cellular automata generalisation of the weisbuch atlan model for immune response taking into account spatial fluctuations in the discrete model for certain types of immune response the model of weisbuch and atlan 1988 is studied on a square lattice in general the authors end up with a healthy carrier state 29 where at all sites all cell types except activated killer cells are present they study the time it takes to reach this fixed point as well as effects of disorder and noise

a collection of 2d elliptic problems for testing adaptive grid refinement algorithms adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations pdes the development of new algorithms and computer codes for the solution of pdes usually involves the use of proof of concept test problems 2d elliptic problems are often used as the first test bed for new algorithms and codes this paper contains a set of twelve parametrized 2d elliptic test problems for adaptive grid refinement algorithms and codes the problems exhibit a variety of types of singularities near singularities and other difficulties

toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

adaptive computations using material forces and residual based error estimators on quadtree meshes quadtree is a hierarchical data structure that is well suited for h adaptive mesh refinement due to the presence of hanging nodes classical shape functions are non conforming on quadtree meshes in this paper we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes to this end the recently proposed construction of polygonal basis functions is adapted to quadtree elements a fast technique for calculating stiffness matrix on quadtree meshes is introduced residual based error estimators and material force technique are used to estimate the error on quadtree meshes the performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary value problems

analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes the equilibrated residual method is now accepted as the best residual type a posteriori error estimator nevertheless there remains a gap in the theory and practice of the method the present work tackles the problem of existence construction and stability of equilibrated fluxes for hp finite element approximation on hybrid meshes consisting of quadrilateral and triangular elements with hanging nodes a practical algorithm for post processing the finite element approximation is presented and shown to produce equilibrated fluxes for a general one irregular partition the resulting fluxes are shown to be stable in the sense that the associated error estimator provides a lower bound on the local error which does not degenerate with the mesh size numerical examples are included to illustrate the theoretical results

adaptive finite element solution of multiscale pde ode systems we consider adaptive finite element methods for a multiscale system consisting of a macroscale model comprising a system of reaction diffusion partial differential equations coupled to a microscale

element wise a posteriori estimates based on hierarchical bases for non linear parabolic problems we show that the issue of a posteriori estimate the errors in the numerical simulation of non linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right hand side depending on known data of the problem and the computed numerical solution a procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right hand side the residual of the p fem solution is introduced the boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs we prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation when the error being estimated is that of a elliptic problem constant free local lower bounds are obtained the local error estimation procedure is applied to non linear parabolic differential equations in several space dimensions some numerical experiments for both the elliptic and the non linear parabolic cases are provided copyright 2005 john wiley sons ltd

a low order nonconforming method for linear elasticity on general meshes in this work we construct a low order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions the method is obtained by hacking the hybrid high order method of di pietro and ern cmame 2015 283 1 21 that requires the use of polynomials of degree k ge 1 for stability specifically we show that coercivity can be recovered for k 0 by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces this term plays a key role in the fulfillment of a discrete korn inequality on broken polynomial spaces for which a novel proof valid for general polyhedral meshes is provided locking free error estimates are derived for both the energy and the l 2 norms of the error that are shown to convergence for smooth solutions as h and h 2 respectively here h denotes the meshsize a thorough numerical validation on a complete panel of two and three dimensional test cases is provided

a hierarchical domain decomposition preconditioner for h p finite element approximation on locally refined meshes a domain decomposition preconditioner suitable for h p finite element approximation on locally refined meshes with nonuniform polynomial degree is proposed and analysed the preconditioner is highly suited for parallel computation the algorithm generalizes methods proposed by smith domain decomposition algorithms for the partial differential equations of linear elasticity ph d thesis mathematics department new york university new york 1991 for the h version finite element method with piecewise linear basis functions and by mandel comput methods appl mech engrg 80 1990 pp 117 128 for the p version finite element method the analysis shows that the condition number of the preconditioned system grows at most as min h h 1 log 2p 1 log 2 hp h this result generalizes the known estimates in each of the special cases mentioned above numerical examples are given confirming the theoretical analysis

