an anisotropic h type mesh refinement strategy abstract an anisotropic h type mesh refinement strategy for finite element approximations is presented the strategy consists of subsecting quadrilateral elements in one of two possible directions the decision which elements should be subsected and in which direction is based on minimization of interpolation errors the method is especially suitable for approximate solutions which in some regions display almost one dimensional behavior for instance solutions with boundary layers

an adaptive finite element method for viscoplastic fluid flows in pipes the numerical modelling of the fully developed poiseuille flow of a yield stress fluid in a square section is presented the dead regions in outer corners and the plug region in the center are exhibited numerical computations cover the complete range of the dimensionless number describing the yield stress effect from a newtonian flow to a fully stopped flow the resolution of variational inequalities describing the flow is based on the augmented lagrangian method and a specific mixed finite element method the localization of yield surfaces is approximated by an anisotropic auto adaptive mesh procedure the limit load analysis and the associated limit yield surface are obtained by an extrapolation procedure

quality local refinement of tetrahedral meshes based on bisection we present a three dimensional 3 d local refinement algorithm for tetrahedral meshes based on a bisection procedure the quality of refined meshes is guaranteed in terms of a tetrahedron shape measure eta specifically it is proved that the algorithm creates a finite number which only depends on the number of tetrahedra in the initial mesh of classes of similar tetrahedra in all refined tetrahedra furthermore if bf t is a tetrahedron in the original mesh and bf t i n is any refined tetrahedron of bf t then eta bf t i n geq c eta bf t where c is a positive constant independent of bf t and the number of refinements it is also proved that for any interior face in the final mesh the absolute value of the difference of the bisection levels of the two adjacent tetrahedra incident on the face is leq 2 which indicates that local refinements on tetrahedra can be smoothly extended to their neighbors the expected time complexity of the algorithm is o n wh

adaptive cfd analysis for rotorcraft aerodynamics abstract an adaptive parallel computational fluid dynamics cfd technique is developed to compute rotor blade aerodynamics for this purpose an error indicator is formulated and coded into an adaptive finite element framework it is shown that the error indicator is effective in resolving the global features of the flow field furthermore for efficiency and problem size considerations once the interpolation errors are reduced to acceptable levels the adaptive refinement is done only in regions affected by the vortical flows to accomplish this a novel vortex core detection technique is used to capture vortex tubes the combination of interpolation error estimate and vortex core detection technique proved to be very effective in computing vortical flow field of rotor blades example adaptive parallel calculations of hovering rotor blades requiring 1 3 million tetrahedral elements are presented

surface shape from the deformation of apparent contours the spatiotemporal analysis of deforming silhouettes apparent contours is here extended using the mathematics of perspective projections and tools from differential geometry analysis of the image motion of a silhouette or apparent contour enables computation of local surface curvature along the corresponding contour generator on the surface assuming viewer motion is known to perform the analysis a spatiotemporal parameterization of image curve motion is needed but is underconstrained a manifestation of the well known aperture problem it is shown that an epipolar parameterization is most naturally matched to the recovery of surface curvature

parallel adaptive finite element euler flow solver for rotary wing aerodynamics we discuss the development of a code for the automated reliable numerical simulation of rotor aerodynamics the code performs all the stages of the analysis finite element solution error indication mesh adaptation in a parallel multiple instruction multiple data environment making use of efficient scalable parallel algorithms the main features of all the building blocks of the implemented approach are detailed and discussed the capabilities and performance characteristics of the parallel automated adaptive procedure are demonstrated with regard to subsonic and transonic compressible flow problems in particular different hovering rotor problems are analyzed

metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing metric tensors play a key role to control the generation of unstructured anisotropic meshes in practice the most well established error analysis enables to calculate a metric tensor on an element basis in this paper we propose to build a metric field directly at the nodes of the mesh for a direct use in the meshing tools first the unit mesh metric is defined and well justified on a node basis by using the statistical concept of length distribution tensors then the interpolation error analysis is performed on the projected approximate scalar field along the edges the error estimate is established on each edge whatever the dimension is it enables to calculate a stretching factor providing a new edge length distribution its associated tensor and the corresponding metric the optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh several examples of interpolation error are proposed as well as preliminary results of anisotropic adaptation for interface and free surface problem using a level set method

