interval process model and non random vibration analysis abstract this paper develops an interval process model for time varying or dynamic uncertainty analysis when information of the uncertain parameter is inadequate by using the interval process model to describe a time varying uncertain parameter only its upper and lower bounds are required at each time point rather than its precise probability distribution which is quite different from the traditional stochastic process model a correlation function is defined for quantification of correlation between the uncertain but bounded variables at different times and a matrix decomposition based method is presented to transform the original dependent interval process into an independent one for convenience of subsequent uncertainty analysis more importantly based on the interval process model a non random vibration analysis method is proposed for response computation of structures subjected to time varying uncertain external excitations or loads the structural dynamic responses thus can be derived in the form of upper and lower bounds providing an important guidance for practical safety analysis and reliability design of structures finally two numerical examples and one engineering application are investigated to demonstrate the feasibility of the interval process model and corresponding non random vibration analysis method

comparison of dynamic response of structures with uncertain but bounded parameters using non probabilistic interval analysis method and probabilistic approach abstract now by combining the finite element analysis and interval mathematics a new non probabilistic set theoretical models that is interval analysis method is being developed in scientific and engineering communities to predict the variability or uncertainty resulting from the unavoidable scatter in structural parameters and the external excitations as an alternative to the classical probabilistic approaches interval analysis methods of uncertainty were developed for modeling uncertain parameters of structures in which bounds on the magnitude of uncertain parameters are only required not necessarily knowing the probabilistic distribution densities instead of conventional optimization studies where the minimum possible response is sought here an uncertainty modeling is developed as an anti optimization problem of finding the least favorable response and the most favorable response under the constraints within the set theoretical description in this study we presented the non probabilistic interval analysis method for the dynamical response of structures with uncertain but bounded parameters under the condition of the interval vector which contains the uncertain but bounded parameters determined from probabilistic statistical information or stochastic sample test through comparing between the non probabilistic interval analysis method and the probabilistic approach in the mathematical proof and the numerical examples we can see that the region of the dynamical response of structures with uncertain but bounded parameters obtained by the interval analysis method contains that produced by the probabilistic approach in other words the width of the maximum or upper and minimum or lower bounds on the dynamical responses yielded by the probabilistic approach is tighter than those produced by the interval analysis method this kind of results is coincident with the meaning of the probabilistic theory and interval mathematics

products of involutions in o v abstract let v be a finite dimensional vector space over a commutative field of characteristic distinct from 2 let v carry a symmetric nondegenerate bilinear form the special orthogonal group is o v o v det 1 main result an element o v is a product of two involutions in o v ifand only if dim v 2 mod 4 or an orthogonal decomposition of v into orthogonally indecomposable modules contains a module of odd dimension

correlation analysis of non probabilistic convex model and corresponding structural reliability technique compared with the probability approach the non probabilistic convex model only requires a small amount of samples to obtain the variation bounds of the imprecise parameters and whereby makes the reliability analysis very convenient and economical in this paper we attempt to propose and create a correlation analysis technique mathematically for the non probabilistic convex model and based on it develop an effective method to construct the multidimensional ellipsoids on the uncertainty a marginal convex model is defined to describe the variation range of each uncertain parameter and a covariance is defined to represent the correlation degree of two uncertain parameters for a multidimensional problem the covariance matrix and correlation matrix can be created through all marginal convex models and covariances based on which the required ellipsoid on the uncertainty can be conveniently achieved by combining the correlation analysis technique and the reliability index approach a non probabilistic reliability analysis method is also developed for uncertain structures six numerical examples are presented to demonstrate the effectiveness of the present method