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

the dpg star method abstract this article introduces the dpg star from now on denoted dpg finite element method it is a method that is in some sense dual to the discontinuous petrov galerkin dpg method the dpg methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem in the same vein the dpg methodology is a means to solve an underdetermined discretization these two viewpoints are developed by embedding the same operator equation into two different saddle point problems the analyses of the two problems have many common elements comparison to other methods in the literature round out the newly garnered perspective notably dpg and dpg methods can be seen as generalizations of l l and least squares methods respectively a priori error analysis and a posteriori error control for the dpg method are considered in detail reports of several numerical experiments are provided which demonstrate the essential features of the new method a notable difference between the results from the dpg and dpg analyses is that the convergence rates of the former are limited by the regularity of an extraneous lagrange multiplier variable

unsymmetric multi level hanging nodes and anisotropic polynomial degrees in h1 conforming higher order finite element methods abstract the implementation of higher order finite element schemes that can handle multi level hanging nodes is known to be a difficult task in fact most of the available literature on hanging nodes in finite element schemes restricts to one level hanging nodes resulting from symmetric bisections the intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a h 1 conforming higher order finite element method the corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric multi level hanging nodes and arbitrary anisotropic polynomial degree distributions where special care is given to possible orientation problems the meshes may even result from non matching refinements given these data structures and algorithms an extension for serendipity spaces is described in detail along with some other techniques to improve computational efficiency numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher order finite element schemes have to offer to the best of our knowledge this paper offers the most comprehensive numerical results so far for various three dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as multi level hanging nodes most notably the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates however the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen full tensor product or serendipity finite element spaces in addition to these numerical results we find that systems of equations defined via finite element spaces using the nodal lagrange polynomials with gauss lobatto quadrature points as support points yield drastically improved condition numbers

systematics of the models of immune response and autoimmune disease a dynamical model of normal immune response has been formulated in terms of cellular automata by kaufmanet al we generalize this model incorporating the antigens as a dynamical variable this generalized model not only describes the kinetics of primary and secondary responses of humoral immunity together with the appropriate memory cells but also describes the vaccinated state as well as the states of low dose and high dose paralysis recently models of autoimmune response have also been developed in terms of discrete automata but the models are underdetermined by the experimental facts i e several models can account for the same set of observed biological facts with an aim to find out how large this underdeterminacy is and how it can be reduced systematically we have carried out an exhaustive computer aided search of all those discrete three cell and five cell models of autoimmune response which at present cannot be ruled out by the existing biological informations out of the 325 possible five cell models only one fulfilled our criteria we also carried out simulations of the dynamics of some of these models on a discrete lattice we discuss the relevance of random interactions in the context of autoimmune disease

control of the immune response by a threshold automata model on a lattice abstract the kut mean field model for the immune response is generalized on a lattice using boolcan and threshold automata it is shown that the lattice model can exhibit properties that the mean field model cannot even if the biological meaning of the lattice geometry is dubious the attractors of the mean field model become unstable under mixing of populations and the system may reach the paralysis state instead of the memory vaccinated the probability of vaccination is calculated analytically and checked numerically a threshold automata model which includes interactions with the antigen and uses random neighbours and random time delays is proposed allowing control of the immune response by the concentration of antigen finally quantitative results of the dynamics are given using the model itself unlike previous works that used differential equations the numerical results show low and high dose paralysis versus medium dose vaccination

non uniform order mixed fem approximation implementation post processing computable error bound and adaptivity the present work provides a straightforward and focused set of tools and corresponding theoretical support for the implementation of an adaptive high order finite element code with guaranteed error control for the approximation of elliptic problems in mixed form the work contains details of the discretisation using non uniform order mixed finite elements of arbitrarily high order a new local post processing scheme for the primary variable the use of the post processing scheme in the derivation of new fully computable bounds for the error in the flux variable and an hp adaptive refinement strategy based on the a posteriori error estimator numerical examples are presented illustrating the results obtained when the procedure is applied to a challenging problem involving a ten pole electric motor with singularities arising from both geometric features and discontinuities in material properties the procedure is shown to be capable of producing high accuracy numerical approximations with relatively modest numbers of unknowns

