a simplified analysis of plate structures using trefftz functions this paper presents a method for a quick evaluation of stresses and displacements in a plate structure for elastostatic problems first an approximation is built from the mathematical structure of the solutions for starshaped domains which is representative of long wavelength effects second a variational formulation consistent with this approximation is proposed finally two alternative procedures for coupling adjacent domains are demonstrated

development of special finite elements for free boundaries a modified two dimensional quadrilateral plane stress element and a modified three dimensional brick element have been developed to eliminate the weakness of the conventional finite elements in determining the stress distributions along free boundaries the results obtained by the said modified elements are in good agreement with the theoretical and experimental results and are superior to the conventional finite element results a comparison is made between the proposed approach of improving stress calculations and another approach which employs the least square fitting method the comparison shows that the two approaches provide comparable accuracy but the latter is more difficult to implement

dual hybrid trefftz element formulation based on independent boundary traction frame reference epfl article 102727view record in web of science record created on 2007 04 23 modified on 2016 08 08

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

brownian limits local limits and variance asymptotics for convex hulls in the ball schreiber and yukich ann probab 36 2008 363 396 establish an asymptotic representation for random convex polytope geometry in the unit ball bd d 2 in terms of the general theory of stabilizing functionals of poisson point processes as well as in terms of generalized paraboloid growth processes this paper further exploits this connection introducing also a dual object termed the paraboloid hull process via these growth processes we establish local functional limit theorems for the properly scaled radius vector and support functions of convex polytopes generated by high density poisson samples we show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius vector and support functions generalized paraboloid growth processes coupled with general techniques of stabilization theory yield brownian sheet limits for the defect volume and mean width functionals finally we provide explicit variance asymptotics and central limit theorems for the k face and intrinsic volume functionals

two scale analysis of heterogeneous elastic plastic materials with asymptotic homogenization and voronoi cell finite element model in this paper a multiple scale finite element model vcfem homo has been developed for elastic plastic analysis of heterogeneous porous and composite materials by combining asymptotic homogenization theory with the voronoi cell finite element model vcfem vcfem for microstructural modeling originates from dirichlet tessellation of representative material elements at sampling points in the structure structural modeling is done by the general purpose finite element code abaqus and interfacing with the microscale vcfem analysis is done through the user subroutine in abaqus for material constitutive relation umat asymptotic homogenization in umat generates macroscopic material parameters for abaqus following the macroscopic analysis a local vcfem analysis is invoked to depict the true evolution of microstructural state variables various numerical examples are executed for validating the effectiveness of vcfem homo and the effect of size shape and distribution of heterogeneities on local and global response is examined

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

central limit theorems for gaussian polytopes choose n random independent points in r d according to the standard normal distribution their convex hull k n is the gaussian random polytope we prove that the volume and the number of faces of k n satisfy the central limit theorem settling a well known conjecture in the field

fracture mechanics analysis using the wavelet galerkin method and extended finite element method summary this paper presents fracture mechanics analysis using the wavelet galerkin method and extended finite element method the wavelet galerkin method is a new methodology to solve partial differential equations where scaling wavelet functions are used as basis functions in solid structural analyses the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain to improve accuracy wavelet functions are superposed on the scaling functions within a region having a high stress concentration such as near a hole or notch thus the method can be considered a refinement technique in fixed grid approaches however because the basis functions are assumed to be continuous in applications of the wavelet galerkin method there are difficulties in treating displacement discontinuities across the crack surface in the present research we introduce enrichment functions in the wavelet galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method this paper presents the mathematical formulation and numerical implementation of the proposed technique as numerical examples stress intensity factor evaluations and crack propagation analyses for two dimensional cracks are presented copyright 2012 john wiley sons ltd

