modal analysis and shape optimization of rotating cantilever beams when a cantilever beam rotates about an axis perpendicular to its neutral axis its modal characteristics often vary significantly if the geometric shape and the material property of the beam are given the modal characteristic variations can be accurately estimated following a well established analysis procedure employing assumed mode method or finite element method in many practical design situations however some modal characteristics are usually specified as design requirements and the geometric shape that satisfies the requirements needs to be found in the present study certain modal characteristic requirements such as maximum or minimum slope natural frequency loci are specified and the geometric shapes that satisfy the requirements are obtained through an optimization procedure

uniqueness and nondegeneracy of positive solutions of delta s u u u p rm in mathbb r n when s is close to 1 we consider the equation delta s u u u p with s in 0 1 in the subcritical range of p we prove that if s is sufficiently close to 1 the equation possesses a unique minimizer which is nondegenerate

a new algorithm for high dimensional uncertainty quantification based on dimension adaptive sparse grid approximation and reduced basis methods in this work we develop an adaptive and reduced computational algorithm based on dimension adaptive sparse grid approximation and reduced basis methods for solving high dimensional uncertainty quantification uq problems in order to tackle the computational challenge of curse of dimensionality commonly faced by these problems we employ a dimension adaptive tensor product algorithm 16 and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions to reduce the heavy computational cost of uq problems modelled by partial differential equations pde we adopt a weighted reduced basis method 7 and develop an adaptive greedy algorithm in combination with the previous verified algorithm for efficient construction of an accurate reduced basis approximation the efficiency and accuracy of the proposed algorithm are demonstrated by several numerical experiments

an element free galerkin approach for rigidflexible coupling dynamics in 2d state based on an improved element free galerkin efg method and the geometric nonlinear elastic theory an element free numerical approach for rigidflexible coupling dynamics of a rotating hubbeam system is implemented using the hamilton principle the coupled nonlinear integral differential governing equations are derived in which the dynamic stiffening effect is expressed by the longitudinal shrinking of the beam induced by the transverse displacement by the efg method where the global interpolating moving least squares imls and generalized moving least squares igmls were used for discretizing the longitudinal and transverse deformation variables respectively the spatially element free discretized dynamic equations of the system are obtained and the newmark scheme was selected as the time integration method for the numerical computation the superiority of the global interpolating property makes the ease for imposing both displacement and derivative boundary conditions the transverse bending vibration and structural dynamics of the flexible beam in non inertial system are analyzed in the numerical simulations and influence parameters of the efg method are also fully discussed simulation results verify the feasibility of the improved efg approach for rigidflexible coupling dynamics

gaussian wave packet transform based numerical scheme for the semi classical schrodinger equation with random inputs abstract in this work we study the semi classical limit of the schrodinger equation with random inputs and show that the semi classical schrodinger equation produces o e oscillations in the random variable space with the gaussian wave packet transform the original schrodinger equation is mapped to an ordinary differential equation ode system for the wave packet parameters coupled with a partial differential equation pde for the quantity w in rescaled variables further we show that the w equation does not produce e dependent oscillations and thus it is more amenable for numerical simulations we propose multi level sampling strategy in implementing the gaussian wave packet transform where in the most costly part i e simulating the w equation it is sufficient to use e independent samples we also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm and hopefully shed light on possible future directions

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

least energy solutions for a weakly coupled fractional schrodinger system abstract in this paper we consider the following system of two weakly coupled fractional nonlinear schrodinger equations s u u u 2 p b x u p 1 v p 1 u x r n s v 2 s v v 2 p b x v p 1 u p 1 v x r n by use of the s harmonic extension technique we establish the existence of a nontrivial least energy solution of the system via variational methods especially in the autonomous case i e b x b a positive least energy solution with both nontrivial components is obtained

non linear free vibrations of a rotating flexible arm abstract the non linear moderately large amplitude flexural free vibrations of an arm clamped with a setting angle to a rigid rotating hub are studied the shear deformation and rotary inertia effects are assumed to be negligible but account is taken of axial inertia non linear curvature and the inextensibility condition the lagrangian approach in conjunction with the assumed modes method assuming constant hub rotation speed is used in a consistent manner to obtain the third order non linear uni modal temporal problem because of the strength of the non linearities in the temporal problem which includes elastic and inertial geometric stiffening as well as inertial softening terms a time transformation method is employed to obtain an approximate solution to the frequency amplitude relation of arm free oscillation results in non dimensional form are presented graphically for the effect of hub rotation speed blade setting angle and hub radius on the variation of the natural frequency with vibration amplitude

