uniform rectifiability and harmonic measure ii poisson kernels in l p imply uniform rectifiability the first author was supported by nsf grants dms 0801079 and dms 1101244 the second author was supported by micinn grant mtm2010 16518 and by csic pie 200850i015 the third author was partially supported by grants dms 0901524 career dms 1056965 us nsf sloan research foundation and mtm2010 16232 mtm2009 14694 c02 01 spain

an objective 3d large deformation finite element formulation for geometrically exact curved kirchhoff rods the objective of this work is the development of a new finite element formulation for beams according to the kirchhoff theory of thin rods which includes the deformation states of axial tension torsion and bending the proposed formulation accounts for large deformations in three dimensional problem settings consisting of slender prismatic beams with arbitrarily curved initial geometries and arbitrary cross section shapes like in geometrically exact reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations rotations and strains a novel orthogonal interpolation strategy is applied to the triad field representing the cross section orientation in order to fulfill the kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem the continuity requirements resulting from the corresponding weak form are fulfilled by a cubic hermite interpolation thus leading to a c1 continuous representation of the beam centerline fundamental properties such as objectivity path independence consistency and accuracy of the developed beam element are verified by means of suitable numerical examples

the cauchy integral analytic capacity and uniform rectifiability several explanations concerning notation terminology and background are in order first notation by 7hi we have denoted the one dimensional hausdorff measure i e length and a z r stands for the closed disc with center z and radius r a curve f is called ad regular that is ahlfors david regular if it satisfies 1 2 with e f since the lower bound is automatic for curves this means that hl r n a z r 0 general sets satisfying 1 2 are called ad regular it is simplest to define the l2 boundedness of the cauchy singular integral operator via the truncated integrals we say that ce is bounded in l2 e without really defining the operator ce itself if there is m ox such that

energy consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints the present work deals with the energy consistent numerical integration of mechanical systems with mixed holonomic and nonholonomic constraints the underlying differential algebraic equations daes with index three are directly discretized this approach makes possible the development of a new energy consistent time stepping scheme for general nonholonomic systems in particular both nonholonomic problems from rigid body dynamics as well as flexible multibody dynamics can be treated in a unified manner in this connection specific constrained formulations of rigid bodies and geometrically exact beams are presented moreover the newly developed discrete null space method is applied to achieve a size reduction and an improved conditioning of the discrete system the numerical examples deal with a nonholonomic rigid body system and a flexible multibody system

stress resultant geometrically nonlinear shell theory with drilling rotations part i a consistent formulation abstract in this work we present a consistent theoretical framework for a novel stress resultant geometrically nonlinear shell theory the main feature of the present shell theory development which stands in contrast with the classical developments in the shell theory is the presence of a rotation component around the shell normal so called drilling rotation in the description of shell finite rotations the relationship of the proposed theory with a finite deformation theory of a three dimensional continuum with independent rotation field is clearly identified the latter is shown to be an important link which facilitates the proper choice of the shell constitutive model and a proper construction of the regularized form of the theory capable of supporting the drilling rotations the corresponding linearized form of the present shell theory is discussed in the closure

bifurcations and internal resonances in space curved rods abstract the paper discusses a geometrically non linear model of space curved beams the timoshenko type model is used which includes the shear effects and rotatory inertia the motion of the system is described by a non linear matrix equation which accounts for non linearities up to the second order to solve the non linear problem discretization methods based on the galerkin method are used and the resulting non linear eigenvalue problem is then solved by the continuation methods the analysis of the discretized problem allows the study of the bifurcations of solutions the post bifurcation analysis is used to explain the phenomenon of internal resonance which is of much importance in the prediction of the dynamic response of the analyzed systems the approach is illustrated by numerical examples which consider the free vibration of beams and circular arches with different boundary conditions

square function non tangential maximal function estimates and the dirichlet problem for non symmetric elliptic operators a we consider divergence form elliptic operators l div a x defined in the half space rn 1 n 2 where the coefficient matrix a x is bounded measurable uniformly elliptic t independent and not necessarily symmetric we establish square function non tangential maximal function estimates for solutions of the homogeneous equation lu 0 and we then combine these estimates with the method of approximability to show that l harmonic measure is absolutely continuous with respect to surface measure i e n dimensional lebesgue measure on the boundary in a scale invariant sense more precisely that it belongs to the class a with respect to surface measure equivalently that the dirichlet problem is solvable with data in lp for some p previously these results had been known only in the case n 1

