isogeometric variational multiscale modeling of wall bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes abstract in this work we combine i nurbs based isogeometric analysis ii residual driven turbulence modeling and iii weak imposition of no slip and no penetration dirichlet boundary conditions on unstretched meshes to compute wall bounded turbulent flows while the first two ingredients were shown to be successful for turbulence computations at medium to high reynolds number i akkerman y bazilevs v m calo t j r hughes s hulshoff the role of continuity in residual based variational multiscale modeling of turbulence comput mech 41 2008 371 378 y bazilevs v m calo j a cottrell t j r hughes a reali g scovazzi variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows comput methods appl mech engrg 197 2007 173 201 it is the weak imposition of no slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher reynolds number y bazilevs t j r hughes weak imposition of dirichlet boundary conditions in fluid mechanics comput fluids 36 2007 12 26 y bazilevs c michler v m calo t j r hughes weak dirichlet boundary conditions for wall bounded turbulent flows comput methods appl mech engrg 196 2007 4853 4862 these three ingredients form a basis of a possible practical strategy for computing engineering flows somewhere between rans and les in complexity we demonstrate this by solving two challenging incompressible turbulent benchmark problems channel flow at friction velocity reynolds number 2003 and flow in a planar asymmetric diffuser we observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data this lends some credence to the proposed approach which we believe may become a viable engineering tool

multiscale approach for optimal design in conductivity of composite materials this paper discusses the multiscale approach for optimal design in conductivity of composite materials the homogenization method and the multiscale asymptotic method are presented the associated numerical algorithms and the convergence analysis are provided finally numerical examples are carried out to confirm the validity of the algorithm

assouad type dimensions and homogeneity of fractals we investigate several aspects of the assouad dimension and the lower dimension which together form a natural dimension pair in particular we compute these dimensions for certain classes of self affine sets and quasiself similar sets and study their relationships with other notions of dimension such as the hausdorff dimension for example we also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity

isogeometric structural shape optimization an old dilemma in structural shape optimization is the needed tight link between design model or geometric description and analysis model the intention of this paper is to show that isogeometric analysis offers a potential and promising way out of this dilemma to this end we show a structural shape optimization framework based on the isogeometric analysis approach with the discretization based on non uniform rational b splines nurbs the analysis model represents the structural geometry exactly furthermore nurbs enable efficient geometry control together with smooth boundaries they are the de facto standard in cad systems but are also widely used in a shape optimal design context to define the geometry representation and the design variables with the presented isogeometric approach to shape optimization the analysis model is inherently merged with the design model omitting the typically involved interplay between both we derive analytical sensitivities for nurbs discretizations which allow application of efficient gradient based optimization algorithms the present contribution is restricted to two dimensional problems of linear elasticity but the extension to three dimensions and other problem classes is straightforward some representative examples demonstrate and validate the methodology further the potential of boundary continuity control within isogeometric structural shape optimization is explored to trigger smooth or less smooth angular designs

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

orbits and phase transitions in the multifractal spectrum we consider the one dimensional classical ising model in a symmetric dichotomous random field the problem is reduced to a random iterated function system rifs for an effective field the dq spectrum of the invariant measure of this effective field exhibits a sharp drop of all dq with q 0 at some critical strength of the random field we introduce the concept of orbits which naturally group the points of the support of the invariant measure we then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so called offshoots as well as for generic orbits in the non overlapping case the sharp drop in the dq spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field which is explicitly given a similar drastic change near the points of a special family of periodic orbits explains a second hitherto unnoticed transition in the dq spectrum as it turns out a decisive role in this mechanism is played by a specific offshoot we furthermore give rigorous upper and or lower bounds on all dq in a wide parameter range in most cases the numerically obtained dq coincide with either the upper or the lower bound the results in this paper are relevant for the understanding of rifss in the case of moderate overlap in which periodic orbits with weak singularity can play a decisive role

