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Recent Developments in CAD/analysis Integration
, 2012
"... For linear elastic problems, it is wellknown that mesh generation dominates the total analysis time. Different types of methods have been proposed to directly or indirectly alleviate this burden associated with mesh generation. We review in this paper a subset of such methods centred on tighter cou ..."
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For linear elastic problems, it is wellknown that mesh generation dominates the total analysis time. Different types of methods have been proposed to directly or indirectly alleviate this burden associated with mesh generation. We review in this paper a subset of such methods centred on tighter coupling between computer aided design (CAD) and analysis (finite element or boundary element methods). We focus specifically on frameworks which rely on constructing a discretisation directly from the functions used to describe the geometry of the object in CAD. Examples include Bspline subdivision surfaces, isogeometric analysis, NURBSenhanced FEM and parametricbased implicit boundary definitions. We review recent advances in these methods and compare them to other paradigms which also aim at alleviating the burden of mesh generation in computational mechanics.
OPTIMALLY CONVERGENT HIGHORDER XFEM FOR PROBLEMS WITH VOIDS AND INCLUSIONS
"... Abstract. Solution of multiphase problems shows discontinuities across the material interfaces, which are usually weak. Using the eXtended Finite Element Method (XFEM), these problems can be solved even for meshes that do not match the geometry. The basic idea is to enrich the interpolation space ..."
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Abstract. Solution of multiphase problems shows discontinuities across the material interfaces, which are usually weak. Using the eXtended Finite Element Method (XFEM), these problems can be solved even for meshes that do not match the geometry. The basic idea is to enrich the interpolation space by means of a ridge function that is able to reproduce the discontinuity inside the elements. This approach yields excellent results for linear elements, but fails to be optimal if highorder interpolations are used. In this work, we propose a formulation that ensures optimal convergence rates for bimaterial problems. The key idea is to enrich the interpolation using a Heaviside function that allows the solution to represent polynomials on both sides of the interface and, provided the interface is accurately approximated, it yields optimal convergence rates. Although the interpolation is discontinuous, the desired continuity of the solution is imposed modifying the weak form. Moreover, in order to ensure optimal convergence, an accurate description of the interface (which also defines an integration rule for the elements cut by the interface) is needed. Here, we comment on different options that have been successfully used to integrate highorder XFEM elements, and describe a general algorithm based on approximating the interface by piecewise polynomials of the same degree that the interpolation functions. E. SalaLardies, S. FernándezMéndez, and A. Huerta 1
The use of hybrid meshes to improve the efficiency of a discontinuous Galerkin method for the solution of Maxwell’s equation
, 2014
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SUMMARY
, 2013
"... A NURBS Enhanced eXtended Finite Element Approach is proposed for the unfitted simulation of structures defined by means of CAD parametric surfaces. In contrast to classical XFEM that uses levelsets to define the geometry of the computational domain, exact CAD description is considered here. Follow ..."
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A NURBS Enhanced eXtended Finite Element Approach is proposed for the unfitted simulation of structures defined by means of CAD parametric surfaces. In contrast to classical XFEM that uses levelsets to define the geometry of the computational domain, exact CAD description is considered here. Following the ideas developed in the context of the NURBSEnhanced Finite Element Method, NURBSEnhanced subelements are defined to take into account the exact geometry of the interface inside an element. In addition, a highorder approximation is considered to allow for large elements compared to the size of the geometrical details (without loss of accuracy). Finally, a geometrically implicit/explicit approach is proposed for efficiency purpose in the context of fracture mechanics. In this paper, only 2D examples are considered: It is shown that optimal rates of convergence are obtained without the need to consider shape functions defined in the physical space. Moreover, thanks to the flexibility given by the Partition of Unity, it is possible to recover optimal convergence rates in the case of reentrant corners, cracks and embedded material interfaces.
Numerical integration over 2D NURBSshaped domains with applications to NURBSenhanced FEM
"... This paper focuses on the numerical integration of polynomial functions along nonuniform rational Bsplines (NURBS) curves and over 2D NURBSshaped domains, i.e. domains with NURBS boundaries. The integration of the constant function f = 1 is of special interest in computer aided design software a ..."
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This paper focuses on the numerical integration of polynomial functions along nonuniform rational Bsplines (NURBS) curves and over 2D NURBSshaped domains, i.e. domains with NURBS boundaries. The integration of the constant function f = 1 is of special interest in computer aided design software and the integration of very highorder polynomials is a key aspect in the recently proposed NURBSenhanced finite element method (NEFEM). Several wellknown numerical quadratures are compared for the integration of polynomials along NURBS curves, and two transformations for the definition of numerical quadratures in triangles with one edge defined by a trimmed NURBS are proposed, analyzed and compared. When exact integration is feasible, explicit formulas for the selection of the number of integration points are deduced. Numerical examples show the influence of the number of integration points in NEFEM computations.
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"... generation of arbitrary order curved meshes for 3D finite element analysis ..."
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generation of arbitrary order curved meshes for 3D finite element analysis
Exact 3D boundary representation in finite element analysis based on Cartesian grids independent of the geometry
"... Abstract This paper proposes a novel Immersed Boundary Method where the embedded domain is exactly described by using its CAD boundary representation with NURBS or TSplines. The common feature with other immersed methods is that the current approach substantially reduces the burden of mesh generat ..."
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Abstract This paper proposes a novel Immersed Boundary Method where the embedded domain is exactly described by using its CAD boundary representation with NURBS or TSplines. The common feature with other immersed methods is that the current approach substantially reduces the burden of mesh generation. In contrast, the exact boundary representation of the embedded domain allows to overcome the major drawback of existing immersed methods that is the inaccurate representation of the physical domain. A novel approach to perform the numerical integration in the region of the cut elements that is internal to the physical domain is presented and its accuracy and performance evaluated using numerical tests. The applicability, performance and optimal convergence of the proposed methodology is assessed by using numerical examples in three dimensions. It is also shown that the accuracy of the proposed methodology is independent on the CAD technology used to describe the geometry of the embedded domain.
Treatment of nearlysingular problems with the XFEM
"... available at the end of the article In this paper, the behaviour of nonconforming methods is studied in the case of the approximation of nearly singular solutions. Such solutions appear when problems involve singularities whose center are located outside (but close) of the domain of interest. These ..."
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available at the end of the article In this paper, the behaviour of nonconforming methods is studied in the case of the approximation of nearly singular solutions. Such solutions appear when problems involve singularities whose center are located outside (but close) of the domain of interest. These solutions are common in industrial structures that usually involve rounded reentrant corners. If these structures are treated with nonconforming finite element methods such as the XFEM (without any enrichment) or the Finite Cell, it is demonstrated that despite being regular, the convergence of the approximation can be bounded to an algebraic rate that depends on the solution. Reasons for such behaviour are presented, and two complementary strategies are proposed and validated in order to recover optimal convergence rates. The first strategy is based on a proper enrichment of the approximation thanks to the XFEM, while the second is based on a proper mesh design that follows a geometric progression. Performances of these approaches are compared both in 1D and 2D, and enable to recover optimal convergence rates.