Tetrahedral Finite Element Meshes Research Articles

Electron Optics Simulator: A Three-Dimensional Finite-Element Electron Gun and Collector Design Tool

This paper reports the development of a fully 3-D finite-element (FE) design tool for the design of electron guns and collectors called the electron optics simulator (EOS). It is a module of the microwave tube simulator suite (MTSS) developed by the University of Electronic Science and Technology of China. EOS specifically targets problem classes including axisymmetric gun, gridded gun, multibeam gun, and multistage depressed collector. EOS is a steady-state trajectory code, which is a solver of electrostatic field and trajectory that takes into account secondary electrons, space-charge effects, and the external magnetic field. The simulator is dependent on the following two front-end features in MTSS: preprocessing that includes a solid modeling tool and a fully 3-D tetrahedral FE mesh automatic generator, and post processing for analysis and display of the results. A friendly interface is provided to design, test, optimize, and validate the electron optics systems. The aforementioned theoretical models are discussed in detail in this paper. In the end, some comparisons between simulation and experiment are presented.

A finite element algorithm for particle/droplet trajectory tracking, tested in a liquid–liquid system in the presence of an external electric field

In this paper we are concerned with modeling a dispersion of electrically charged droplets in motion in a second immiscible liquid phase continuum in the presence of an external electric field. The system is highly relevant to solvent extraction processes and to liquid–liquid reactions such as phase transfer catalysis. Electrostatic enhancement of liquid–liquid contacting processes is well known and relies upon intensified drop breakup and drop acceleration due to the presence of significant electrical forces. The enhanced drop motion results in greatly increased rates of mass transfer. The scope of the current work is the mathematical modeling and experimental validation of the trajectories of multiple charged drops in three-dimensional motion in a liquid–liquid system in the presence of an externally applied field. A new particle tracking algorithm using a tetrahedral finite element mesh to solve the relevant system of differential equations is described in detail. Such elements are shown to be especially convenient for real word simulation because of their simplicity, their high stability with strictly linear interpolation, and their flexibility for dealing with complex computational domains. The calculations of trajectory tracking for several hundred particles taking in a mesh consisting of 50,000 tetrahedral elements was shown only to require a few seconds of computational time using a PC class computer. Using a double step stepping algorithm, with intermediate velocity interpolation, second-order convergence was proven in a rigorous benchmark test. Comparison of simulations of individual drop motion with experimental measurements proved to be very accurate. The full three-dimensional simulation, having greater requirements and involving more degrees of freedom than the earlier two-dimensional case was successfully demonstrated, with greater accuracy and economic use of computational time. Preliminary simulation of swarming motion of droplets using a cloud model is also presented. This shows significant promise for the evaluation of the mass distribution of very small droplets (∼1 to 100 μm) in a real contactor for optimal shape design. The model, when coupled with the models for mass transfer kinetics and interfacial reaction kinetics, should provide a very valuable tool for liquid–liquid reactor design.

Three‐dimensional numerical simulation of injection molding filling of optical lens and multiscale geometry using finite element method

AbstractThis article presents the development, verification, and validation of three‐dimensional (3‐D) numerical simulation for injection molding filling of 3‐D parts and parts with microsurface features. For purpose of verification and comparison, two numerical models, the mixed model and the equal‐order model, were used to solve the Stokes equations with three different tetrahedral elements (Taylor‐Hood, MINI, and equal‐order). The control volume scheme with tetrahedral finite element mesh was used for tracking advancing melt fronts and the operator splitting method was selected to solve the energy equation. A new, simple memory management procedure was introduced to deal with the large sparse matrix system without using a huge amount of storage space. The numerical simulation was validated for mold filling of a 3‐D optical lens. The numerical simulation agreed very well with the experimental results and was useful in suggesting a better processing condition. As a new application area, a two‐step macro–micro filling approach was adopted for the filling analysis of a part with a micro‐surface feature to handle both macro and micro dimensions while avoiding an excessive number of elements. POLYM. ENG. SCI., 46:1263–1274, 2006. © 2006 Society of Plastics Engineers

