The problem of developing an adaptive isogeometric method aigm for solving elliptic secondorder partial differential equations with truncated hierarchical bsplines of arbitrary degree and different order of continuity is addressed. We consider an adaptive algorithm for finite element methods for the isogeometric analysis igafem of elliptic possibly nonsymmetric secondorder partial differential equations in arbitrary. The different representations in finite element analysis and isogeometric analysis of. The adaptive isogeometric method is a numerical method that combines concepts of isogeometric analysis, itself a hybrid research. Mesh refinement strategies for the adaptive isogeometric method.
In order to obtain reliable results nonetheless, adaptive mesh refinement utilizing. Isogeometric analysis is an approximation method exploiting the existing geometrical representation of the cad model. Mesh refinement strategies for the adaptive isogeometric. We develop a fast solver for the fractional di erential equation fdes involving riesz fractional derivative. Multilevel bezier extraction for hierarchical local refinement. By following 10,12,11, optimal convergence rates of the aigm can be proved when suitable approximation classes are considered. Adaptive isogeometric analysis with hierarchical splines has already been studied using re. Isogeometric analysis is a computational approach that offers the possibility of integrating finite element analysis fea into conventional nurbsbased cad design tools. The role of bezier extraction in adaptive isogeometric.
Theory, implementation, and practice november 9, 2010 springer. Adaptive isogeometric analysis with hierarchical box splines. This contribution presents bezier extraction of truncated hierarchical bsplines. However, the lack of satisfaction of the isoparametric concept led to theoretical questions which were addressed in later versions 46 t. Siam journal on numerical analysis society for industrial. Adaptive finite element methods for optimal contr ol pr oblems karin kraft c karin kraft, 2008 no 2008. This survey presents an uptodate discussion of adaptive.
Pdf bezier extraction and adaptive refinement of truncated. This work presents a local finite element enrichment technique based on isogeometric. Read goaladaptive isogeometric analysis with hierarchical splines, computer methods in applied mechanics and engineering on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. We consider an adaptive algorithm for finite element methods for the isogeometric analysis igafem of elliptic possibly nonsymmetric second. We consider an adaptive isogeometric method aigm based on truncated hierarchical bsplines and present the study of its numerical properties. Goaladaptive isogeometric analysis with hierarchical. A hierarchical approach to adaptive local refinement in. The proposed methods are applied to the solution of exemplary elliptic problems of poissons equation and a prototypical freesurface. Mathematical analysis of variational isogeometric methods.
Isogeometric finite elements combine the numerical solution of partial differential equations and the description of the computational domain given by rational splines from computer aided geometric design. Suitably graded thbspline refinement and coarsening. Thereby a hierarchical approach is adapted to the numerical requirements and the relevant theoretical properties of the basis are ensured. Cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations, computer methods in applied mechanics and engineering on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. It is based on the use of hierarchical matrices hmatrices. In the context of the fcm, the bspline version is introduced, which applies highorder and highcontinuity bspline bases within the fcm concept. Pdf the p and bspline versions of the geometrically. The substantial idea behind isogeometric methods is the use of the same geometry representation throughout the whole engineering process. Note that the complexity analysis of the refinement algorithms is a fundamental ingredient to prove the optimality of hierarchical isogeometric methods 6,7.
Surely argyris in germany and england, and martin and clough in. The adaptivity analysis holds in any space dimensions. We compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector, the cost of direct and iterative solvers, the accuracy versus degrees of freedom and the accuracy versus computing time. Siebert and andreas veeser abstractthisisasurveyonthetheoryofadaptive.
Isogeometric analysis, just like fem, is a method for solving partial differential equations and we use the same models introduced in sec. This paper deals with an adaptive finite element method originally developed by prof. On this basis, we show that isogeometric collocation has the potential to increase the. The thesis at hand addresses two recently introduced concepts intended to support an efficient interaction between geometrical models and finite element analysis. Isogeometric finite element enrichment for problems dominated by. We prove that our aigm is optimal in the sense tha. Isogeometric analysis iga bridges the gap between computer aided geometric design cad and finite element analysis fea. In this paper we present numerical simulations of soil plasticity using isogeometric analysis comparing the results to the solutions from conventional finite element method. It is based on the use of hierarchical matrices hmatrices for the. In this chapter we will present the method called isogeometric analysis based on the previous sections about finite element analysis and computer aided geometric design. Adaptive finite element methods afem is welldeveloped today.
However, to the best of our knowledge, the thorough mathematical analysis of adaptive isogeometric finite element methods igafem is so far restricted to hierarchical splines bg16,bg17,ghp17. This thesis explains and discusses several mesh re. We consider an adaptive isogeometric method aigm based on truncated hierarchical bsplines and continue the study of its numerical properties. Refinement algorithms for adaptive isogeometric methods with. Adaptive hierarchical isogeometric finite element methods core. Isogeometric finite element analysis of nonlinear structural. Currently, it is necessary to convert data between cad and fea packages to analyse new designs during development, a difficult task since the two computational geometric approaches are diffe.