foundations of semi differential invariants this paper elaborates the theoretical foundations of a semi differential framework for invariance semi differential invariants combine coordinates and their derivatives with respect to some contour parameter at several points of the image contour thus allowing for an optimal trade off between identification of points and the calculation of derivatives a systematic way of generating complete and independent sets of such invariants is presented it is also shown that invariance under reparametrisation can be cast in the same framework the theory is illustrated by a complete analysis of 2d affine transformations in a companion paper pauwels et al 1995 these affine semi differential invariants are implemented in the computer program form flat object recognition method for the recognition of planar contours under pseudo perspective projection

efficient solution algorithms for the riemann problem for real gases a procedure for constructing solutions to the riemann problem for gas dynamics with a general convex equation of state is given approximate procedures involving a local parametrization of the equation of state are introduced in order to calculate numerical fluxes in conservative finite difference schemes this leads to difference schemes which are as accurate and almost as fast as the analogous schemes for polytropic gases numerical results in one and two space variables are presented

bounded and periodic solutions of differential equations in banach space abstract we study the equation u t au t f t t 0 1 in a banach space x with a the generator of an analytic or a strongly continuous semigroup s and prove that if solutions of 1 are bounded and ultimate bounded with f t periodic and s t compact then 1 has a t periodic solution we also show that the existence of a proper liapunov function implies the boundedness and ultimate boundedness of solutions of 1 these results extend earlier results in finite dimensional spaces we then apply the results to a parabolic partial differential equation u t t x 2m c x d u t x f t x 2

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

monotone hopf harmonics we introduce the concept of monotone hopf harmonics in 2d as an alternative to harmonic homeomorphisms much of the foregoing is motivated by the principle of non interpenetration of matter in the mathematical theory of nonlinear elasticity ne the question we are concerned with is whether or not a dirichlet energy minimal mapping between jordan domains with a prescribed boundary homeomorphism remains injective in the domain the classical theorem of rado kneser choquet asserts that this is the case when the target domain is convex an alternative way to deal with arbitrary target domains is to minimize the dirichlet energy subject to only homeomorphisms and their limits this leads to the so called hopf laplace equation among its solutions some rather surreal are continuous monotone mappings of sobolev class mathscr w 1 2 text loc called monotone hopf harmonics it is at the heart of the present paper to show that such solutions are correct generalizations of harmonic homeomorphisms and in particular are legitimate deformations of hyperelastic materials in the modern theory of ne we make this clear by means of several examples

solution adaptation of unstructured grids for two dimensional aerodynamic computations a solution adaptive gridding technique applicable to the numerical solution of the navier stokes equations using unstructured triangular grids is presented and evaluated grid adaptation is directed by a parameter based on undivided differences of density and mach number normalized with their respective grid extrema a new solution dependent retriangulation algorithm locally restructures the grid to recover an anisotropic grid following adaptation the artificial dissipation operator is modified to provide improved accuracy on irregular high aspect ratio triangular grids that may be created in some regions by the adaptive process the technique is applied to several subsonic and transonic turbulent aerodynamic flows including the flow over a three element high lift configuration grid convergence studies are presented that demonstrate that the grid adaptation procedure leads to significant error reduction in lift and drag coefficient in most circumstances the effectiveness of the adaptation is significantly improved as a result of the solution dependent retriangulation and the improved artificial dissipation operator

differential and numerically invariant signature curves applied to object recognition we introduce a new paradigm the differential invariant signature curve or manifold for the invariant recognition of visual objects a general theorem of e cartan implies that two curves are related by a group transformation if and only if their signature curves are identical the important examples of the euclidean and equi affine groups are discussed in detail secondly we show how a new approach to the numerical approximation of differential invariants based on suitable combination of joint invariants of the underlying group action allows one to numerically compute differential invariant signatures in a fully group invariant manner applications to a variety of fundamental issues in vision including detection of symmetries visual tracking and reconstruction of occlusions are discussed

