An analysis of simplex shape measures for anisotropic meshes

This paper analyses and generalises several simplex shape measures documented in the literature and currently used for mesh adaptation and mesh optimisation. It first summarises important properties of simplices and their degeneration in Euclidean space. Different simplex shape measures are then defined and validated according to a validity criterion. The shape measures are generalised to Riemannian spaces in order to extend their use to anisotropic meshes. They are then analysed and compared using complementary approaches: a visualisation method helping to show their regularity, some theoretical relations establishing their equivalence, and a discussion on the evaluation of the global quality of a mesh. Conclusions are drawn on the choice of a simplex shape measure to guide mesh optimisation.     

Discretizations for weighted condition number smoothing on general unstructured meshes

New method for weighted condition number smoothing of general unstructured computational meshes is presented. Its core, proper discretization of weighted smoothness functional, is detailed, options of particular implementation are discussed and demonstrated on general convex polygonal cells in two dimensions. Possible applications of this algorithm are suggested, namely solution-sensitive mesh adaptation (respecting the variation of some variable) and prevention of unwanted smoothing effects on polar meshes.     

User controllable anisotropic shape distribution on 3D meshes

Computational Visual Media - This paper presents an automatic method for computing an anisotropic 2D shape distribution on an arbitrary 2-manifold mesh. Our method allows the user to specify the...     

Mortar finite elements for a heat transfer problem on sliding meshes

We consider a heat transfer problem with sliding bodies, where heat is generated on the interface due to friction. Neglecting the mechanical part, we assume that the pressure on the contact interface is a known function. Using mortar techniques with Lagrange multipliers, we show existence and uniqueness of the solution in the continuous setting. Moreover, two different mortar formulations are analyzed, and optimal a priori estimates are provided. Numerical results illustrate the flexibility of the approach. The work was supported by the EU-IHP Breaking Complexity project, CEE HPRN-CT-2002-00286.     

A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes

We consider residual-based a posteriori error estimators for the heterogeneous Maxwell equations using isotropic as well as anisotropic meshes. The continuous problem is approximated by using conforming approximated spaces with minimal assumptions. Lower and upper bounds are obtained under standard assumptions on the meshes. The lower bound holds unconditionally, while the upper bound depends on alignment properties of the meshes with respect to the solution. In particular for isotropic meshes the upper bound also holds unconditionally. A numerical test is presented which confirms our theoretical results.     

High order finite elements for shells

In this article the p-version finite element method is applied to thin-walled structures. Two different hierarchic element formulations are compared, a shell approach as well as a shell-like, solid formulation. Both approaches are compared for linear elastic and elastoplastic problems. Special emphasis is placed on the efficiency as well as on determining the area of application for both formulations.     

Superconvergence analysis of nonconforming finite element method for time-fractional nonlinear parabolic equations on anisotropic meshes

In this paper, we prove a novel result of the consistency error estimate with order $O\left(h^2\right)$ for $E{Q}_1^{rot}$ element (see Lemma 2) on anisotropic meshes. Then, a linearized fully discrete Galerkin finite element method (FEM) is studied for the time-fractional nonlinear parabolic problems, and the superclose and superconvergent estimates of order $O(\tau +h^2)$ in broken $H^1$-norm on anisotropic meshes are derived by using the proved character of $E{Q}_1^{rot}$ element, which improve the results in the existing literature. Numerical results are provided to confirm the theoretical analysis.     

Towards practical anisotropic adaptive FEM on triangular meshes: a new refinement paradigm

Adaptive anisotropic refinement of finite element meshes allows one to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows one to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.     

Computer determination of low order models for high order systems

An investigation directed at finding the best low-order model which approximates a given high-order system is presented. New insight is gained into the cost paid for the simplicity of the model and in the accuracy of the transient response of the model related to the magnitude of a cost function.The problem is solved in the time domain by finding the best pole and zero locations of the model which minimize a defined error criterion. The computer is used to estimate these parameters, via a parameter minimization program. A number of examples are included.     

A simple construction of very high order non-oscillatory compact schemes on unstructured meshes

In [Abgrall R, Roe PL. High order fluctuation schemes on triangular meshes. J Sci Comput 2003;19(1-3):3-36] have been constructed very high order residual distribution schemes for scalar problems. They were using triangle unstructured meshes. However, the construction was quite involved and was not very flexible. Here, following [Abgrall R. Essentially non-oscillatory residual distribution schemes for hyperbolic problems. J Comput Phys 2006;214(2):773-808], we develop a systematic way of constructing very high order non-oscillatory schemes for such meshes. Applications to scalar and systems problems are given.     

A general topology-based framework for adaptive insertion of cohesive elements in finite element meshes

Large-scale simulation of separation phenomena in solids such as fracture, branching, and fragmentation requires a scalable data structure representation of the evolving model. Modeling of such phenomena can be successfully accomplished by means of cohesive models of fracture, which are versatile and effective tools for computational analysis. A common approach to insert cohesive elements in finite element meshes consists of adding discrete special interfaces (cohesive elements) between bulk elements. The insertion of cohesive elements along bulk element interfaces for fragmentation simulation imposes changes in the topology of the mesh. This paper presents a unified topology-based framework for supporting adaptive fragmentation simulations, being able to handle two- and three-dimensional models, with finite elements of any order. We represent the finite element model using a compact and “complete” topological data structure, which is capable of retrieving all adjacency relationships needed for the simulation. Moreover, we introduce a new topology-based algorithm that systematically classifies fractured facets (i.e., facets along which fracture has occurred). The algorithm follows a set of procedures that consistently perform all the topological changes needed to update the model. The proposed topology-based framework is general and ensures that the model representation remains always valid during fragmentation, even when very complex crack patterns are involved. The framework correctness and efficiency are illustrated by arbitrary insertion of cohesive elements in various finite element meshes of self-similar geometries, including both two- and three-dimensional models. These computational tests clearly show linear scaling in time, which is a key feature of the present data-structure representation. The effectiveness of the proposed approach is also demonstrated by dynamic fracture analysis through finite element simulations of actual engineering problems.

Prismatic mixed finite elements for second order elliptic problems

In this paper, three families of mixed finite elements based on prisms are introduced. These spaces are analogues to those based on simplices and cubes in three space variables. Error estimates in L² and H⁻⁵ are given. This work is supported in part by the National Science Foundation.     

Some remarks on the discrete maximum-principle for finite elements of higher order

The discrete maximum principle for finite element approximations of standard elliptic problems in the plane is discussed. Even in the case Δu=0 a slightly stronger version of the principle does not hold with piecewise quadratic elements for all but some very special triangularisation geometries.     

Transition-path theory calculations on non-uniform meshes in two and three dimensions using finite elements

Rare events between states in complex systems are fundamental in many scientific fields and can be studied by building reaction pathways. A theoretical framework to analyze reaction pathways is provided by transition-path theory (TPT). The central object in TPT is the committor function, which is found by solution of the backward-Kolmogorov equation on a given potential. Once determined, the committor can be used to calculate reactive fluxes and rates, among other important quantities. We demonstrate here that the committor can be calculated using the method of finite elements on non-uniform meshes. We show that this approach makes it feasible to perform TPT calculations on 3D potentials because it requires many fewer degrees of freedom than a regular-mesh finite-difference approach. In various illustrative 2D and 3D problems, we calculate the committor function and reaction rates at different temperatures, and we discuss effects of temperatures and simple entropic barriers on the structure of the committor and the reaction rate constants.