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

a hierarchical approach to adaptive local refinement in isogeometric analysis abstract adaptive local refinement is one of the key issues in isogeometric analysis in this article we present an adaptive local refinement technique for isogeometric analysis based on extensions of hierarchical b splines we investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity furthermore we use concepts well established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting this also allows us to access a posteriori error estimation techniques numerical results for several different examples are given and they turn out to be very promising

a dimension formula for bernoulli convolutions we present a dynamical approach to the study of the singularity of infinitely convolved bernoulli measuresv for the golden section we introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square the fat baker s transformation our approach strongly relies on the markov structure of the underlying dynamical system indeed if golden mean the fat baker s transformation has a very simple markov coding the ambiguity of order two of this coding which appears when projecting on the line due to passages for the central overlapping zone can be expressed by means of products of matrices of order two this product has a markov distribution inherited by the markov structure of the map the dimension of the projected measure is therefore associated to the growth of this product our dimension formula appears in a natural way as a version of the furstenberg guivarch formula our technique provides an explicit dimension formula and most important provides a formalism well suited for the multifractal analysis of this measure as we will show in a forthcoming paper

convolution of multifractals and the local magnetization in a random field ising chain the local magnetization in the one dimensional random field ising model is essentially the sum of two effective fields with multifractal probability measure the probability measure of the local magnetization is thus the convolution of two multifractals in this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution the pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures the generalized box dimensions of the convolution of cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization returning to the study of this measure we apply the general inequalities and present numerical approximations of the dq spectrum for the first time we are able to obtain results on multifractal properties of a physical quantity in the one dimensional random field ising model which in principle could be measured experimentally the numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h

multi scale asymptotic analysis and computation of the elliptic eigenvalue problems in curvilinear coordinates abstract a novel second order two scale asymptotic method is presented for the eigenvalue problems of the second order elliptic operator in the general composite domain the eigenvalue equation is firstly reformulated in curvilinear coordinates with periodic configuration using proper coordinate transformation and by applying the asymptotic expansion technique the eigenfunctions of the system are expanded to the second order terms using the argument of the so called corrector equations the eigenvalues are expressed in terms of the homogenized eigenfunctions and the cell functions are defined in the representative cell domain the feature of the proposed model is that some homogenized material coefficients and all the microscopic cell functions are dependent on the macroscopic coordinates various reduced expressions of the eigenfunctions and eigenvalues are discussed under specific coordinate transformations and the conditions that the cell functions and homogenized coefficients are decoupled from the macroscopic coordinates are elaborated the finite element algorithm is developed and three numerical experiments are carried out which demonstrate the effectiveness of our proposed method in simulating and predicting the vibration behavior of the composite structures it is also indicated that the second order correctors are of necessity to capture the locally oscillating behavior within a periodicity of the eigenfunctions by the coordinate transformation the asymptotic analysis method can be generalized to more general composite domain with quasi periodic and non periodic configurations

the hole filling method and multiscale algorithm for the heat transfer performance of periodic porous materials in this paper we consider the transient heat transfer problem with rapidly oscillating coefficients in periodic porous materials the hole filling method through filling all holes with a very compliant material is investigated and as the thermal conductivity parameters of the weak phase go to zero the proof of the limiting process is derived in detail then the multiscale analysis and computations based on the hole filling method for the associated multiphase problem in a domain without holes are proposed finally some numerical results are given which show a good agreement and verify the feasibility of the hole filling method a multiscale analysis and computation based on hole filling method is proposed transient heat transfer problem of periodic porous materials is considered proof of the limiting process as the parameters of the weak phase go to zero is derived some numerical results are given in detail to validate the multiscale method developed

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

first order corrections to the homogenised eigenvalues of a periodic composite medium a convergence proof let e be a dirichlet eigenvalue of the periodically rapidly oscillating elliptic operator a x e and let be a corresponding simple eigenvalue of the homogenised operator a we characterise the possible limit points of the ratio e e as e 0 our characterisation is quite explicit when the underlying domain is a planar convex classical polygon with sides of rational or infinite slopes in particular in this case it implies that there is often a continuum of such limit points

multiscale finite element algorithm of the eigenvalue problems for the elastic equations in composite materials in this paper we discuss the multiscale computation of the eigenvalue problems for the elastic equations in composite materials the multiscale finite element method for solving these problems is presented numerous numerical results validate the proposed method of this paper

