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.     

Higher‐resolution convection schemes for flow in porous media on highly distorted unstructured grids

Higher‐resolution schemes are presented for convective flow approximation on highly distorted unstructured grids. The schemes are coupled with continuous full‐tensor Darcy‐flux approximations. A sequence of non‐uniform and distorted grid formulations are developed and compared for a range of unstructured meshes with variable grid spacing. The higher‐order schemes are constructed using non‐uniform grid slope limiters such that they are stable with a local maximum principle, ensuring that solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full‐tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher‐order formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts. Copyright © 2008 John Wiley & Sons, Ltd.     

Hierarchical tree‐based finite element mesh generation

Hierarchical grid generation and its use as a basis for finite element mesh generation are considered in this paper. The hierarchical grids are generated by recursive subdivision using quadtrees in two dimensions and octrees in three dimensions. A numbering system for efficient storage of the quadtree grid information is examined, tree traversal techniques are devised for neighbour finding, and accurate boundary representation is considered. It is found that hierarchical grids are straightforward to generate from sets of seeding points which lie along domain boundaries. Quadtree grids are triangularized to provide finite element meshes in two dimensions. Three‐dimensional tetrahedral meshes are generated from octree grids. The meshes can be generated automatically to model complicated geometries with highly irregular boundaries and can be adapted readily at moving boundaries. Examples are given of two‐ and three‐dimensional hierarchical tree‐based finite element meshes and their application to modelling free surface waves. Copyright © 1999 John Wiley & Sons, Ltd.     

The partition of unity finite element method for short wave acoustic propagation on non‐uniform potential flows

A novel numerical method is proposed for modelling time‐harmonic acoustic propagation of short wavelength disturbances on non‐uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non‐uniform. Many wavelengths can be included within a single element leading to ultra‐sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid‐based schemes. Numerical results for lined uniform two‐dimensional ducts and for non‐uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.     

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.     

A node‐to‐node scheme with the aid of variable‐node elements for elasto‐plastic contact analysis

A strategy for a two‐dimensional contact analysis involving finite strain plasticity is developed with the aid of variable‐node elements. The variable‐node elements, in which nodes are added freely where they are needed, make it possible to transform the non‐matching meshes into matching meshes directly. They thereby facilitate an efficient analysis, maintaining node‐to‐node contact during the contact deformation. The contact patch test, wherein the contact patch is constructed out of variable‐node elements, is thus passed, and iterations for equilibrium solutions reach convergence faster in this scheme than in the conventional approach based on the node‐to‐surface contact. The effectiveness and accuracy of the proposed scheme are demonstrated through several numerical examples of elasto‐plastic contact analyses. Copyright © 2015 John Wiley & Sons, Ltd.     

A semi‐Lagrangean time‐integration approach for extended finite element methods

Many computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi‐Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapolated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one‐dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two‐dimensional and a three‐dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche‐type formulation for interface problems with embedded strong discontinuities. Copyright © 2014 John Wiley & Sons, Ltd.     

New two‐dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes

In this paper, we introduce an extension of Van Leer's slope limiter for two‐dimensional discontinuous Galerkin (DG) methods on arbitrary unstructured quadrangular or triangular grids. The aim is to construct a non‐oscillatory shock capturing DG method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion. Unlike some splitting techniques that are limited to linear approximations on rectangular grids, in this work, the solution is approximated by means of piecewise quadratic functions. The main idea of this new reconstructing and limiting technique follows a well‐known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution. Numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of our proposed slope limiters. Copyright © 2004 John Wiley & Sons, Ltd.     