an anisotropic h adaptive finite element method for compressible navier stokes equations abstract application of the finite element method with significantly stretched elements to solve compressible navier stokes equations is presented stretched elements are introduced in an adaptive process as a result of selective directional subsecting of elements an adaptive strategy of such anisotropic h type refinements was developed it is based on minimization of the interpolation errors reduction of the residual error indicators and an adaptive recovery of the viscous fluxes with the possibility to control their accuracy an iterative solver with the convergence characteristics independent of the aspect ratio of elements is presented the solver is the gmres algorithm with the domain decomposition schwarz type preconditioner the proposed techniques are used to solve a benchmark problem a hypersonic high reynolds number flow with strong shock boundary layer interaction

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

using extremal boundaries for 3 d object modeling the extremal boundaries of 3 d curved objects are the images of special curves drawn on the object and are called rims they are viewpoint dependent and characterized by the fact that the optical rays of their points are tangential to the surface of the object the mathematics of the relationship between the extremal boundaries and the surface of the object is studied this study makes it possible to design an algorithm for detecting those boundaries in the images that are likely to be extremal once this has been done one can reconstruct the rims and compute the differential properties of the surface of the object along them up to the second order if a qualitative description is sufficient the sign of the gaussian curvature of the surface along the rim can be computed in a much simpler way experimental results are presented on synthetic and real images the work provides a better understanding of the relationship between the apparent and real shape of a 3 d object as well as algorithms for reconstructing the local shape of such an object along the rims

the fundamental matrix theory algorithms and stability analysis in this paper we analyze in some detail the geometry of a pair of cameras i e a stereo rig contrarily to what has been done in the past and is still done currently for example in stereo or motion analysis we do not assume that the intrinsic parameters of the cameras are known coordinates of the principal points pixels aspect ratio and focal lengths this is important for two reasons first it is more realistic in applications where these parameters may vary according to the task active vision second the general case considered here captures all the relevant information that is necessary for establishing correspondences between two pairs of images this information is fundamentally projective and is hidden in a confusing manner in the commonlyused formalism of the essential matrix introduced by longuet higgins 1981 this paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 x 3 matrix of rank 2 which we propose to call the fundamental matrix after this theoretical analysis we embark on the task of estimating the fundamental matrix from point corre spondences a task which is of practical importance we analyze theoretically and compare experimentally using synthetic and real data several methods of estimation the problem of the stability of the estimation is studied from two complementary viewpoints first we show that there is an interesting relationship between the fundamental matrix and three dimensional planes which induce homographies between the images and create unstabilities in the estimation procedures second we point to a deep relation between the unstability of the estimation procedure and the presence in the scene of so called critical surfaces which have been studied in the context of motion analysis finally we conclude by stressing the fact that we believe that the fundamental matrix will play a crucial role in future applications of three dimensional computer vision by greatly increasing its versatility robustness and hence applicability to real difficult problems

schwarps locally projective image warps based on 2d schwarzian derivatives image warps or just warps capture the geometric deformation existing between two images of a deforming surface the current approach to enforce a warp s smoothness is to penalize its second order partial derivatives bookstein in ieee trans pattern anal mach intell 11 567 585 1989 rueckert et al in ieee trans med imaging 18 712 721 1999 because this favors locally affine warps this fails to capture the local projective component of the image deformation this may have a negative impact on applications such as image registration and deformable 3d reconstruction we propose a novel penalty designed to smooth the warp while capturing the deformation s local projective structure our penalty is based on equivalents to the schwarzian derivatives which are projective differential invariants exactly preserved by homographies we propose a methodology to derive a set of partial differential equations with only homographies as solutions we call this system the schwarzian equations and we explicitly derive them for 2d functions using differential properties of homographies we name as schwarp a warp which is estimated by penalizing the residual of schwarzian equations experimental evaluation shows that schwarps outperform existing warps in modeling and extrapolation power and lead to better results in three deformable reconstruction methods namely shape reconstruction in shape from template camera calibration in shape from template and non rigid structure from motion