In this paper, we investigate the extension of the adaptive algorithm for isogeometric analysis performed with bspline basis functions. Primer of adaptive finite element methods, in multiscale and adaptivity. Adaptive hierarchical isogeometric finite element methods isogeometric finite elements combine the numerical solution of partial differential equations and the description of the computational domain given by rational splines from computer aided geometric design. The full text of this article hosted at is unavailable due to technical difficulties. In this contribution, an adaptive isogeometric analysis framework. Computer methods in applied mechanics and engineering 364, 112925. Pdf adaptive isogeometric analysis with hierarchical box. Existence and uniqueness again follows by the laxmilgram theorem. Ices report 1205 an isogeometric designthroughanalysis. Siam journal on numerical analysis siam society for. Mar 01, 2014 read goal adaptive isogeometric analysis with hierarchical splines, computer methods in applied mechanics and engineering on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Evans, isogeometric triangular bernsteinbezier discretizations. Isogeometric analysis is a numerical method that uses nonuniform rational bsplines nurbs as basis functions instead of the lagrangian polynomials often used in the finite element method. Adaptive isogeometric analysis with hierarchical box.
Moreover, standard procedures of adaptive finite element analysis for error. Goaladaptive isogeometric analysis with hierarchical splines. On this basis, we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods, when a specified level of accuracy is to be achieved with minimum computational cost. Computer methods in applied mechanics and engineering, vol. Sorry, we are unable to provide the full text but you may find it at the following locations. We present 2 adaptive refinement techniques, namely, adaptive local refinement and adaptive hierarchical refinement, for isogeometric analysis. A nearoptimal hierarchical estimate based adaptive finite element method for obstacle problems.
The baker group uses adaptive finite element methods to solve problems in continuum electrostatics and diffu. It thus bridges the gap between numerical analysis and geometry, and moreover it allows to tackle new cutting edge applications at the frontiers of research in. The classical isoparametric approach of the finite element method. Experience is the traditional method of determining whether or not the mesh and basis will be optimal or even adequate for the analysis at hand. This work gives a wellfounded introduction to this topic and then extends isogeometric. However, though being well understood and available in various academic codes. Integrated structural analysis using isogeometric finite element. Isogeometric finite elements combine the numerical solution of partial. Adaptive isogeometric analysis on twodimensional trimmed. An isogeometric designthroughanalysis methodology based on adaptive hierarchical refinement of nurbs, immersed boundary methods and tspline cad surfaces, computer methods in applied mechanics and engineering, 249252, pp. Adaptive finite element method for fractional differential equations using hierarchical matrices xuan zhao y, xiaozhe huz, wei caix, and george em karniadakisabstract. Automatic mesh generation and geometrically exact finite element analysis. Sep 14, 2017 we present 2 adaptive refinement techniques, namely, adaptive local refinement and adaptive hierarchical refinement, for isogeometric analysis.
Adaptive hierarchical isogeometric finite element methods. Since the approximation functions are used as the same as the functions representing the geometry bsplines, the method. The important output of our analysis is the definition of classes of admissibility for meshes underlying hierarchical splines and the design of an optimal adaptive strategy based on. Development of a finite model is considered as a design problem similar to structural optimization problems. Natural hierarchical refinement for finite element methods. Application of the isogeometric concept on the immersed. Projection and transfer operators in adaptive isogeometric analysis with hierarchical bsplines. An isogeometric designthroughanalysis methodology based on adaptive hierarchical refinement of nurbs, immersed boundary methods, and tspline cad surfaces 5a. Isogeometric analysis of the dual boundary element method. Isogeometric analysis iga is a recently established paradigm based on finite element analysis fea that replaces standard simple building blocks in geometry and solution space with more complex and geometricallyoriented compounds coming from computeraided design cad, see. Download citation adaptive hierarchical isogeometric finite element methods during the last chapter several aspects of isogeometric analysis were. Section 5 describes the appr oach of dual weighted residuals to an optimal contr ol pr oblem and includes the description of the adaptive nite element method.
These results are in line with the estimates obtained in the context of adaptive finite element methods 21,22. We then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis. Adaptive finite element methods in flow computations. Isogeometric analysis is a groundbreaking computational approach that promises the possibility of integrating the finite element method into conventional splinebased cad design tools. This paper is concerned with an introduction of a concept of adaptive grid design for finite element analysis by combining numerical gridgeneration methods and adaptive finite element methods. Adaptive finite element methods for optimal control problems. The err or estimate which is used for the adaptive method is also given. Analysissuitability, bezier extraction, and application as an adaptive basis for isogeometric analysis. This concept bases on adaptive hierarchical refinement of bsplines, immersed boundary methods and tspline computer aided design cad surfaces. Domain decomposition methods in science and engineering xix, 317324.
In order to overcome the gap between computeraided design, numerical simulation and manufacturing, isogeometric analysis iga was introduced by hughes et al. Jul 14, 2006 2020 adaptive isogeometric analysis on twodimensional trimmed domains based on a hierarchical approach. The research community has made important steps towards the ambitious. This work gives a wellfounded introduction to this topic and then extends isogeometric finite elements by a local refinement technique, which is essential for an efficient adaptive simulation. Major steps of adaptive finite element the usual finite element analysis would proceed from the selection of a mesh and basis to the generation of a solution to an accuracy appraisal and analysis. Currently, it is necessary to convert data between cad and fea packages to analyse new designs during development, a difficult task since the two computational geometric approaches are different.
Isogeometric analysis, adaptivity, hierarchical splines, thbsplines. Adaptive isogeometric methods with hierarchical splines. Adaptive isogeometric analysis with hierarchical box splines article pdf available in computer methods in applied mechanics and engineering 316. Both problems are posed on 2d domains, yet the methodology is formulated in general dimen. Refinement algorithms for adaptive isogeometric methods.
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