an isogeometric analysis for elliptic homogenization problems a novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed these problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods the isogeometric analysis heterogeneous multiscale method iga hmm investigated in this paper is regarded as an alternative approach to the standard finite element heterogeneous multiscale method fe hmm which is currently an effective framework to solve these problems the method utilizes non uniform rationalb splines nurbs in both macro and micro levels instead of standard lagrange basis besides the ability to describe exactly the geometry it tremendously facilitates high order macroscopic microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by nurbs basis functions furthermore the nearly optimal quadrature rule for iga auricchio et al 2012 introduced recently is utilized to reduce significantly the number of micro problems which is the main factor contributing into the computational cost in heterogeneous multiscale method a priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro micro nurbs basis functions of arbitrary orders are derived an efficient coupling between degrees of macro and micro basis functions is introduced numerical results show the excellent performance of the proposed method

multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains the multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains are discussed in this paper a multiscale asymptotic analysis formulation for this problem is presented by constructing properly the boundary layer a multiscale numerical algorithm and a postprocessing technique are given finally numerical results show that the method presented in this paper is effective and reliable

multi scale modal analysis for axisymmetric and spherical symmetric structures with periodic configurations abstract a new modal analysis method with second order two scale sots asymptotic expansion is presented for axisymmetric and spherical symmetric structures the symmetric structures considered are periodically distributed with homogeneous and isotropic constituent materials by the asymptotic expansion of the eigenfunctions the homogenized modal equations the effective materials coefficients the first and second order correctors are obtained the derived homogenized constitutive relationships are the same as the ones which serve to homogenize the corresponding static problems the eigenvalues are also expanded to the second order terms and using the so called corrector equation the correctors of the eigenvalues are expressed in terms of the first and second order correctors of the eigenfunctions the anisotropic materials are obtained by homogenization with different properties in the circumferential direction especially for the two dimensional axisymmetric layered structure the one dimensional plane axisymmetric and spherical symmetric structures the homogenized eigenfunctions and eigenvalues as well as their corresponding correctors are all solved analytically the finite element algorithm is established three typical numerical experiments are carried out and the necessity of the second order correctors is discussed based on the numerical results it is validated that the sots asymptotic expansion homogenization method is effective to identify the eigenvalues of the axisymmetric and spherical symmetric structures with periodic configurations and the original eigenfunctions with periodic oscillation can be reproduced by adding the correctors to the homogenized eigenfunctions

second order two scale analysis and numerical algorithm for the damped wave equations of composite materials with quasi periodic structures in this paper we perform a second order two scale analysis and introduce a numerical algorithm for the damped wave equations of composite materials with a quasi periodic structure firstly second order two scale asymptotic expansion solutions for these problems are constructed by a multiscale asymptotic analysis in addition we explain the importance of the second order two scale solutions by the error analysis in the pointwise sense moreover explicit convergence rates of these second order two scale solutions are obtained in the integral sense then a second order two scale numerical method based on a newmark scheme is presented to solve these multiscale problems finally some numerical examples show the effectiveness and efficiency of the multiscale numerical method we proposed

multiscale asymptotic method for maxwell s equations in composite materials in this paper we discuss the multiscale analysis of maxwell s equations in composite materials with a periodic microstructure the new contributions in this paper are the determination of higher order correctors and the explicit convergence rate for the approximate solutions see theorem 2 3 consequently we present the multiscale finite element method and derive the convergence result see theorem 4 1 the numerical results demonstrate that higher order correctors are essential for solving maxwell s equations in composite materials

a short and versatile finite element multiscale code for homogenization problems we describe a multiscale finite element fe solver for elliptic or parabolic problems with highly oscillating coefficients based on recent developments of the so called heterogeneous multiscale method hmm the algorithm relies on coupled macro and microsolvers the framework of the hmm allows to design a code whose structure follows the classical finite elements implementation at the macro level to account for the fine scales of the problem elementwise numerical integration is replaced by micro fe methods on sampling domains we discuss a short and flexible fe implementation of the multiscale algorithm which can accommodate simplicial or quadrilateral fe and various coupling conditions for the constrained micro simulations extensive numerical examples including three dimensional and time dependent problems are presented illustrating the efficiency and the versatility of the computational strategy

