The role of the Bézier extraction operator for T‐splines of arbitrary degree: linear dependencies,partition of unity property,nesting behaviour and local refinement

We determine linear dependencies and the partition of unity property of T‐spline meshes of arbitrary degree using the Bézier extraction operator. Local refinement strategies for standard, semi‐standard and non‐standard T‐splines – also by making use of the Bézier extraction operator – are presented for meshes of even and odd polynomial degrees. A technique is presented to determine the nesting between two T‐spline meshes, again exploiting the Bézier extraction operator. Finally, the hierarchical refinement of standard, semi‐standard and non‐standard T‐spline meshes is discussed. This technique utilises the reconstruction operator, which is the inverse of the Bézier extraction operator. Copyright © 2015 John Wiley & Sons, Ltd.     

Isogeometric finite element data structures based on Bézier extraction of T‐splines

We develop finite element data structures for T‐splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so‐called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T‐junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T‐splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.     

Powell–Sabin B‐splines and unstructured standard T‐splines for the solution of the Kirchhoff–Love plate theory exploiting Bézier extraction

The equations that govern Kirchhoff–Love plate theory are solved using quadratic Powell–Sabin B‐splines and unstructured standard T‐splines. Bézier extraction is exploited to make the formulation computationally efficient. Because quadratic Powell–Sabin B‐splines result in ‐continuous shape functions, they are of sufficiently high continuity to capture Kirchhoff–Love plate theory when cast in a weak form. Unlike non‐uniform rational B‐splines (NURBS), which are commonly used in isogeometric analysis, Powell–Sabin B‐splines do not necessarily capture the geometry exactly. However, the fact that they are defined on triangles instead of on quadrilaterals increases their flexibility in meshing and can make them competitive with respect to NURBS, as no bending strip method for joined NURBS patches is needed. This paper further illustrates how unstructured T‐splines can be modified such that they are ‐continuous around extraordinary points, and that the blending functions fulfil the partition of unity property. The performance of quadratic NURBS, unstructured T‐splines, Powell–Sabin B‐splines and NURBS‐to‐NURPS (non‐uniform rational Powell–Sabin B‐splines, which are obtained by a transformation from a NURBS patch) is compared in a study of a circular plate. Copyright © 2015 John Wiley & Sons, Ltd.     

A geometrically exact isogeometric Kirchhoff plate: Feature‐preserving automatic meshing and C1 rational triangular Bézier spline discretizations

The analysis of the Kirchhoff plate is performed using rational Bézier triangles in isogeometric analysis coupled with a feature‐preserving automatic meshing algorithm. Isogeometric analysis employs the same basis function for geometric design as well as for numerical analysis. The proposed approach also features an automatic meshing algorithm that admits localized geometric features (eg, small geometric details and sharp corners) with high resolution. Moreover, the use of rational triangular Bézier splines for domain triangulation significantly increases the flexibility in discretizing spaces bounded by complicated nonuniform rational B‐spline curves. To raise the global continuity to C¹ for the solution of the plate bending problem, Lagrange multipliers are leveraged to impose continuity constraints. The proposed approach also manipulates the control points at domain boundaries in such a way that the geometry is exactly described. A number of numerical examples consisting of static bending and free vibration analysis of thin plates bounded by complicated nonuniform rational B‐spline curves are used to demonstrate the advantage of the proposed approach.     

An isogeometric solid‐like shell element for nonlinear analysis

An isogeometric solid‐like shell formulation is proposed in which B‐spline basis functions are used to construct the mid‐surface of the shell. In combination with a linear Lagrange shape function in the thickness direction, this yields a complete three‐dimensional representation of the shell. The proposed shell element is implemented in a standard finite element code using Bézier extraction. The formulation is verified using different benchmark tests. Copyright © 2013 John Wiley & Sons, Ltd.     

Lagrange extraction and projection for NURBS basis functions: A direct link between isogeometric and standard nodal finite element formulations

We introduce Lagrange extraction and projection that link a C⁰ nodal basis with a smooth B‐spline basis. Our technology is equivalent to Bézier extraction and projection but offers an alternative implementation based on the interpolatory property of nodal basis functions. The Lagrange extraction operator can be constructed by simply evaluating B‐spline basis functions at nodal points and eliminates the need for introducing Bernstein polynomials as new shape functions. The Lagrange projection operator is defined as the inverse of the Lagrange extraction operator and directly relates function values at nodal points to element‐level B‐spline coefficients of a local interpolant. For geometries based on polynomial B‐splines, our technology allows the implementation of isogeometric analysis in standard nodal finite element codes with simple algorithms and minimal intrusion. Copyright © 2016 John Wiley & Sons, Ltd.     

A hybrid variational‐collocation immersed method for fluid‐structure interaction using unstructured T‐splines

We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.     

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

In this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B -bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher-order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B -bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses. The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real-world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison.     

Validity of Lagrange (Bézier) and rational Bézier quads of degree 2

Finite elements of degree two or more are needed to solve various PDE problems. This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. The way in which a Bézier quad patch and a Q2 finite element quad are related is introduced. The two possible quads are discussed, the 9‐node (or complete) quad together with the 8‐node (or Serendipity) quad. A validity condition, the positivity of the Jacobian, is exhibited for these two elements. The discussion continues with a rational Bézier quad patch that can be used as a finite element. Extension to arbitrary degrees is given. Copyright © 2014 John Wiley & Sons, Ltd.     

