% of full scale.
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Rivas, Carlos E., Paul E. Barbone, and Assad A. Oberai. "A Stabilized B-Splines FEM Formulation for the Solution of an Inverse Elasticity Problem Arising in Medical Imaging." In ASME 2008 International Mechanical Engineering Congress and Exposition. ASMEDC, 2008. http://dx.doi.org/10.1115/imece2008-66700.

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Soft tissue pathologies are often associated with changes in mechanical properties. For example, breast and other tumors usually present as stiff lumps. Imaging the spatial distribution of the mechanical properties of tissues thus reveals information of diagnostic value. Doing so, however, typically requires the solution of an inverse elasticity problem. In this work we consider the inverse elasticity problem for an incompressible material in plane stress, formulated and solved as a constrained optimization problem. We formulate this inverse problem enforcing high order continuity for our variables. Driven by the requirements for the strong and weak solutions to this problem, we assume that our data field (i.e. the measured displacement) is in H2 and our parameter distribution (i.e. the sought shear modulus distribution) is in H1. This high order regularity requirement for the data is incompatible with standard FEM. We solve this problem using a FEM formulation that is novel in two respects. First, we employ quadratic b-splines that enforce C1 continuity in our displacement field, consistent with the variational requirements of the continuous problem. Second, we include Galerkin-least-squares (GLS) stabilization in the iterative optimization formulation. GLS adds consistent stability to the discrete formulation that otherwise violates an ellipticity condition that is satisfied by the continuous problem. Computational examples validate this formulation and demonstrate numerical convergence with mesh refinement.
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Arefeen, Asif, and Yujiang Xiang. "Two-Dimensional Team Lifting Prediction With Different Box Weights." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22115.

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Abstract A novel multibody dynamics modeling method is proposed for two-dimensional (2D) team lifting prediction. The box itself is modeled as a floating-base rigid body in Denavit-Hartenberg representation. The interactions between humans and box are modeled as a set of grasping forces which are treated as unknowns (design variables) in the optimization formulation. An inverse-dynamics-based optimization method is used to simulate the team lifting motion where the dynamic effort of two humans is minimized subjected to physical and task-based constraints. The design variables are control points of cubic B-splines of joint angle profiles of two humans and the box, and the grasping forces between humans and the box. Two numerical examples are successfully simulated with different box weights (20 Kg and 30 Kg, respectively). The humans’ joint angle, torque, ground reaction force, and grasping force profiles are reported. The joint angle profiles are validated with the experimental data.
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Steuben, John, and Cameron J. Turner. "Robust Optimization Exploration Using NURBs-Based Metamodeling Techniques." In ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/detc2010-28226.

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Non-uniform rational B-splines (NURBs) demonstrate properties that are highly conductive to performing metamodeling for engineering design purposes. Previous research has resulted in the development of algorithms capable of fitting NURBs metamodels to design spaces of many input variables and performance indicies, and performing various discreet optimizations upon these metamodels. In the present research we expand upon this basis by illustrating the development of robust optimization algorithms that leverages the unique properties of NURBs metamodels. This optimization is conducted in a general fashion by considering both optimality and various robustness metrics as global or local model properties, and illustrates the tradeoffs between them using a novel graphical approach. The appeal of this approach is demonstrated by a series of test functions of one performance index and one or two performance indicies. A case study in designing composite structures of specific stiffness, of four and five design variables, follows. We proceed to discuss the future of NURBs metamodeling techniques and the potential for considering model properties besides optimality and robustness during optimization.
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Yu, Deqi, Xiaojun Zhang, Jiandao Yang, et al. "An Optimization Design Platform for Fir-Tree Root and Groove for Steam Turbine." In ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/gt2018-76595.