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

potts ferromagnet transformations and critical exponents in planar hierarchical lattices we prove that the duality transformation for a potts ferromagnet on two rooted planar hierarchical lattices hl preserves the thermal eigenvalue this leads to a relation between the correlation length critical exponents of a hl and its corresponding dual lattice using hyperscaling we show that their specific heat critical exponents coincide for a smaller class of hl namely of diamond and tress types we prove that another transformation also preserves and

a finite element method with mesh separation based approximation technique and its application in modeling crack propagation with adaptive mesh refinement summary this paper presents a fem with mesh separation based approximation technique that separates a standard element into three geometrically independent elements a dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation the novel technique makes it very easy for mesh generation of problems with complex or solution dependent varying geometry it offers a flexible way to construct displacement approximations and provides a unified framework for the fem to enjoy some of the key advantages of the hansbo and hansbo method the meshfree methods the semi analytical fems and the smoothed fem for problems with evolving discontinuities the method enables the devising of an efficient crack tip adaptive mesh refinement strategy to improve the accuracy of crack tip fields both the discontinuities due to intra element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level the effectiveness and robustness of the present method are benchmarked with several numerical examples the numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test and it is essential to apply local adaptive mesh refinement for low fracture energy problems copyright 2014 john wiley sons ltd

multi level hp adaptivity for cohesive fracture modeling summary discretization induced oscillations in the load displacement curve are a well known problem for simulations of cohesive crack growth with finite elements the problem results from an insufficient resolution of the complex stress state within the cohesive zone ahead of the crack tip this work demonstrates that the hp version of the finite element method is ideally suited to resolve this complex and localized solution characteristic with high accuracy and low computational effort to this end we formulate a local and hierarchic mesh refinement scheme that follows dynamically the propagating crack tip in this way the usually applied static a priori mesh refinement along the complete potential crack path is avoided which significantly reduces the size of the numerical problem studying systematically the influence of h refinement p refinement and hp refinement we demonstrate why the suggested hp formulation allows to capture accurately the complex stress state at the crack front preventing artificial snap through and snap back effects this allows to decrease significantly the number of degrees of freedom and the simulation runtime furthermore we show that by combining this idea with the finite cell method the crack propagation within complex domains can be simulated efficiently without resolving the geometry by the mesh copyright 2016 john wiley sons ltd

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

finite difference methods and spatial a posteriori error estimates for solving parabolic equations in three space dimensions on grids with irregular nodes adaptive methods for solving systems of partial differential equations have become widespread much of the effort has focused on finite element methods in this paper modified finite difference approximations are obtained for grids with irregular nodes the modifications are required to ensure consistency and stability asymptotically exact a posteriori error estimates of the spatial error are presented for the finite difference method these estimates are derived from interpolation estimates and are computed using central difference approximations of second derivatives of the solution at grid nodes the interpolation error estimates are shown to converge for irregular grids while the a posteriori error estimates are shown to converge for uniform grids computational results demonstrate the convergence of the finite difference method and a posteriori error estimates for cases not covered by the theory

an hp finite element adaptive scheme to solve the laplace model for fluid solid vibrations abstract in this paper we introduce an hp finite element method to solve a two dimensional fluid structure spectral problem this problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid we prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues we define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair we show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree we present an hp adaptive algorithm and several numerical tests which show the performance of the scheme including some numerical evidence of exponential convergence

irrationality proof of certain lambert series using little q jacobi polynomials we apply the pade technique to find rational approximations to h q 1 q 2 email protected k 1 q 1 k1 q 2 k 0

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

essential boundary conditions and multi point constraints in finite element analysis the application of boundary conditions and other constraints to the stiffness matrix and load vector is an integral part of any finite element code this process is usually trivial but can present difficulties to the point where some codes disallow certain combinations of commonly occurring boundary conditions a general approach to this problem is presented and applied to representative examples necessary and sufficient conditions are obtained under which constraints can be applied sequentially bounds are obtained for the condition number of the reduced system showing it is essentially the same as for the original problem

