a conforming finite element method for overlapping and nonmatching grids in this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity both overlapping and nonoverlapping cases are considered we prove that the new method admits an optimal convergence rate the error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains our results are valid for multiple subdomains and any spatial dimensions

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

multiscale coupling in function space weak coupling between molecular dynamics and continuum mechanics we present a function space oriented coupling approach for the multiscale simulation of non linear processes in mechanics using finite elements and molecular dynamics concurrently the key idea is to construct a transfer operator between the different scales on the basis of weighted local averaging instead of using point wise taken values the local weight functions are constructed by assigning a partition of unity to the molecular degrees of freedom shepard s approach this allows for decomposing the micro scale displacements into a low frequency and a high frequency part by means of a weighted l2 projection numerical experiments illustrating the stabilizing effect of our coupling approach are given copyright 2009 john wiley sons ltd

using a signed distance function for the simulation of metal forming processes formulation of the contact condition and mesh adaptation from a lagrangian approach to an eulerian approach this paper proposes to use the metric properties of the distance function between two bodies in contact or gap function in simulations involving contact problems first the normal vectors which are involved in the formulation of the contact condition are defined through the gradient of this distance function this definition avoids to deal with the numerical penetration parameter which is generally introduced otherwise furthermore it allows the contact problem to be extended in a simple way to an eulerian formulation second this paper investigates two mesh adaptation strategies based on the properties of the distance function the first strategy consists in building a size map according to the values of this function in order to refine locally the mesh and consequently to improve the description of the contact surface the second strategy consists in adapting locally the mesh to the geometry of the contact surface this anisotropic adaptation is performed by constructing a metric map that allows the mesh size to be imposed in the direction of the distance function gradient a lot of elements are saved when compared with the isotropic case throughout this paper many numerical simulations are presented in the context of the forging process the deformable material is pressed between two rigid tools furthermore the algorithm used to calculate the signed distance to a surface mesh is detailed in appendix of this paper

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

an immersed finite element method with integral equation correction summary we propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions the solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution for computing the finite element solution the physical domain is embedded into a slightly larger cartesian box shaped domain and is discretized using a block structured mesh the defect in the essential boundary conditions which occurs along the physical domain boundaries is subsequently corrected with an integral equation method in order to facilitate the mapping between the finite element and integral equation solutions the physical domain boundary is represented with a signed distance function on the block structured mesh as a result only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated except for the non boundary conforming block structured mesh the overall approach is first presented for the poisson equation and then generalized to incompressible viscous flow equations as an example of fluid structure coupling the settling of a heavy rigid particle in a closed tank is considered copyright 2010 john wiley sons ltd

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

coupled atomistic continuum simulations using arbitrary overlapping domains we present a formulation for coupling atomistic and continuum simulation methods for application to both quasistatic and dynamic analyses in our formulation a coarse scale continuum discretization is assumed to cover all parts of the computational domain with atomistic crystals introduced only in regions of interest the geometry of the discretization and crystal are allowed to overlap arbitrarily our approach uses interpolation and projection operators to link the kinematics of each region which are then used to formulate a system potential energy from which we derive coupled expressions for the forces acting in each region a hyperelastic constitutive formulation is used to compute the stress response of the defect free continuum with constitutive properties derived from the cauchy born rule a correction to the cauchy born rule is introduced in the overlap region to minimize fictitious boundary effects features of our approach will be demonstrated with simulations in one two and three dimensions

from molecular models to continuum mechanics we present here a limiting process allowing us to write some continuum mechanics models as a natural asymptotic of molecular models the approach is based on the hypothesis that the macroscopic displacement is equal to the microscopic one we carry out the corresponding calculations in the case of two body energies including higher order terms and also in the case of thomas fermi type models

continuum approach to discreteness we study analytically and numerically continuum models derived on the basis of pade approximations and their effectiveness in modeling spatially discrete systems we not only analyze features of the temporal dynamics that can be captured through these continuum approaches e g shape oscillations radiation effects and trapping but also point out ones that cannot be captured such as peierls nabarro barriers and bloch oscillations we analyze the role of such methods in providing an effective homogenization of spatially discrete as well as of heterogeneous continuum equations finally we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations as well as to compare them with other semicontinuum approximations

