an isoparametric approach to high order curvilinear boundary layer meshing abstract the generation of high order curvilinear meshes for complex three dimensional geometries is presently a challenging topic particularly for meshes used in simulations at high reynolds numbers where a thin boundary layer exists near walls and elements are highly stretched in the direction normal to flow in this paper we present a conceptually simple but very effective and modular method to address this issue we propose an isoparametric approach whereby a mesh containing a valid coarse discretization comprising of high order triangular prisms near walls is refined to obtain a finer prismatic or tetrahedral boundary layer mesh the validity of the prismatic mesh provides a suitable mapping that allows one to obtain very fine mesh resolutions across the thickness of the boundary layer we describe the method in detail for a high order approximation using modal basis functions discuss the requirements for the splitting method to produce valid prismatic and tetrahedral meshes and provide a sufficient criterion of validity in both cases by considering two complex aeronautical configurations we demonstrate how highly stretched meshes with sufficient resolution within the laminar sublayer can be generated to enable the simulation of flows with reynolds numbers of 10 6 and above

gibbs measures on brownian currents motivated by applications to quantum field theory we consider gibbs measures for which the reference measure is wiener measure and the interaction is given by a double stochastic integral and a pinning external potential in order properly to characterize these measures through dlr equations we are led to lift wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields we prove existence and basic properties of such gibbs measures in the small coupling regime by means of cluster expansion

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

the optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron abstract the main result of this paper is the computation of the mean optimal symmetrical interpolation points in the tetrahedron up to degree 9 this interpoation set has the smallest lebesgue constant known today

euclidean gibbs measures on loop lattices existence and a priori estimates we present a new method to prove existence and uniform a priori estimates for euclidean gibbs measures corresponding to quantum anharmonic crystals it is based first on the alternative characterization of gibbs measures in terms of their logarithmic derivatives through integration by parts formulas and second on the choice of appropriate lyapunov functionals

renormalization group and analyticity in one dimension a proof of dobrushin s theorem we consider one dimensional systems described by many body potentials with finite first moment and prove that the correlation functions are analytic in the interaction parameters this result is not new dobrushin 1973 but our proof is simpler and physically more transparent we show that by introducing suitable blocks and averaging over the variables associated to a subset of the blocks decimation procedure the resulting effective interaction is such that the system can always be dealt with as a high temperature system

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows ii efficient flux quadrature a new and efficient quadrature rule for the flux integrals arising in the space time discontinuous galerkin discretization of the euler equations in a moving and deforming space time domain is presented and analyzed the quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme the local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product gauss quadrature rules are used details of the approximation of the dissipation in the numerical flux are presented which render the scheme consistent and stable the method is successfully applied to the simulation of a three dimensional transonic flow over a deforming wing

on the rational derivation of a hierarchy of dynamic equations for a homogeneous isotropic elastic plate abstract flexural equations of motion for a homogeneous isotropic elastic plate are derived by an antisymmetric expansion in the thickness coordinate of the displacement components all but the lowest order expansion functions are eliminated with the help of the three dimensional equations of motion and are plugged into the boundary conditions eliminating between these an equation is obtained for the mean plane vertical displacement which also includes arbitrary loading on the plate surface this equation can be truncated to any order in the thickness and it is in particular noted that the corresponding dispersion relation seems to correspond to a power series expansion of the exact rayleigh lamb dispersion relation to all orders various truncations of the equation are discussed and are compared numerically with each other the exact three dimensional solution and mindlin s plate equation both the dispersion relation and the corresponding displacement components as well as an excitation problem are used for the comparisons the theories are reasonably close to each other and in order to be on the safe side none of them should in fact be used for frequencies above the cutoff of the second flexural mode

stable spectral methods on tetrahedral elements a framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented although most of the developments are of a general nature an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron in the first part the question of well behaved multivariate polynomial interpolation on the tetrahedron is addressed and it is shown how to extend the electrostatic analogy of the jacobi polynomials to problems beyond the line this allows for the identification of nodal sets suitable for polynomial interpolation within the tetrahedron and subsequently for the formulation of stable spectral schemes on such unstructured nodal sets the second part of this work is devoted to a discussion of weakly imposed boundary conditions and energy stable schemes are formulated for a wide class of problems exemplified by advection problems advection diffusion problems and linear symmetric hyperbolic systems finally in the third part issues related to computational efficiency and implementation of the schemes are discussed the spectral accuracy of the approximation is confirmed through an example and factorization methods for the efficient computation of derivatives on the general nodal sets within the d simplex are developed ensuring that the proposed schemes are competitive with tensor product based methods in this last part we also show that the advective operator results in an cal o n 2 restriction on the time step similar to that of spectral collocation methods employing a tensor product based approximation the performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain confirming the expected accuracy and stability