Tetrahedral versus hexahedral finite elements in numerical modelling of the proximal femur

The objective of this study was to evaluate and compare tetrahedral and hexahedral finite element meshes of simplified and realistic proximal intact femur geometries. Theoretical expressions of stresses and strains were derived for the simplified model of the femur. For the analysis of the realistic geometry of the proximal femur, a CAD model of the third generation composite femur (Pacific Research Labs, Vashion Island, WA) was used and simulations were performed with Hyperworks ® (Altair Engineering, Inc., Troy, MI) finite element analysis software. Convergence tests with hexahedral (8- and 20-node brick-elements) and tetrahedral (4- and 10-node tetrahedrons) elements were analysed by comparing the periosteal von Mises stresses and principal strains at a selected point of the femur. The numerical periosteal strains were also compared with experimental ones to determine the accuracy of the finite elements. Overall, we concluded, for the simplified femur, that tetrahedral linear element allowed results more closely to theoretical ones, but hexahedral quadratic elements seem to be more stable and less influenced to the degree of refinement (NDOF) of the mesh. It was not possible to correlate these findings with those observed for the realistic proximal femur simulations where no significant differences were seen using refined meshes, with more than NDOF = 20 000 for hexahedral elements.

A new element-by-element method for trajectory calculations with tetrahedral finite element meshes

A new method to track massless particles in a three-dimensional flow field is presented. The method is based on an element-by-element approach coupled with a predictor–corrector shooting scheme and does not use any time step. By analogy with time-dependent schemes, the number of shootings is related to an equivalent number of time steps. The method has been implemented in a finite element framework using unstructured tetrahedral finite element meshes. However, it is general enough so that it can be implemented in finite difference and finite volume frameworks as well. It has been tested on a variety of flow systems namely: the Poiseuille flow in an empty circular pipe, the rotating flow in a stirred tank, the shear flow in a square tank and the flow through a static mixer. Accuracy has been found to depend on the accuracy of the velocity computation, the number of points per element and the level of mesh refinement. Copyright © 2006 John Wiley & Sons, Ltd.

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L 2 norm error estimator is proposed for the Poisson equation. The error estimator can be applied to anisotropic tetrahedral or triangular finite element meshes. The estimator is rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropic bubble functions and the corresponding inverse inequalities. The upper error bound utilizes non‐standard anisotropic interpolation estimates. Its proof requires H 2 regularity of the Poisson problem, and its quality depends on how good the anisotropic mesh resolves the anisotropy of the problem. This is measured by a so‐called ‘matching function’. A numerical example supports the anisotropic error analysis.

Variational Delaunay approach to the generation of tetrahedral finite element meshes

AbstractWe describe an algorithm which generates tetrahedral decomposition of a general solid body, whose surface is given as a collection of triangular facets. The principal idea is to modify the constraints in such a way as to make them appear in an unconstrained triangulation of the vertex set à priori. The vertex set positions are randomized to guarantee existence of a unique triangulation which satisfies the Delaunay empty‐sphere property. (Algorithms for robust, parallelized construction of such triangulations are available.) In order to make the boundary of the solid appear as a collection of tetrahedral faces, we iterate two operations, edge flip and edge split with the insertion of additional vertex, until all of the boundary facets are present in the tetrahedral mesh. The outcome of the vertex insertion is another triangulation of the input surfaces, but one which is represented as a subset of the tetrahedral faces. To determine if a constraining facet is present in the unconstrained Delaunay triangulation of the current vertex set, we use the results of Rajan which re‐formulate Delaunay triangulation as a linear programming problem. Copyright © 2001 John Wiley & Sons, Ltd.

