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.     

h‐Adaptive FE methods applied to single‐ and multiphase problems

A major problem in using the finite element method for solving numerous engineering problems in the framework of single‐ and multiphase materials is the assessment of discretization errors and the design of suitable meshes. To overcome this problem, adaptive finite element methods have been developed. Based on the error indicator by Zienkiewicz and Zhu, it is the goal of the present paper to present a new error indicator which is especially designed for multiphase problems. Furthermore, efficient h‐adaptive strategies concerning both the generation of new meshes in the framework of independent and hierarchical remeshing strategies and the data transfer between old and new meshes are pointed out. Finally, numerical examples are given to exhibit the efficiency and the quality of the presented h‐adaptive methods and to compare the different strategies to each other. Copyright © 2002 John Wiley & Sons, Ltd.     

An automatic adaptive refinement finite element procedure for 2D elastostatic analysis

An automatic adaptive refinement procedure for finite element analysis is presented. The procedure is applied to two-dimensional elastostatic problems to obtain solutions of prescribed accuracy. Through the combined use of new mesh generator using contour developed by Lo¹ and the concept of strain energy concentration, high-quality graded finite element meshes are generated. The whole process is fully automatic and no user intervention is required during the successive cycles of the mesh refinements. The Zienkiewicz and Zhu² error estimator is found to be effective and has been adopted for the present implementation. In the numerical examples tested, the error estimator gives an accurate error norm estimation and the effectivity index of the estimator converges to a value close to unity.     

A locally anisotropic fluid–structure interaction remeshing strategy for thin structures with application to a hinged rigid leaflet

An immersed finite element fluid–structure interaction algorithm with an anisotropic remeshing strategy for thin rigid structures is presented in two dimensions. One specific feature of the algorithm consists of remeshing only the fluid elements that are cut by the solid such that they fit the solid geometry. This approach allows to keep the initial (given) fluid mesh during the entire simulation while remeshing is performed locally. Furthermore, constraints between the fluid and the solid may be directly enforced with both an essential treatment and elements allowing the stress to be discontinuous across the structure. Remeshed elements may be strongly anisotropic. Classical interpolation schemes – inf–sup stable on isotropic meshes – may be unstable on anisotropic ones. We specifically focus on a proper finite element pair choice. As for the time advancing of the fluid–structure interaction solver, we perform a geometrical linearization with a sequential solution of fluid and structure in a backward Euler framework. Using the proposed methodology, we extensively address the motion of a hinged rigid leaflet. Numerical tests demonstrate that some finite element pairs are inf–sup unstable with our algorithm, in particular with a discontinuous pressure. Copyright © 2015 John Wiley & Sons, Ltd.     

Selective refinement: A new strategy for automatic node placement in graded triangular meshes

Automating triangular finite element mesh generation involves two interrelated tasks: generatine a distribution of well-placed nodes on the boundary and in the interior of a domain, and constructing a triangulation of these nodes. For a given distribution of nodes, the Delaunay triangulation generally provides a suitable mesh, and Watson's algorithm²⁶ provides a flexible means of constructing it. In this paper, a new method is described for automating node placement in a Delaunay triangulation by seieclive refinement of an initial triangulation. Grading of the mesh is controlled by an explicit or implicit node spacing function. Although this paper describes the technique only in the planar context, the method generalizes to three dimensions as well.     

A predictor-corrector contouring algorithm for isoparametric 3D elements

This paper describes a new contouring algorithm for isoparametric elements which can be used to map three-dimensional scalar fields. The contours are generated on arbitrary planes intersecting finite element structures. Tracing element contour lines is accomplished by an accurate predictor-corrector technique. A method of finding starting points for the algorithm on the boundary of the elements is also given.     