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

a high precision element with a central circular hole an eight node rectangular membrane element with a central circular hole has been developed using complex potentials to define stress and strain functions this new element type can be combined with the conventional elements e g isoparametric elements with no modifications required two numerical examples were employed to illustrate the superiority and versatility of the proposed element type the results obtained using the said element type are closer to the theoretical results as compared with those obtained by similar types of elements

toward a hybrid trefftz element with a hole for elasto plasticity the paper deals with the modelling of riveted assemblies for full scale complete aircraft crashworthiness many comparisons between experiments and fe computations of bird impacts onto aluminium riveted panels have shown that macroscopic plastic strains were not sufficiently developed and localised in the riveted shell fe in the impact area consequently fe models never succeed in initialising and propagating the rupture in the sheet metal plates and along rivet rows as shown by experiments without calibrating the input data especially the damage and failure properties of the riveted shell fe to model the assembly correctly it appears necessary to investigate on fe techniques such as hybrid trefftz finite element method h t fem indeed perforated fe plates developed for elastic problems based on a hybrid trefftz formulation have been found in the open literature our purpose is to find a way to extend this formulation so that the super element can be used for crashworthiness to reach this aim the main features of an elastic hybrid trefftz plate are presented and are then followed by a discussion on the possible extensions finally the interpolation functions of the element are evaluated numerically

a boundary element algorithm using compatible boundary displacements and tractions with the aid of muskhelishvili s complex plane elasticity solution representation compatible displacement and stress fields are constructed the complex functions in these formulas are represented by cauchy integrals which are discretized along the boundary with the aid of complex shape functions for each boundary element the constructed displacement and stress functions satisfy the navier equations and the equilibrium equations respectively the use of fifth order complex basis functions with continuous second complex derivatives gives numerical results of high accuracy

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

a law of large numbers for nearest neighbour statistics in practical data analysis methods based on proximity near neighbour relationships between sample points are important because these relations can be computed in time n log n as the number of points n associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample if the sample points are independent and identically distributed the associated random variables will also be identically distributed but not independent despite this we show that random variables of this type satisfy a strong law of large numbers in the sense that their sample means converge to their expected values almost surely as the number of sample points n

a central limit theorem for projections of the cube we prove a central limit theorem for the volume of projections of the cube 1 1 n onto a random subspace of dimension n when n is fixed and n rightarrow infty randomness in this case is with respect to the haar measure on the grassmannian manifold

hybrid trefftz finite element method for reissner plates on an elastic foundation abstract a modified variational principle for analysis of reissner plates on an elastic foundation prevented the foundation may be of winkler type or pasternak type a hybrid trefftz ht finite element fe model has been developed based on the principle and a new set of t complete functions several numerical examples are considered to assess the effectiveness of the proposed method

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

central limit theorems for random polygons in an arbitrary convex set we study the probability distribution of the area and the number of vertices of random polygons in a convex set kr 2 the novel aspect of our approach is that it yields uniform estimates for all convex sets kr 2 without imposing any regularity conditions on the boundary k our main result is a central limit theorem for both the area and the number of vertices settling a well known conjecture in the field we also obtain asymptotic results relating the growth of the expectation and variance of these two functionals

reply to comments by r piltner on development of two dimensional elements with a central circular hole comput methods appl mech engrg 188 2000 431 440

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

intrinsic volumes and f vectors of random polytopes on appelle polytope aleatoire k n l enveloppe convexe de n points choisis de facon aleatoire dans un corps convexe k on s interesse a l approximation stochastique de k par k n plus particulierement on obtient des resultats d approximation pour les esperances de v s k v s k n ou v s est le s ieme volume intrinseque du nombre de faces s dimensionnelles de k n et de la distance de hausdorff entre k et k n

a multi level wave based numerical modelling framework for the steady state dynamic analysis of bounded helmholtz problems with multiple inclusions the wave based method wbm is an alternative numerical prediction method for both interior and exterior steady state dynamic problems which is based on an indirect trefftz approach it applies wave functions which are exact solutions of the governing differential equation to describe the dynamic field variables the smaller system of equations and the absence of pollution errors make the wbm very suitable for the treatment of helmholtz problems in the mid frequency range where element based methods are no longer feasible due to the associated computational costs a sufficient condition for convergence of the method is the convexity of the considered problem domain as a result only problems of moderate geometrical complexity can be considered and some geometrical features cannot be handled at all in this paper these limitations are alleviated through the development of a general modelling framework based on existing wbm methodologies which allows for the efficient introduction of inclusion configurations in bounded wbm models for problems governed by one or more helmholtz equations the feasibility and efficiency of the method is illustrated by means of numerical verification studies in which the methodology is applied to two types of dynamic problems on the one hand a single helmholtz equation associated with the steady state dynamic behaviour of acoustic cavities is studied on the other hand the framework is applied to the solution of the navier system of partial differential equations that describe the elastodynamic response of two dimensional perforated solids