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

graded parametric cutfem and cutiga for elliptic boundary value problems in domains with corners we develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neig

pyramid algorithms for bernstein bezier finite elements of high nonuniform order in any dimension the archetypal pyramid algorithm is the de casteljau algorithm which is a standard tool for the evaluation of bezier curves and surfaces pyramid algorithms replace an operation on a single high order polynomial by a recursive sequence of self similar affine combinations and are ubiquitous in computer aided geometric design for computations involving high order curves and surfaces pyramid algorithms have received no attention whatsoever from the high or low order finite element community we develop and analyze pyramid algorithms for the efficient handling of all of the basic finite element building blocks including the assembly of the element load vectors and element stiffness matrices the complexity of the algorithm for generating the element stiffness matrix is optimal a new nonuniform order variant of the de casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm the work provides the

a locking free hp dpg method for linear elasticity with symmetric stresses we present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size h and polynomial degree p this is achieved within the recently developed discontinuous petrov galerkin dpg framework in this framework both the stress and the displacement approximations are discontinuous across element interfaces we study locking free convergence properties and the interrelationships between the two dpg methods

a polytree based adaptive approach to limit analysis of cracked structures abstract we in this paper present a novel adaptive finite element scheme for limit analysis of cracked structures the key idea is to develop a general refinement algorithm based on a so called polytree mesh structure the method is well suited for arbitrary polygonal elements and furthermore traditional triangular and quadrilateral ones which are considered as special cases also polytree meshes are conforming and can be regarded as a generalization of quadtree meshes for the aim of this paper we restrict our main interest in plane strain limit analysis to von mises type materials yet its extension to a wide class of other solid mechanics problems and materials is completely possible to avoid volumetric locking we propose an approximate velocity field enriched with bubble functions using wachspress coordinates on a primal mesh and design carefully strain rates on a dual mesh level an adaptive mesh refinement process is guided by an l 2 norm based indicator of strain rates through numerical validations we show that the present method reaches high accuracy with low computational cost this allows us to perform large scale limit analysis problems favorably

a preconditioner based on domain decomposition for h p finite element approximation on quasi uniform meshes the problem of solving the algebraic systems arising from the discretization of a symmetric elliptic boundary value problem using h p finite element methods in two dimensions is addressed a preconditioning technique similar to those of bramble pasciak and schatz math comp 47 1986 pp 103 135 based on domain decomposition also known as substructuring is developed the method is applicable to problems on general domains involving differential operators with quite general coefficients the algorithm reduces to that of bramble pasciak and schatz if linear elements are used it is shown that the condition number of the preconditioned system which determines the rate of convergence of the iterative method grows at most as 1 log 2 p 1 log 2 hp h where p is the polynomial degree h is the size of the elements and h is the size of the subdomains

biorthogonal basis functions in hp adaptive fem for elliptic obstacle problems abstract in this paper the discretization of a non symmetric elliptic obstacle problem with h p adaptive h 1 conforming finite elements is discussed for this purpose a higher order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions the h p adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper the use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi smooth newton scheme which can be modified to ensure global convergence h p adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the h 1 conforming finite element discretization this is typically done by using so called connectivity matrices in this paper a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions finally the applicability of the theoretical findings is demonstrated with several numerical experiments

higher order conformal decomposition fem cdfem abstract a higher order accurate finite element method is proposed which uses automatically generated meshes based on implicit level set data for the description of boundaries and interfaces in two and three dimensions the method is an alternative for fictitious domain and extended finite element methods the domain of interest is immersed in a background mesh composed by higher order elements the zero level sets are identified and meshed followed by a decomposition of the cut background elements into conforming sub elements adaptivity is a crucial ingredient of the method to guarantee the success of the mesh generation it ensures the successful decomposition of cut elements and enables improved geometry descriptions and approximations it is confirmed that higher order accurate results with optimal convergence rates are achieved with the proposed conformal decomposition finite element method cdfem