interval k l expansion of interval process model for dynamic uncertainty analysis abstract the interval process model describes a time variant or dynamic uncertain parameter by the upper and lower bounds rather than the precise probability distribution at each time point providing an effective structural dynamic uncertainty quantification model with insufficient sample information by reference to the karhunen loeve k l expansion for stochastic process and random field models a novel expansion method for the interval process model namely the interval k l expansion is proposed in this work the interval k l expansion describes the continuous uncertainty of the interval process in time domain by superposition of infinite deterministic time related functions with uncorrelated interval coefficients which makes the interval process model much more convenient to use by introducing a splicing technique the interval k l expansion method for multiple correlated interval processes is also presented based on the interval k l expansion the vibration of structures subject to interval process excitations is analyzed where the analytic formulation of dynamic response bounds for linear elastic systems and continuum structures are derived finally several numerical examples are investigated to demonstrate the effectiveness of the proposed method

nonprobabilistic convex theoretic modeling of scatter in material properties nonprobabilistic convex modeling of uncertain material properties for viscoelastic structures is developed in this paper in particular the problem of forced vibrations of viscoelastic beams is studied first the analytic solution by inman is generalized for a deterministic set of variables describing material properties next these variables are treated as varying in a solid ball in the four dimensional space thus modeling the scatter in material properties the least favorable response needed for the design of the structure is determined

stability of elastic bars on uncertain foundations using a convex model a method is presented for evaluating the buckling load of weightless prismatic rigid bars resting on uncertain foundations which are modeled using linear extensional elastic springs the uncertainty in the spring elements is expressed in terms of a nonprobabilistic convex model an ellipsoidal bound is used which defines the uncertainty of the foundation in terms of a size parameter and the deviations of the elastic spring constants from their nominal values the size parameter represents the size of the ellipsoid and is analogous to the standard deviation magnitude in probabilistic analyses the semiaxes of the ellipsoid are the deviations of the foundation spring constants from their nominal values and they determine the shape of the ellipsoid a first order analysis shows that the reduction in the buckling load when uncertainty in the foundation s spring stiffnesses is present is a linear function of the size parameter and a nonlinear function of the semiaxes of the uncertainty ellipsoid for the same uncertainty in the spring elements different reductions in the buckling load result for beams with multimode buckling

optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval an optimization method for uncertain structures is suggested based on convex model and a satisfaction degree of interval in the investigated problem the uncertainty only exists in constraints convex model is used to describe the uncertainty in which the intervals of the uncertain parameters are only needed not necessarily to know the precise probability distributions a satisfaction degree of interval which represents the possibility that one interval is smaller than another is employed to deal with the uncertain constraints based on a predetermined satisfaction degree level the uncertain constraints are transformed to deterministic ones and the transformed optimization problem can be solved by traditional optimization methods for complex structural problems that the optimization model cannot be expressed in an explicit form the interval analysis method is adopted to calculate the intervals of the constraints efficiently and whereby eliminate the optimization nesting two numerical examples have been presented to demonstrate the efficiency of the suggested method

a sequential nonlinear interval number programming method for uncertain structures a sequential nonlinear interval number programming sninp method is suggested to deal with the uncertain optimization problems a general uncertain optimization model is investigated in which the objective function and constraints are both nonlinear and uncertain a nonlinear interval number programming ninp method is employed to transform the uncertain optimization problem into a deterministic two objective optimization problem then using the linear combination method and the constraint penalty function method a deterministic single objective and non constraint optimization problem is formulated in terms of a penalty function combining this ninp method with a modified approximation management framework amf an efficient sninp method is then constructed at each iterative step an approximation optimization problem is created based on the latin hypercube design lhd and the quadratic polynomial response surface approximation rsa and it can then be solved by the ninp method efficiently the trust region method is used to manage the sequence of the approximation optimization problems based on a reliability index an efficient method is suggested to calculate the actual penalty function and whereby the reliability index and based on it the current design space can be updated two numerical examples are presented to demonstrate the effectiveness of the present method