optimal discretization strategy for boundary element based shape optimization problem this paper deals with the development of a discretization strategy that minimizes the total computing time for solving shape optimization problems in linearly elastic systems it employs a combination of boundary elements for integration of the governing differential equations and a first order optimization algorithm the proposed optimal discretization strategy is implemented and tested using a representative example problem and its effectiveness is assessed in comparison to alternative strategies

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

variational formulations for tfem of piezoelectricity abstract the paper presents a family of variational formulations of trefftz finite elements wherein the assumed internal displacement and electric potential fields a priori fulfil the governing differential equations of the problem over the element sub domain while the inter element continuity and the boundary conditions are enforced using a modified variational principle together with an independent frame field defined on each element boundary it is based on four free energy densities each with two kinds of independent variables as basic independent variables i e d e e e d and e based on the assumed intra element and frame fields an element stiffness matrix equation is obtained which is implemented into computer programs for numerical analysis some numerical examples are considered to show the application of the proposed formulation

variance asymptotics for random polytopes in smooth convex bodies let k subset mathbb r d be a smooth convex set and let mathcal p lambda be a poisson point process on mathbb r d of intensity lambda the convex hull of mathcal p lambda cap k is a random convex polytope k lambda as lambda rightarrow infty we show that the variance of the number of k dimensional faces of k lambda when properly scaled converges to a scalar multiple of the affine surface area of k similar asymptotics hold for the variance of the number of k dimensional faces for the convex hull of a binomial process in k

consistent approximations and approximate functions and gradients in optimal control because of the unavoidable use of numerical integration methods such as runge kutta or finite elements the numerical solution of optimal control problems with either ode or pde dynamics is governed by a discretization parameter such as the integration mesh size usually when explicit integration techniques are used function and derivative values can be computed exactly for the discretized problems recently we have come across some examples where function and derivative values of the explicitly discretized problems had to be approximated by the outcome of n iterations of a solver consequently the discretization of these problems is controlled by two parameters the mesh size and the number of iterations of the solver referring to e polak optimization algorithms and consistent approximations springer verlag 1997 we find a theory for solving optimization problems that require discretization it deals with two situations in the first which is referred to as that of consistent approximations it is assumed that an infinite dimensional optimization problem can be suitably approximated by a family of progressively higher dimensional optimization problems in this case strategies in the form of algorithm models are presented for diagonalizing the solution process in the second situation it is assumed that numerical solution of the dynamic equations does not result in a family of finite dimensional consistent approximations e g when implicit integration methods are used for this case the theory provides models for the implementation of conceptual algorithms unfortunately neither of these situations envisions the possibility of two discretization parameters in this paper we present new algorithm models that can be used with two discretization parameters the first one controls the mesh size of an explicit integration scheme and the second one controls the precision with which functions and gradients associated with a fixed mesh size are computed the result can be seen as a framework of quasi consistent approximations we implemented these new algorithm models using an approximate steepest descent method for the solution of two problems a two point boundary value problem in which the discretized linear ode dynamics are solved approximately using the gauss seidel method and a distributed control problem in which the discretized dynamics are solved using a domain decomposition algorithm which can be implemented on parallelized computers our numerical results show that these new algorithms perform quite well and are fairly insensitive to the selection of user set parameters also they appear to be superior to some alternative ad hoc schemes