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

symmetric quadrature rules on a triangle abstract we present a class of quadrature rules on triangles in r 2 which somewhat similar to gaussian rules on intervals in r 1 have rapid convergence positive weights and symmetry by a scheme combining simple group theory and numerical optimization we obtain quadrature rules of this kind up to the order 30 on triangles this scheme essentially a formalization and generalization of the approach used by lyness and jespersen over 25 years ago can be easily extended to other regions in r 2 and surfaces in higher dimensions such as squares spheres we present example formulae and relevant numerical results

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

singularities of electromagnetic fields in polyhedral domains in this paper we investigate the singular solutions of time harmonic maxwell equations in a domain which has edges and polyhedral corners it is now well known that in the presence of non convex edges the solution fields have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the laplace operator 4 5 we show how this type of result can be derived from the classical mellin analysis and how this analysis leads to sharper results concerning the singular parts which belong to h 1 for the singular functions we exhibit simple and explicit formulas based on generalized dirichlet and neumann singularities for the laplace operator these formulas are more explicit than the results announced in our note 10

gibbs measures relative to brownian motion we consider brownian motion perturbed by the exponential of an action the action is the sum of an external one body potential and a two body interaction potential which depends only on the increments under suitable conditions on these potentials we establish existence and uniqueness of the corresponding gibbs measure we also provide an example where uniqueness fails because of a slow decay in the interaction potential

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle the main results of this paper are the analysis of the quality of approximation of polynomial interpolation and the computation of the approximate optimal interpolation points in the triangle we introduce various norms for the interpolation operator computational results indicate that for a given polynomial degree the set that minimizes the mean l2 norm of the interpolation operator is close to the smallest lebesgue constant interpolation set in particular for the triangle this set gives the smallest lebesgue constant currently known

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

robust untangling of curvilinear meshes this paper presents a technique that allows to untangle high order curvilinear meshes the technique makes use of unconstrained optimization where element jacobians are constrained to lie in a prescribed range through moving log barriers the untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded jacobians bounds on jacobians are computed using the results of johnen et al 2012 2013 1 2 the technique is applicable to any kind of polynomial element for surface volume hybrid or boundary layer meshes a series of examples demonstrate both the robustness and the efficiency of the technique the final example involving a time explicit computation shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure

evidence of exponential convergence in the computation of maxwell eigenvalues abstract this paper describes a finite element approach for the approximation of maxwell eigenvalues for three dimensional cavity problems hierarchical h 1 and h curl conforming discretisations are employed in conjunction with a technique that reduces the size of the problem to be solved numerical results are given on both polyhedral and curvilinear domains these results indicate that exponential rates of convergence can be obtained with the appropriate combination of h and p refinements

the effect of numerical quadrature in the p version of the finite element method we investigate the use of numerical quadrature in the p version of the finite element method we describe a set of minimal conditions that the quadrature rules should satisfy for various types of elements under sufficient assumptions of smoothness we establish optimality of the asymptotic rate of convergence some computational results are presented which illustrate under what conditions overintegration may be useful

optimizing the geometrical accuracy of curvilinear meshes this paper presents a method to generate valid high order meshes with optimized geometrical accuracy the high order meshing procedure starts with a linear mesh that is subsequently curved without taking care of the validity of the high order elements an optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two dimensional meshes but they prove computationally too costly for optimization purposes a fast estimate of the geometrical accuracy based on taylor expansions of the curves is introduced an unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes as measured by the standard hausdorff distance between the geometrical model and the mesh several examples illustrate the beneficial impact of this method on cfd solutions with a particular role of the enhanced mesh boundary smoothness