the local projective shape of smooth surfaces and their outlines this article examines projectively invariant local geometric properties of smooth curves and surfaces oriented projective differential geometry is proposed as a general framework for establishing such invariants and characterizing the local projective shape of surfaces and their outlines it is applied to two problems 1 the projective generalization of koenderink s famous characterization of convexities concavities and inflections of the apparent contours of solids bounded by smooth surfaces and 2 the image based construction of rim meshes which provide a combinatorial description of the arrangement induced on the surface of an object by the contour generators associated with multiple cameras observing it

adaptive multivlevel methods in three space dimensions we consider the approximate solution of self adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non uniform tetrahedral mesh which is generated adaptively the arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner here the accuracy of the iterative solution is coupled with the discretization error as the performance of hierarchical bases preconditioners deteriorates in three space dimensions the bpx preconditioner is used taking special care of an efficient implementation reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements to illustrate the theoretical results we consider a familiar model problem involving reentrant corners and a real life problem arising from hyperthermia a recent clinical method for cancer therapy

anisotropic adaptive simulation of transient flows using discontinuous galerkin methods an anisotropic adaptive analysis procedure based on a discontinuous galerkin finite element discretization and local mesh modification of simplex elements is presented the procedure is applied to transient two and three dimensional problems governed by euler s equation a smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified the mesh metric field in smooth portions of the domain is controlled by a hessian matrix constructed using a variational procedure to calculate the second derivatives the transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain copyright c 2004 john wiley sons ltd

tetrahedral mesh optimisation and adaptivity for steady state and transient finite element calculations abstract a method for optimising a pre existing mesh of tetrahedral finite elements is described it is based on a series of mesh connectivity and node position searches of the landscape defining mesh quality a riemannian metric reflecting the a posteriori error measure is used to calculate element size and shape a functional is defined which embodies both shape and size quality of an element with respect to the metric and is used to gauge mesh quality a heuristic based local search strategy is adopted local in the sense that it has no hill climbing abilities the paper presents applications of the method to complex steady state and time dependent problems which highlight its anisotropic feature capturing abilities numerical evidence is provided which suggests that the computational complexity time of the proposed algorithm varies linearly with the number of elements or nodes of the finite element mesh

projective visual hulls this article presents a novel method for computing the visual hull of a solid bounded by a smooth surface and observed by a finite set of cameras the visual hull is the intersection of the visual cones formed by back projecting the silhouettes found in the corresponding images we characterize its surface as a generalized polyhedron whose faces are visual cone patches edges are intersection curves between two viewing cones and vertices are frontier points where the intersection of two cones is singular or intersection points where triples of cones meet we use the mathematical framework of oriented projective differential geometry to develop an image based algorithm for computing the visual hull this algorithm works in a weakly calibrated setting that is it only requires projective camera matrices or equivalently fundamental matrices for each pair of cameras the promise of the proposed algorithm is demonstrated with experiments on several challenging data sets and a comparison to another state of the art method

high order discontinuous galerkin schemes on general 2d manifolds applied to the shallow water equations an innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous galerkin method this new approach combines the advantages of the usual approaches described in the literature the vectorial fields are expressed in a unit non orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements while the convective flux functions are written is the usual 3d cartesian coordinate system the number of vectorial components is therefore minimum and the tangency constraint is naturally ensured while the method remains robust and general since not relying on a particular parametrization of the manifold the discontinuous galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition the possible discontinuities of this basis are then taken into account in the riemann solver on inter element interfaces the approach is validated on the sphere using the shallow water equations for computing standard atmospheric benchmarks in particular the williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error the propagation of gravity waves is eventually computed on non conventional irregular curved manifolds to illustrate the robustness and generality of the method

adaptive mesh generation for viscous flows using delaunay triangulation abstract a method for generating an unstructured triangular mesh in two dimensions suitable for computing high reynolds number flows over arbitrary configurations is presented the method is based on a delaunay triangulation which is perfored in a locally stretched space in order to obtain very high aspect ratio tiangles in the boundary layer and wake regions it is shown how the method can be coupled with an unstructured navier stokes solver to produce a solution adaptive mesh generation procedure for viscous flows