Data Structures for Geometric and Topological Aspects of Finite Element Algorithms - Abstract

This paper uses simplicial complexes and simplicial (co)homology theory to expose a foundation for data structures for tetrahedral finite element meshes. Identifying tetrahedral meshes with simplicial complexes leads, by means of Whitney forms, to the connection between simplicial cochains and fields in the region modeled by the mesh. Furthermore, lumped field parameters are tied to matrices associated with simplicial (co)homology groups. The data structures described here are sparse, and the computational complexity of constructing them is O(n) where n is the number of vertices in the finite element mesh. Non-tetrahedral meshes can be handled by an equivalent theory. These considerations lead to a discrete form of Poincaré duality which is a powerful tool for developing algorithms for topological computations on finite element meshes. This duality emerges naturally in the data structures. We indicate some practical applications of both data structures and underlying theory.

Requirements for Mesh Resolution in 3D Computational Hemodynamics

Computational techniques are widely used for studying large artery hemodynamics. Current trends favor analyzing flow in more anatomically realistic arteries. A significant obstacle to such analyses is generation of computational meshes that accurately resolve both the complex geometry and the physiologically relevant flow features. Here we examine, for a single arterial geometry, how velocity and wall shear stress patterns depend on mesh characteristics. A well-validated Navier-Stokes solver was used to simulate flow in an anatomically realistic human right coronary artery (RCA) using unstructured high-order tetrahedral finite element meshes. Velocities, wall shear stresses (WSS), and wall shear stress gradients were computed on a conventional "high-resolution" mesh series (60,000 to 160,000 velocity nodes) generated with a commercial meshing package. Similar calculations were then performed in a series of meshes generated through an adaptive mesh refinement (AMR) methodology. Mesh-independent velocity fields were not very difficult to obtain for both the conventional and adaptive mesh series. However, wall shear stress fields, and, in particular, wall shear stress gradient fields, were much more difficult to accurately resolve. The conventional (nonadaptive) mesh series did not show a consistent trend towards mesh-independence of WSS results. For the adaptive series, it required approximately 190,000 velocity nodes to reach an r.m.s. error in normalized WSS of less than 10 percent. Achieving mesh-independence in computed WSS fields requires a surprisingly large number of nodes, and is best approached through a systematic solution-adaptive mesh refinement technique. Calculations of WSS, and particularly WSS gradients, show appreciable errors even on meshes that appear to produce mesh-independent velocity fields.

An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes

A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only. The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be a useful tool. With its help anisotropic interpolation estimates and subsequently the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes. A numerical example supports the anisotropic error analysis.

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II?A framework for volume mesh optimization and the condition number of the Jacobian matrix

Three-dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid, high-quality meshes. The approach uses matrices and matrix norms to extend the work in Part I to build suitable 3D objective functions. Because certain matrix norm identities which hold for 2×2 matrices do not hold for 3×3 matrices, significant differences arise between surface and volume mesh optimization objective functions. It is shown, for example, that the equality in two dimensions of the smoothness and condition number of the Jacobian matrix objective functions does not extend to three dimensions and further, that the equality of the Oddy and condition number of the metric tensor objective functions in two dimensions also fails to extend to three dimensions. Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions. The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements. Non-dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization. The performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems, focusing particularly on all-hexahedral ‘whisker-weaved’ meshes. Performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes. Although several objective functions are competitive, the condition number objective function is particularly attractive. The objective functions are closely related to mesh quality measures. To illustrate, it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure. Published in 2000 by John Wiley & Sons, Ltd.