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

frame indifferent beam finite elements based upon the geometrically exact beam theory this paper describes a new beam finite element formulation based upon the geometrically exact beam theory in contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame indifference or objectivity of the underlying beam theory the space interpolation of rotational degrees of freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory copyright 2002 john wiley sons ltd

uniform rectifiability calderon zygmund operators with odd kernel and quasiorthogonality in this paper we study some questions in connection with uniform rectifiability and the l boundedness of calderon zygmund operators we show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the jones numbers we also use these new coefficients to prove that n dimensional calderonzygmund operators with odd kernel of type c2 are bounded in l if is an n dimensional uniformly rectifiable measure

energy conserving decaying implicit time stepping scheme for nonlinear dynamics of three dimensional beams undergoing finite rotations in this work we present an extension of time integration energy conserving scheme which introduces desirable properties of controllable energy decay as well as numerical dissipation of high frequency contribution to total response finite element implementation details are given for the chosen model problem of geometrically exact beam undergoing finite rotations several numerical simulations illustrate a very satisfying performance of the proposed time stepping scheme

mode localization phenomena in large space structures the possibility of localization or confinement of vibratory modes in large space structures is investigated theoretically and numerically these structures belong to a class of periodic structures that have recently been shown to be sensitive to periodicity b reaking disorder or imperfections when localization occurs the modal amplitude of a global mode becomes confined to a local region of the structure with serious implications for the control problem the results of this study indicate that mode localization is most likely to occur in structures consisting of a large number of weakly coupled substructures certain large space structures with high modal densities fall in this category and it is therefore important to include the effect of structural imperfections and disorder when designing control systems for shape or directional control of such structures k kh kw ku k k m m m n n p qi qt t u v w uf 8fj e 6

an analyst s traveling salesman theorem for sets of dimension larger than one in his 1990 inventiones paper p jones characterized subsets of rectifiable curves in the plane via a multiscale sum of beta numbers these beta numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location jones result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point moreover computing this square sum for a curve returns the length of the curve up to multiplicative constant k okikiolu extended his result from subsets of the plane to subsets of euclidean space g david and s semmes extended the discussion to include sets of integer dimension larger than one under the assumption of ahlfors regularity and using a variant of jones beta numbers this variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set in this paper we give a version of p jones theorem for sets of arbitrary integer dimension lying in euclidean space our main result is a lower bound for the d dimensional hausdorff measure of a set in terms of an analogous sum of beta type numbers we also show an upper bound of this type the combination of these results gives a jones theorem for higher dimensional sets while there is no assumption of ahlfors regularity or of a measure on the underlying set there is an assumption of a lower bound on the hausdorff content we adapt david and semmes version of jones beta numbers by redefining them using a choquet integral allowing them to be defined for arbitrary sets and not just sets of locally finite measure a key tool in the proof is g david and t toro s parametrization of reifenberg flat sets with holes

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

numerical integration of non linear elastic multi body systems this paper is concerned with the modelling of nonlinear elastic multi body systems discretized using the finite element method the formulation uses cartesian co ordinates to represent the position of each elastic body with respect to a single inertial frame the kinematic constraints among the various bodies of the system are enforced via the lagrange multiplier technique the resulting equations of motion are stiff non linear differential algebraic equations the integration of these equations presents a real challenge as most available techniques are either numerically unstable or present undesirable high frequency oscillations of a purely numerical origin an approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system whereas the forces of constraint are discretized so that the work they perform vanishes exactly the combination of these two features of the discretization guarantees the stability of the numerical integration process for non linear elastic multi body systems examples of the procedure are presented

on the hausdorff dimension of harmonic measure in higher dimension assume a i n e a domain in f n where e is a compact set denote o a a x the harmonic measure for a of a evaluated at x r d according to 0ksendal s theorem o o e o a x is singular with respect to d dimensional lebesgue measure for d 2 and general domains this result seemed to be so far the only known localization property recently for d 2 it has been shown by p jones and t wolff j w that s o9 has dimension at most 1 this result completes previous work due to n g makarov and l carleson see i m and c2 let g be the green s function of a r with pole at some point of c r o9 in both the carleson and jones wolff arguments the integral g g nn log nn as boundary

on updating the structure of sparse matrix factors structural analysis often involves the solution of a sparse system of linear equations using matrix factorization for structural reanalysis problems the system of equations is progressively changing and matrix factorization is required at each iteration step in this paper we study the problem of updating the structure of sparse matrix factors using an ordered tree model the ordered tree model provides the information needed to symbolically compute the non zero structure of the matrix factor of a sparse matrix furthermore for matrix factor modification problems this tree model can be used to determine the numerically modified entries in the matrix factor examples for modification of a structure and h adaptive refinement of a finite element model are given to illustrate the potential application of the procedures developed in this study