free vibration analysis of rotating cantilever plates linearized equations of motion for the free vibration analysis of rotating cantilever plates are derived two inplane stretch variables are introduced and approximated to obtain the ordinary differential equations of motion the use of the two in plane stretch variables enables one to obtain the equations of motion which include proper motion induced stiffness variation terms the equations of motion are transformed into dimensionless forms in which dimensionless parameters are identieed the effects of the dimensionless parameters on the modal characteristics of rotating cantilever plates are investigated through numerical study the eigenvalue loci veering and crossing along with the associated mode shape variations are also presented and discussed

the probabilistic approach for rotating timoshenko beams a stochastic modal analysis is developed for reliability studies of rotating beams with uncertain material and sectional properties uncertain geometric parameters and random rotating speed the formulation of the stochastic modal analysis is based on mean centered second order perturbation technique and nonlinear eigenvalue analysis a consistent linearization of the fully geometrically nonlinear beam theory and the virtual work principle are used to derive the governing equations of a rotating timoshenko beam and a power series method is employed for the nonlinear eigenvalue analysis to obtain the natural frequencies and vibration modes for free vibration the effects of variations in material and sectional properties geometric parameters and rotating speed on variation in natural frequency are investigated a sensitivity analysis is performed to identify the important factors on the variation of frequency responses here a definition of resonant failure of a rotating beam is introduced and considered as the limit state function in the reliability assessment of the rotating beams to establish the validity of the present probabilistic approach numerical examples with results obtained by using the monte carlo method are given for comparison

stochastic convergence acceleration through basis enrichment of polynomial chaos expansions given their mathematical structure methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory this is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition this enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non continuous dependence of the solution on the stochastic parameters in particular given the polynomial structure of these approximations the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated an enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non smooth functions by integrating a priori knowledge about their behaviour the proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues it is observed that through judicious selection of enrichment functions the spectrum of such a random system can be more efficiently characterized even for systems with closely spaced eigenvalues copyright 2007 john wiley sons ltd

validation of the greater trochanter method with radiographic measurements of frontal plane hip joint centers and knee mechanical axis angles and two other hip joint center methods several motion capture methods exist for predicting hip joint centers hjc these methods include regression models functional joints and projections from greater trochanters while regression and functional methods have been compared to imaging techniques the troch method has not been previously validated the purpose of this study was to compare frontal plane hjcs and knee mechanical axis angles estimated using the greater trochanter method with a regression bell and a functional method against those obtained using radiographs thirty five participants underwent a long standing anteroposterior radiograph and performed static and functional motion capture trials the bell functional and trochanter hjcs were constructed to predict mechanical axes and compare hjc locations one way repeated measures anovas were used to compare mechanical axes and hjc locations estimated by motion capture methods and measured using radiographs p 0 05 all methods overestimated mechanical axes compared to radiographs 2 but were not different mediolateral hjc locations and inter hjc widths were similar between methods however inter hjc widths were underestimated average 3 7 compared to radiographs the bell hjc was more superior and anterior to both functional and trochanter methods the trochanter hjc was more posterior to both methods the bell method outperformed the other methods in leg length predictions compared to radiographs although differences existed between methods all frontal plane hjc location differences were 1 7 cm this study validated the trochanter hjc prediction method mediolaterally and vertically with small respective correction factors therefore all hjc methods seem to be viable in predicting mechanical axes and frontal plane hjc locations compared with radiographs

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

modal characteristic of a rotating rectangular cantilever plate modal characteristics of a rotating cantilever plate are investigated in the present work a dynamic modelling method for rectangular plates undergoing prescribed overall motion is employed to derive the equations of motion the general equations are particularized for the modal analysis of a rotating cantilever plate and dimensionless parameters are identified through dimensional analysis the effects of the dimensionless parameters on the modal characteristics of the rotating plate are investigated incidentally eigenvalue loci veering and crossing phenomena along with the corresponding modeshape variations are exhibited and discussed