Singularity enrichment for complete sliding contact using the partition of unity finite element method

In this paper, the numerical modelling of complete sliding contact and its associated singularity is carried out using the partition of unity finite element method. Sliding interfaces in engineering components lead to crack nucleation and growth in the vicinity of the contact zone. To accurately capture the singular stress field at the contact corner, we use the partition of unity framework to enrich the standard displacement‐based finite element approximation by additional (enriched) functions. These enriched functions are derived from the analytical expression of the asymptotic displacement field in the vicinity of the contact corner. To characterize the intensity of the singularity, a domain integral formulation is adopted to compute the generalized stress intensity factor (GSIF). Numerical results on benchmark problems are presented to demonstrate the improved accuracy and benefits of this technique. We conduct an investigation on issues pertaining to the extent of enrichment, accurate numerical integration of weak‐form integrals and the rate of convergence in energy. The use of partition of unity enrichment leads to accurate estimations of the GSIFs on relatively coarse meshes, which is particularly beneficial for modelling non‐linear sliding contacts. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

Solving the advection–diffusion equation on unstructured meshes with discontinuous/mixed finite elements and a local time stepping procedure

Explicit schemes are known to provide less numerical diffusion in solving the advection–diffusion equation, especially for advection‐dominated problems. Traditional explicit schemes use fixed time steps restricted by the global CFL condition in order to guarantee stability. This is known to slow down the computation especially for heterogeneous domains and/or unstructured meshes. To avoid this problem, local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local CFL condition have been developed. In this paper, a local time stepping approach is used with a numerical model based on discontinuous Galerkin/mixed finite element methods to solve the advection–diffusion equation. The developments are detailed for general unstructured triangular meshes. Numerical experiments are performed to show the efficiency of the numerical model for the simulation of (i) the transport of a solute on highly unstructured meshes and (ii) density‐driven flow, where the velocity field changes at each time step. The model gives stable results with significant reduction of the computational cost especially for the non‐linear problem. Moreover, numerical diffusion is also reduced for highly advective problems. Copyright © 2009 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Blurred derivatives and meshless methods

In this work we first introduce and describe the concept of blurred derivatives. It is shown how they can be used both to approximate differential equations and to re‐express them in alternative ways. In particular, formulations in terms of functional integrals can be obtained using blurred derivatives and extended to non‐linear problems. Blurred derivatives are shown to provide higher flexibility for selection of approximation functions than strong and weak formulations. Some computational implementations of one‐dimensional problems are discussed and the relationship between several well‐known numerical methods is analysed. Finally a meshless numerical scheme for the Poisson equation is described in detail. Its performance is compared with linear finite elements and generalized finite differences on unstructured meshes of points. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient solution method for incompressible N‐S equations using non‐orthogonal collocated grid

An efficient strategy for the solution of N‐S Equations using collocated, non‐orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co‐ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non‐orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non‐orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid‐driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti‐symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.     

Interpolation functions in the immersed boundary and finite element methods

In this paper, we review the existing interpolation functions and introduce a finite element interpolation function to be used in the immersed boundary and finite element methods. This straightforward finite element interpolation function for unstructured grids enables us to obtain a sharper interface that yields more accurate interfacial solutions. The solution accuracy is compared with the existing interpolation functions such as the discretized Dirac delta function and the reproducing kernel interpolation function. The finite element shape function is easy to implement and it naturally satisfies the reproducing condition. They are interpolated through only one element layer instead of smearing to several elements. A pressure jump is clearly captured at the fluid–solid interface. Two example problems are studied and results are compared with other numerical methods. A convergence test is thoroughly conducted for the independent fluid and solid meshes in a fluid–structure interaction system. The required mesh size ratio between the fluid and solid domains is obtained.     

Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization

We consider the numerical simulation of non‐linear multi‐body contact problems in elasticity on complex three‐dimensional geometries. In the case of warped contact boundaries and non‐matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the non‐penetration conditions at the contact interface. We enforce the discrete contact constraints by means of a non‐conforming domain decomposition method, which allows for optimal error estimates. Here, we develop an efficient method to assemble the discrete coupling operator by computing the triangulated intersection of opposite element faces in a locally adjusted projection plane but carrying out the required quadrature on the faces directly. Our new element‐based algorithm does not use any boundary parameterizations and is also suitable for isoparametric elements. The emerging non‐linear system is solved by a monotone multigrid method of optimal complexity. Several numerical examples in 3D illustrate the effectiveness of our approach. Copyright © 2008 John Wiley & Sons, Ltd.