the hole filling method and the multiscale computation for the wave equations in perforated domains in this paper we discuss the multiscale computation of the wave equations with rapidly oscillating coefficients in a perforated domain the hole filling method through filling all holes with a very compliant material is introduced as the material parameters of the weak phase go to zero the proof of this limiting process is derived based on the above hole filling method the multiscale asymptotic method and the convergence results for the associated multiphase problem in a domain without holes are presented the numerical experiments are carried out to validate the theoretical results

second order asymptotic algorithm for heat conduction problems of periodic composite materials in curvilinear coordinates a new second order two scale sots asymptotic analysis method is presented for the heat conduction problems concerning composite materials with periodic configuration under the coordinate transformation the heat conduction problems are solved on the transformed regular domain with quasi periodic structure in the general curvilinear coordinate system by the asymptotic expansion the cell problems effective material coefficients and homogenized heat conduction problems are obtained successively the main characteristic of the approximate model is that each cell problem defined on the microscopic cell domain is associated with the macroscopic coordinate the error estimation of the asymptotic analysis method is established on some regularity hypothesis some common coordinate transformations are discussed and the reduced sots solutions are presented especially by considering the general one dimensional problem the explicit expressions of the sots solutions are derived and stronger error estimation is presented finally the corresponding finite element algorithms are presented and numerical results are analyzed the numerical errors presented agree well with the theoretical prediction which demonstrate the effectiveness of the second order asymptotic analysis method by the coordinate transformation the asymptotic analysis method can be extended to more general domain with periodic microscopic structures

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

multiscale analysis and numerical algorithm for the schrodinger equations in heterogeneous media abstract in solid state physics the most widely used techniques to calculate the electronic levels in nanostructures are the effective masses approximation ema and its extension the multiband k p method see 9 they have been particularly successful in the case of heterostructures see e g 4 9 11 this paper discusses the multiscale analysis of the schrodinger equation with rapidly oscillating coefficients the new contributions obtained in this paper are the determination of the convergence rate for the approximate solutions the definition of boundary layer solutions and higher order correctors consequently a multiscale finite element method and some numerical results are presented as one of the main results of this paper we give a reasonable interpretation why the effective mass approximation is very accurate for calculating the band structures in semiconductor in the vicinity of point from the viewpoint of mathematics

asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the dirichlet problem for second order elliptic equations in perforated domains in this paper we study the spectral properties of dirichlet problems for second order elliptic equation with rapidly oscillating coefficients in a perforated domain the asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained and the multiscale finite element algorithms and numerical results are proposed

uniformly fat sets in this paper we study closed sets e which are locally uniformly fat with respect to a certain nonlinear riesz capacity we show that e is actually locally uniformly fat with respect to a weaker riesz capacity two applications of this result are given the first application is concerned with proving sobolev type inequalities in domains whose complements are uniformly fat the second application is concerned with the fekete points of e introduction let x x x2 x be a point in euclidean n space rn with jxl the norm of x for a 0 let i x ixi a n denote the riesz kernel of order a and put i f x f ix y f n f y dy x e r rn ic u x fr yj c n d u y x e r rn when f is a lebesgue integrable function and u a borel measure on rn here dy denotes lebesgue measure on rn if e c rn 0 1 define the a p outer riesz capacity of e by ra p e inf jfllfp ia f 1 on el where if lip is the lebesgue p norm of f we note that a 1 p 2 is the classical newtonian capacity if ap n we let rc p e inf jfjjf j f 1 on el where jc is the truncated riesz kernel defined by je x ixi a n 100 a n x 100 0 lxl 100 we shall say that a property holds a p quasi everywhere abbreviated a p q e on a set e if it holds on e except perhaps for a set of rop capacity zero next we list some properties of riesz capacities to simplify the writing we state these properties only for 0 u l p l 1 a p q e on e and p 1 a p q e on supp u c e moreover e jjli pi ujj rop e received by the editors march 11 1987 1980 mathematic3 subject cla33ification 1985 revi3ion primary 31c15 31b15 1988 american mathematical society 0002 9947 88 1 00 25 per page