multi fidelity non intrusive polynomial chaos based on regression abstract in this paper we present a multi fidelity mf extension of non intrusive polynomial chaos based on regression point collocation for uncertainty quantification purposes the proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients due to its usage of regression to estimate the coefficients the present method offers high flexibility in the sampling and generation of the polynomial basis the method takes advantage of a nested sampling plan to create the samples for the low fidelity lf and correction expansions where the high fidelity hf samples are a subset of the lf ones to build the polynomial basis a total order or hyperbolic truncation strategy is used with a highly flexible combination of the lf and correction polynomial expansions the method is demonstrated on some artificial test problems and aerodynamic problems of the euler flow around an airfoil and common three dimensional research models in order to derive the strategies for successful mf approximation the effect of the correlation and the errors between the lf and hf functions is also studied the results show that high correlation and moderately low errors are important to improve the mf approximation s accuracy on a common research model problem the mf approach with partially converged simulations as the lf samples can successfully reduce the computational cost to about 40 for similar accuracy compared to an approach using a single hf expansion

semiclassical states for fractional schrodinger equations with critical growth abstract in this paper we consider the following fractional nonlinear schrodinger equations e 2 s s u v x u p x g u q x u 2 s 2 u x r n and prove the existence and concentration of positive solutions under suitable assumptions on the potentials v x p x and q x we show that the semiclassical solutions u e with maximum points x e concentrating at a special set s p characterized by v x p x and q x moreover for any sequence x e x 0 s p v e x u e e x x e convergence strongly in h s r n to a ground state solution v of s v v x 0 v p x 0 g v q x 0 v 2 s 2 v x r n

random field finite elements the probabilistic finite element method pfem is formulated for linear and non linear continua with inhomogeneous random fields analogous to the discretization of the displacement field in finite element methods the random field is also discretized the formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization furthermore it is shown that a reduced set of the uncorrelated variables is sufficient for the second moment analysis based on the linear formulation of the pfem the method is then extended to transient analysis in non linear continua the accuracy and efficiency of the method is demonstrated by application to a one dimensional elastic plastic wave propagation problem and a two dimensional plane stress beam bending problem the moments calculated compare favourably with those obtained by monte carlo simulation also the procedure is amenable to implementation in deterministic fem based computer programs

a posteriori error estimation and adaptivity in stochastic galerkin fem for parametric elliptic pdes beyond the affine case abstract we consider a linear elliptic partial differential equation pde with a generic uniformly bounded parametric coefficient the solution to this pde problem is approximated in the framework of stochastic galerkin finite element methods we perform a posteriori error analysis of galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations practical versions of this error estimate are discussed and tested numerically for a model problem with non affine parametric representation of the coefficient furthermore we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric pde problem the performance of the adaptive algorithm is tested numerically for model problems with two different non affine parametric representations of the coefficient

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

effects of hip center location on the moment generating capacity of the muscles we have developed a three dimensional biomechanical model of the human lower extremity to study how the location of the hip center affects the moment generating capacity of four muscle groups the hip abductors adductors flexors and extensors the model computes the maximum isometric force and the resulting joint moments that each of 25 muscle tendon complexes develops at any body position abduction adduction flexion and extension moments calculated with the model correspond closely with isometric joint moments measured during maximum voluntary contractions we used the model to determine 1 the hip center locations that maximize and minimize the moment generating capacity of each muscle group and 2 the effects of superior inferior anterior posterior and medial lateral displacement of the hip center on the moment arms maximum isometric muscle forces and maximum isometric moments generated by each muscle group we found that superior inferior displacement of the hip center has the greatest effect on the force and moment generating capacity of the muscles a 2 cm superior displacement decreases abduction force 44 moment arm 12 and moment 49 while a 2 cm inferior displacement increases abduction force 20 moment arm 7 and moment 26 similarly a 2 cm superior displacement decreases flexion force 27 moment arm 6 and moment 22 while inferior displacement increases all three variables anterior posterior displacement alters the moment generating capacity of the flexors and extensors considerably primarily due to moment arm changes medial lateral displacement has a large effect on the moment generating capacity of the adductors only a 2 cm medial displacement decreases adduction moment arm 20 force 26 and moment 40 these results demonstrate that the force and moment generating capacities of the muscles are sensitive to the location of the hip center

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

the pohozaev identity for the fractional laplacian in this paper we prove the pohozaev identity for the semilinear dirichlet problem delta s u f u in omega u equiv0 in mathbb r n backslash omega here s in 0 1 s is the fractional laplacian in mathbb r n and is a bounded c1 1 domain to establish the identity we use among other things that if u is a bounded solution then u delta s omega is c up to the boundary where x dist x in the fractional pohozaev identity the function u delta s partial omega plays the role that u plays in the classical one surprisingly from a nonlocal problem we obtain an identity with a boundary term an integral over which is completely local as an application of our identity we deduce the nonexistence of nontrivial solutions in star shaped domains for supercritical nonlinearities