Converting an unstructured quadrilateral/hexahedral mesh to a rational T-spline

This paper presents a novel method for converting any unstructured quadrilateral or hexahedral mesh to a generalized T-spline surface or solid T-spline, based on the rational T-spline basis functions. Our conversion algorithm consists of two stages: the topology stage and the geometry stage. In the topology stage, the input quadrilateral or hexahedral mesh is taken as the initial T-mesh. To construct a gap-free T-spline, templates are designed for each type of node and applied to elements in the input mesh. In the geometry stage, an efficient surface fitting technique is developed to improve the surface accuracy with sharp feature preservation. The constructed T-spline surface and solid T-spline interpolate every boundary node in the input mesh, with C ²-continuity everywhere except the local region around irregular nodes. Finally, a Bézier extraction technique is developed and linear independence of the constructed T-splines is studied to facilitate T-spline based isogeometric analysis.     

T‐spline finite element method for the analysis of shell structures

A T‐spline surface is a nonuniform rational B‐spline (NURBS) surface with T‐junctions, and is defined by a control grid called T‐mesh. The T‐mesh is similar to a NURBS control mesh except that in a T‐mesh, a row or column of control points is allowed to terminate in the inner parametric space. This property of T‐splines makes local refinement possible. In the present study, shell formulation based on the T‐spline finite element method (FEM) is presented. Shell formulation based on NURBS or T‐splines has fundamental limitations because rotational DOFs, which are necessary in the shell formulation, cannot be defined on control points. In this study, the simple mapping scheme, in which every control point is mapped into one geometric point on the surface, is employed to eliminate the limitations. Using this mapping scheme, T‐spline FEM can be easily extended to the analysis of shells. The proposed shell formulation is verified through various benchmarking problems. This study is a part of the efforts by the authors for the integration of CAD–CAE processes. Copyright © 2009 John Wiley & Sons, Ltd.     

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Extension to nearly incompressible implicit and explicit elastodynamics in finite strains

We present a novel unified finite element framework for performing computationally efficient large strain implicit and explicit elastodynamic simulations using triangular and tetrahedral meshes that can be generated using the existing mesh generators. For the development of a unified framework, we use Bézier triangular and tetrahedral elements that are directly amenable for explicit schemes using lumped mass matrices and employ a mixed displacement-pressure formulation for dealing with the numerical issues arising due to volumetric and shear locking. We demonstrate the accuracy of the proposed scheme by studying several challenging benchmark problems in finite strain elastostatics and nonlinear elastodynamics modelled with nearly incompressible hyperelastic and von Mises elastoplastic material models. We show that Bézier elements, in combination with the mixed formulation, help in developing a simple unified finite element formulation that is accurate, robust, and computationally very efficient for performing a wide variety of challenging nonlinear elastostatic and implicit and explicit elastodynamic simulations.     

Toxicity of some non‐metal ions to activated sludge

Abstract

In this paper a more flexible subdivision theorem for nonuniform B‐spline curves is given; this theorem is then used to derive a theorem for the arbitrary subdivision of Bézier curves.     

Construction of tetrahedral meshes of degree two

There is a need for finite elements of degree two or more to solve various PDE problems. This paper discusses a method to construct such meshes in the case of tetrahedral element of degree two. The first section of this paper returns to Bézier curves, Bézier triangles and then Bézier tetrahedra of degree two. The way in which a Bézier tetrahedron and a P2 finite element tetrahedron are related is introduced. A validity condition is then exhibited. Extension to arbitrary degree and dimension is given. A construction method is then proposed and demonstrated by means of various concrete application examples. Copyright © 2012 John Wiley & Sons, Ltd.     

Subdivision of the Bézier curve

Any segment between two points on a Bézier curve is itself a Bézier curve whose Bézier polygon is expressed explicitly in terms of the sides of the Bézier polygon associated with the original curve.     

Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions

Adaptive local refinement is one of the main issues for isogeometric analysis (IGA). In this paper, an adaptive extended IGA (XIGA) approach based on polynomial splines over hierarchical T‐meshes (PHT‐splines) for modeling crack propagation is presented. The PHT‐splines overcome certain limitations of nonuniform rational B‐splines–based formulations; in particular, they make local refinements feasible. To drive the adaptive mesh refinement, we present a recovery‐based error estimator for the proposed method. The method is based on the XIGA method, in which discontinuous enrichment functions are added to the IGA approximation and this method does not require remeshing as the cracks grow. In addition, crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors under linear elastic fracture mechanics. The proposed method has been used to analyze numerical examples, and the stress intensity factors results were compared with reference results. The findings demonstrate the accuracy and efficiency of the proposed method.     

Shape optimal design of structures: an efficient shape representation concept

This paper describes a shape representation concept that may be used in general shape optimization procedures. It relies on the assumption that the finite element mesh is defined as a convective mesh following automatically the shape changes of a conveniently parameterized body. Suitable parameterization of the body is achieved by combining the design element technique and a convenient design element. The design element is defined as a rational Bézier body. It represents a general‐purpose design element that may serve as the geometrical data provider for the response and sensitivity analysis for virtually any finite element type. Practical implementation of the proposed approach for space frame structures is discussed in detail. The validity of the proposed approach as well as the use of a body‐like design elements are illustrated by three numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.