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

a multiscale approach and a hybrid fe be algorithm for heterogeneous scattering of maxwell s equations abstract in this paper we discuss the multiscale approach for the scattering problem of maxwell s equations in a heterogeneous material with a periodic microstructure the new contributions in this paper are the determination of the multiscale correctors and the strong convergence in the norm of the space h curl with an explicit rate for the approximate solutions see theorem 2 4 consequently we present a multiscale hybrid finite element method boundary element method fe be the numerical examples are carried out to validate the theoretical results of this paper

multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains abstract this study develops a novel multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains firstly the second order two scale sots solutions for these multiscale problems are successfully obtained based on multiscale asymptotic analysis then the error analysis of sots solutions in the pointwise sense is given to illustrate the importance of developing the sots solutions furthermore the error estimate for the sots approximate solutions in the integral sense is presented in addition a sots numerical algorithm is proposed to effectively solve these problems based on finite element method fem finally some numerical examples verify the feasibility and effectiveness of the sots numerical algorithm we proposed in this paper a unified two scale computational framework is established for heat conduction problems of composite structures with diverse periodic configurations in different subdomains

asymptotic expansions of one dimensional second order helmholtz equation with rapidly oscillating coefficient in this paper we discuss one dimensional second order helmholtz equation with rapidly oscillating coefficient and present an asymptotic expansion method for the problem which will play an important role in numerical computation of course the convergence result and its proof will be given then its corresponding finite element algorithm is proposed finally some numerical experiments are given to investigate the asymptotic expansion method and the finite element algorithm

multiscale asymptotic method for steklov eigenvalue equations in composite media in this paper we consider the multiscale analysis of a steklov eigenvalue equation with rapidly oscillating coefficients arising from the modeling of a composite media with a periodic microstructure there are mainly two new results in the present paper first we obtain the convergence rate with varepsilon 1 2 for the multiscale asymptotic expansions of the eigenvalues and the eigenfunctions of the steklov eigenvalue problem second the boundary layer solution is defined numerical simulations are then carried out to validate the above theoretical results

reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems a new finite element method for the efficient discretization of elliptic homogenization problems is proposed these problems characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed recently the finite element heterogeneous multiscale method fe hmm has been proposed for such problems based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh departing from the approach used in the fe hmm we show that interpolation techniques based on the reduced basis methodology an offline online strategy allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions this new method called the reduced basis finite element heterogeneous multiscale method rb fe hmm is significantly more efficient than the fe hmm for high order macroscopic discretizations and for three dimensional problems when the repeated computation of micro problems over the whole computational domain is expensive a priori error estimates of the rb fe hmm are derived numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method

iterated two scale asymptotic method and numerical algorithm for the elastic structures of composite materials in this paper we shall discuss the elasto static problem of composite materials with two or more different length scales and shall present iterated two scale asymptotic expansion of the solution for considering problem the mathematical proofs of some convergence theorems will be given under some assumptions finally some numerical results are reported

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

a multiscale approach for optimal control problems of linear parabolic equations this paper discusses multiscale analysis for optimal control problems of linear parabolic equations with rapidly oscillating coefficients that depend on spatial and temporal variables there are mainly three new results in the present paper first we obtain the convergence results with an explicit convergence rate for the multiscale asymptotic expansions of the solution of the optimal control problem in the case without constraints second for a general bounded lipschitz polygonal domain the boundary layer solution is defined and the corresponding convergence results are also derived finally an explicit convergence rate varepsilon 1 2 in the presence of constraint is reported

active impedance control of linear one dimensional wave equations active impedance control of linear one dimensional wave equations is investigated the proposed control algorithms are based on the concepts of wave propagation and impedance matching two control objectives are considered the first objective is to obtain total reflection and the second objective is to achieve total absorption e g matched impedance both control laws utilize some interesting properties of the wave type partial differential equation and do not require information about the disturbance and boundary conditions the resulting controllers although boundary independent contain an infinite number of poles on the imaginary axis and the closed loop systems are not internally stable a simple modification is added to the control laws and the stability is analysed an example of active sound cancellation in ducts with a moving medium is given to demonstrate one of the control algorithms