stability analysis for the flapwise motion of a cantilever beam with rotary oscillation abstract the flapwise motion of a cantilever beam with rotary oscillation is analyzed to investigate the dynamic stability of the beam the cantilever beam is regarded as a system subjected to parametric excitation because the angular speed of the cantilever beam varies harmonically to consider the stiffening effect due to the centrifugal force this study adopts the linear partial differential equation of flapwise motion which is derived by a modelling method using the stretch deformation instead of the conventional longitudinal deformation after the partial differential equation is discretized by the galerkin method the method of multiple scales is applied using this method the stability of the beam is analyzed for the variations of the oscillating frequency and the maximum angular speed in addition to verify the stability results the time responses of flapwise motion are computed by the generalized time integration method

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

dynamics of rectangular plates undergoing prescribed overall motion a set of linear equations of motion for rectangular plates undergoing prescribed overall motion is derived in this paper two in plane stretch variables with which the in plane strain energy of the plate can be expressed in an exact quadratic form are employed and approximated to derive the equations of motion the equations of motion include motion induced stiffness variation terms which are expressed as explicit functions of the prescribed overall motion the effects of the motion induced stiffness variation terms which are neglected in the conventional linear modelling method on the dynamic response of the plate are investigated the reliability and the accuracy of the equations of motion which are derived in the present work are examined through the numerical study

dynamics of flexible beams undergoing overall motions abstract a modelling method for straight beams undergoing large overall motions as well as small elastic deformations is presented in this paper different from the classical linear cartesian modelling method which employs three cartesian deformation variables the present modelling method uses a non cartesian variable along with two cartesian variables to describe the elastic deformation a quadratic form of the strain energy expressed with the hybrid set of deformation variables is used to obtain the generalized active forces and a geometric constraint equation relating the non cartesian variable and cartesian variables is used to obtain the generalized inertia forces in the equations of motion the present modelling method not only provides accurate simulation results but also clarifies the limit of validity of the classical linear cartesian modelling method

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

bound state solutions for the supercritical fractional schrodinger equation abstract we prove the existence of positive solutions for the supercritical nonlinear fractional schrodinger equation s u v x u u p 0 in r n with u x 0 as x where p n 2 s n 2 s for s 0 1 n 2 s we show that if v x o x 2 s as x then for p n 2 s 1 n 2 s 1 this problem admits a continuum of solutions more generally for p n 2 s n 2 s conditions for solvability are also provided this result is the extension of the work by davila del pino musso and wei to the fractional case our main contributions are the existence of a smooth radially symmetric entire solution of s w w p in r n and the analysis of its properties the difficulty here is the lack of phase plane analysis for a nonlocal ode instead we use conformal geometry methods together with schaaf s argument as in the paper by ao chan delatorre fontelos gonzalez and wei on the singular fractional yamabe problem

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

dynamic analysis of flexible beams undergoing overall motion employing linear strain measures a modeling method for the dynamic analysis of a eexible beam undergoing free overall motion is proposed the overall motion is measured at a reference frame that is a rigid cross section of the beam the elastic deformation is measured with respect to the reference frame and is expressed with deformation variables including a stretch variable theuseof astretch variable leadsto asimplelinear strain measure and acorrespondingquadraticstrain energy expression in spite of the simplicity motion induced stiffness variation effects which are often critically important to the dynamic analysis of e exible structures undergoing overall motion can be captured accurately by the modeling method to demonstrate the accuracy of the modeling method test problems are presented and solved the present modeling method proves to be as accurate as a nonlinear e nite element modeling method applicable ranges of the present modeling method are investigated through dimensional analysis and numerical study

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

uncertainty propagation in finite deformations a spectral stochastic lagrangian approach this work proposes a method for quantifying uncertainty propagation in finite deformation problems using the spectral stochastic finite element method ssfem a spectral expansion of the current configuration of a deforming body is proposed to compute the stochastic deformation gradient which is in turn used to compute the stochastic analogs of the various quantities which appear in large deformation analysis such as strain and stress measures and consistent moduli a total lagrangian approach to the stochastic large deformation problem is presented model problems in large deformation elasto plasticity are considered highlighting the features of the methodology developed rigorous comparisons with monte carlo solutions are presented it is shown that the proposed approach can estimate the probability density function and response statistics of the field variables with satisfactory accuracy

