3D NURBS‐enhanced finite element method (NEFEM)

This paper presents the extension of the recently proposed NURBS‐enhanced finite element method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the computational domain by means of its CAD boundary representation with non‐uniform rational B‐splines (NURBS) surfaces. Specific strategies for interpolation and numerical integration are presented for those elements affected by the NURBS boundary representation. For elements not intersecting the boundary, a standard finite element rationale is used, preserving the efficiency of the classical FEM. In 3D NEFEM special attention must be paid to geometric issues that are easily treated in the 2D implementation. Several numerical examples show the performance and benefits of NEFEM compared with standard isoparametric or cartesian finite elements. NEFEM is a powerful strategy to efficiently treat curved boundaries and it avoids excessive mesh refinement to capture small geometric features. Copyright © 2011 John Wiley & Sons, Ltd.     

Comparison of high‐order curved finite elements

Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points on the accuracy of the computation is also studied. Two‐dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBS‐enhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element method (FEM). Moreover, NEFEM outperforms Cartesian FEM and p‐FEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Quality improvement of non‐manifold hexahedral meshes for critical feature determination of microstructure materials

This paper describes a novel approach to improve the quality of non‐manifold hexahedral meshes with feature preservation for microstructure materials. In earlier works, we developed an octree‐based isocontouring method to construct unstructured hexahedral meshes for domains with multiple materials by introducing the notion of material change edge to identify the interface between two or more materials. However, quality improvement of non‐manifold hexahedral meshes is still a challenge. In the present algorithm, all the vertices are categorized into seven groups, and then a comprehensive method based on pillowing, geometric flow and optimization techniques is developed for mesh quality improvement. The shrink set in the modified pillowing technique is defined automatically as the boundary of each material region with the exception of local non‐manifolds. In the relaxation‐based smoothing process, non‐manifold points are identified and fixed. Planar boundary curves and interior spatial curves are distinguished, and then regularized using B‐spline interpolation and resampling. Grain boundary surface patches and interior vertices are improved as well. Finally, the optimization method eliminates negative Jacobians of all the vertices. We have applied our algorithms to two beta titanium data sets, and the constructed meshes are validated via a statistics study. Finite element analysis of the 92‐grain titanium is carried out based on the improved mesh, and compared with the direct voxel‐to‐element technique. Copyright © 2009 John Wiley & Sons, Ltd.     

Clustered generalized finite element methods for mesh unrefinement,non‐matching and invalid meshes

In spite of significant advancements in automatic mesh generation during the past decade, the construction of quality finite element discretizations on complex three‐dimensional domains is still a difficult and time demanding task. In this paper, the partition of unity framework used in the generalized finite element method (GFEM) is exploited to create a very robust and flexible method capable of using meshes that are unacceptable for the finite element method, while retaining its accuracy and computational efficiency. This is accomplished not by changing the mesh but instead by clustering groups of nodes and elements. The clusters define a modified finite element partition of unity that is constant over part of the clusters. This so‐called clustered partition of unity is then enriched to the desired order using the framework of the GFEM. The proposed generalized finite element method can correctly and efficiently deal with: (i) elements with negative Jacobian; (ii) excessively fine meshes created by automatic mesh generators; (iii) meshes consisting of several sub‐domains with non‐matching interfaces. Under such relaxed requirements for an acceptable mesh, and for correctly defined geometries, today's automated tetrahedral mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user. A detailed technical discussion of the proposed generalized finite element method with clustering along with numerical experiments and some implementation details are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupling of non‐conforming meshes in a component mode synthesis method

A common mesh refinement‐based coupling technique is embedded into a component mode synthesis method, Craig–Bampton. More specifically, a common mesh is generated between the non‐conforming interfaces of the coupled structures, and the compatibility constraints are enforced on that mesh via L2‐minimization. This new integrated method is suitable for structural dynamic analysis problems where the substructures may have non‐conforming curvilinear and/or surface interface meshes. That is, coupled substructures may have different element types such as shell, solid, and/or beam elements. The proposed method is implemented into a commercial finite element software, B2000++, and its demonstration is carried out using an academic and industry oriented test problems. Copyright © 2013 John Wiley & Sons, Ltd.     

