Mixed finite element methods for linear elasticity with weakly imposed symmetry

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

     

Shortest Path Problem on a Network with Imprecise Edge Weight

A network with its arc lengths as imprecise number, instead of a real number, namely, interval number and triangular fuzzy number is considered here. Existing ideas on addition and comparison between two imprecise numbers of same type are introduced. To obtain a fuzzy shortest path from a source vertex to all other vertices, a common algorithm is developed which works well on both types of imprecise numbers under consideration. In the proposed algorithm, a decision-maker is to negotiate with the obtained fuzzy shortest paths according to his/her view only when the means are same but the widths are different of the obtained paths. Otherwise, a fuzzy optimal path is obtained to which the decision-maker always satisfies with different grades of satisfaction. All pairs fuzzy shortest paths can be found by repeated use of the proposed algorithm.     

Interior penalty preconditioners for mixed finite element approximations of elliptic problems

It is established that an interior penalty method applied to second-order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method. This relation can be utilized in order to construct preconditioners for the discrete mixed system. As an example, a family of additive Schwarz preconditioners for these systems is constructed. Numerical examples which confirm the theoretical results are also presented.

     

SinaPlot: An Enhanced Chart for Simple and Truthful Representation of Single Observations Over Multiple Classes

Recent developments in data-driven science have led researchers to integrate data from several sources, over diverse experimental procedures, or databases. This alone poses a major challenge in truthfully visualizing data, especially when the number of data points varies between classes. To aid the representation of datasets with differing sample size, we have developed a new type of plot overcoming limitations of current standard visualization charts. SinaPlot is inspired by the strip chart and the violin plot and operates by letting the normalized density of points restrict the jitter along the x-axis. The plot displays the same contour as a violin plot but resembles a simple strip chart for a small number of data points. By normalizing jitter over all classes, the plot provides a fair representation for comparison between classes with a varying number of samples. In this way, the plot conveys information of both the number of data points, the density distribution, outliers and data spread in a very simple, comprehensible, and condensed format. The package for producing the plots is available for R through the CRAN network using base graphics package and as geom for ggplot through ggforce. We also provide access to a web-server accepting excel sheets to produce the plots (http://servers.binf.ku.dk:8890/sinaplot/).     

Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

A robust nonconforming -element

Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming -element which is -conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.

     

Tractable approximations for probabilistic models: the adaptive Thouless-Anderson-Palmer mean field approach

We develop an advanced mean field method for approximating averages in probabilistic data models that is based on the Thouless-Anderson-Palmer (TAP) approach of disorder physics. In contrast to conventional TAP, where the knowledge of the distribution of couplings between the random variables is required, our method adapts to the concrete couplings. We demonstrate the validity of our approach, which is so far restricted to models with nonglassy behavior, by replica calculations for a wide class of models as well as by simulations for a real data set.