A new element bisection algorithm for unstructured adaptive tetrahedral mesh generation

A new h‐refinement adaptive tetrahedral mesh generation algorithm is presented. Three‐dimensional domains, to be analysed by the finite element method, are initially modelled by a coarse background mesh of tetrahedral elements. This mesh forms the input for finite element analysis and error estimation by the Zienkiewicz‐Zhu simple error estimator. Adaptive mesh refinement proceeds by selecting an element for remeshing whose longest edge is shared by elements that also require refinement. This group of elements is refined by inserting a new node at the mid‐point of the shared edge thereby bisecting all elements within the group. Adaptive parameters are calculated for the new node and elements. Refinement then proceeds until no further group of elements can be found for refinement or no elements within the current mesh require further refinement. The shape quality of the current mesh is then enhanced by the iterative application of nodal relaxation plus three topological transformations. The entire refinement process is repeated iteratively until the required degree of mesh refinement is reached. Ten‐noded linear strain tetrahedral finite element meshes have been used for the finite element and error estimation analyses. Four examples of adaptive tetrahedral mesh generation for linear elastic stress/displacement analysis are presented which show that this algorithm is robust and efficient in terms of reduction of the domain error with a minimum number of degrees of freedom being generated, number of iterations, and therefore finite element analyses, required and computational time for refinement when compared to the advancing front method and Delaunay triangulation.

Adaptive finite element mesh refinement techniques in three-dimensional micromagnetic modeling

Two different methods to adaptively refine tetrahedral finite element meshes are presented. The influence of the mesh refinement on the obtained results of micromagnetic calculations is demonstrated for the case of magnetization patterns in soft magnetic platelets. The specific addition of vertices and finite elements in inhomogeneous regions is used to resolve the magnetic structure on small length scales. A method discussed in detail is the shrinking of finite elements in regions with strong inhomogeneities.

Tetrahedral finite element mesh generation from NURBS solid models

In this paper, the development and the implementation of a tetrahedral meshing algorithm for generation of finite element meshes from NURBS solid models is presented. The meshing algorithm is based on a Delaunay technique, and makes use of some spatial data structures. The algorithm is capable of generating both uniform and varying size four-node and ten-node tetrahedral meshes. The algorithm has been implemented in a building block approach as part of a software library. It has been used as a practical tool in engineering design processes. Several representative test cases illustrate the effectiveness of the automatic solid mesh generator.

WETD - a finite element time-domain approach for solving Maxwell's equations

A family of finite element time-domain methods, WETD(/spl Theta/), is derived to solve the time-varying Maxwell's equations. The proposed methodology is based upon the application of the Faedo-Galerkin procedure and the use of the Whitney 1-forms as bases to result in an ordinary differential equation in time for the electric field. Moreover, the resultant ordinary differential equation is solved by employing central and/or backward difference approximations. Since the WETD methods presented here are used in conjunction with tetrahedral finite element meshes, it imposes no limitations on the problem geometry. Also, in this contribution, a general stability condition has been derived for the WETD(/spl Theta/) method of which the central and backward differences are special cases corresponding to /spl Theta/=1 and /spl Theta/=0, respectively. >

GEOMPACK — a software package for the generation of meshes using geometric algorithms

We describe the mathematical software package GEOMPACK, which contains standard Fortran 77 routines for the generation of two-dimensional triangular and three-dimensional tetrahedral finite element meshes using efficient geometric algorithms. This package results from our research into mesh generation and geometric algorithms. It contains routines for constructing two- and three-dimensional Delaunay triangulations, decomposing a general polygonal region into simple or convex polygons, constructing the visibility polygon of a simple polygon from a viewpoint, and other geometric algorithms, from which our mesh generation method is built and others can be implemented. Our method generates meshes in polygonal or polyhedral regions specified by their boundary representation and possible interfaces between subregions.

Graded tetrahedral finite element meshes

AbstractVertices in the body centred cubic (bcc) lattice are used to create a tetrahedral spatial decomposition. With this spatial decomposition an octree approach is combined with Delaunay triangulations to decompose solids into tetrahedral finite element meshes. Solids must have their surfaces triangulated and the vertices in the triangulation are finite element nodes. Local densities of interior tetrahedra are controlled by the densities of surface triangles. Accuracy of the decomposition into finite elements depends on the accuracy of the surface triangulation which can be constructed with state of the art computer aided design systems.