first order corrections to the homogenized eigenvalues of a periodic composite medium this paper studies the eigenvalue problem associated with the vibration of a com posite medium assumed to have a periodic microstructure the investigation concentrates on the first order correction to the homogenized eigenvalues a major difficulty arises due to the interaction of the periodic microstructure with the boundary of the medium the way in which the boundary intersects the underlying periodic structure strongly influences the value of the correction a rigorous derivation is given of the eigenvalue correction formula the properties of the eigenvalue correction are studied in one and two dimensional numerical calculations

second order two scale asymptotic analysis for axisymmetric and spherical symmetric structure with periodic configurations abstract a new second order two scale sots analysis finite element algorithm is developed for the axisymmetric and spherical symmetric elastic problems with small periodic configurations the axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction and the spherical symmetric structure is only periodic in radial direction and homogeneous in other two directions the sots asymptotic expansions for the space problem plane axisymmetric problem and spherical symmetric problem are presented and the main feature is that the anisotropic material is obtained by the homogenization the analytical expressions of the cell functions and homogenized solutions for plane axisymmetric and spherical symmetric problems are obtained and the error estimations of the expansions are established the second order asymptotic analysis finite element algorithm is presented and the numerical examples are solved including the hollow cylinder rotating disk and hollow sphere composed of periodic composite materials the computational results demonstrate the effectiveness and accuracy of the sots asymptotic analysis algorithm and the converging behavior of the asymptotic analysis algorithm agrees well with the theoretical prediction it is also indicated that the stress distributions can be correctly computed only by adding the second order correctors

second order two scale computational method for damped dynamic thermo mechanical problems of quasi periodic composite materials abstract in this paper a novel second order two scale sots analysis method and related numerical algorithm are developed for damped dynamic thermo mechanical problems of quasi periodic composite materials the formal sots solutions for these problems are constructed by the multiscale asymptotic analysis then we theoretically explain the importance of the sots solutions by the error analysis in the pointwise sense furthermore the convergence result with an explicit rate for the sots solutions is obtained in the integral sense in addition a sots numerical algorithm is proposed to solve these problems effectively finally some numerical examples verify the feasibility and effectiveness of the sots numerical algorithm we proposed

combining multifractal additive and multiplicative chaos the purpose of this article is the study of the new class of multifrac tal measures which combines additive and multiplicative chaos defined by x j 1 b j j 2 x

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

the metric entropy of diffeomorphisms soit m une variete de riemann compacte c sans bord soit f un diffeomorphisme c 2 de m et soit m une mesure de probabilite de borel f invariante sur m soit h m f l entropie de f on donne des relations entre h m f et les proprietes geometriques de f et de m

approximate boundary controllability for the wave equation in perforated domains this paper considers the wave equation in a perforated domain with holes of size r epsilon distributed with epsilon periodicity with the assumption that there exists a neighborhood of the exterior boundary without holes the following question is asked is it possible to approximately control the wave equation in the perforated domain in such a way that when epsilon goes to zero the exact controllability of the limit system is obtained two main theorems give a positive answer to this question when r epsilon is the critical size that transforms at the limit the operator partial 2 partial t 2 delta into partial 2 partial t 2 delta mu where mu is a positive measure in the first theorem in a suitable sense l 2 omega times h 1 omega is approximated by the space of the admissible data for the exact controllability in the perforated domain in the second it is shown that the limit control supported only by the exterior boundary of the perforated domain is such that the related state in the perforated domain goes at the limit to the equilbrium state at the time t

multiscale asymptotic expansions methods and numerical algorithms for the wave equations in perforated domains abstract in this paper we are concerned with the wave equations in perforated domains with a homogeneous neumann condition on the boundary of the holes the multiscale asymptotic expansion of the solution for the problem are constructed and associated explicit convergence rates are obtained a multiscale numerical method is introduced finally we present some numerical results which support strongly the convergence theorem

in between the inequalities of sobolev and hardy abstract we establish both sufficient and necessary conditions for the validity of the so called hardy sobolev inequalities on open sets of the euclidean space these inequalities form a natural interpolating scale between the weighted sobolev inequalities and the weighted hardy inequalities the assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