Hierarchical tree‐based finite element mesh generation

Hierarchical grid generation and its use as a basis for finite element mesh generation are considered in this paper. The hierarchical grids are generated by recursive subdivision using quadtrees in two dimensions and octrees in three dimensions. A numbering system for efficient storage of the quadtree grid information is examined, tree traversal techniques are devised for neighbour finding, and accurate boundary representation is considered. It is found that hierarchical grids are straightforward to generate from sets of seeding points which lie along domain boundaries. Quadtree grids are triangularized to provide finite element meshes in two dimensions. Three‐dimensional tetrahedral meshes are generated from octree grids. The meshes can be generated automatically to model complicated geometries with highly irregular boundaries and can be adapted readily at moving boundaries. Examples are given of two‐ and three‐dimensional hierarchical tree‐based finite element meshes and their application to modelling free surface waves. Copyright © 1999 John Wiley & Sons, Ltd.     

An extended finite element method approach for structural‐acoustic problems involving immersed structures at arbitrary positions

Noise reduction for passengers' comfort in transport industry is now an important constraint to be taken into account during the design process. This process involves to study several configurations of the structures immersed in a given acoustic cavity in the context of an optimization, uncertainty, or reliability study for instance. The finite element method may be used to model this coupled fluid–structure problem but needs an interface conforming mesh for each studied configuration that may become time consuming. This work aims at avoiding this remeshing step by using noncompatible meshes between the fluid and the structures. The immersed structures are supposed to be thin shells and are localized in the fluid domain by a signed distance level‐set. To take into account the pressure discontinuity from one side of the structures to the other one, the fluid pressure approximation is enriched according to the structures positions by a Heaviside function using a partition of unity strategy (extended finite element method). The same fluid mesh of the empty cavity is then used during the whole parametric study. The method is implemented for a three‐dimensional fluid and tested on academic examples before being applied to an industrial‐like case. Copyright © 2012 John Wiley & Sons, Ltd.     

On continuous,discontinuous, mixed,and primal hybrid finite element methods for second‐order elliptic problems

Finite element formulations for second‐order elliptic problems, including the classic H¹‐conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three‐dimensional test problem with smooth solution, the simulations are performed with h‐refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two‐dimensional problem, the results are for approximation spaces based on given sets of hp‐refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L²‐convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.     

A dynamic finite element preanalyser

In a mixed-mode dynamic crack propagation problem using remeshing, certain elements in the neighbour-hood of the propagating crack-tips may have dimensions that do not comform to the allowable limit. Hence, they need to be refined by recursively bisecting the exclusive sides until a global conformity is achieved. ADFEP (a dynamic finite element preanalyser) has been written for this purpose. Apart from this objective, ADFEP has a wide spectrum of salient applications to various discretization related problems, such as automatic irregular generation, adaptive refinement and others. A simple energy criterion to determine the cut-off frequency of a transient loading, and the concept of preprocessing the input loading using digital filters introduced by Valliappan and Murti¹ are employed. They constitute the first module of ADFEP. The second (main) module consists of element splitting procedures where algorithms based on graph theory and element splitting templates have been extensively used. The notion of a ‘cross digraph’ introduced to carry a multiple ‘weights’ and to avoid duplication in node-generation, is especially useful in three-dimensional element splitting techniques. Finally, numerical inverse isoparametric mapping techniques presented by Murti and Valliappan² are also used to interpolate the nodal quantity vectors at the newly generated nodes. These vectors are needed as initial values in subsequent iterations for a dynamic restart. This aspect, together with the node renumbering scheme, is included in the third module of ADFEP. Some illustrative examples are included to elucidate the effectiveness of ADFEP.     