a procedure for a posteriori error estimation for h p finite element methods abstract a new approach to a posteriori error estimation is outlined which is applicable to general h p finite element approximations of general classes of boundary value problems the approach makes use of duality arguments and is based on the element residual method erm important aspects of the method are that it provides a systematic approach toward deriving element boundary conditions for the erm it leads to an upper bound for the global error in an appropriate energy norm and it is valid for non uniform and irregular h p meshes in the present exposition a brief outline of the theoretical foundations of the method is given together with the results of its application to several representative problems these results show that the approach is applicable to general linearly elliptic systems including unsymmetrical operators and that the method is valid for broad classes of linear and non linear problems

a polytree based adaptive polygonal finite element method for topology optimization summary we propose a highly effective approach using a novel adaptive methodology to perform topology optimization with polygonal meshes called polytree meshes polytree is a hierarchical data structure based on the principle of recursive spatial decomposition of each polygonal element with n nodes into n 1 arbitrary new polygonal elements enabling more efficient utilization of unstructured meshes and arbitrary design domains in topology optimization in order to treat hanging nodes after each optimization loop we define the wachspress coordinate on a reference element and then utilize an affine map to obtain shape functions and their gradients on arbitrary polygons with n vertices and m hanging nodes called side nodes as the polygonal element with n m vertices the polytree meshes do not only improve the boundary description quality of the optimal result but also reduce the computational cost of optimization process in comparison with the use of uniformly fine meshes several numerical examples are investigated to show the high effectiveness of the proposed method this article is protected by copyright all rights reserved

spline element methods allowing multiple level hanging nodes abstract in this paper a systematic algorithm of spline element method allowing multiple level hanging nodes on triangular mesh is proposed the classical smoothness condition of the piecewise polynomials on triangular mesh is generalized to the case of meshes with hanging nodes and is treated as the linear constraints of the system equations we then derive a concise formulation for such linear constraints according to the special hierarchical configurations of the hanging nodes moreover an efficient linear equation solver is proposed when applied to solve the elliptic partial differential equations numerical examples are illustrated to show the accuracy of the proposed methods and the comparisons are made between different levels of hanging nodes

goal oriented hp adaptivity for elliptic problems we propose and test a fully automatic goal oriented hp adaptive strategy for elliptic problems the method combines two techniques the standard goal oriented adaptivity based on a simultaneous solution of a dual problem and a recently proposed hp strategy based on minimizing the projection based interpolation error of a reference solution the proposed strategy is illustrated with two numerical examples laplace equation in l shape domain and an axisymmetric maxwell problem involving radiation of a loop antenna wrapped around a metallic cylinder into a conductive medium

a lowest order weak galerkin method for linear elasticity abstract the lowest order weak galerkin wg method is considered for linear elasticity based on the displacement formulation the new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution a priori error estimates of optimal order in the discrete h 1 and l 2 norms for the displacement are proved when the solution is smooth the error estimates are independent of the lame constant thus the performance of the new method does not deteriorate as the elastic material becomes incompressible further a simple post processing technique to obtain a numerical approximation of the stress is presented a careful error analysis reveals that the l 2 norm error in the stress is also optimal and independent of several numerical experiments confirm the locking free property and optimal convergence rates of the new method

implementation of continuous h p adaptive finite element spaces without limitations on hanging sides and distribution of approximation orders adaptive techniques using h p refinement are known to be one of the most efficient methodologies to accelerate the convergence of finite element algorithms however the implementation of computational tools for the development of h p adaptive algorithms is intricate and depends strongly on the data structure there exist few computational environments available to the scientific finite element community capable to implement h p adaptive approximation spaces for the complete family of finite element topologies and which implement hanging sides this article describes a methodology for the development of continuous h p adaptive finite element approximation spaces without constraints on the refinement strategy concerning the difference of levels and approximation orders between neighboring elements the shape functions are hierarchical and the coefficient constraints associated with hanging sides that can occur in non conformal geometric meshes are defined using l 2 projections the topological and functional aspects of the construction are described in one two and three dimensions for a variety of geometric entities line triangle quadrilateral tetrahedron pyramid prism and hexahedron the implementation is demonstrated in the object oriented scientific computational environment neopz http github com labmec neopz neopz is a general finite element approximation software which incorporates a variety of variational formulations validation of the refinement methodology is demonstrated by two and three dimensional numerical experiments