The generation of arbitrary order curved meshes for 3D finite element analysis

A procedure for generating curved meshes, suitable for high-order finite element analysis, is described. The strategy adopted is based upon curving a generated initial mesh with planar edges and faces by using a linear elasticity analogy. The analogy employs boundary loads that ensure that nodes representing curved boundaries lie on the true surface. Several examples, in both two and three dimensions, illustrate the performance of the proposed approach, with the quality of the generated meshes being analysed in terms of a distortion measure. The examples chosen involve geometries of particular interest to the computational fluid dynamics community, including anisotropic meshes for complex three dimensional configurations.     

Unconstrained plastering—Hexahedral mesh generation via advancing‐front geometry decomposition

The generation of all‐hexahedral finite element meshes has been an area of ongoing research for the past two decades and remains an open problem. Unconstrained plastering is a new method for generating all‐hexahedral finite element meshes on arbitrary volumetric geometries. Starting from an unmeshed volume boundary, unconstrained plastering generates the interior mesh topology without the constraints of a pre‐defined boundary mesh. Using advancing fronts, unconstrained plastering forms partially defined hexahedral dual sheets by decomposing the geometry into simple shapes, each of which can be meshed with simple meshing primitives. By breaking from the tradition of previous advancing‐front algorithms, which start from pre‐meshed boundary surfaces, unconstrained plastering demonstrates that for the tested geometries, high quality, boundary aligned, orientation insensitive, all‐hexahedral meshes can be generated automatically without pre‐meshing the boundary. Examples are given for meshes from both solid mechanics and geotechnical applications. Copyright © 2009 John Wiley & Sons, Ltd.     

An adaptive remeshing procedure for discontinuous finite element limit analysis

An adaptive remeshing procedure is proposed for discontinuous finite element limit analysis. The procedure proceeds by iteratively adjusting the element sizes in the mesh to distribute local errors uniformly over the domain. To facilitate the redefinition of element sizes in the new mesh, the interelements discontinuous field of elemental bound gaps is converted into a continuous field, ie, the intensity of bound gap, using a patch‐based approximation technique. An analogous technique is subsequently used for the approximation of element sizes in the old mesh. With these information, an optimized distribution of element sizes in the new mesh is defined and then scaled to match the total number of elements specified for each iteration in the adaptive remeshing process. Finally, a new mesh is generated using the advancing front technique. This adaptive remeshing procedure is repeated several times until an optimal mesh is found. Additionally, for problems involving discontinuous boundary loads, a novel algorithm for the generation of fan‐type meshes around singular points is proposed explicitly and incorporated into the main adaptive remeshing procedure. To demonstrate the feasibility of our proposed method, some classical examples extracted from the existing literary works are studied in detail.     

Volumetric coupling approaches for multiphysics simulations on non‐matching meshes

In finite element analysis of volume coupled multiphysics, different meshes for the involved physical fields are often highly desirable in terms of solution accuracy and computational costs. We present a general methodology for volumetric coupling of different meshes within a monolithic solution scheme. A straightforward collocation approach is compared to a mortar‐based method for nodal information transfer. For the latter, dual shape functions based on the biorthogonality concept are used to build the projection matrices, thus further reducing the evaluation costs. We give a detailed explanation of the integration scheme and the construction of dual shape functions for general first‐order and second‐order Langrangian finite elements within the mortar method, as well as an analysis of the conservation properties of the projection operators. Moreover, possible incompatibilities due to different geometric approximations of curved boundaries are discussed. Numerical examples demonstrate the flexibility of the presented mortar approach for arbitrary finite element combinations in two and three dimensions and its applicability to different multiphysics coupling scenarios. Copyright © 2016 John Wiley & Sons, Ltd.     

