The micromechanics of three‐dimensional collagen‐I gels

We study the micromechanics of collagen‐I gel with the goal of bridging the gap between theory and experiment in the study of biopolymer networks. Three‐dimensional images of fluorescently labeled collagen are obtained by confocal microscopy, and the network geometry is extracted using a 3D network skeletonization algorithm. Each fiber is modeled as an elastic beam that resists stretching and bending, and each crosslink is modeled as torsional spring. The stress–strain curves of networks at three different densities are compared with rheology measurements. The model shows good agreement with experiment, confirming that strain stiffening of collagen can be explained entirely by geometric realignment of the network, as opposed to entropic stiffening of individual fibers. The model also suggests that at small strains, crosslink deformation is the main contributer to network stiffness, whereas at large strains, fiber stretching dominates. As this modeling effort uses networks with realistic geometries, this analysis can ultimately serve as a tool for understanding how the mechanics of fibers and crosslinks at the microscopic level produce the macroscopic properties of the network. © 2010 Wiley Periodicals, Inc. Complexity 16: 22‐28, 2011     

Well‐posedness of the equations governing the motions of a one‐dimensional hybrid thermo‐elastic structure

In this paper a model for the vibrations of a one‐dimensional hybrid thermo‐elastic structure consisting of an extensible thermo‐elastic beam which is hinged at one end, with a rigid body attached to its free end, is studied with a view to establishing the existence of a unique solution in a weak sense. The model takes account of the effect of stretching on bending and rotational inertia. By treating eigenvalue problems with the spectral parameter also in the boundary conditions, we are able to employ the method of Faedo–Galerkin approximations. Copyright © 2004 John Wiley & Sons, Ltd.     

On‐site surrogates for large‐scale calibration

Motivated by a computer model calibration problem from the oil and gas industry, involving the design of a honeycomb seal, we develop a new Bayesian methodology to cope with limitations in the canonical apparatus stemming from several factors. We propose a new strategy of on‐site design and surrogate modeling for a computer simulator acting on a high‐dimensional input space that, although relatively speedy, is prone to numerical instabilities, missing data, and nonstationary dynamics. Our aim is to strike a balance between data‐faithful modeling and computational tractability in a calibration framework—tailoring the computer model to a limited field experiment. Situating our on‐site surrogates within the canonical calibration apparatus requires updates to that framework. We describe a novel yet intuitive Bayesian setup that carefully decomposes otherwise prohibitively large matrices by exploiting the sparse blockwise structure. Empirical illustrations demonstrate that this approach performs well on toy data and our motivating honeycomb example.     

Uniform stabilization of a one‐dimensional hybrid thermo‐elastic structure

This paper is concerned with the stabilization of a one‐dimensional hybrid thermo‐elastic structure consisting of an extensible thermo‐elastic beam which is hinged at one end with a rigid body attached to its free end. The model takes account of the effect of stretching on bending and rotational inertia. The property of uniform stability of the energy associated with the model is asserted by constructing an appropriate Lyapunov functional for an abstract second order evolution problem. Critical use is made of a multiplier of an operator theoretic nature, which involves the fractional power A^?1/2 of the bi‐harmonic operator pair A acting in the abstract evolution problem. An explicit decay rate of the energy is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

CR Structures on the 3‐Sphere and the Blow‐Up of the Projective Plane

Motivated by a problem of characterizing CR‐structures on the 3‐sphere, we give a geometric construction of formal deformations of a complex surface, which is the complement of a ball in the projective plane. They are described by cohomology groups of the blow‐up X of the projective plane. Moreover it will be shown that the space of these formal deformations is an infinite dimensional space with a natural stratification by finite dimensional subspaces. This stratification re ects algebro‐geometric properties of X. It is expected that our construction will clarify the complex geometric nature of the space of non‐embeddable CR‐structures on the 3‐sphere.     

Computation of periodic orbits in three‐dimensional Lotka‐Volterra systems

This paper deals with an adaptation of the Poincaré‐Lindstedt method for the determination of periodic orbits in three‐dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three‐dimensional Lotka‐Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

Analytic solutions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system

Based on a Riccati equation and one of its new generalized solitary solutions constructed by the Exp‐function method, new analytic solutions with free parameters and arbitrary functions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system are obtained. These free parameters and arbitrary functions reveal that the (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system has rich spatial structures. As an illustrative example, two new spatial structures are shown by setting the arbitrary functions as different Jacobi elliptic functions. Compared with tanh‐function method and its extensions, the method proposed in this paper is more powerful and it can be applied to other nonlinear evolution equations. Copyright © 2010 John Wiley & Sons, Ltd.     