conditioning of hierarchic p version nedelec elements on meshes of curvilinear quadrilaterals and hexahedra the conditioning of a set of hierarchic basis functions for p version edge element approximation of the space h curl is studied theoretical bounds are obtained on the location of the eigenvalues and on the growth of the condition numbers for the mass curl curl and stiffness matrices that naturally arise from galerkin approximation of maxwell s equations the theory is applicable to meshes of curvilinear quadrilaterals or hexahedra in two and three dimensions respectively including the case in which the local order of approximation is nonuniform throughout the theory is illustrated with numerical examples that show that the theoretical asymptotic bounds are sharp and are attained within the range of practical computation

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

piecewise polynomial shape functions for hp finite element methods abstract high order piecewise polynomial shape functions for quadrilateral finite elements are proposed the shape functions are tensor products of one dimensional piecewise functions based on legendre polynomials this allows the construction of h 1 conforming approximations on meshes with arbitrary levels of hanging nodes without the necessity for constrained approximation techniques furthermore meshes can be refined simply by inserting nodes on element edges application examples including the l shape domain a plate problem and a tie rod connection plate illustrate the applicability of the method

adaptive hpq finite element methods for the analysis of 3d based models of complex structures part 1 hierarchical modeling and approximations this is a paper out of a series of three papers devoted to model and hpq adaptive finite element methods assigned for modeling and analysis of complex structures in this paper we focus our attention on hierarchical models and hierarchical approximations while the issues of error estimation and adaptivity control will be presented in the next papers of the series in all the papers we consider the elastic structures of complex mechanical description we apply 3d or 3d based mechanical models hierarchical modeling and hierarchical approximations within our finite element formulation in the proposed approach the mechanical model and discretization parameters such as the size h of the element and the longitudinal and transverse approximation orders p and q can vary locally i e they can be different in each finite element the a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors the error controlled adaptive procedures are derived from a four step strategy with possible iterations on h and p steps

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

robustness of the network models of immune response we introduce a quantitative measure of the robustness of the network models of immune response against damage of inter cellular interactions as an illustrative example we compute the robustness of a model which is equivalent to the one proposed recently by chowdhury and stauffer j stat phys 59 1990 1019 we also study the basins of attraction of the fixed points of this model with damaged inter cellular interactions

statistical physics of immune networks abstract several discrete models of immune networks have been developed in the last few years to describe the dynamics of the various aspects of immune response the populations of the various types of cells involved in immune response are represented in these theries by automata and the population dynamics of these cells are formulated in terms of discrete maps in discrete time in contrast to the classical approaches where differential equations describe the time evolution of the populations which are taken to be real variables we summarize here the main results of these investigations and present a critical analysis of the successes and limitations of this discrete approach by comparing with the results of corresponding continuum models recently interclonal interactions have been taken into account by implementing jerne s idea of a functional network using a shape space approach we report some of the progress made recently in this area of interdisciplinary research and also outline a simple theory of t cell anergy

new irrationality measures for q logarithms the three main methods used in diophantine analysis of q series are combined to obtain new upper bounds for irrationality measures of the values of the q logarithm function when p 1 q z 0 1 and z q

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

a robust a posteriori error estimate for the fortin soulie finite element method the subject of a posteriori error estimation is widely studied and a variety of such error estimates have been used for elasticity problems in recent years of particular interest is the work carried out in 1 2 in this paper we derive a new a posteriori error estimator for the quadratic nonconforming fortin soulie element for the error in an energy like norm then we illustrate the new error bound by presenting some numerical examples and show an example of a sequence of adaptively refined meshes

a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations we analyze an a posteriori error estimator for nonlinear parabolic differential equations in several space dimensions the spatial discretization is carried out using the p version of the finite element method the error estimates are obtained by solving an elliptic problem at the desired times when the estimation is wanted some numerical experiments prove the efficiency of the error estimation

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985