Automatic mesh generation from solid models based on recursive spatial decompositions

This paper introduces a two-stage algorithm for the automatic conversion of solid models into finite element meshes. In Stage 1 the solid is approximated by a collection of variably sized cells generated by recursive spatial decomposition and stored in a logical tree. In Stage 2 the approximating cell structure, which includes cells that are wholly inside the solid (IN) as well as cells that are neither inside nor outside (NIO), is transformed into a finite element model. IN cells are directly mapped into finite elements while NIO cells are decomposed according to their topologically complexity through either template mapping or recursive element extraction. Although specifically designed for adaptive remeshing, the algorithm is of general use and can be implemented in any Solid Modelling System that supports Boolean operations on solids and maintains a complete boundary representation. Core algorithms for Stages i and 2 are rigorously developed to insure their applicability within a genuinely automatic procedure. Specific issues related to boundary evaluation and decomposition procedures are identified and discussed. The implementation of the algorithm into an experimental system based on the PADL-2 solid modeller is described. The paper concludes with a comparative study of existing meshing algorithms based on recursive spatial decompositions.     

Subdivision schemes for smooth contact surfaces of arbitrary mesh topology in 3D

This paper presents a strategy to parameterize contact surfaces of arbitrary mesh topology in 3D with at least C¹‐continuity for both quadrilateral and triangular meshes. In the regular mesh domain, four quadrilaterals or six triangles meet in one node, even C²‐continuity is attained. Therefore, we use subdivision surfaces, for which non‐physical pressure jumps are avoided for contact interactions. They are usually present when the contact kinematics is based on facet elements discretizing the interacting bodies. The properties of subdivision surfaces give rise to basically four different implementation strategies. Each strategy has specific features and requires more or less efforts for an implementation in a finite element program. One strategy is superior with respect to the others in the sense that it does not use nodal degrees of freedom of the finite element mesh at the contact surface. Instead, it directly uses the degrees of freedom of the smooth surface. Thereby, remarkably, it does not require an interpolation. We show how the proposed method can be used to parameterize adaptively refined meshes with hanging nodes. This is essential when dealing with finite element models whose geometry is generated by means of subdivision techniques. Three numerical 3D problems demonstrate the improved accuracy, robustness and performance of the proposed method over facet‐based contact surfaces. In particular, the third problem, adopted from biomechanics, shows the advantages when designing complex contact surfaces by means of subdivision techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

An updated Lagrangian finite element sensitivity analysis of large deformations using quadrilateral elements

A continuum parameter and shape sensitivity analysis is presented for metal forming processes using the finite element method. The sensitivity problem is posed in a novel updated Lagrangian framework as suitable for very large deformations when remeshing operations are performed during the analysis. In addition to exploring the issue of transfer of variables between meshes for finite deformation analysis, the complex problem of transfer of design sensitivities (derivatives) between meshes for large deformation inelastic analyses is also discussed. A method is proposed that is shown to give accurate estimates of design sensitivities when remeshing operations are performed during the analysis. Sensitivity analysis for the consistent finite element treatment of near incompressibility within the context of the assumed strain methods is also proposed. In particular, the performance of four‐noded quadrilateral elements for the sensitivity analysis of large deformations is studied. The results of the continuum sensitivity analysis are validated by a comparison with those obtained by a finite difference approximation (i.e. using the solution of a perturbed deformation problem). The effectiveness of the method is demonstrated by applications in the design optimization of metal forming processes. Copyright © 2001 John Wiley & Sons, Ltd.     

Intersecting and trimming parametric meshes on finite element shells

We present an algorithm for intersecting finite element meshes defined on parametric surface patches. The intersection curves are modelled precisely and both meshes are adjusted to the newly formed borders, without unwanted reparametrizations. The algorithm is part of an interactive shell modelling program that has been used in the design of large offshore oil structures. To achieve good interactive response, we represent meshes with a topological data structure that stores its entities in spatial indexing trees instead of linear lists. These trees speed up the intersection computations required to determine points of the trimming curves; moreover, when combined with the topological information, they allow remeshing using only local queries. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient isostatic mixed shell element for coarse mesh solution

