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Null Space (null + space)
Selected AbstractsA new approach to reduce membrane and transverse shear locking for one-point quadrature shell elements: linear formulationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2006Rui P. R. Cardoso Abstract In the last decade, one-point quadrature shell elements attracted many academic and industrial researchers because of their computational performance, especially if applied for explicit finite element simulations. Nowadays, one-point quadrature finite element technology is not only applied for explicit codes, but also for implicit finite element simulations, essentially because of their efficiency in speed and memory usage as well as accuracy. In this work, one-point quadrature shell elements are combined with the enhanced assumed strain (EAS) method to develop a finite element formulation for shell analysis that is, simultaneously, computationally efficient and more accurate. The EAS method is formulated to alleviate locking pathologies existing in the stabilization matrices of one-point quadrature shell elements. An enhanced membrane field is first constructed based on the quadrilateral area coordinate method, to improve element's accuracy under in-plane loads. The finite element matrices were projected following the work of Wilson et al. (Numerical and Computer Methods in Structural Mechanics, Fenven ST et al. (eds). Academic Press: New York, 1973; 43,57) for the incompatible modes approach, but the present implementation led to more accurate results for distorted meshes because of the area coordinate method for quadrilateral interpolation. The EAS method is also used to include two more displacement vectors in the subspace basis of the mixed interpolation of tensorial components (MITC) formulation, thus increasing the dimension of the null space for the transverse shear strains. These two enhancing vectors are shown to be fundamental for the Morley skew plate example in particular, and in improving the element's transverse shear locking behaviour in general. Copyright © 2005 John Wiley & Sons, Ltd. [source] On robust stability of uncertain systems with multiple time-delaysINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 15 2010Tong ZhouArticle first published online: 27 NOV 200 Abstract On the basis of an infinite to one mapping and the structure of the null space of a multivariate matrix polynomial (MMP), a novel sufficient condition is derived in this paper for the robust stability of a linear time-invariant system with multiple uncertain time-delays, parametric modelling errors and unmodelled dynamics. This condition depends on time-delay bounds and is less conservative than the existing ones. An attractive property is that this condition becomes also necessary in some physically meaningful situations, such as the case that there is only one uncertain time-delay and neither parametric perturbations nor unmodelling errors exist. Moreover, using ideas of representing a positive-definite MMP through matrix sum of squares, an asymptotic necessary and sufficient condition is derived for the robust stability of this system. All the conditions can be converted to linear matrix inequalities. Copyright © 2009 John Wiley & Sons, Ltd. [source] Null model analysis of communities on gradientsJOURNAL OF BIOGEOGRAPHY, Issue 6 2004James G. Sanderson Abstract Aim, I employed a novel null model and metric to uncover unusual species co-occurrence patterns in a herpetofaunal assemblage of 49 species collected at discrete elevations along a gradient. Location, Mount Kupe, Cameroon. Methods, Using a construction algorithm that started from a matrix of 0s, a sample null space of 25,000 unique null matrices was generated by simultaneously conserving (1) the number of occurrences of each species, (2) site richness and (3) species range spans derived from the observed incidence matrix. I then compared the number of times each pair of confamilial species co-occurred in the null space with the same number derived from the observed incidence matrix. Two cases dealing with embedded absences in species ranges were tested: (1) embedded absences were maintained, and (2) embedded absences were assumed to be sampling omissions and were replaced by presences. Results, In the observed absence/presence assemblage there were 147 possible confamilial species pairs. Therefore, 5% or eight were expected by chance alone to have co-occurrence patterns that differed from chance expectations by chance alone. Of these confamilial species pairs, 38 were congeneric and so 5% or two were expected to differ from chance expectations. For case (1) 16, and for case (2) 17 confamilial species pairs' co-occurrence patterns differed significantly from chance expectations. For case (1) nine congeneric species pairs, and for case (2) 10 congeneric pairs differed significantly from chance expectations. For case (1) four, and for case (2) five congeneric species pairs formed checkerboards (patterns of mutual exclusion). Results from case (1) were a proper subset of case (2) indicating that sampling omissions did not alter greatly the results. Main conclusions, I have demonstrated that null models are valuable tools to analyse ecological communities provided that proper models are employed. The choice of the appropriate null space to analyse distributions is critical. The null model employed to analyse birds on islands of an archipelago can be adapted to analyse species along gradients provided an additional range constraint is added to the null model. Moreover, added precision to results can be obtained by analysing each species pair separately, particularly those in the same family or genus, as opposed to applying a community-wide metric to the faunal assemblage. My results