a new method for estimating joint parameters from motion data joint centers and axes of rotation joint parameters are central to all branches of movement analysis in gait analysis the standard protocol used to determine hip and knee joint parameters is prone to errors arising from palpation anthropometric regression equations and misplaced alignment devices several alternative methods have been proposed but to date none have been shown to be accurate and reliable enough for use in the clinical setting this article describes a new method for joint parameter estimation the new method can be summarized as follows i the motions of two adjacent segments spanning a single joint are tracked ii the axis of rotation between every pair of observed segment configurations is computed iii the most likely intersection of all axes effective joint center and most likely orientation of the axes effective joint axis is found initial validation of the method was conducted on a hinged mechanical analog and a single healthy adult subject for the analog the center was found to be within 3 8 mm of the geometric center and 2 0 degrees of the geometric axis standard deviation for the adult subject hip centers varied on the order of 1 3 mm knee centers by 3 9 mm and knee axes by 2 0 degrees the results suggest that the new method is an objective precise and practical alternative to the standard clinical approach

stochastic modeling and vibration analysis of rotating beams considering geometric random fields abstract geometric parameters such as the thickness and width of a beam are random for various reasons including manufacturing tolerance and operation wear due to these random parameter properties the vibration characteristics of the structure are also random in this paper we derive equations of motion to conduct stochastic vibration analysis of a rotating beam using the assumed mode method and stochastic spectral method the accuracy of the proposed method is first verified by comparing analysis results to those obtained with monte carlo simulation mcs the efficiency of the proposed method is then compared to that of mcs finally probability densities of various modal and transient response characteristics of rotating beams are obtained with the proposed method

on the non linear vibrations of an inextensible rotating arm with setting angle and flexible hub abstract this work is concerned with the development of a dynamic model for a slender flexible arm undergoing relatively large planar flexural deformations and clamped with a setting angle to a compliant rotating hub the model is derived assuming an inextensible euler bernoulli beam but takes into account the axial inertia and nonlinear curvature two coordinate transformations were used to obtain the inertial position vector of a typical material point along the beam span the inextensibility constraint is used to relate the axial and transverse displacements as well as velocities of the material point in the beam body coordinate system the displacement and velocity vectors obtained are used in the system kinetic energy and exact curvature to eliminate dependence of the system lagrangian on the beam axial deflection and velocity the assumed mode method is used to discretize the system lagrangian and to derive directly the nonlinear equivalent temporal solutions to the problem the resulting dynamic model is a set of four strongly coupled nonlinear ordinary differential equations which in the present work is solved only numerically examples of the results of the numerical solutions for the effects of setting angle and defined physical parameter ratios on the system dynamic response characteristics for a sinusoidal hub torque profile are presented and discussed

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

dynamic analysis of a rotating cantilever beam by using the finite element method a finite element analysis for a rotating cantilever beam is presented in this study based on a dynamic modelling method using the stretch deformation instead of the conventional axial deformation three linear partial differential equations are derived from hamilton s principle two of the linear differential equations are coupled through the stretch and chordwise deformations the other equation is an uncoupled one for the flapwise deformation from these partial differential equations and the associated boundary conditions are derived two weak forms one is for the chordwise motion and the other is for the flapwise motion the weak forms are spatially discretized with newly defined two node beam elements with the discretized equations the behaviours of the natural frequencies are investigated for the variation of the rotating speed in addition the time responses and distributions of the deformations and stresses are computed when the rotating speed is prescribed the effects of the rotating speed profile on the vibrations of the beam are also investigated

unified polynomial expansion for interval and random response analysis of uncertain structure acoustic system with arbitrary probability distribution abstract for structure acousticsystem with uncertainties the interval model the random model and the hybrid uncertain model have been introduced in the interval model and the random model the uncertain parameters are described as either the random variable with well defined probability density function pdf or the interval variable without any probability information whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously for response analysis of these three uncertain models of structure acoustic problem involving arbitrary pdfs a unified polynomial expansion method named as the interval and random arbitrary polynomial chaos method irapcm is proposed in irapcm the response of the structure acoustic system is approximated by apc expansion in a unified form particularly only the weight function of polynomial basis is required to be changed to construct the apc expansion for the response of different uncertain models through the unified apc expansion the uncertain properties of the response of three uncertain models can be efficiently obtained as the apc expansion can provide a free choice of the polynomial basis the optimal polynomial basis for the random variable with arbitrary pdfs can be obtained by using the proposed irapcm the irapcm has been employed to solve a mathematical problem and a structure acoustic problem and the effectiveness of the unified irapcm for response analysis of three uncertain models is demonstrated by fully comparing it with the hybrid first order perturbation method and several existing polynomial chaos methods