on discretely entropy stable weight adjusted discontinuous galerkin methods curvilinear meshes abstract we construct entropy conservative and entropy stable discontinuous galerkin dg discretizations for time dependent nonlinear hyperbolic conservation laws on curvilinear meshes the resulting schemes preserve a semi discrete quadrature approximation of a continuous global entropy inequality the proof requires the satisfaction of a discrete geometric conservation law which we enforce through an appropriate polynomial approximation we extend the construction of entropy conservative and entropy stable dg schemes to the case when high order curvilinear mass matrices are approximated using low storage weight adjusted approximations and describe how to retain global conservation properties under such an approximation for certain types of curvilinear meshes these weight adjusted approximations deliver optimal rates of convergence finally the high order accuracy local conservation and discrete conservation or dissipation of entropy for under resolved solutions are verified through numerical experiments for the compressible euler equations on triangular and tetrahedral meshes

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

spin boson model through a poisson driven stochastic process we give a functional integral representation of the semigroup generated by the spin boson hamiltonian by making use of a poisson point process and a euclidean field we present a method of constructing gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities using these tools we then show existence and uniqueness of the ground state of the spin boson and analyze ground state properties in particular we prove super exponential decay of the number of bosons gaussian decay of the field operators derive expressions for the positive integer fractional and exponential moments of the field operator and discuss the field fluctuations in the ground state

functional integral representations of the pauli fierz model with spin 1 2 a feynman kac type formula for a levy and an infinite dimensional gaussian random process associated with a quantized radiation field is derived in particular a functional integral representation of e thpf generated by the pauli fierz hamiltonian with spin 1 2 in non relativistic quantum electrodynamics is constructed when no external potential is applied hpf turns translation invariant and it is decomposed as a direct integral hpf r3 hpf p dp the functional integral representation of e thpf p is also given although all these hamiltonians include spin nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions as an application of the functional integral representations energy comparison inequalities are derived

fiber hamiltonians in non relativistic quantum electrodynamics abstract a translation invariant hamiltonian h of non relativistic quantum electrodynamics is studied this hamiltonian is decomposed with respect to the total momentum p t h r d h p d p where the self adjoint fiber hamiltonian h p is defined for arbitrary values of coupling constants the relationship between rotation invariance of h p and polarization vectors is discussed and functional integral representations of n point euclidean green functions of h p are given from these applications into the degeneracy of ground states ground state energy and expectation values of suitable observables with respect to ground states are given

geometry representation issues associated with p version finite element computations this paper addresses issues related to accurate geometry representation for p version finite elements on curved three dimensional domains specific options to account for domain geometry information during element level computation are identified accuracy requirements on the geometry related approximations to preserve the optimal rate of finite element error convergence for second order elliptic boundary value problems are given an element geometric mapping scheme based on blending the exact shape of the domain boundary is described that can either be used directly during element integrations or used to construct element level geometric approximations of required accuracy smoothness issues of the rational blends on simplex topologies are discussed and a numerical example based on the solution of poisson s equation in three dimensions is presented to illustrate the impact of the rational blends on the optimal rate of finite element error convergence

the approximation theory for the p version of the finite element method the purpose of this article is to present the approximation theory underlying the p version of the finite element method by exploiting the relationship between polynomial approximation and certain weighted sobolev spaces which are identified as the domains of positive real powers of the legendre operator this one dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted sobolev spaces on triangulated domains of r n it is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions inverse results are given for the special case of approximation in l 2

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

mesh generation in curvilinear domains using high order elements the ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high order algorithms in very complex geometries compact high order algorithms such as spectral element or p type finite element techniques offer the potential of high accuracy if the solution is smooth and a well behaved mapping exists between the local sub domains and a standard region in this paper we address the problem of coupling high order algorithms with unstructured mesh generation and cad representation techniques we present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings the first strategy investigates the influence of an anisometric parametric surface representation on the quality of high order unstructured meshes we show that a straightforward splitting of a standard linear mesh for use in a high order method can lead to the generation of distorted possibly invalid meshes even for simple computational domains such as cubes the second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries in the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher order information about the surface into the mesh generation using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high order discretization of the computational domain copyright 2001 john wiley sons ltd

gibbs measures for brownian paths under the effect of an external and a small pair potential we consider brownian motion in the presence of an external and a weakly coupled pair interaction potential and show that its stationary measure is a gibbs measure uniqueness of the gibbs measure for two cases is shown also the typical path behaviour the degree of mixing and some further properties are derived we use cluster expansion in the small coupling parameter