The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids

Isogeometric analysis has recently become very popular for the numerical modeling of structures and fluids. Among other potential features, advantages of using a non‐uniform rational B‐splines (NURBS)‐based isogeometric analysis over the traditional finite element method include the possibility of using higher‐order polynomials for the basis functions of the approximation space, which may be easily built on a recursive (hierarchical) fashion as well as higher convergence ratio. Nevertheless, NURBS‐based isogeometric analysis suffers from the same problems depicted by other methods when it comes to reproduce isochoric deformations, that is, it shows volumetric locking, especially for low‐order basis functions. Similar remedies as those that have been proposed for the finite element method may be appropriate for integration in the NURBS‐based isogeometric analysis and some have already been tried with success. In this work, the analysis of the underlying space of incompressible deformations of a NURBS‐based isogeometric approximation is performed with the main objective of understanding the likelihood of volumetric locking. As a remedy, the enhanced assumed strain methodology is blended with the NURBS‐based isogeometric analysis to alleviate the volumetric locking associated with incompressible deformations. The solution includes a stabilization term derived directly from a penalized form of the classical Veubeke–Hu–Washizu three‐field variational principle. Copyright © 2012 John Wiley & Sons, Ltd.     

A fixed‐mesh finite element thermally coupled flow formulation for the numerical analysis of melting processes

In the framework of the finite element method, a temperature‐based thermally coupled flow formulation including phase‐change effects is proposed to study melting processes. The governing equations of the problem, written in terms of its primitive variables, are solved using a generalized streamline operator technique that enables the use of equal interpolation functions for the unknowns: velocity, pressure and temperature. Moreover, a unique fixed finite element mesh is used to avoid the difficulties related to moving meshes. This methodology is applied and assessed in the numerical analysis of a benchmark problem known as the melting process of gallium in a differentially heated recipient using distinct geometric aspect ratios. Copyright © 2001 John Wiley & Sons, Ltd.     

An isogeometric extension of Trefftz method for elastostatics in two dimensions

In this paper, an approach to blend the Hybrid‐Trefftz Finite Element Method (HTFEM) and the Isogeometric Analysis (IGA) called the Isogeometric Trefftz (IGAT) method is presented. The structure of the isogeometric extension of the Trefftz method is formally the same as for its conventional counterpart, except the approximation of the boundary displacements and geometry that are carried out using the Non‐Uniform Rational B‐Splines (NURBS) instead of polynomials. In other words, only the element boundaries are approximated using NURBS basis while the Trefftz approximation is used in the interior of the elements. For that reason, IGAT can be ranked alongside recently developed Isogeometric Boundary Element Method (IGABEM), the NURBS‐Enhanced Finite Element Method (NEFEM), the Isogeometric Local Maximum Entropy (IGA‐LME) method, and the Isogeometrically enhanced Scaled‐Boundary element method (SBFEM), which all use NURBS approximation at the domain boundary only. Theoretical conjectures made in this paper are accompanied by three examples that show that IGAT leads to excellent results using only a few elements.     

The generation of hexahedral meshes for assembly geometry: survey and progress

The finite element method is being used today to model component assemblies in a wide variety of application areas, including structural mechanics, fluid simulations, and others. Generating hexahedral meshes for these assemblies usually requires the use of geometry decomposition, with different meshing algorithms applied to different regions. While the primary motivation for this approach remains the lack of an automatic, reliable all‐hexahedral meshing algorithm, requirements in mesh quality and mesh configuration for typical analyses are also factors. For these reasons, this approach is also sometimes required when producing other types of unstructured meshes. This paper will review progress to date in automating many parts of the hex meshing process, which has halved the time to produce all‐hex meshes for large assemblies. Particular issues which have been exposed due to this progress will also be discussed, along with their applicability to the general unstructured meshing problem. Published in 2001 by John Wiley & Sons, Ltd.     

An approach to refining three-dimensional tetrahedral meshes based on Delaunay transformations

A technique for refining three-dimensional tetrahedral meshes is proposed in this paper. The proposed technique is capable of treating arbitrary unstructured tetrahedral meshes, convex or non-convex with multiple regions resulting in high quality constrained Delaunay triangulations. The tetrahedra generated are of high quality (nearly equilateral). Sliver tetrahedra, which present a real problem to many algorithms are not produced with the new method. The key to the generation of high quality tetrahedra is the iterative application of a set of topological transformations based on the Voronoi–Delaunay theory and a reposition of nodes technique. The computational requirements of the proposed technique are in linear relationship with the number of nodes and tetrahedra, making it ideal for direct employment in a fully automatic finite element analysis system for 3-D adaptive mesh refinement. Application to some test problems is presented to show the effectiveness and applicability of the new method.     

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

A framework to validate and generate curved nodal high‐order meshes on Computer‐Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin‐shell and 3D finite element analysis with unstructured high‐order methods. First, we define a distortion (quality) measure for high‐order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high‐order), and handles with low‐quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high‐order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes

We present a robust method for generating high‐order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high‐order nodes, second, displacing the boundary nodes to ensure that they are on the computer‐aided design surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high‐order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching, and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high‐order finite element analyses. Copyright © 2015 John Wiley & Sons, Ltd.     

A parallel implicit/explicit hybrid time domain method for computational electromagnetics

The numerical solution of Maxwell's curl equations in the time domain is achieved by combining an unstructured mesh finite element algorithm with a cartesian finite difference method. The practical problem area selected to illustrate the application of the approach is the simulation of three‐dimensional electromagnetic wave scattering. The scattering obstacle and the free space region immediately adjacent to it are discretized using an unstructured mesh of linear tetrahedral elements. The remainder of the computational domain is filled with a regular cartesian mesh. These two meshes are overlapped to create a hybrid mesh for the numerical solution. On the cartesian mesh, an explicit finite difference method is adopted and an implicit/explicit finite element formulation is employed on the unstructured mesh. This approach ensures that computational efficiency is maintained if, for any reason, the generated unstructured mesh contains elements of a size much smaller than that required for accurate wave propagation. A perfectly matched layer is added at the artificial far field boundary, created by the truncation of the physical domain prior to the numerical solution. The complete solution approach is parallelized, to enable large‐scale simulations to be effectively performed. Examples are included to demonstrate the numerical performance that can be achieved. Copyright © 2009 John Wiley & Sons, Ltd.     

Fully automatic two dimensional mesh generation using normal offsetting

A technique, based on a normal offsetting procedure, for the fully automatic generation of two dimensional meshes suitable for finite element analysis is presented. The method positions nodes by first meshing the geometric entities that compose the object boundary, then offsetting those nodal locations along vectors normal to the boundary geometry. The offset row of nodes is processed to ensure a good nodal spacing appropriate for generating well shaped elements. Following processing, the new row is offset again and the cycle is repeated until the entire area is filled with nodes. The boundary based technique ensures good quality element shapes for analysis in critical boundary regions and facilitates applications involving integration of mesh generation with design geometry databases. Nodal locations are calculated based on local parameters avoiding the higher order time complexities associated with global calculations. A technique for controlling mesh density by overlaying an independent mesh density function on the geometry is also presented as part of the method. This approach allows mesh density to be automatically controlled by a variety of factors, such as previous analysis results, that are external to the actual mesh generation process. The independent nature of the function method allows different sources of density information to be used interchangeably without modification to the mesh generation procedure.     

A refined global-local finite element analysis method

A refined global-local method was proposed to improve the efficiency of finite element analysis. The proposed method was based on the regular finite element method in conjunction with three basic step, i.e. the global analysis, the local analysis and the refined global analysis. In the first two steps, a coarse finite element mesh was used to analyse the entire structure to obtain the nodal displacements which were subsequently used as displacement boundary conditions for local regions of interest. These local regions with the prescribed boundary conditions were then analysed with refined meshes to obtain more accurate stresses. In the third step, a new global displacement distribution based on the results of the previous two steps was assumed for the analysis, from which much improved solutions for both stresses and displacements were produced. Numerical examples showed that the proposed method yielded accurate solutions with significant savings in computing time compared with the regular finite element method. Further, this method is suitable for parallel computation.