interval field model and interval finite element analysis uncertain parameters with inherent spatial variability are commonly encountered in engineering modeling of this kind of spatial uncertainty plays a fundamental role in structural uncertainty analysis which provides a necessary basis for subsequent uncertainty propagation through the system an interval field model for quantification of spatial uncertain parameters is proposed by which only the upper and lower bounds of the spatial uncertain parameters rather than their precise probability distributions are required the dependency can be fully considered by the proposed interval field model with the information of dependency an interval k l expansion is presented as a combination of deterministic functions with uncorrelated standard interval variables through which the continuous spatial interval field with dependency can be expressed only by very limited intervals necessary mathematical illustrations are provided for the proposed interval field model and the interval k l expansion the sampling method for the interval field model is given providing a robust numerical analysis basis for subsequent structural uncertainty analysis when the interval field model is applied in finite element analysis of structures with spatial uncertain parameters the non deterministic equilibrium equations with interval factors is then formulated the mcs method and the perturbation method are developed for solution of the derived interval equilibrium equations the feasibility and validity of the proposed interval field model and corresponding interval finite element methods are verified by numerical examples where the upper and lower bounds of the responses such as the displacements and the stresses of structures with spatial uncertain parameters are computed and compared

commutator groups of orthogonal groups over the reals with emphasis on lorentz groups abstract let be the commutator subgroup of the n dimensional lorentz group we give a criterion when an element of is a product of 2 or 3 involutions of we prove that a real element of is 2 reflectional then we study orthogonal groups over the reals with arbitrary signature in this situation each real element in the kernel of the spinorial norm is 2 reflectional in the kernel of the spinorial norm a main result states that the commutator group p q of an orthogonal group o p q over the reals is 2 reflectional if and only if the signature p q satisfies p q p q 2 mod 4 for all special orthogonal groups over arbitrary fields we prove that real elements are 2 reflectional

robust topology optimization under multiple independent unknown but bounded loads abstract the optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable or even lead to catastrophic failures a two level optimization formulation is often used in the robust topology optimization rto under uncertain loads various approaches have been reported to identify the critical loads associated with the worst structure responses because convex model approaches apply convex approximations to the original non convex model at the lower level the optimal designs obtained by these methods are greatly dependent on the quality of the approximation in this paper a new formulation based on the wolfe duality for the rto problems with multiple independent unknown but bounded loads is proposed following the two level formulation the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the wolfe duality both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non convexity associated with the lower level formulation numerical examples are also presented to demonstrate the proposed approach

the quasi complementarity problem abstract variational inequality techniques are used to suggest a unified and general iterative algorithm for computing the approximate solution of a new class of complementarity problems the convergence properties of this algorithm are also discussed

forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions abstract in this study a general frame of the forward and inverse structural uncertainty propagations ups based on the dimension reduction dr method and the derivative lambda probability density function pdf is proposed to handle arbitrary probability distribution for the forward up a dr method is applied to decompose a multivariable system into multiple univariate subsystems and a derivative pdf is adopted to transform the arbitrary probability distribution of each subsystem then the statistical moments and a fitting region are mathematically derived to analyze the fitting ability of the derivative pdf according to whether the kurtosis skewness point lies in or out the fitting region two different strategies combining the gauss gegenbauer quadrature are proposed to implement the forward propagation compared with the conventional methods the proposed method has advantages in higher accuracy stability and efficiency for the inverse propagation because the unknown variable may be arbitrary distribution the general frame based on the derivative pdf and the gauss gegenbauer quadrature are utilized to convert the uncertainty propagation into multiple deterministic calculations based on this optimization method is adopted to accurately obtain the statistical moments and pdfs of the unknown stochastic variables five examples are provided to verify the accuracy and efficiency of the proposed general frame for the forward and inverse ups

on products of two involutions in the orthogonal group of a vector space let v be a finite dimentional vector space over a commutative field of characteristic distinct from 2 let v carry a symmetric nondegenerate bilinear form results a let where o v and are involutions there exists an orthogonal decomposition of v into orthogonally indecomposable modules which are simultaneously invariant under and b let o v one can find involutions o v such that and b b b holds if and only if an orthogonal decomposition of v into orthogonally indecomposable modules does not contain a term whose minimum polynomial is x 1 where is even

