Unstructured Finite Element Meshes Research Articles

An adaptive mesh adjoint data assimilation method

An important issue in the numerical modelling of ocean dynamics is how to improve the predictive capabilities of models which include the representation of complex local flows around and over coastal topography. Four-dimensional (space and time) variational data assimilation represents a means to achieve this and can be realised using adjoint methods. An advantage of this approach is that it is able to fit a numerical simulation to observational data over both space and time. In this work, an adjoint method for an adaptive mesh forward model is developed and its correctness and feasibility verified. Here a forward and adjoint model, both based upon adaptive unstructured mesh finite element techniques, is tested by inverting boundary conditions in two-dimensional barotropic flow near a coastal headland. The inversion method uses independent adapting meshes for the forward and adjoint solution fields. This means that the transient characteristics of the different solution fields may be taken into account in the optimised computational meshes. The issues and advantages of this new data assimilation method are discussed. The computational efficiency of the method is presented. The robustness of the inversion (adjoint) model is tested by assimilating noisy and sparse observations into the model. It is seen that the inversion model can extract the mean observation information and produce good quality inversion results.

A dynamic variational multiscale method for large eddy simulations on unstructured meshes

A dynamic version of the variational multiscale (VMS) method for large eddy simulation (LES) on unstructured meshes of turbulent compressible flows is presented. Its key features include two variational analogues of Germano’s algebraic identity, and a computationally efficient, agglomeration-based numerical procedure. Agglomeration is used both for separating a priori the scales and computing dynamically a Smagorinsky parameter and the turbulent Prandtl number on unstructured finite volume or finite element meshes. Numerical predictions of low-speed, 20° angle of attack, turbulent flows past a 6:1 prolate spheroid and a forward swept scaled wing are presented. They are contrasted with counterpart results obtained using similar implementations on unstructured meshes of the VMS-LES, traditional LES, and traditional dynamic LES methods. They are also compared to experimental data when available.

A geometric nonlinear triangular finite element with an embedded discontinuity for the simulation of quasi-brittle fracture

This paper is aimed to model the appearance and evolution of discrete cracks in quasi-brittle materials using triangular finite elements with an embedded interface in a geometric nonlinear setting. The kinematics for the discontinuous displacement field is presented and the standard variational formulation with respect to the reference configuration is extended to a body with an internal discontinuity. Special attention is paid to the algorithmic treatment. The discontinuity is modeled by additional global degrees of freedom and the continuity of the displacements across the element boundaries is enforced. Finally, representative numerical examples for mode-I and mixed-mode fracture, namely a tension test, different three-point bending tests and a single edge notched beam with structured and unstructured finite element meshes are discussed to study the evolving crack pattern and to show the ability of the model.

An assumed-gradient finite element method for the level set equation

The level set equation is a non-linear advection equation, and standard finite-element and finite-difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed-distance function. For some interface-evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity-capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed-gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level-set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton–Jacobi equation with convex/non-convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite-element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd.

Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.

Partitioning finite element meshes using space-filling curves

Using space-filling curves to partition unstructured finite element meshes is a widely applied strategy when it comes to distributing load among several computation nodes. Compared to more elaborated graph partitioning packages, this geometric approach is relatively easy to implement and very fast. However, results are not expected to be as good as those of the latter. In this paper we present results of our experiments comparing the quality of partitionings computed with different types of space-filling curves to those generated with the graph partitioning package Metis.

Finite-Element Model for Modified Boussinesq Equations. II: Applications to Nonlinear Harbor Oscillations

The finite-element model for modified Boussinesq equations developed in Part I of this paper is used to study harbor resonance problems. The internal wavemaker technique is implemented on an unstructured finite-element mesh, and a sponge layer is used as an open boundary condition. Both linear and nonlinear harbor oscillations in rectangular and circular basins are examined, and experimental measurements and analytical solutions are used to validate the model. Finally, the wavelet transform is employed to analyze transient features of the nonlinear resonance problem.

Wave–structure interaction using coupled structured–unstructured finite element meshes

The interaction of inviscid gravity waves with submerged fixed horizontal structures is modelled in two-dimensions using a finite element numerical wave tank. An adaptive hybrid coupled mesh is utilised that tracks the free surface through vertical movement of the free surface nodes in a Lagrange–Eulerian scheme. The hybrid mesh consists of a combination of structured and Voronoi unstructured meshes, with the submerged structure located in the Voronoi mesh region. Validation tests include free and forced sloshing in a rectangular tank, regular progressive wave propagation in a flume, and regular wave loading on a horizontal cylinder. The results are found to be in close agreement with analytical potential theory solutions for waves of small amplitude. Non-linear effects are noted for steeper waves. The wave-induced force components on the horizontal cylinder match the expected results from Ogilvie's [J Fluid Mech 16 (1963) 451] linear theory and Vada's [J Fluid Mech 174 (1987) 23] second order model. Wave interactions with a pair of submerged horizontal cylinders spaced at half the wavelength of undisturbed regular waves are examined.