on the use of consistent approximations in boundary element based shape optimization in the presence of uncertainty this work addresses certain aspects of shape optimization for linearly elastic systems in the presence of uncertainty the goal is to formulate and test a set of computationally tractable methods for incorporating uncertainty in the system the boundary element method is employed in solving the underlying elasticity equations the resulting minimax problem is studied in the context of the so called consistent approximations which allow for adaptive control of the discretization error during the iterative optimization process

conforming boundary elements in plane elasticity for shape design sensitivity the structural design sensitivity analysis of a two dimensional continuum using conforming continuous boundary elements is investigated implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices a singular formulation of the boundary element method is used implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis the behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated a rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices thus avoiding the problems associated with their numerical integration a formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented the design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results numerical data comparing the performance of conforming and non conforming formulations in the calculation of design sensitivities are also presented the accuracy of the present results is demonstrated through comparisons with existing analytical results

extended voronoi cell finite element model for multiple cohesive crack propagation in brittle materials this paper introduces an extended voronoi cell finite element model x vcfem for modelling cohesive crack propagation in brittle materials with multiple cracks the cracks are modelled by a cohesive zone model and their incremental directions and growth lengths are determined in terms of the cohesive energy near the crack tip extension to vcfem is achieved through enhancements in stress functions in the assumed stress hybrid formulation in addition to polynomial terms the stress functions include branch functions in conjunction with level set methods and multi resolution wavelet functions in the vicinity of crack tips the wavelet basis functions are adaptively enriched to accurately capture crack tip stress concentrations conditions and methods of stability are enforced in x vcfem for improved convergence with propagating cracks two classes of problems are solved and compared with existing solutions in the literature for validation of the x vcfem algorithms the first set corresponds to results for static cracks while in the latter set the propagation of cohesive cracks are considered comparison of x vcfem simulation results with results in literature for several fracture mechanics problems validates the effectiveness of x vcfem copyright 2005 john wiley sons ltd

beta polytopes and poisson polyhedra f vectors and angles abstract we study random polytopes of the form x 1 x n defined as convex hulls of independent and identically distributed random points x 1 x n in r d with one of the following densities f d x c d 1 x 2 x 1 beta distribution 1 or f d x c d 1 x 2 x r d beta distribution d 2 here c d and c d are normalizing constants this setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases we derive exact and asymptotic formulae for the expected number of k faces of x 1 x n for arbitrary k 0 1 d 1 we prove that for any such k this expected number is strictly monotonically increasing with n also we compute the expected internal and external angles of these polytopes at faces of every dimension and more generally the expected conic intrinsic volumes of their tangent cones by passing to the large n limit in the beta case we compute the expected f vector of the convex hull of poisson point processes with power law intensity function using convex duality we derive exact formulae for the expected number of k faces of the zero cell for a class of isotropic poisson hyperplane tessellations in r d this family includes the zero cell of a classical stationary and isotropic poisson hyperplane tessellation and the typical cell of a stationary poisson voronoi tessellation as special cases in addition we prove precise limit theorems for this f vector in the high dimensional regime as d finally we relate the d dimensional beta and beta distributions to the generalized pareto distributions known in extreme value theory

stochastical approximation of convex bodies enveloppe convexe de points aleatoires sur la sphere enveloppe convexe de points aleatoires sur la frontiere d un corps convexe lisse

risk averse shape optimization risk averse optimization has attracted much attention in finite dimensional stochastic programming in this paper we propose a risk averse approach in the infinite dimensional context of shape optimization we consider elastic materials under stochastic loading as measures of risk awareness we investigate the expected excess and the excess probability the developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets we incorporate topological derivatives to allow for topological changes in the shape optimization procedure numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios

postbuckling analysis of thin plates by a hybrid trefftz finite element method abstract a hybrid trefftz finite element ht fe model for postbuckling analysis of thin plates is presented based on a modified variational principle with the aid of incremental form of equilibrium equations all non linear terms may be viewed as pseudo loads and then the in plane and out of plane equations can be uncoupled as a consequence the solution procedure for the postbuckling plate becomes very simple two numerical examples are considered to illustrate the correctness and accuracy of the proposed formulation