A nonconforming exponentially fitted finite element method for two‐dimensional drift‐diffusion models in semiconductors

A new nonconforming exponentially fitted finite element for a Galerkin approximation of convection–diffusion equations with a dominating advective term is considered. The attention is here focused on the drift‐diffusion current continuity equations in semiconductor device modeling. The scheme extends to the two‐dimensional case, the well known Scharfetter–Gummel method, by imposing a divergence‐free current over each element of the triangulation. Convergence of the method in the energy norm is proved and some numerical results are included. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 133–150, 1999     

Three‐point bending tests—Part I: Mathematical study and asymptotic analysis

The goal of this work is to study the static behaviour of a three‐dimensional elastic beam when is subjected to a three‐point bending test. In the first part, under suitable compatibility conditions on the applied forces and on the geometry of the beam, we will prove the existence of a unique solution for the associated contact elastic problem; these conditions of compatibility on the data come from the absence of a Dirichlet condition on the beam boundary. In the second part, we will study the asymptotic behaviour of this problem; in particular, we will deduce the one‐dimensional models associated with the displacement components, and we will give the existence and uniqueness of solution for them. Moreover, we will give an expression for the normal axial stress in the beam which is related to the modulus of rupture of brittle materials. In the final part of the work, we will deal with the regularity of the solution for the bending problem and we will prove some properties of the coincidence set. Copyright © 2011 John Wiley & Sons, Ltd.     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter identification based on FE‐models by genetic algorithms

A realistic and reliable model is an important precondition for the simulation of revitalization tasks as well as for the estimation of properties of existing buildings. Within one theory the parameters of the model should be approximated best by gradually performed experiments and their analysis. Usually this kind of optimization problems leads into non‐convex non‐differentiable objective function spaces with high dimensions. Normally ore complex structures are modeled using finite element method. We present a method of identifying Young's modulus for a beam and a plate by using FE‐models and genetic optimization algorithms for parameter identification.     

A modal approach to flatness‐based control of flexible structures

Modal coordinates are used to construct a state and input parameterizing flat output for flexible structures and to design open‐loop controls. This is illustrated for the boundary control of a Timoshenko beam model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonstandard methods for the convective‐dispersive transport equation with nonlinear reactions

A new nonstandard Eulerian‐Lagrangian method is constructed for the one‐dimensional, transient convective‐dispersive transport equation with nonlinear reaction terms. An “exact” difference scheme is applied to the convection‐reaction part of the equation to produce a semi‐discrete approximation with zero local truncation errors with respect to time. The spatial derivatives involved in the remaining dispersion term are then approximated using standard numerical methods. This approach leads to significant, qualitative improvements in the behavior of the numerical solution. It suppresses the numerical instabilities that arise from the incorrect modeling of derivatives and nonlinear reaction terms. Numerical experiments demonstrate the scheme's ability to model convection‐dominated, reactive transport problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 617–624, 1999     

Clustering electricity consumers using high‐dimensional regression mixture models

A massive amount of data about individual electrical consumptions are now provided with new metering technologies and smart grids. These new data are especially useful for load profiling and load modeling at different scales of the electrical network. A new methodology based on mixture of high‐dimensional regression models is used to perform clustering of individual customers. It leads to uncovering clusters corresponding to different regression models. Temporal information is incorporated in order to prepare the next step, the fit of a forecasting model in each cluster. Only the electrical signal is involved, slicing the electrical signal into consecutive curves to consider it as a discrete time series of curves. Interpretation of the models is given on a real smart meter dataset of Irish customers.     

Bilinear Form and N‐Shock‐Wave Solutions for a (2+1)‐Dimensional Breaking Soliton Equation in Certain Fluids with the Bell Polynomials and Auxiliary Function

In this paper, we will investigate a (2+1)‐dimensional breaking soliton (BS) equation for the (2+1)‐dimensional collision of a Riemann wave with a long wave in certain fluids. Using the Bell polynomials and an auxiliary function, we derive a new bilinear form for the (2+1)‐dimensional BS equation, which is different from those in the previous literatures. One‐, two‐ and N‐shock‐wave solutions are obtained with the Hirota method and symbolic computation. One shock wave is found to be able to stably propagate. Two shock waves are observed to have the parallel collision, oblique collision, and stable propagation of the V‐type structure. In addition, we present the collision between one shock wave and V‐type structure, and the collision between two V‐type structures.     

The existence of periodic solutions for nonlinear beam equations on by a para‐differential method

This paper focuses on the construction of periodic solutions of nonlinear beam equations on the d‐dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para‐differential conjugation. Given the nonresonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.     

Simulation of flow and electrokinetic transport in a micro‐analysis‐system

The aim of this work is the investigation of the flow and electrokinetic transport in a micro‐analysis‐system. A mathematical model for electroosmotic and electrophoretic phenomena is introduced in order to perform two‐dimensional, time‐dependent Finite Element simulations for an existing device. The model includes the feature of the flow field, the mass transfer, the external applied electric field and the involved chemistry. The results of the simulation are validated against experimental data and show good agreement. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)