support some of the speculations of previous authors who were unable to demonstrate their suspicions analytically. [source] Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equationMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2007Ali Demir Abstract This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi-linear parabolic equation ut(x,t)=(k(u(x,t))ux(x,t))x, with Dirichlet boundary conditions u(0,t)=,0, u(1,t)=,1. The main purpose of this paper is to investigate the distinguishability of the input,output mappings ,[,]:,, ,C1[0,T], ,[,]:,,,C1[0,T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input,output mappings ,[,] and ,[,] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))ux(0,t) or/and h(t):=k(u(1,t))ux(1,t), the values k(,0) and k(,1) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values ku(,0) and ku(,1) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically by an integral representation. Hence the input,output mappings ,[,]:,,, C1[0,T], ,[,]:,,,C1[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd. [source] Linear system solution by null-space approximation and projection (SNAP)NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2007M. Ili Abstract Solutions of large sparse linear systems of equations are usually obtained iteratively by constructing a smaller dimensional subspace such as a Krylov subspace. The convergence of these methods is sometimes hampered by the presence of small eigenvalues, in which case, some form of deflation can help improve convergence. The method presented in this paper enables the solution to be approximated by focusing the attention directly on the ,small' eigenspace (,singular vector' space). It is based on embedding the solution of the linear system within the eigenvalue problem (singular value problem) in order to facilitate the direct use of methods such as implicitly restarted Arnoldi or Jacobi,Davidson for the linear system solution. The proposed method, called ,solution by null-space approximation and projection' (SNAP), differs from other similar approaches in that it converts the non-homogeneous system into a homogeneous one by constructing an annihilator of the right-hand side. The solution then lies in the null space of the resulting matrix. We examine the construction of a sequence of approximate null spaces using a Jacobi,Davidson style singular value decomposition method, called restarted SNAP-JD, from which an approximate solution can be obtained. Relevant theory is discussed and the method is illustrated by numerical examples where SNAP is compared with both GMRES and GMRES-IR. Copyright © 2006 John Wiley & Sons, Ltd. [source] Uncovering a Latent Multinomial: Analysis of Mark,Recapture Data with MisidentificationBIOMETRICS, Issue 1 2010William A. Link Summary Natural tags based on DNA fingerprints or natural features of animals are now becoming very widely used in wildlife population biology. However, classic capture,recapture models do not allow for misidentification of animals which is a potentially very serious problem with natural tags. Statistical analysis of misidentification processes is extremely difficult using traditional likelihood methods but is easily handled using Bayesian methods. We present a general framework for Bayesian analysis of categorical data arising from a latent multinomial distribution. Although our work is motivated by a specific model for misidentification in closed population capture,recapture analyses, with crucial assumptions which may not always be appropriate, the methods we develop extend naturally to a variety of other models with similar structure. Suppose that observed frequencies,f,are a known linear transformation,f=A,x,of a latent multinomial variable,x,with cell probability vector,,=,(,). Given that full conditional distributions,[, | x],can be sampled, implementation of Gibbs sampling requires only that we can sample from the full conditional distribution,[x | f, ,], which is made possible by knowledge of the null space of A,. We illustrate the approach using two data sets with individual misidentification, one simulated, the other summarizing recapture data for salamanders based on natural marks. [source] Linear system solution by null-space approximation and projection (SNAP)NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2007M. Ili Abstract Solutions of large sparse linear systems of equations are usually obtained iteratively by constructing a smaller dimensional subspace such as a Krylov subspace. The convergence of these methods is sometimes hampered by the presence of small eigenvalues, in which case, some form of deflation can help improve convergence. The method presented in this paper enables the solution to be approximated by focusing the attention directly on the ,small' eigenspace (,singular vector' space). It is based on embedding the solution of the linear system within the eigenvalue problem (singular value problem) in order to facilitate the direct use of methods such as implicitly restarted Arnoldi or Jacobi,Davidson for the linear system solution. The proposed method, called ,solution by null-space approximation and projection' (SNAP), differs from other similar approaches in that it converts the non-homogeneous system into a homogeneous one by constructing an annihilator of the right-hand side. The solution then lies in the null space of the resulting matrix. We examine the construction of a sequence of approximate null spaces using a Jacobi,Davidson style singular value decomposition method, called restarted SNAP-JD, from which an approximate solution can be obtained. Relevant theory is discussed and the method is illustrated by numerical examples where SNAP is compared with both GMRES and GMRES-IR. Copyright © 2006 John Wiley & Sons, Ltd. [source] | |||||||||||||