Unstructured Finite Element Meshes Research Articles

Interior point tracking in shape evolving unstructured finite element meshes

An algorithm is presented for the tracking of interior points in a shape evolving unstructured FE mesh. Evolution of the boundary shape may be associated with a governing equation, as in moving boundary problems, or may be prescribed, as in structural shape optimisation. In the latter SSO case the point tracking algorithm may be used in conjunction with a FD approximation to determine geometric sensitivities: in this case the boundary deformation is a small perturbation. For meshes undergoing gross deformations of the boundary an incremental method is used. Reversibility tests are undertaken to assess the robustness and accuracy of the algorithm and examples are given to illustrate the general utility of the method.

Magnetohydrodynamic effects on pellet injection in tokamaks

The location at which pellets are injected into a plasma can have a significant effect on what fraction of the pellet mass remains in the plasma for refueling purposes. Magnetohydrodynamic (MHD) simulations presented here, confirm the results of pellet injection experiments: toroidal curvature makes it favorable to inject pellets from the inboard side or from the top or bottom, rather than from the outboard side. Sufficiently large pellets injected at the inboard edge can reach the plasma center, and in the process drive magnetic reconnection to produce negative central shear. Injection at the top (or bottom) of the tokamak causes relatively little displacement of the pellet. A scaling law is obtained for pellet displacement which agrees well with the simulations. The MHD simulations were carried out with a new unstructured mesh finite element version of the MH3D full MHD code.

Topological improvement procedures for quadrilateral finite element meshes

This paper presents a set of procedures for improving the topology of unstructured quadrilateral finite element meshes. These procedures are based on the topology of the finite element mesh, and all operations act only on local regions of the mesh. The goal is to optimize the topology such that the smoothing process can produce the best possible element quality. Topological improvement procedures are presented both for elements that are interior to the mesh and for elements connected to the boundary. Also presented is a discussion of efficiency and optimal ordering of the procedures. Several example meshes are included to show the effectiveness of the current approach in improving element qualities in a finite element mesh.

Adaptive refinement of unstructured finite-element meshes

The finite-element method used in conjunction with adaptive mesh refinement algorithms can be an efficient tool in many scientific and engineering applications. In this paper we review algorithms for the adaptive refinement of unstructured simplicial meshes (triangulations and tetra hedralizations). We discuss bounds on the quality of the meshes resulting from these refinement algorithms. Unrefinement and refinement along curved surfaces are also discussed. Finally, we give an overview of recent developments in parallel refinement algorithms.

Topological refinement procedures for triangular finite element meshes

This paper presents a topological approach to improve the quality of unstructured triangular finite element meshes. Topological improvement procedures are presented both for elements that are interior to the mesh and for elements connected to the boundary. Optimal ordering of the topology improvement operations and their efficient implementation is also discussed. Several example meshes are included to demonstrate the effectiveness of the approach in improving element quality in a finite element mesh.

AN EULERIAN FINITE ELEMENT METHOD FOR TIME-DEPENDENT FREE SURFACE PROBLEMS IN HYDRODYNAMICS

A numerical method based on the finite element method is presented for simulating the two-dimensional transient motion of a viscous liquid with free surfaces. For ease of numerical treatment of the free surface expressed by a multiple-valued function, the marker particle method is employed. Numerous virtual particles are spread over all regions occupied by liquid. They move about on a fixed finite element mesh with the liquid velocity at their positions. These particles contribute nothing to the dynamics of the liquid and only serve as markers of liquid regions. The velocity field within liquid regions is calculated by solving the Navier– Stokes equations and the equation of continuity by the finite element method based on quadrilateral elements. A detailed discussion is given of the methodological problems arising in the implementation of the marker particle method on an unstructured finite element mesh and of the solutions to these problems. The proposed method is demonstrated on three sample problems: the broken dam problem, the impact of a falling liquid drop on a still liquid and the entry of a rigid block into water. Good agreement has been obtained in the comparison of the present numerical results with available experimental data.