three dimensional hybrid trefftz stress elements the stress model of the hybrid trefftz finite element formulation is applied to the linear elastostatic analysis of solids the stresses are approximated in the domain of the element and displacements on its boundary complete linearly independent hierarchical polynomial approximation functions are used in both domain and boundary approximations the displacement basis is defined independently on each inter element surface continuity at the edges and on the corners of the elements is not enforced a priori the stress basis is constrained to solve locally the beltrami governing differential equation it is derived from the associated papkovitch neuber elastic displacement solution generalized variables are used to ensure that the approximations are independent of the geometric description of the elements the solving system is derived directly from the fundamental relations of elastostatics the solving system is symmetric when the same property applies to the local elasticity condition sparse described by boundary integral arrays and well suited to p refinement and parallel processing the numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the finite element formulation copyright 2000 john wiley sons ltd

comments on development of two dimensional elements with a central circular hole a k soh z f long comput methods appl mech engrg 188 2000 431 440

the spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points we show that the random point measures induced by vertices in the convex hull of a poisson sample on the unit ball when properly scaled and centered converge to those of a mean zero gaussian field we establish limiting variance and covariance asymptotics in terms of the density of the poisson sample similar results hold for the point measures induced by the maximal points in a poisson sample the approach involves introducing a generalized spatial birth growth process allowing for cell overlap

variational principles for softening gradient dependent plasticity abstract a gradient dependent plasticity model is analysed in which the yield function not only depends on the stress tensor and an invariant plastic strain measure but also on the spatial derivatives of the latter quantity a minimum variational principle is obtained by analysing the structure discretized into sufficiently small plastic elements and thus the uniqueness of fe solutions can be ensured for softening plasticity the positive definiteness of the variational functional is determined only by the sign of the diffuse coefficient associated with the highest order derivative of the plastic multiplier in the yield function in order to adopt mix hybrid elements several general variational principles are obtained by relaxing the associated constraints on the displacement strain stress and or plastic multiplier gradient radiation fields

probabilistic approach to free form airfoil shape optimization under uncertainty free form shape optimization of airfoils to minimize drag directly poses unexpected difficulties practical experience has indicated that a deterministic optimization for discrete operating conditions can result in dramatically inferior performance when the actual operating conditions are different from the somewhat arbitrarily selected design conditions used during the optimization extensions from single point to multipoint optimization have proven unable to remedy this problem of localized optimization adequately near the sampled operating conditions an intrinsically statistical approach is presented and how the shortcomings of traditional optimization methods can be overcome is demonstrated we discuss two algorithms the first one is based on a numerical evaluation of the expectation integral during each optimization step the second one is based on a closed form second order analytic approximation of the integration the airfoil geometry optimization example inviscid euler flow also reveals how the relative likelihood of each of the operating conditions is automatically taken into consideration during the optimization process this is a key advantage over the use of multipoint methods

limit theorems for functionals of convex hulls in 4 a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived this is done by approximating the process of vertices of the convex hull by the process of extreme points of a poisson point process and by considering the latter process of extreme points as a markov process for a particular parametrization we show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull this extents results of renyi and sulanke 1963 and buchta 1984 and shows that the boundary length and the area have a strikingly different probabilistic behavior

statistics of partial minima we study pseudo optimal solutions to multi objective optimization problems by introducing partial minima defined as follows point x k dominates x when at least k of the coordinates of x are smaller than the corresponding coordinates of x a point not k dominated by any other point in the set is a k minimum or a partial minimum generalizing the global minimum we study statistical properties of partial minima for a set of n points independently distributed inside the d dimensional unit hypercube using exact probabilistic methods and heuristic scaling techniques the average number of partial minima a decays algebraically with the total number of points a n d k k when 1 k d interestingly there are k 1 distinct scaling laws characterizing the largest coordinates the distribution p yj of the jth largest coordinate yj decays algebraically with for 1 j k 1 the average number of partial minima grows logarithmically when k d the full distribution of the number of minima is obtained in closed form in two dimensions