a multiscale algorithm for radiative heat transfer equation with rapidly oscillating coefficients this paper as a continued work of huang and cao 2014 discusses the multiscale computation of the radiative heat transfer in composite materials or porous media a novel multiscale asymptotic expansion is presented and an explicit rate of convergence is derived we develop a multiscale algorithm for solving this kind of problem a fully implicit scheme is carefully studied and an iterative algorithm is given the convergence of the iterative algorithm is proved by the fixed point method numerical results confirm the efficiency and accuracy of this approach and show that the novel multiscale asymptotic expansion is essential for the radiative dominated cases

multiscale asymptotic method of optimal control on the boundary for heat equations of composite materials in this paper we study the optimal control on the boundary for parabolic equations with rapidly oscillating coefficients arising from the heat transfer problems and the optimal control on the boundary of composite materials or porous media the multiscale asymptotic expansion of the solution for the problem in the case without any constraints is presented we derive the proofs of all convergence results

on the multifractal analysis of bernoulli convolutions ii dimensions we show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved bernoulli measures for the golden mean in this second part we show how the hausdorff dimension of the set where the measure has a power law singularity of strength is related to the large deviation function given in part i

the local microscale problem in the multiscale modeling of strongly heterogeneous media effects of boundary conditions and cell size many multiscale methods are based on the idea of extracting macroscopic behavior of solutions by solving an array of microscale models over small domains a key ingredient in such multiscale methods is the boundary condition and the size of the computational domain over which the microscale problems are solved this problem is systematically investigated in the present paper in the context of modeling strongly heterogeneous media three different boundary conditions are considered the periodic boundary condition dirichlet boundary condition and the neumann boundary condition each is applied to several benchmark problems the random checker board problem periodic problem with isotropic macroscale behavior periodic problem with anisotropic macroscale behavior and periodic laminated media in each case convergence studies are conducted as the domain size for the microscale problem is changed convergence rates as well as the size of fluctuations in the computed effective coefficients are compared for the different formulations in addition we will discuss a mixed dirichlet neumann boundary condition that is often used in porous medium modeling we explain why that leads to unsatisfactory results and how it can be corrected also discussed are the different averaging methods used in extracting the effective coefficients

convergence for elliptic equations in periodic perforated domains abstract convergence for the solutions of elliptic equations in periodic perforated domains is concerned let denote the size ratio of the holes of a periodic perforated domain to the whole domain it is known that by energy method the gradient of the solutions of elliptic equations is bounded uniformly in in l 2 space also when approaches 0 the elliptic solutions converge to a solution of some simple homogenized elliptic equation in this work above results are extended to general w 1 p space for p 1 more precisely a uniform w 1 p estimate in for p 1 and a w 1 p convergence result for p n n 2 for the elliptic solutions in periodic perforated domains are derived here n is the dimension of the space domain one also notes that the l p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in

multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains in this paper we will discuss the mixed boundary value problems for the second order elliptic equation with rapidly oscillating coefficients in perforated domains and will present the higher order multiscale asymptotic expansion of the solution for the problem which will play an important role in the numerical computation the convergence theorems and their rigorous proofs will be given finally a multiscale finite element method and some numerical results will be presented

theoremes limite pour les systemes lineaires a coefficients markoviens nous montrons un theoreme central limite et une estimee des grandes deviations pour les solutions d equations differentielles stochastiques lineaires a coefficients markoviens ces resultats sont obtenus dans le cadre general des processus markoviens multiplicatifs ils s appliquent alors par exemple au flot derive d un flot stochastique sur une variete compacte et aux produits de matrices aleatoires en dependance markovienne

a simple algorithm for obtaining nearly optimal quadrature rules for nurbs based isogeometric analysis we develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem a simple and robust algorithm is developed to determine the rules which are exact for important b spline spaces of uniform and geometrically stretched knot spacings we consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems we find that the rules are almost optimally efficient but much easier to obtain than optimal rules which require the solution of global nonlinear problems that are often ill posed

the equivalence between pointwise hardy inequalities and uniform fatness we prove an equivalence result between the validity of a pointwise hardy inequality in a domain and uniform capacity density of the complement this result is new even in euclidean spaces but our methods apply in general metric spaces as well we also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the hardy inequality in the setting of metric spaces in addition we consider the relations between the above concepts and certain hausdorff content conditions