An effective data parallel self‐starting explicit methodology for computational structural dynamics on the connection machine CM‐5

AbstractThis paper discusses the implementation aspects and our experiences towards a data parallel explicit self‐starting finite element transient methodology with emphasis on the Connection Machine (CM‐5) for linear and non‐linear computational structural dynamic applications involving structured and unstructured grids. The parallel implementation criteria that influence the efficiency of an algorithm include the amount of communication, communication routing, and load balancing. To provide simplicity, high level of accuracy, and to retain the generality of the finite element implementation for both linear and non‐linear transient explicit problems on a data parallel computer which permit optimum amount of communications, we implemented the present self‐starting dynamic formulations (in comparison to the traditional approaches) based on nodal displacements, nodal velocities, and elemental stresses on the CM‐5. Data parallel language CMFortran is employed with virtual processor constructs and with:SERIAL and:PARALLEL layout directives for arrays. The communications via the present approach involve only one gather operation (extraction of element nodal displacements or velocities from global displacement vector) and one scatter operation (dispersion of element forces onto global force vector) for each time step. These gather and scatter operations are implemented using the Connection Machine Scientific Software Library communication primitives for both structured and unstructured finite element meshes. The implementation aspects of the present self‐starting formulations for linear and elastoplastic applications on serial and data parallel machines are discussed. Numerical test models for linear and non‐linear one‐dimensional applications and a two‐dimensional unstructured finite element mesh are then illustrated and their performance studies are discussed.

Parallel finite element algorithms applied to computational rheology

We review the work of our research group over the last 4 years towards the development of efficient parallel finite element algorithms. Target applications are physical problems described by means of non-linear sets of partial differential or integro-differential equations of mixed type, and solved in complex geometries using unstructured finite element meshes. A typical example considered in this paper is the flow of viscoelastic fluids. The complexity of the governing equations is such that it prevents the use of established parallel numerical algorithms developed for elliptic problems. After a brief discussion of viscoelastic governing equations and related sequential numerical techniques, we describe a generic parallel approach to the assembly and solution of finite element equation sets. Automatic load balancing schemes and mesh partitioning methods are discussed. Finally, the proposed algorithms are evaluated in the simulation of viscoelastic flows described by integral and differential constitutive equations. Results are reported for various distributed memory MIMD parallel computers, including the INTEL IPSC/860 hypercube, the CONVEX Meta Series, and a heterogeneous network of engineering workstations.

Parallel finite element algorithms applied to computational rheology

We review the work of our research group over the last 4 years towards the development of efficient parallel finite element algorithms. Target applications are physical problems described by means of non-linear sets of partial differential or integro-differential equations of mixed type, and solved in complex geometries using unstructured finite element meshes. A typical example considered in this paper is the flow of viscoelastic fluids. The complexity of the governing equations is such that it prevents the use of established parallel numerical algorithms developed for elliptic problems. After a brief discussion of viscoelastic governing equations and related sequential numerical techniques, we describe a generic parallel approach to the assembly and solution of finite element equation sets. Automatic load balancing schemes and mesh partitioning methods are discussed. Finally, the proposed algorithms are evaluated in the simulation of viscoelastic flows described by integral and differential constitutive equations. Results are reported for various distributed memory MIMD parallel computers, including the INTEL IPSC/860 hypercube, the CONVEX Meta Series, and a heterogeneous network of engineering workstations.

Optimized partitioning of unstructured finite element meshes

AbstractWe address the problem of automatic partitioning of unstructured finite element meshes in the context of parallel numerical algorithms based on domain decomposition. A two‐step approach is proposed, which combines a direct partitioning scheme with a non‐deterministic procedure of combinatorial optimization. In contrast with previously published experiments with non‐deterministic heuristics, the optimization step is shown to produce high‐quality decompositions at a reasonable compute cost. We also show that the optimization approach can accommodate complex topological constraints and minimization objectives. This is illustrated by considering the particular case of topologically one‐dimensional partitions, as well as load balancing of frontal subdomain solvers. Finally, the optimization procedure produces, in most cases, decompositions endowed with geometrically smooth interfaces. This contrasts with available partitioning schemes, and is crucial to some modern numerical techniques based on domain decomposition and a Lagrange multiplier treatment of the interface conditions.