monotone sobolev mappings of planar domains and surfaces an approximation theorem of youngs duke math j 15 87 94 1948 asserts that a continuous map between compact oriented topological 2 manifolds surfaces is monotone if and only if it is a uniform limit of homeomorphisms analogous approximation of sobolev mappings is at the very heart of geometric function theory gft and nonlinear elasticity ne in both theories the mappings in question arise naturally as weak limits of energy minimizing sequences of homeomorphisms as a result of this the energy minimal mappings turn out to be monotone in the present paper we show that conversely monotone mappings in the sobolev space fancyscript w 1 p 1 p infty are none other than fancyscript w 1 p weak also strong limits of homeomorphisms in fact these are limits of diffeomorphisms by way of illustration we establish the existence of traction free energy minimal deformations for p harmonic type energy integrals

global invertibility for orientation preserving sobolev maps via invertibility on or near the boundary by a result of ball proc r soc edinb sect a math 88 315 328 1981 https doi org 10 1017 s030821050002014x a locally orientation preserving sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension as shown here for any dimension the conclusions of ball s theorem and related results can be reached while completely avoiding the problem of homeomorphic extension for suitable domains it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps an application in nonlinear elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed as a tool in the proofs strictly orientation preserving maps and their global invertibility properties are studied from a purely topological point of view

a theory of the motion fields of curves this article reports a study of the motion field generated by moving 3 d curves that are observed by a camera we first discuss the relationship between optical flow and motion field and show that the assumptions made in the computation of the optical flow are a bit difficult to defend

the visual hull concept for silhouette based image understanding many algorithms for both identifying and reconstructing a 3 d object are based on the 2 d silhouettes of the object in general identifying a nonconvex object using a silhouette based approach implies neglecting some features of its surface as identification clues the same features cannot be reconstructed by volume intersection techniques using multiple silhouettes of the object this paper addresses the problem of finding which parts of a nonconvex object are relevant for silhouette based image understanding for this purpose the geometric concept of visual hull of a 3 d object is introduced this is the closest approximation of object s that can be obtained with the volume intersection approach it is the maximal object silhouette equivalent to s i e which can be substituted for s without affecting any silhouette only the parts of the surface of s that also lie on the surface of the visual hull can be reconstructed or identified using silhouette based algorithms the visual hull depends not only on the object but also on the region allowed to the viewpoint two main viewing regions result in the external and internal visual hull in the former case the viewing region is related to the convex hull of s in the latter it is bounded by s the internal visual hull also admits an interpretation not related to silhouettes algorithms for computing visual hulls are presented and their complexity analyzed in general the visual hull of a 3 d planar face object turns out to be bounded by planar and curved patches

univalent σ harmonic mappings we study mappings from 2 into 2 whose components are weak solutions to the elliptic equation in divergence form div u 0 which we call harmonic mappings we prove sufficient conditions for the univalence i e injectivity of such mappings moreover we prove local bounds in bmo on the logarithm of the jacobian determinant of such univalent mappings thus obtaining the a e nonvanishing of their jacobian in particular our results apply to harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization

on the shape of tetrahedra from bisection we present a procedure for bisecting a tetrahedron t successively into an infinite sequence of tetrahedral meshes j 0 91 g2 which has the following properties 1 each mesh sfn is conforming 2 there are a finite number of classes of similar tetrahedra in all the 3n n 0 3 for any tetrahedron t7 in s7n jj t cl1 t where ti is a tetrahedron shape measure and cl is a constant 4 3 j c2 1 2 ni3j t where d t denotes the diameter of tetrahedron t1 and c2 is a constant estimates of cl and c2 are provided properties 2 and 3 extend similar results of stynes and adler and of rosenberg and stenger respectively for the 2 d case the diameter bound in property 4 is better than one given by kearfott

on p harmonic functions in the plane and their stream functions on considere l equation p harmonique p div p 2 0 1 p dans le plan on developpe un certain nombre d analogies avec la theorie de base des fonctions complexes dans c

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians