A Geometric Toolbox for Tetrahedral Finite Element Partitions

In this work we present a survey of some geometric results on tetrahedral partitions and their refinements in a unified manner. They can be used for mesh generation and adaptivity in practical calculations by the finite element method (FEM), and also in theoretical finite element (FE) analysis. Special emphasis is laid on the correspondence between relevant results and terminology used in FE computations, and those established in the area of discrete and computational geometry (DCG)

On nonobtuse refinements of tetrahedral finite element meshes

It is known that piecewise linear continuous finite element (FE) approximations on nonobtuse tetrahedral FE meshes guarantee the validity of discrete analogues of various maximum principles for a wide class of elliptic problems of the second order. Such analogues are often called discrete maximum principles (or DMPs in short). In this work we present several global and local refinement techniques which produce nonobtuse conforming (i.e. face-to-face) tetrahedral partitions of polyhedral domains. These techniques can be used in order to compute more accurate FE approximations (on finer and/or adapted tetrahedral meshes) still satisfying DMPs

On simplicial red refinement in three and higher dimensions

summary:We show that in dimensions higher than two, the popular "red refinement" technique, commonly used for simplicial mesh refinements and adaptivity in the finite element analysis and practice, never yields subsimplices which are all acute even for an acute father element as opposed to the two-dimensional case. In the three-dimensional case we prove that there exists only one tetrahedron that can be partitioned by red refinement into eight congruent subtetrahedra that are all similar to the original one

Adaptive finite element method assisted by stochastic simulation of chemical systems

Stochastic models of chemical systems are often analysed by solving the corresponding\ud Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density

Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain $\Omega \subset R^d$, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. We perform a rigorous passage to the limit as first the spatial discretization parameter, and then the temporal discretization parameter tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system. The passage to the limit is performed under minimal regularity assumptions on the data: a square-integrable and divergence-free initial velocity datum $u_0$ for the Navier-Stokes equation and a nonnegative initial probability density function $\psi_0$ for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian M

Numerical simulation of fracture pattern development and implications for fuid flow

Simulations are instrumental to understanding flow through discrete fracture geometric representations that capture the large-scale permeability structure of fractured porous media. The contribution of this thesis is threefold: an efficient finite-element finite-volume discretisation of the advection/diffusion flow equations, a geomechanical fracture propagation algorithm to create fractured rock analogues, and a study of the effect of growth on hydraulic conductivity. We describe an iterative geomechanics-based finite-element model to simulate quasi-static crack propagation in a linear elastic matrix from an initial set of random flaws. The cornerstones are a failure and propagation criterion as well as a geometric kernel for dynamic shape housekeeping and automatic remeshing. Two-dimensional patterns exhibit connectivity, spacing, and density distributions reproducing en echelon crack linkage, tip hooking, and polygonal shrinkage forms. Differential stresses at the boundaries yield fracture curving. A stress field study shows that curvature can be suppressed by layer interaction effects. Our method is appropriate to model layered media where interaction with neighbouring layers does not dominate deformation. Geomechanically generated fracture patterns are the input to single-phase flow simulations through fractures and matrix. Thus, results are applicable to fractured porous media in addition to crystalline rocks. Stress state and deformation history control emergent local fracture apertures. Results depend on the number of initial flaws, their initial random distribution, and the permeability of the matrix. Straightpath fracture pattern simplifications yield a lower effective permeability in comparison to their curved counterparts. Fixed apertures overestimate the conductivity of the rock by up to six orders of magnitude. Local sample percolation effects are representative of the entire model flow behaviour for geomechanical apertures. Effective permeability in fracture dataset subregions are higher than the overall conductivity of the system. The presented methodology captures emerging patterns due to evolving geometric and flow properties essential to the realistic simulation of subsurface processes