discrete variational lie group formulation of geometrically exact beam dynamics the goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics by using a lie group variational integrator both spatial and temporal discretization are implemented in a geometry preserving manner the resulting scheme preserves both the discrete momentum maps and symplectic structures and exhibits almost perfect energy conservation comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically

computational aspects of vector like parametrization of three dimensional finite rotations theoretical and computational aspects of vector like parametrization of three dimensional finite rotations which uses only three rotation parameters are examined in detail in this work the relationship of the proposed parametrization with the intrinsic representation of finite rotations via an orthogonal matrix is clearly identified careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of reissner s non linear beam theory pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2 node three dimensional beam element a number of numerical simulations in three dimensional finite rotation analysis are presented in order to illustrate the proposed approach

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

geometrically exact 3d beam theory implementation of a strain invariant finite element for statics and dynamics geometrically exact 3d beam theory has been used as a basis for development of a variety of finite element formulations it has recently become apparent that the important requirement of objectivity of adopted strain measures although provided by the theory itself does not automatically extend to a finite element formulation in this paper we present a new finite element formulation of the geometrically exact 3d beam theory specifically designed to preserve the objectivity of the adopted strain measures in order to do so the current local rotations are interpolated in a manner similar to that adopted in co rotational approaches however no approximations typical for co rotational approaches are introduced into the procedure so in contrast to co rotational formulations the present formulation fully preserves the geometric exactness of the theory a range of numerical examples serves to illustrate the problem and to assess the formulation

a kinematically exact space finite strain beam model finite element formulation by generalized virtual work principle the present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by simo and vu quoc a geometrically exact rod model incorporating shear and torsion warping deformation int j solids structures 27 3 1991 371 3931 along the lines of the work on the planar beam theory presented by saje a variational principle for finite planar deformation of straight slender elastic beams internat j solids and structures 26 1990 887 9001 we apply exact non linear kinematic relationships of the space finite strain beam theory assuming the bernoulli hypothesis and neglecting the warping deformations of the cross section finite displacements and rotations as well as finite extensional shear torsional and bending strains are accounted for in the formulation a deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame rigidly attached to the cross section of the beam the position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step also the incremental rotational vector is introduced which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points because in 3 d space both the axial and the follower moments are non conservative not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure similar to that of lagrangian multipliers this makes possible the elimination of the displacement vector field from the principle so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle other researchers on the other hand employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field as a result more accurate and efficient family of beam finite elements for the non linear analysis of space frames has been obtained a one field formulation results in the fact that in the present finite elements the locking never occurs any combination of deformation states is described equally precisely this is in contrast with the elements developed in literature where in order to avoid the locking a reduced numerical integration has to be applied which unfortunately diminishes the accuracy of the solution polynomials have been chosen for the approximation of the components of the rotational vector in this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value the newton method is used for the iterative solution of the non linear equilibrium equations in an non equilibrium configuration the tangent stiffness matrix obtained by the linearization of governing equations using the directional derivative is non symmetric even for conservative loadings only upon achieving an equilibrium state the tangent stiffness matrix becomes symmetric thus obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the newton method for non conservative loadings however the tangent stiffness matrix is always non symmetric the numerical examples demonstrate capability of the present formulation to determine accurately the non linear behaviour of space frames in numerical examples the out of plane buckling loads are determined and the whole pre and post critical load displacement paths of a cantilever and a right angle frame are traced these in the analysis of space beams standard verification example problems show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself the rotations of 2n and extensional strains much beyond the realistic values of linear elastic material

kahler einstein metrics with edge singularities this article considers the existence and regularity of kahler einstein metrics on a compact kahler manifold m with edge singularities with cone angle 2 along a smooth divisor d we prove existence of such metrics with negative zero and some positive cases for all cone angles 2 2 the results in the positive case parallel those in the smooth case we also establish that solutions of this problem are polyhomogeneous i e have a complete asymptotic expansion with smooth coecients along d for all 2 2

interpolation of rotational variables in nonlinear dynamics of 3d beams the formulation of dynamic procedures for three dimensional 3 d beams requires extensive use of the algebra pertaining to the non linear character of the rotation group in space the corresponding extraction procedure to obtain the rotations that span a time step has certain limitations which can have a detrimental eect on the overall stability of a time integration scheme the paper describes two algorithms for the dynamics of 3 d beams which dier in their manifestation of the above limitation the rst has already been described in the literature and involves the interpolation of iterative rotations while an alternative formulation which eliminates the above eect by design requires interpolation of incremental rotations theoretical arguments are backed by numerical results similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one 1998 john wiley sons ltd

objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite element implementation the paper discusses the issue of discretization of the strainconfiguration relationships in the geometrically exact theory of threedimensional 3d beams which has been at the heart of most recent

a new approach to absolute continuity of elliptic measure with applications to non symmetric equations in the late 1950s and early 1960s the work of de giorgi degi and nash n and then moser mo initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients weak solutions in a domain 0 a priori only in a sobolev space w 1 loc 0 were shown to be ho lder continuous of some order depending just on ellipticity and maximum principles and harnack inequalities were established the dirichlet problem for such operators with continuous data on the boundary was established in lsw this in turn paved the way for a more systematic and detailed study of the properties of the elliptic measures d l associated to l div a on a domain 0 the classical properties of existence of non tangential limits of solutions fatou type theorems and comparison principles appeared in cfms but owed a great deal to the earlier work of carleson ca and hunt and wheeden h w on harmonic functions in lipschitz domains all the results mentioned above were carried out for elliptic operators l div a where the matrix a aij has bounded measurable coefficients and is symmetric however it turns out that the symmetry of the matrix is not needed to get these results morrey mor first observed this in connection with the de giorgi nash moser theory for the results in cfms this fact has not been formally observed until now with appropriate reformulation in terms of adjoint operators and adjoint green s functions the results of cfms are valid without the symmetry assumption see section 1 doi 10 1006 aima 1999 1899 available online at http www idealibrary com on

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

eigenvalue reanalysis of locally modified structures using a generalized rayleigh s method in this paper we introduce approximate reanalysis methods based on the generalized rayleigh s quotient with these methods the fundamental eigenvalue of a locally modified system can be computed readily using the fundamental mode as well as the static solution of the original system following a general development of theory several special cases of in terest in design are derived here and then tabulated numerical examples illustrate the accuracy of the new formulations

a consistent 3d corotational beam element for nonlinear dynamic analysis of flexible structures abstract the purpose of the paper is to present a corotational beam element for the nonlinear dynamic analysis of 3d flexible frames the novelty of the formulation lies in the use of the corotational framework i e the decomposition into rigid body motion and pure deformation to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix as a consequence cubic interpolations are adopted to formulate both inertia and internal local terms in the derivation of the dynamic terms an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained to enhance the efficiency of the iterative procedure an approximate expression of the tangent dynamic matrix is adopted four numerical examples are considered to assess the performance of the new formulation against the one suggested by simo and vu quoc 1988 48 it was observed that the proposed formulation proves to combine accuracy with efficiency in particular the present approach achieves the same level of accuracy as the formulation of simo and vu quoc but with a significantly smaller number of elements

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells

square functions nontangential limits and harmonic measure in codimension larger than 1 we characterize the rectifiability both uniform and not of an ahlfors regular set e of arbitrary codimension by the behavior of a regularized distance function in the complement of that set in particular we establish a certain version of the riesz transform characterization of rectifiability for lower dimensional sets we also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of e this allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that in sharp contrast with the usual setting of codimension 1 a converse to dahlberg s theorem must be false on lower dimensional boundaries without additional assumptions

locking free isogeometric collocation formulation for three dimensional geometrically exact shear deformable beams with arbitrary initial curvature abstract this paper presents a displacement based and a mixed isogeometric collocation formulations for three dimensional geometrically exact shear deformable beams with arbitrarily curved initial geometry strong forms of both formulations are consistently linearized with respect to the underlying configuration manifold high efficiency is achieved by using a geometrically consistent kinematic model in which incremental rotations are parametrized through the spatial rotation vector several numerical tests demonstrate that the proposed mixed formulation is totally locking free for any choice of approximation degree mixed formulation also exhibits higher robustness in the incremental iterative algorithm elimination of shear locking and low computational cost assured by the combination of isogeometric collocation with a consistent kinematic model result in a very efficient accurate and geometrically powerful method able to handle geometrically complex beam problems