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

concentrating standing waves for the fractional nonlinear schrodinger equation abstract we consider the semilinear equation e 2 s s u v x u u p 0 u 0 u h 2 s r n where 0 s 1 1 p n 2 s n 2 s v x is a sufficiently smooth potential with inf r v x 0 and e 0 is a small number letting w be the radial ground state of s w w w p 0 in h 2 s r n we build solutions of the form u e x i 1 k w i x i e e where i v i e and the i e approach suitable critical points of v via a lyapunov schmidt variational reduction we recover various existence results already known for the case s 1 in particular such a solution exists around k nondegenerate critical points of v for s 1 this corresponds to the classical results by floer and weinstein 13 and oh 24 25

equilibrium and modal analyses of rotating multibeam structures employing multiple reference frames a single reference frame is frequently employed to derive equations of motion in structural dynamic analysis for simple structures like a single beam and a single plate the method of employing a single reference frame usually provides well converged and accurate results for equilibrium and modal analyses however for structures having more general configurations such as multibeam structures use of a single reference frame often results in slow convergence and even erroneous analysis results in the present study a modeling method employing multiple reference frames for dynamic equilibrium and modal analyses of rotating structures is presented the superiority of the proposed modeling method over the single reference frame modeling method is verified with numerical examples

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

dynamic modelling and stability analysis of axially oscillating cantilever beams dynamic stability of an axially oscillating cantilever beam is investigated in this paper equations of motion for the axially oscillating beam are derived and transformed into dimensionless forms the equations include harmonically oscillating parameters which are related to the motion induced stiffness variation stability diagrams of the first and the second order approximate solutions are obtained by using the multiple scale perturbation method the stability diagrams show that there exist significant difference between the first and the second order approximate solutions it is also found that relatively large unstable regions exist around the first bending natural frequency twice the first bending natural frequency and twice the second being natural frequency the validity of the stability diagram is verified by direct numerical integrations of the equations of motion of the system

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

dynamic modeling and stability analysis of an axially oscillating beam undergoing periodic impulsive force a dynamic modeling method of an axially oscillating beam undergoing periodic impulsive force is proposed in the present work an element specific modeling method using a stretch deformation variable is employed to derive the linear equations of motion the stiffness variation effect induced by the periodic impulsive force as well as the axially oscillating motion is included in the linear equations of motion the accuracy of the proposed modeling method is verified by comparing its numerical solutions to those of a nonlinear finite element code the effects of the impulse magnitude the oscillating frequency the oscillating speed amplitude and the modal damping ratio on the dynamic stability of the beam are investigated with the modeling method

explanation on the importance of narrow band random excitation characters in the response of a cantilever beam a detailed theoretical investigation into the first and second mode response of a parametrically excited slender cantilever beam where the narrow band random excitation characters are taken into consideration is presented the method of multiple scales is used to determine a uniform first order expansion of the solution of the nonlinear integro differential equations the first order moment frequency and force response data curves of a specimen beam tested by other investigators are obtained further comparisons have been made and results show that whether the first order moment frequency response data curves or the first order moment force response data curves of the first two modes are all in agreement with other investigators experimental results furthermore the stochastic jump and bifurcation have been investigated for the first modal parametric principal resonance by using the stationary joint probability of amplitude and phase results show that stochastic jump occurs mainly in the region of triple valued solution for the frequency response domain if the bandwidth is a variable and others keep constant the basic phenomena indicate that the most probable motion is around the higher branch when the bandwidth is smaller whereas the most probable motion gradually approaches the lower one when the bandwidth becomes higher if the excitation central frequency f is a variable and others keep constant the basic phenomena imply that the higher is f the more probable is the jump from the higher branch to the lower one once f exceeds an certain value for the force response domain there is a region of excitation acceleration a within which the joint probability density has two peaks an upper peak and a lower peak results show that the upper peak decreases while the lower peak increases as the value of a decreases

asymptotic behavior of solutions for nonlinear elliptic problems with the fractional laplacian abstract in this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional laplacians and the critical exponents this work can be seen as a nonlocal analog of the results of han 1991 24 and rey 1990 35

numerical studies of the stochastic korteweg de vries equation we present numerical solutions of the stochastic korteweg de vries equation for three cases corresponding to additive time dependent noise multiplicative space dependent noise and a combination of the two we employ polynomial chaos for discretization in random space and discontinuous galerkin and finite difference for discretization in physical space the accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and monte carlo simulation results of particular interest is the interplay of spatial discretization error with the stochastic approximation error which is examined for different orders of spatial and stochastic approximation