A novel mixed shell finite element (FE) is presented. The element is obtained from the Hellinger–Reissner variational principle and it is based on an elastic solution of the generalized stress field, which is ruled by the minimum number of variables. As such, the new FE is isostatic because the number of stress parameters is equal to the number of kinematical parameters minus the number of rigid body motions. We name this new FE MISS-8. MISS-8 has generalized displacements and rotations interpolated along its contour and drilling rotation is also considered as degree of freedom. The element is integrated exactly on its contour, it does not suffer from rank defectiveness and it is locking-free. Furthermore, it is efficient for recovering both stress and displacement fields when coarse meshes are used. The numerical investigation on its performance confirms the suitability, accuracy, and efficiency to recover elastic solutions of thick- and thin-walled beam-like structures. Numerical results obtained with the proposed FE are also compared with those obtained with isogeometric high-performance solutions. Finally, numerical results show a rate of convergence between h² and h⁴ .     

Assessment of discretized errors and adaptive refinement with quadrilateral finite elements

This paper studies discretized errors, and their estimation in conjunction with quadrilateral finite element meshes which are generated by the intelligent mesh generator XFORMQ.¹ The exact energy error is used to evaluate the distortion effect of the quadrilateral mesh. The Zienkiewicz–Zhu² error estimate and actaptive procedure are applied to the short cantilever and the square plate problems using the quadrilateral mesh generator XFORMQ. It is shown that the multistage quadrilateral element refinement produces results superior to the triangular element refinement in the test cases.     

Parallel terrain rendering using a cluster of computers

This article presents a distributed parallel processing technique for rendering massive terrain using a cluster of machines consisting of one designated rendering node and 20 computing nodes. With a novel approach, the presented technique achieves an increase in rendering speed and an improvement in rendering capability. Adaptive terrain mesh constructions are done in parallel at computing nodes and the resulting meshes are combined and subsequently rendered at the rendering node. This study uses a height field of the United States at 30-m resolution spacing. It is divided into smaller blocks consisting of 4096?×?4096 vertices. Each computing node is assigned one or four blocks and tasked with creating the level-of-detail mesh that corresponds to view-dependent parameters provided by the rendering node. These individual terrain meshes are subsequently combined and rendered as seamless terrain meshes with a continuous terrain surface. The high rendering capacity of the presented technique is essential to the high-resolution large display system.     

FINITE ELEMENT REMESHING: A METAL FORMING APPROACH FOR QUADRILATERAL MESH GENERATION AND REFINEMENT

The paper presents an automatic finite element remeshing system for quadrilateral elements consisting of modules for mesh generation, densification, smoothing and interpolation of field variables. The mesh generator takes into account the contour of the old mesh, eventual interference with dies and the plastic deformation of the material. An initial coarse mesh is created by utilizing a grid-based approach for creating well-shaped internal elements, in conjunction with a nodal connection approach based on constrained Delaunay triangulation, for linking with the boundary. Subsequent local mesh refinement is performed according to parameters depending on past, present and predicted future deformation related field variables; being, respectively, the strain gradient and strain rate distribution in relation with the velocity field, element size and quality. Smoothing is accomplished using an iterative Laplacian repositioning method. As illustrated in the presented examples this overall strategy ensures a robust and efficient remeshing scheme for finite element simulation of bulk metal-forming processes. © 1997 by John Wiley & Sons, Ltd.     

An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation

A new hybrid algorithm for automatically generating either an all-quadrilateral or an all-triangular element mesh within an arbitrarily shaped domain is described. The input consists of one or more closed loops of straight-line segments that bound the domain. Internal mesh density is inferred from the boundary density using a recursive spatial decomposition (quadtree) procedure. All-triangular element meshes are generated using a boundary contraction procedure. All-quadrilateral element meshes are generated by modifying the boundary contraction procedure to produce a mixed element mesh at half the density of the final mesh and then applying a polygon-splitting procedure. The final meshes exhibit good transitioning properties and are compatible with the given boundary segments which are not altered. The algorithm can support discrete crack growth simulation wherein each step of crack growth results in an arbitrarily shaped region of elements deleted about each crack tip. The algorithm is described and examples of the generated meshes are provided for a representative selection of cracked and uncracked structures.