development of two dimensional elements with a central circular hole abstract two new two dimensional 2d elements one is a eight node rectangular and the other is a eight node circular element each with a central circular hole have been developed using complex potentials to define stress and strain functions these new elements can be combined with the conventional elements e g isoparametric elements with no modifications required two numerical examples were employed to illustrate the potential accuracy and versatility of the proposed elements the results obtained using these two elements are in good agreement with the theoretical results moreover the proposed rectangular element performs better than the proposed circular element

trefftz indirect methods for plane piezoelectricity in this paper a unified theory for the general solutions of three dimensional elasticity is employed to transform the governing equations of plane piezoelectricity into three generalized laplace equations the latter equations are uncoupled and can be pursued separately by introducing three generalized complex variables the complete solution sets for plane piezoelectricity are constructed by adopting the constituents in the solution sets as the trial functions the trefftz collocation and the trefftz galerkin methods are formulated both methods fall into the category of trefftz indirect methods as the coefficients of their trial functions are non physical numerical examples are presented to illustrate the efficacy of the formulations copyright 2005 john wiley sons ltd

variance asymptotics and scaling limits for random polytopes abstract let k be a convex set in r d and let k be the convex hull of a homogeneous poisson point process p of intensity on k when k is a simple polytope we establish scaling limits as for the boundary of k in a vicinity of a vertex of k and we give variance asymptotics for the volume and k face functional of k k 0 1 d 1 resolving an open question posed in 17 the scaling limit of the boundary of k and the variance asymptotics are described in terms of a germ grain model consisting of cone like grains pinned to the extreme points of a poisson point process on r d 1 r having intensity d e d h d h d v

trefftz solutions for piezoelectricity by lekhnitskii s formalism and boundary collocation method in this paper a solution procedure for plane piezoelectricity is developed by trefftz boundary collocation method starting with the general plane piezoelectricity solution derived by lekhnitskii s formalism the basic sets of trefftz functions which satisfy the homogeneous governing equations are derived moreover special sets of trefftz functions are derived for impermeable elliptical voids impermeable sharp cracks and permeable sharp cracks with arbitrary orientations with respect to the material poling direction the functions in the special sets satisfy not only the homogeneous governing equations but also the boundary conditions at the peripheries of the pertinent defects by adopting trefftz functions as the trial functions multi domain trefftz boundary collocation method is formulated numerical examples are presented to illustrate the efficacy of the formulation copyright 2005 john wiley sons ltd

limit theorems for the typical poisson voronoi cell and the crofton cell with a large inradius in this paper we are interested in the behavior of the typical poisson voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity we prove a law of large numbers for its number of vertices and the area of the cell outside the disk moreover for the latter we establish a central limit theorem as well as moderate deviation type results the proofs deeply rely on precise connections between poisson voronoi tessellations convex hulls of poisson samples and germ grain models in the unit ball besides we derive analogous facts for the crofton cell of a stationary poisson line process in the plane

a quadrilateral mixed finite element with two enhanced strain modes summary an improved plane strain stress element is derived using a hu washizu variational formulation with bilinear displacement interpolation seven strain and stress terms and two enhanced strain modes the number of unknowns of the four node element is increased from eight to ten degrees of freedom for linear and non linear applications the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element which is denoted by qe2 the excellent performance of the proposed element is demonstrated using several linear and non linear examples

random points on the boundary of smooth convex bodies the convex hull of n independent random points chosen on the boundary of a convex body k c r d according to a given density function is a random polytope the expectation of its i th intrinsic volume for i 1 d is investigated in the case that the boundary of k is sufficiently smooth asymptotic expansions for these expected intrinsic volumes as n oo are derived

comparison of shape design sensitivity analysis formulations via material derivative adjoint variable and implicit differentiation techniques for 3 d and 2 d curved boundary element this paper reviews and compares two different approaches to continuum structural shape design sensitivity analysis the material derivative adjoint variable method and the implicit differentiation method using 3 d and 2 d bem employing identical bea models curved element technology numerical integration and equation solving techniques the similarities and differences of both methods are highlighted significant theoretical and computational aspects associated with both methods are presented and accuracy and computational efficiency characteristics are demonstrated advantages and disadvantages of both techniques are identified the characterization provided can aid in the prudent selection between these sensitivity analysis techniques based on the characteristics of the proposed application