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

deformation modes in the finite element absolute nodal coordinate formulation abstract the objective of this study is to provide interpretation of the deformation modes in the finite element absolute nodal coordinate formulation using several strain definitions in this finite element formulation the nodal coordinates consist of absolute position coordinates and gradients that can be used to define a unique rotation and deformation fields within the element as well as at the nodal points the results obtained in this study clearly show cross section deformation modes eliminated when the number of the finite element nodal coordinates is systematically and consistently reduced using the procedure discussed in this paper one can obtain a reduced order dynamic model eliminate position vector gradients that introduce high frequencies to the solution of some problems achieve the continuity of the remaining gradients at the nodal points and obtain a formulation that automatically satisfies the principle of work and energy furthermore the resulting dynamic model unlike large rotation finite element formulations leads to a unique rotation field and as a consequence the obtained formulation does not suffer from the problem of coordinate redundancy that characterizes existing large deformation finite element formulations in order to accurately define strain components that can have easy physical interpretation a material coordinate system is introduced to define the material element rotation and deformation using the material coordinate system the timoshenko reissner and euler bernoulli beam models can be systematically obtained as special cases of the absolute nodal coordinate formulation beam models while a constraint approach is used in this study to eliminate the cross section deformation modes it is important to point out as mentioned in this paper that lower order finite elements some of which already presented in previous investigations can be efficiently used in thin and stiff structure applications

approximate eigensolution of laplacian matrices for locally modified graph products laplacian matrices and their spectrum are of great importance in algebraic graph theory there exist efficient formulations for eigensolutions of the laplacian matrices associated with a special class of graphs called product graphs in this paper the problem of determining a few approximate smallest eigenvalues and eigenvectors of large scale product graphs modified through the addition or deletion of some nodes and or members is investigated the eigenproblem associated with a modified graph model is reduced using the set of master eigenvectors and linear approximated slave eigenvectors from the original model implicitly restarted lanczos method is employed to obtain the required eigenpairs of the reduced problem examples of large scale models are included to demonstrate the efficiency of the proposed method compared to the direct application of the irl method

a lanczos based method for structural dynamic reanalysis problems in this paper a new solution method for the modified eigenvalue problem with specific application to structural dynamic reanalysis is presented the method which is based on the block lanczos algorithm is developed for multiple low rank modifications to a system and calculates a few selected eigenpairs given the solution to the original system ax x procedures are developed for the modified standard eigenvalue problem a a x x where 1 a jbsjbt where sj s p p p n and b n p is constant for all the perturbations sj 2 a i jbisjbit where bi n p may vary with the pertubations sj the procedures are then extended for the reciprocal and generalized eigenvalue problems so that they are directly applicable to the structural dynamic reanalysis problem numerical examples are given to demonstrate the applications of the method

on rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model abstract in this work we discuss the finite element implementation of the internal constraints in a three dimensional 3d geometrically exact beam model which can be formulated as holonomic constraint relationships model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component appropriate modifications of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time integration schemes for nonlinear dynamics consequently only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved thus avoiding the use of the extra variables the lagrange multipliers and the pertinent difficulties in integrating the system of differential algebraic equations a number of numerical simulations considering dynamic analysis of multibody systems with rigid flexible components and joint constraints are presented in order to illustrate versatility of the proposed procedure

sparse matrix factor modification in structural reanalysis structural reanalysis problems such as in nonlinear finite element analysis or optimum design involve progressive changes in the global stiffness matrix and its matrix factors although many studies have been devoted to the subject of matrix factor modification most investigations have dealt with the problem separately from sparse matrix methods this paper introduces a graph theoretic model for the forward solution procedure which is applicable for identifying the modified entries of the matrix factors due to changes in the original matrix applications of this graph theoretic model to existing refactorization methods are presented the relation between substructuring and sparse matrix ordering strategies and their effects on reanalysis are discussed modification of a sparse matrix associated with an n n finite element grid ordered by the nested dissection scheme is analysed

on the uniform rectifiability of ad regular measures with bounded riesz transform operator the case of codimension 1 we prove that if is a d dimensional ahlfors david regular measure in mathbb r d 1 then the boundedness of the d dimensional riesz transform in l2 implies that the non baup david semmes cells form a carleson family combined with earlier results of david and semmes this yields the uniform rectifiability of

improved first order approximation of eigenvalues and eigenvectors a method based on reduced basis approximation concepts is presented for improved first order approximation of eigenvalues and eigenvectors of modified structural dynamic systems the terms of a local approximation based on taylor or matrix power series are used as basis vectors for approximating the perturbed eigenparameters for each eigenmode a reduced eigensystem is generated by using the baseline eigenvector and the first order approximation term as ritz vectors the solution of the reduced eigensystem leads to two possible estimates of each perturbed eigenvalue and eigenvector criteria for selection of the best approximation are presented the zero and first order rayleigh quotient approximation can be directly recovered as special cases of the present method results are presented for approximate dynamic reanalysis of a 25 bar planar truss structure it is shown that high quality approximation of the perturbed eigenparameters can be obtained for moderate perturbations in the stiffness and mass matrices for very large local perturbations of the structure including deletion of some structural members it is shown that the present method yields reasonable quality approximations