Finite Element Based Riemann Solvers for Time-Dependent and Steady-State Radiation Transport

A high-order, nonoscillatory scheme is described which solves the transient and steady-state Boltzmann transport equation on unstructured Finite Element (FE) meshes. Flux limiters are applied in the space and time domains resulting in a scheme which is both free from oscillations and globally high-order accurate in space and time. The method described is finite volume based and uses a consistent FE representation of the solution variables to obtain a high-order solution along the control volume boundaries. Careful inspection of the eigenstructure of the Riemann problem allows one to switch smoothly between a high-order and a low-order nonoscillatory solution.

An Optimal Parallel Nonoverlapping Domain Decomposition Iterative Procedure

A nonoverlapping domain decomposition iterative procedure is developed and analyzed for second order elliptic problems in $\mathbb R^N$. Its convergence is proved. The method is based on a Robin-type consistency condition with two parameters, called a transmission coefficient and a penalty coefficient, as a transmission condition together with a derivative-free transmission data updating technique on the artificial interfaces. Then the method is applied to the nonconforming finite element problems. A nonoverlapping domain decomposition iterative procedure for solving the nonconforming finite element problems of second order partial differential equations is developed and analyzed, which is directly presented to the nonconforming finite element problems without introducing any Lagrange multipliers. Its convergence is demonstrated, and the convergence rate is derived. The convergence analyses imply that the convergence rate is independent of the finite element meshes size while choosing the right parameters. Furthermore, the conclusions are extended to the unstructured finite element meshes. For both continuous problems and discrete problems, the method of this paper can be applied to general multisubdomain decompositions and implemented on parallel machines with local communications naturally. The method also allows choosing subdomains very flexibly, even as small as an individual element for finite element problems.

Sparse matrix element topology with application to AMG(e) and preconditioning

AbstractThis paper defines topology relations of elements treated as overlapping lists of nodes. In particular, the element topology makes use of element faces, element vertices and boundary faces which coincide with the actual (geometrical) faces, vertices and boundary faces in the case of true finite elements. The element topology is used in an agglomeration algorithm to produce agglomerated elements (a non‐overlapping partition of the original elements) and their topology is then constructed, thus allowing for recursion. The main part of the algorithms is based on operations on Boolean sparse matrices and the implementation of the algorithms can utilize any available (parallel) sparse matrix format. Applications of the sparse matrix element topology to AMGe (algebraic multigrid for finite element problems), including elementwise constructions of coarse non‐linear finite element operators are outlined. An algorithm to generate a block nested dissection ordering of the nodes for generally unstructured finite element meshes is given as well. The coarsening of the element topology is illustrated on a number of fine‐grid unstructured triangular meshes. Published in 2002 by John Wiley & Sons, Ltd.

The Front-Tracking ALE Method: Application to a Model of the Freezing of Cell Suspensions

A new front-tracking method to compute discontinuous solutions on unstructured finite element meshes is presented. Using an arbitrary Lagrangian–Eulerian formulation, the mesh is continuously adapted by moving the nearest nodes to the interface. Thus, the solution is completely sharp at the interface and no smearing takes place. The dynamic node adjustment is confined to global nodes near the front, rendering remeshing unnecessary. The method has been applied to the osmotic motion of a two-dimensional cell arising from a concentration gradient generated by a moving solidification front. The engulfment of one cell by an advancing solidification front, which rejects the solutes in a binary salt solution, is then computed. The results indicate that the ice increases the solute gradient around the cell. Furthermore, the presence of the cell, which prevents diffusion of the solute, leads to large changes in the morphology of the ice front.

Derived data structure algorithms for unstructured finite element meshes

AbstractA set of derived data structure algorithms for unstructured finite element meshes is presented. Both serial and parallel algorithms are described for each data structure. Colouring groups for the elements are used to facilitate parallelization on shared memory architectures. Scaling studies indicate that the parallel algorithms are most efficient when the number of elements per processor is on the order of 106 or higher, and overall efficiencies of 60–70% are achieved down to 0.5×106 elements per processor. Although the meshes under consideration are tetrahedral, the algorithms are general in nature and can be extended to arbitrary element types with minimal effort. Copyright © 2002 John Wiley & Sons, Ltd.

A parallel nodal-based evolutionary structural optimization algorithm

This paper is concerned with the minimum weight design of structures using Finite Element Analysis (FEA). A new evolutionary structural optimization (ESO) algorithm is presented. This method departs from previous studies of ESO in that it exploits the movements of the nodes in an unstructured finite element mesh in an appropriate way. An attractive feature of the scheme presented is that it carries out topology optimization in the interior of the domain concurrently with shape optimization of the exterior of the domain. Circular cavities are inserted into the interior of the domain from which the internal topology is then revealed by migration of the cavity edge nodes. Due to the complexity of the resulting cavity geometry the FE mesh tends to be refined internally. A scheme for maintaining a roughly uniform density unstructured finite element mesh throughout the optimization in a two-dimensional domain is presented. The designs produced posses smooth internal and external boundaries. The method uses iterative finite element analysis and re-meshing to correct for any element distortion. The benchmark “Michell Arch" problem is used to demonstrate the approach.