construction of gibbs measures for 1 dimensional continuum fields we study 1 dimensional continuum fields of ginzburg landau type under the presence of an external and a long range pair interaction potentials the corresponding gibbs states are formulated as gibbs measures relative to brownian motion 17 in this context we prove the existence of gibbs measures for a wide class of potentials including a singular external potential as hard wall ones as well as a non convex interaction our basic methods are i to derive moment estimates via integration by parts and ii in its finite volume construction to represent the hard wall gibbs measure on c in terms of a certain rotationally invariant gibbs measure on c 3

curved boundary treatments for the discontinuous galerkin method applied to aeroacoustic propagation a discontinuous galerkin method is applied to unstructured grids to simulate aeroacoustic propagation modeled by the linearized euler equations on triangular and tetrahedral elements with straight edges the quadrature free form of the discontinuous galerkin method is used in addition to the classical linear treatment of wall boundaries two treatments involving a second order representation of the geometry are presented the simulation of acoustic scattering problems shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved elements overcomes this restriction the benefits of higher order treatments are also assessed for more realistic geometries namely a high lift airfoil and an elliptical muffler

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

approximation of the flexural velocity branch in plates the fitting of the flexural velocity branch by means of the pade approximation is examined its use for free isotropic plates is elaborated and compared to other approaches the flexural velocity expansion into power series which underlies the pade implementation is detailed the generalization for fluid loaded plates and inhomogeneous plates is outlined and the impact of anisotropy is discussed in some detail

arbitrary order edge elements for electromagnetic scattering simulations using hybrid meshes and a pml the use of arbitrary order edge elements for the simulation of two dimensional electromagnetic scattering problems on hybrid meshes of triangles and quadrilaterals is described single frequency incident waves generated by a source in the far field are considered and the solution is determined in the frequency domain for numerical simulation the solution domain is truncated at a finite distance from the perfectly conducting scatterer and the non reflecting boundary condition at the truncated boundary is imposed by the use of a perfectly matched layer pml several examples are included to demonstrate the performance of the proposed procedure copyright 2002 john wiley sons ltd

nurbs enhanced finite element method nefem the development of nurbs enhanced finite element method nefem is revisited this technique allows a seamless integration of the cad boundary representation of the domain and the finite element method fem the importance of the geometrical model in finite element simulations is addressed and the benefits and potential of nefem are discussed and compared with respect to other curved finite element techniques

on the so called variational consistency of plate models i indefinite plates evaluation of dispersive behaviour abstract four groups of refined shear deformation plate theories are examined including the so called variationally inconsistent theories of schmidt levinson and touratier whose inconsistence is in fact weak after a unified presentation of these two dimensional theories all are deduced from the three dimensional theory of elasticity through a suitable variation procedure which disconnects the choice of the so called virtual motions from the choice of an approximation for the small displacement field this procedure makes obvious in a natural way the importance of the stress boundary conditions on the faces of the pate for any two dimensional theory as a test comparison a wave dispersion evaluation of these theories is performed for the propagation of flexural waves in an indefinite plate a detailed comparison leads to the conclusion that the inconsistent theories may offer some advantages in the domain of short waves

a quantum crystal with multidimensional anharmonic oscillators the quantum anharmonic crystal is made up of a large number of multidimensional anharmonic oscillators arranged in a periodic spatial lattice with a nearest neighbor coupling if the coupling coefficient is sufficiently small then there is a convergent expansion for the ground state of the crystal the estimates on the convergence are independent of the size of the crystal the proof uses the path integral representation of the ground state in terms of diffusion processes the convergence of the cluster expansion depends on the ergodicity properties of these processes

geometrical validity of curvilinear finite elements in this paper we describe a way to compute accurate bounds on jacobian determinants of curvilinear polynomial finite elements our condition enables to guarantee that an element is geometrically valid i e that its jacobian determinant is strictly positive everywhere in its reference domain it also provides an efficient way to measure the distortion of curvilinear elements the key feature of the method is to expand the jacobian determinant using a polynomial basis built using bezier functions that has both properties of boundedness and positivity numerical results show the sharpness of our estimates