a hybrid finite element method for heterogeneous materials with randomly dispersed rigid inclusions a hybrid finite element approach is proposed for the mechanical response of two dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes in conventional finite element methods many elements must be used to represent one inclusion in this work each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion in numerically approximating stress and displacement distributions around the inclusion classical elasticity solutions for a multiply connected region are employed a modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense the accuracy and efficiency of the proposed method are demonstrated by two boundary value problems in one example the results based on the proposed method with only 64 hybrid elements 450 degrees of freedom are shown to be almost identical to those based on the traditional method with 2928 conventional elements 5526 degrees of freedom

equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points a general formula is proved which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points this formula is applied to improve some well known results in geometric probability for example an estimate which was established for a special case by l c g rogers is obtained with the best possible bound and an asymptotic formula due to a renyi and r sulanke is extended to an asymptotic expansion

analysis of stress concentrations in plates with rectangular openings by a combined conformal mapping finite element approach plates with rectangular openings develop stress concentrations under bending while these stresses can be determined using finite elements in many problems this would be difficult because a high density mesh would be needed in the neighborhood of every opening corner in this paper it is shown how a complex variable conformal mapping approach can be numerically coupled with the finite element method to analyze these corner stresses this approach can be used even with relatively coarse meshes where the finite element results by themselves do not resolve the stress concentrations in essence the method relies on finite element analysis to obtain information on the stress field in a region surrounding the plate opening this information is subsequently used to set up the parameters of the conformal mapping approach to obtain the near field stresses at the opening corners

central limit theorems for random polytopes in a smooth convex set let k be a smooth convex set with volume one in choose n random points in k independently according to the uniform distribution the convex hull of these points denoted by kn is called a random polytope we prove that several key functionals of kn satisfy the central limit theorem as n tends to infinity

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

three dimensional hybrid trefftz stress finite elements for plates and shells three dimensional hybrid trefftz stress finite elements for plates and shells are proposed two independent fields are approximated stresses within the element and displacement on their boundary the required stress field derived from the papkovitch neuber solution of the navier equation which a priori satisfies the trefftz constraint is generated using homogeneous harmonic polynomials restriction on the polynomial degree in the coordinate measured along the thickness direction is imposed to reduce the number of independent terms explicit expressions of the corresponding independent polynomials are listed up to the fifth order illustrative applications to evaluate displacements and stresses are carried out by hexahedral hybrid trefftz stress element models the hierarchical p refinement and h refinement strategy are exploited in the numerical tests

hybrid trefftz plane elasticity elements with p method capabilities a family of p method plane elasticity elements is derived based on the hybrid trefftz formulation 1 exact solutions of the lame navier equations are used for the intra element displacement field together with an independent displacement frame function field along the element boundary the final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom at corner nodes and an optional number of hierarchic degrees of freedom associated with the mid side nodes since the element matrices do not involve integration over the element area the elements have a polygonal contour with an optional number of curved sides the quadrilateral element has the same external appearance as the conventional p method plane elasticity element 2 3 but unlike in the conventional p method approach suitable special purpose trefftz functions are generally used to handle the singularity and or stress concentration problems rather than a local mesh refinement the practical efficiency of the new elements is assessed through a series of examples

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

a variational principle for gradient plasticity abstract we elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by aifantis and co workers the generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher order spatial gradients of the equivalent plastic strain in the yield condition first it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands due to the presence of higher order gradients additional boundary conditions for the equivalent plastic strain are required this question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work we demonstrate here that upon assuming a certain type of additional boundary conditions the structural symmetries of the gradient dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain the variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given

