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Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT) ; Tomar, Satyendra ; et al in International Journal for Numerical Methods in Engineering (2018) This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution ... [more ▼] This paper presents an approach to generalize the concept of isogeometric analysis (IGA) by allowing different spaces for parameterization of the computational domain and for approximation of the solution field. The method inherits the main advantage of isogeometric analysis, i.e. preserves the original, exact CAD geometry (for example, given by NURBS), but allows pairing it with an approximation space which is more suitable/flexible for analysis, for example, T-splines, LR-splines, (truncated) hierarchical B-splines, and PHT-splines. This generalization offers the advantage of adaptive local refinement without the need to re-parameterize the domain, and therefore without weakening the link with the CAD model. We demonstrate the use of the method with different choices of the geometry and field splines, and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for non-nested spaces. [less ▲] Detailed reference viewed: 229 (8 UL)Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization ; ; et al in Computer Methods in Applied Mechanics and Engineering (2018), 328 In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries. Instead of the computational domain bounded by four B ... [more ▼] In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries. Instead of the computational domain bounded by four B-spline curves, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B´ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B´ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1/G1-continuity constraints on the interface of neighboring Bezier patches with respect to each quad in the quadrangulation, the high-quality Bezier patch parameterization is obtained by a local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach. [less ▲] Detailed reference viewed: 153 (7 UL)Weakening the tight coupling between geometry and simulation in isogeometric analysis Tomar, Satyendra ; ; et al Presentation (2016, June 07) In the standard paradigm of isogeometric analysis, the geometry and the simulation spaces are tightly integrated, i.e. the same non-uniform rational B-splines (NURBS) space, which is used for the geometry ... [more ▼] In the standard paradigm of isogeometric analysis, the geometry and the simulation spaces are tightly integrated, i.e. the same non-uniform rational B-splines (NURBS) space, which is used for the geometry representation of the domain, is employed for the numerical solution of the problem over the domain. However, there are situations where this tight integration is a bane rather than a boon. Such situations arise where, e.g., (1) the geometry of the domain is simple enough to be represented by low order NURBS, whereas the unknown (exact) solution of the problem is sufficiently regular, and thus, the numerical solution can be obtained with improved accuracy by using NURBS of order higher than that required for the geometry, (2) the constraint of using the same space for the geometry and the numerical solution is particularly undesirable, such as in the shape and topology optimization, and (3) the solution of the problem has low regularity but for the curved boundary of the domain one can employ higher order NURBS. Therefore, we propose to weaken this constraint. An extensive study of patch tests on various combinations of polynomial degree, geometry type, and various cases of varying degrees and control variables between the geometry and the numerical solution will be discussed. It will be shown, with concrete reasoning, that why patch test fails in certain cases, and that those cases should be avoided in practice. Thereafter, selective numerical examples will be presented to address some of the above-mentioned situations, and it will be shown that weakening the tight coupling between geometry and simulation offers more flexibility in choosing the numerical solution spaces, and thus, improved accuracy of the numerical solution. [less ▲] Detailed reference viewed: 187 (9 UL)Weakening the tight coupling between geometry and simulation in isogeometric analysis Bordas, Stéphane ; Tomar, Satyendra ; et al Scientific Conference (2016, June 05) In the standard paradigm of isogeometric analysis, the geometry and the simulation spaces are tightly integrated, i.e. the same non-uniform rational B-splines (NURBS) space, which is used for the geometry ... [more ▼] In the standard paradigm of isogeometric analysis, the geometry and the simulation spaces are tightly integrated, i.e. the same non-uniform rational B-splines (NURBS) space, which is used for the geometry representation of the domain, is employed for the numerical solution of the problem over the domain. However, there are situations where this tight integration is a bane rather than a boon. Such situations arise where, e.g., (1) the geometry of the domain is simple enough to be represented by low order NURBS, whereas the unknown (exact) solution of the problem is sufficiently regular, and thus, the numerical solution can be obtained with improved accuracy by using NURBS of order higher than that required for the geometry, (2) the constraint of using the same space for the geometry and the numerical solution is particularly undesirable, such as in the shape and topology optimization, and (3) the solution of the problem has low regularity but for the curved boundary of the domain one can employ higher order NURBS. Therefore, we propose to weaken this constraint. An extensive study of patch tests on various combinations of polynomial degree, geometry type, and various cases of varying degrees and control variables between the geometry and the numerical solution will be discussed. It will be shown, with concrete reasoning, that why patch test fails in certain cases, and that those cases should be avoided in practice. Thereafter, selective numerical examples will be presented to address some of the above-mentioned situations, and it will be shown that weakening the tight coupling between geometry and simulation offers more flexibility in choosing the numerical solution spaces, and thus, improved accuracy of the numerical solution. Powered by [less ▲] Detailed reference viewed: 159 (5 UL)Generalizing the isogeometric concept: weakening the tight coupling between geometry and simulation in IGA Tomar, Satyendra ; ; et al Presentation (2016, June 02) In the standard paradigm of isogeometric analysis [2, 1], the geometry and the simulation spaces are tightly integrated, i.e. the non-uniform rational B-splines (NURBS) space, which is used for the ... [more ▼] In the standard paradigm of isogeometric analysis [2, 1], the geometry and the simulation spaces are tightly integrated, i.e. the non-uniform rational B-splines (NURBS) space, which is used for the geometry representation of the domain, is also employed for the numerical solution of the problem over the domain. However, in certain situations, such as, when the geometry of the domain can be represented by low order NURBS but the numerical solution can be obtained with improved accuracy by using NURBS of order higher than that required for the geometry; or in the shape and topology optimization where the constraint of using the same space for the geometry and the numerical solution is not favorable, this tight coupling is disadvantageous. Therefore, we study the effect of decoupling the spaces for the geometry representation and the numerical solution, though still using the prevalent functions in CAD/CAGD. To begin with, we perform the patch tests on various combinations of polynomial degree, geometry type, and various cases of varying degrees and control variables between the geometry and the numerical solution. This shows that certain cases, perhaps intuitive, should be avoided in practice because patch test fails. The above-mentioned situations are further explored with some numerical examples, which shows that weakening the tight coupling between geometry and simulation offers more flexibility in choosing the numerical solution spaces. [1] J. Cottrell, T.J.R. Hughes, and Y. Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA, volume 80. Wiley, Chichester, 2009. [2] T.J.R. Hughes, J. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005. [less ▲] Detailed reference viewed: 192 (11 UL)Generalizing the isogeometric concept: weakening the tight coupling between geometry and simulation in IGA Bordas, Stéphane ; Tomar, Satyendra ; et al Scientific Conference (2016, May 30) In the standard paradigm of isogeometric analysis [2, 1], the geometry and the simulation spaces are tightly integrated, i.e. the non-uniform rational B-splines (NURBS) space, which is used for the ... [more ▼] In the standard paradigm of isogeometric analysis [2, 1], the geometry and the simulation spaces are tightly integrated, i.e. the non-uniform rational B-splines (NURBS) space, which is used for the geometry representation of the domain, is also employed for the numerical solution of the problem over the domain. However, in certain situations, such as, when the geometry of the domain can be represented by low order NURBS but the numerical solution can be obtained with improved accuracy by using NURBS of order higher than that required for the geometry; or in the shape and topology optimization where the constraint of using the same space for the geometry and the numerical solution is not favorable, this tight coupling is disadvantageous. Therefore, we study the effect of decoupling the spaces for the geometry representation and the numerical solution, though still using the prevalent functions in CAD/CAGD. To begin with, we perform the patch tests on various combinations of polynomial degree, geometry type, and various cases of varying degrees and control variables between the geometry and the numerical solution. This shows that certain cases, perhaps intuitive, should be avoided in practice because patch test fails. The above-mentioned situations are further explored with some numerical examples, which shows that weakening the tight coupling between geometry and simulation offers more flexibility in choosing the numerical solution spaces. [less ▲] Detailed reference viewed: 159 (3 UL)Geometry-Independent Field approximaTion: CAD-Analysis Integration, geometrical exactness and adaptivity ; ; et al in Computer Methods in Applied Mechanics and Engineering (2014) In isogeometric analysis (IGA), the same spline representation is employed for both the geometry of the domain and approximation of the unknown fields over this domain. This identity of the geometry and ... [more ▼] In isogeometric analysis (IGA), the same spline representation is employed for both the geometry of the domain and approximation of the unknown fields over this domain. This identity of the geometry and field approximation spaces was put forward in the now classic 2005 paper [20] as a key advantage on the way to the integration of Computer Aided Design (CAD) and subsequent analysis in Computer Aided Engineering (CAE). [20] claims indeed that any change to the geometry of the domain is automatically inherited by the approximation of the field variables, without requiring the regeneration of the mesh at each change of the domain geometry. Yet, in Finite Element versions of IGA, a parameterization of the interior of the domain must still be constructed, since CAD only provides information about the boundary. The identity of the boundary and field representation decreases the flexibility in which this parameterization can be generated and somewhat constrains the modeling and simulation process, because an approximation able to represent the domain geometry accurately need not be adequate to also approximate the field variables accurately, in particular when the solution is not smooth. We propose here a new paradigm called Geometry-Independent Field approximaTion (GIFT) where the spline spaces used for the geometry and the field variables can be chosen and adapted independently while preserving geometric exactness and tight CAD integration. GIFT has the following features: (1) It is possible to flexibly choose between different spline spaces with different properties to better represent the solution of the problem, e.g. the continuity of the solution field, boundary layers, singularities, whilst retaining geometrical exactness of the domain boundary. (2) For multi-patch analysis, where the domain is composed of several spline patches, the continuity condition between neighboring patches on the solution field can be automatically guaranteed without additional constraints in the variational form. (3) Refinement operations by knot insertion and degree elevation are performed directly on the spline space of the solution field, independently of the spline space of the geometry of the domain, which makes the method versatile. GIFT with PHT-spline solution spaces and NURBS geometries is used to show the effectiveness of the proposed approach. Keywords : Super-parametric methods, Isogeometric analysis (IGA), Geometry-independent Spline Space, PHT-splines, local refinement, adaptivity [less ▲] Detailed reference viewed: 1144 (30 UL)GEOMETRY-INDEPENDENT FIELD APPROXIMATION FOR SPLINE-BASED FINITE ELEMENT METHODS ; ; Bordas, Stéphane in Proceedings of the 11th World Congress in Computational Mechanics (2014, July) We propose a discretization scheme where the spline spaces used for the geometry and the field variables can be chosen independently in spline-based FEM. he method is thus applicable to arbitrary domains ... [more ▼] We propose a discretization scheme where the spline spaces used for the geometry and the field variables can be chosen independently in spline-based FEM. he method is thus applicable to arbitrary domains with spline representation. (2) It is possible to flexibly choose between different spline spaces with different properties to better represent the solution of the PDE, e.g. the continuity of the solution field. (3) Refinement operations by knot insertion and degree elevation are performed directly on the spline space of the solution field, independently of the spline space of the geometry of the domain, i.e. the parameterization of the given geometry is not altered during the refinement process. Hence, the initial design can be optimized in the subsequent shape optimization stage without constraining the geometry discretization space to conform to the field approximation space. [less ▲] Detailed reference viewed: 603 (15 UL) |
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