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Fir-tree root and groove profiles are widely used in gas turbine and steam turbine. Normally, the fir-tree root and groove are characterized with straight line, arc or even elliptic fillet and splines, then the parameters of these features were defined as design variables to perform root profile optimization. In ultra-long blades of CCPP and nuclear steam turbines and high-speed blades of industrial steam turbine blades, both the root and groove strength are the key challenges during the design process. Especially, in industrial steam turbines, the geometry of blade is very small but the operation velocity is very high and the blade suffers stress concentration severely. In this paper, two methods for geometry configuration and relevant optimization programs are described. The first one is feature-based using straight lines and arcs to configure the fir-tree root and groove geometry and genetic algorithm for optimization. This method is quite fit for wholly new root and groove design. And the second local optimization method is based on B-splines to configure the geometry where the local stress concentration occurs and the relevant optimization algorithm is used for optimization. Also, several cases are studied as comparison by using the optimization design platform. It can be used not only in steam turbines but also in gas turbines.
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Randell, David, Yanyun Wu, Philip Jonathan, and Kevin Ewans. "Modelling Covariate Effects in Extremes of Storm Severity on the Australian North West Shelf." In ASME 2013 32nd International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/omae2013-10187.

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Careful modelling of covariate effects is critical to reliable specification of design criteria. We present a spline based methodology to incorporate spatial, directional, temporal and other covariate effects in extreme value models for environmental variables such as storm severity. For storm peak significant wave height events, the approach uses quantile regression to estimate a suitable extremal threshold, a Poisson process model for the rate of occurrence of threshold exceedances, and a generalised Pareto model for size of threshold. Multidimensional covariate effects are incorporated at each stage using penalised tensor products of B-splines to give smooth model parameter variation as a function of multiple covariates. Optimal smoothing penalties are selected using cross-validation, and model uncertainty is quantified using a bootstrap resampling procedure. The method is applied to estimate return values for a large spatial neighbourhood of locations off the North West Shelf of Australia, incorporating spatial and directional effects.
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Jauregui, Juan C., Diego Cardenas, Hugo Elizalde, and Oliver Probst. "Dynamic Modelling of Blades Based on a Novel Curved Thin Walled Beam Theory." In ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/gt2018-76016.

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There are several Thin-Walled Beam models for straight beams, but few TWB models consider beams with arbitrary curvatures. Although, a curved beam can be modelled using finite elements, the number of degrees of freedom is too large and a nonlinear dynamic solution is very cumbersome, if not impossible. In this work, a general description of arbitrary three-dimensional curves, based on the Frenet-Serret field frame, is applied to determine the dynamic stresses in wing turbines blades. The dynamic model is developed using the Isogeometric Analysis (IGA) and the in plane and out-of-plane curvature’s gradients are found in an Euler-type formulation, allowing the treatment of cases with highly-curved geometry. An Isogeometrical (IGA) formulation relies on a linear combination of Non-Uniform Rational B-Splines (NURBS) to represent not just the model’s geometry, a standard practice in most Computer-Aided Design (CAD) platforms, but also the unknown solution field of each sought variable. For the unified model hitherto described, these variables are represented by a NURBS curve.
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Dennis, Brian H., George S. Dulikravich, and Zhen-Xue Han. "Constrained Shape Optimization of Airfoil Cascades Using a Navier-Stokes Solver and a Genetic/SQP Algorithm." In ASME 1999 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/99-gt-441.

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The objective in this aerodynamic shape design effort is to minimize total pressure loss across the two-dimensional linear airfoil cascade row while satisfying a number of constraints. They included fixed axial chord, total torque, inlet and exit flow angles, and blade cross-section area, while maintaining thickness distribution greater than a minimum specified value. The aerodynamic shape optimization can be performed by using any available flow-field analysis code. For the analysis of the performance of intermediate cascade shapes we used an unstructured grid based compressible Navier-Stokes flow-field analysis code with k-e turbulence model. A robust genetic optimization algorithm was used for optimization and a constrained sequential quadratic programming was used enforcement of certain constraints. The airfoil geometry was parameterized using conic section parameters and B-splines thus keeping the number of geometric design variables to a minimum while achieving a high degree of geometric flexibility and robustness. Significant reductions of the total pressure loss were achieved using this constrained method for a supersonic exit flow axial turbine cascade.
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