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

hierarchical enrichment for bridging scales and mesh free boundary conditions the finite element method when used with a basis made up of piecewise polynomials often requires the generation of a very fine computational mesh in order to capture localized solution phenomena such as boundary layers or near singularities enrichment of the basis with additional functions obtained through analytical or experimental means can allow for a coarser mesh and more accurate solution we introduce an enrichment scheme in which an interaction or bridging scale term is used to separate the basis formed by the enrichment functions from the original set of basis functions in effect making the enrichment hierarchical this separation of scales allows the simple application of essential boundary conditions it also allows a quantification of the effects of the enrichment leading to strategies for error estimation and control of the stiffness matrix condition number we also find that this formulation allows for the simple application of essential boundary conditions for mesh free shape functions which are notoriously problematic we find that for multiple dimensions care must be taken to derive a weak form which is truly consistent with the strong form on the essential boundary copyright 2001 john wiley sons ltd

isogeometric enriched field approximations boundaries with specified behavior phase boundaries crack surfaces or singular points are geometrically speaking lower dimensional features relative to two or three dimensional geometrical domains often the distinguishing characteristics of the behavior at these features are known a priori and may be exploited to enrich isogeometric models explicit geometrical representations possess parametrically computable tangents normals and curvature while in implicit strategies the geometric exactness of enriching lower dimensional features is not exploited or retrieved only in the limit of mesh refinement in the present work cad inspired hierarchical partition of unity field compositions are extended to modeling explicitly defined enrichments within the isogeometric framework the base approximations are enriched isogeometrically on parametrically defined lower dimensional geometrical features of the base entity and by constructing distance fields from them the efficiency and robustness of distance computations is significantly improved by composing monotonic distance measures defined piecewise on the enriching geometric entity using r functions the procedure allows both the behavioral approximation as well as the material description to be enriched enabling the modeling of material damage or alternatively local stiffening further the enrichments may ensure known function value or its derivative function value enrichments are demonstrated to model dirichlet boundary conditions and propagating cracks the derivative enrichments are used to model neumann boundary conditions as well as strain jumps across material interfaces the material enrichments are demonstrated through the use of a cohesive damage law to model arbitrary crack initiation and propagation within the domain

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

higher dimensional nonnested multigrid methods nonnested multigrid methods are shown to be optimal order solvers for systems of finite element equations arising from elliptic boundary problems in any space dimension results are derived for lagrange type elements of arbitrary degree

an eulerian lagrangian method for fluid structure interaction based on level sets abstract we describe a method for fluid structure interaction with the interface and free surfaces defined by level sets the fluid is treated by an eulerian mesh whereas the solid or structure by a lagrangian mesh the level set method provides a compact framework for writing the weak form and for developing and computing the discretized equations lagrange multiplier methods and penalty methods are used to couple the fluid and structure results are presented for a few compressible fluid problems representative of high pressure waves

algebraic distance estimations for enriched isogeometric analysis abstract in problems with evolving boundaries interfaces or cracks blending functions are used to enrich the underlying domain with the known behavior on the enriching entity the blending functions are typically dependent on the distance from the propagating boundaries for boundaries defined by free form curves or surfaces the distance fields have to be constructed numerically this may require either a polytope approximation to the boundary and or an iterative solution to determine the exact distance to the boundary in this paper a purely algebraic and computationally efficient technique is described for constructing distance measures from non uniform rational b splines nurbs boundaries that retain the geometric exactness of the boundaries while eliminating the need for iterative and non robust distance calculation the constructed distance measures are level sets of the implicitized constituent bezier patches of the nurbs surfaces that are obtained purely algebraically since in general the implicitized functions extend beyond the parametric range of the generating bezier patch algorithmic procedures are developed to trim these global implicit functions to the boundaries of the bezier patch boolean compositions are then carried out between adjoining bezier patches to construct a composite distance field over the domain the compositions rely on r functions that are also algebraic in nature the developed technique is demonstrated by constructing algebraic distance field for complex geometries and by solving a variety of examples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three dimensional cracks