non probabilistic convex model process a new method of time variant uncertainty analysis and its application to structural dynamic reliability problems in this paper we propose a method for time variant uncertainty analysis namely the non probabilistic convex model process which provides an effective mathematical tool for the analysis of structural dynamic uncertainty when lacking relevant information in the convex model process we express the variables at any time with intervals and establish the corresponding auto covariance function and correlation coefficient function to depict the correlation between variables at different times we also define several important characteristic parameters for the uni and bi dimensional convex model processes including the mid value function variance function auto covariance function and cross covariance function we provide the definition for the stationary convex model process and its ergodicity then by combining the convex model process with the first passage failure mechanism we propose a non probabilistic analysis model of structural dynamic reliability and formulate the solving algorithm based on monte carlo simulation finally through the analysis of numerical examples we verify the effectiveness of the convex model process and the model of dynamic reliability analysis proposed in this paper

multidimensional parallelepiped model a new type of non probabilistic convex model for structural uncertainty analysis summary non probabilistic convex models need to be provided only the changing boundary of parameters rather than their exact probability distributions thus such models can be applied to uncertainty analysis of complex structures when experimental information is lacking the interval and the ellipsoidal models are the two most commonly used modeling methods in the field of non probabilistic convex modeling however the former can only deal with independent variables while the latter can only deal with dependent variables this paper presents a more general non probabilistic convex model the multidimensional parallelepiped model this model can include the independent and dependent uncertain variables in a unified framework and can effectively deal with complex multi source uncertainty problems in which dependent variables and independent variables coexist for any two parameters the concepts of the correlation angle and the correlation coefficient are defined through the marginal intervals of all the parameters and also their correlation coefficients a multidimensional parallelepiped can easily be built as the uncertainty domain for parameters through the introduction of affine coordinates the parallelepiped model in the original parameter space is converted to an interval model in the affine space thus greatly facilitating subsequent structural uncertainty analysis the parallelepiped model is applied to structural uncertainty propagation analysis and the response interval of the structure is obtained in the case of uncertain initial parameters finally the method described in this paper was applied to several numerical examples copyright 2015 john wiley sons ltd

antioptimization of structures with large uncertain but non random parameters via interval analysis non probabilistic interval modeling of uncertain but non random parameters for structures is developed in this paper for antioptimization analysis consisting in determining the least favorable responses the uncertain but non random parameter is considered to be a deterministic variable belonging to a set modeled as an interval the least favorable static displacement bound estimation for structures with uncertain but non random parameters is transformed into solving interval linear equations for small interval parameters the width of interval being small the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters by means of the interval matrix central notation and the natural interval extension interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by qiu et al 1 for large interval parameters a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study the numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method

a new nonlinear interval programming method for uncertain problems with dependent interval variables this paper proposes a new nonlinear interval programming method that can be used to handle uncertain optimization problems when there are dependencies among the interval variables the uncertain domain is modeled using a multidimensional parallelepiped interval model the model depicts single variable uncertainty using a marginal interval and depicts the degree of dependencies among the interval variables using correlation angles and correlation coefficients based on the order relation of interval and the possibility degree of interval the uncertain optimization problem is converted to a deterministic two layer nesting optimization problem the affine coordinate is then introduced to convert the uncertain domain of a multidimensional parallelepiped interval model to a standard interval uncertain domain a highly efficient iterative algorithm is formulated to generate an efficient solution for the multi layer nesting optimization problem after the conversion three computational examples are given to verify the effectiveness of the proposed method

bounds analysis of a beam based on the convex model of uncertain foundation a methodology of bounds analysis based on the convex set of uncertainty is presented an infinitely long beam on an uncertain foundation is subjected to a moving force with constant speed the steady state vibration is analysed in the context of the finite element method uncertainty variables representing the uncertain fluctuation of foundation stiffness is confined within a convex hull given as a hyperellipse the fluctuation of dynamic responses of the beam that is the vertical deflection and the bending moment are approximated to a first order with respect to the uncertainty variables the upper and lower bounds of the response fluctuations are identified on the boundary of the convex model by means of the lagrange multiplier method numerical analysis demonstrates the validity of the proposed formulation for various constant speeds of the moving force the analysis is summarized in terms of the worst response which is the maximum absolute value of the response within the identified bounds