Visualization of CFD results in immersive virtual environments

An object-oriented event-driven immersive virtual environment (VE) is described for the visualization of computational fluid dynamics (CFD) results. The VE incorporates the following types of primitive software objects: interface objects, support objects, geometric entities, and finite elements. The fluid domain is discretized using either a multi-block structured grid or an unstructured finite element mesh. The VE allows natural ‘fly-through’ visualization of the model, the CFD grid, and the model's surroundings. In order to help visualize the flow and its effects on the model, the VE incorporates the following objects: stream objects (lines, surface-restricted lines, ribbons, and volumes); colored surfaces; elevation surfaces; surface arrows; global and local iso-surfaces; vortex cores; and separation/attachment surfaces and lines. Most of these objects can be used for dynamically probing the flow. Particles and arrow animations can be displayed on top of stream objects. Primitive response quantities as well as derived quantities can be used. A recursive tree search algorithm is used for real-time point and value search in the CFD grid.

Performance of algebraic multi‐grid solvers based on unsmoothed and smoothed aggregation schemes

AbstractA comparison is made of the performance of two algebraic multi‐grid (AMG0 and AMG1) solvers for the solution of discrete, coupled, elliptic field problems. In AMG0, the basis functions for each coarse grid/level approximation (CGA) are obtained directly by unsmoothed aggregation, an appropriate scaling being applied to each CGA to improve consistency. In AMG1 they are assembled using a smoothed aggregation with a constrained energy optimization method providing the smoothing. Although more costly, smoothed basis functions provide a better (more consistent) CGA. Thus, AMG1 might be viewed as a benchmark for the assessment of the simpler AMG0. Selected test problems for D'Arcy flow in pipe networks, Fick diffusion, plane strain elasticity and Navier–Stokes flow (in a Stokes approximation) are used in making the comparison. They are discretized on the basis of both structured and unstructured finite element meshes. The range of discrete equation sets covers both symmetric positive definite systems and systems that may be non‐symmetric and/or indefinite. Both global and local mesh refinements to at least one order of resolving power are examined. Some of these include anisotropic refinements involving elements of large aspect ratio; in some hydrodynamics cases, the anisotropy is extreme, with aspect ratios exceeding two orders. As expected, AMG1 delivers typical multi‐grid convergence rates, which for all practical purposes are independent of mesh bandwidth. AMG0 rates are slower. They may also be more discernibly mesh‐dependent. However, for the range of mesh bandwidths examined, the overall cost effectiveness of the two solvers is remarkably similar when a full convergence to machine accuracy is demanded. Thus, the shorter solution times for AMG1 do not necessarily compensate for the extra time required for its costly grid generation. This depends on the severity of the problem and the demanded level of convergence. For problems requiring few iterations, where grid generation costs represent a significant penalty, AMG0 has the advantage. For problems requiring a large investment in iterations, AMG1 has the edge. However, for the toughest problems addressed (vector and coupled vector–scalar fields discretized exclusively using finite elements of extreme aspect ratio) AMG1 is more robust: AMG0 has failed on some of these tests. However, but for this deficiency AMG0 would be the preferred linear approximation solver for Navier–Stokes solution algorithms in view of its much lower grid generation costs. Copyright © 2001 John Wiley & Sons, Ltd.

How We solve PDEs

A finite difference method on an unstructured finite element mesh which we call finite difference element method (FDEM) is presented. The FDEM program package will be a black-box solver for nonlinear systems of elliptic and parabolic PDEs with mesh refinement and automatic control of the consistency order in each space grid point. In this paper we present the solution method (with examples) for 2-D systems of elliptic PDEs.

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 algorithm for partitioning of finite element meshes

An efficient algorithm is developed for partitioning unstructured finite element meshes. A new graph model is presented and employed for transforming the connectivity properties of meshes. This method leads to a load balance partitioning in which the number of interface nodes are confined to the smallest possible, and aspect ratios of subdomains are desired values. Examples are included to illustrate the performance and efficiency of the presented method.

Local mesh adaptation technique for front tracking problems

A numerical model is developed for the simulation of moving interfaces in viscous incompressible flows. The model is based on the finite element method with a pseudo-concentration technique to track the front. Since a Eulerian approach is chosen, the interface is advected by the flow through a fixed mesh. Therefore, material discontinuity across the interface cannot be described accurately. To remedy this problem, the model has been supplemented with a local mesh adaptation technique. This latter consists in updating the mesh at each time step to the interface position, such that element boundaries lie along the front. It has been implemented for unstructured triangular finite element meshes. The outcome of this technique is that it allows an accurate treatment of material discontinuity across the interface and, if necessary, a modelling of interface phenomena such as surface tension by using specific boundary elements. For illustration, two examples are computed and presented : the broken dam problem and the Rayleigh-Taylor instability