a family of quadrilateral hybrid trefftz p elements for thick plate analysis abstract this paper presents a family of quadrilateral hybrid trefftz ht p elements based on reissner mindlin theory of moderately thick plates as compared with existing higher order ht thick plate elements j petrolito which make use of an incomplete set of polynomial homogeneous solutions for the internal displacement and rotation fields and of an auxiliary conforming frame field involving independently interpolated boundary distributions of displacements and rotations the new elements differ essentially with respect to the following two points 1 a t complete set of homogeneous solutions including polynomial and nonpolynomial functions has been derived and used to represent the intraelement displacement and rotations fields 2 the displacement component of the auxiliary frame field has been linked with the tangential component of the rotation so as to satisfy a particular constraint derived from equilibrium considerations the practical efficiency of these modifications has been assessed through a series of numerical examples which have shown that the new p elements are robust more accurate in terms of the number of unknowns and of the computational effort and that they do not lock in the thin limit

a class of difference schemes with flexible local approximation solutions of many physical problems have salient local features that are qualitatively known a priori for example singularities at point sources edge and corners boundary layers derivative jumps at material interfaces strong dipole field components near polarized spherical particles cusps of electronic wavefunctions at the nuclei electrostatic double layers around colloidal particles etc the known methods capable of providing flexible local approximation of such features include the generalized finite element partition of unity method special variational difference schemes in broken sobolev spaces and a few other specialized techniques in the proposed new class of flexible local approximation methods flame a desirable set of local approximating functions such as cylindrical or spherical harmonics plane waves harmonic polynomials etc defines a finite difference scheme on a chosen grid stencil one motivation is to minimize the notorious staircase effect at curved and slanted interface boundaries however the new approach has much broader applications as illustrative examples the paper presents arbitrarily high order 3 point schemes for the 1d schrodinger equation and a 1d singular equation schemes for electrostatic interactions of colloidal particles electromagnetic wave propagation and scattering plasmon resonances moreover many classical finite difference schemes including the collatz mehrstellen schemes are direct particular cases of flame

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

a new particular solution strategy for hyperbolic boundary value problems using hybrid trefftz displacement elements summary a new indirect approach to the problem of approximating the particular solution of non homogeneous hyperbolic boundary value problems is presented unlike the dual reciprocity method which constructs approximate particular solutions using radial basis functions polynomials or trigonometric functions the method reported here uses the homogeneous solutions of the problem obtained by discarding all time derivative terms from the governing equation nevertheless what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the trefftz compliant homogeneous solution basis the particular solution trial basis is capable of significantly improving the trefftz solution even when the original equation is genuinely homogeneous an advantage that is lost if the basis is used exclusively for the recovery of the source terms similarly a sufficiently refined trefftz compliant basis is able to compensate for possible weaknesses of the particular solution approximation the method is implemented using the displacement model of the hybrid trefftz finite element method the functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the hermitian sparse and localized structure of the solving system that typifies hybrid trefftz formulations even when domain integrals are present they are generally easy to handle because the integrand presents no singularity copyright 2015 john wiley sons ltd

on the mean width of random polytopes on the boundary of a d dimensional convex body a probability distribution with a positive continuous density function g is given the convex hull of n points chosen independently according to g is a random polytope the asymptotic behaviour n of the expected value of the mean width of the random polytope is determined

convex hulls of random walks expected number of faces and face probabilities abstract consider a sequence of partial sums s i 1 i 1 i n starting at s 0 0 whose increments 1 n are random vectors in r d d n we are interested in the properties of the convex hull c n conv s 0 s 1 s n assuming that the tuple 1 n is exchangeable and a certain general position condition holds we prove that the expected number of k dimensional faces of c n is given by the formula e f k c n 2 k n l 0 n 1 d 2 l d 2 l k 1 for all 0 k d 1 where n m and n m are stirling numbers of the first and second kind respectively further we compute explicitly the probability that for given indices 0 i 1 i k 1 n the points s i 1 s i k 1 form a k dimensional face of conv s 0 s 1 s n this is done in two different settings for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments these results generalize the classical one dimensional discrete arcsine law for the position of the maximum due to e sparre andersen all our formulae are distribution free that is do not depend on the distribution of the increments k s the main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types a n 1 and b n this probability in turn is related to the number of weyl chambers of a product type reflection group that are intersected by a linear subspace in general position

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk