Macromechanical characteristics deduced from tests on glass/epoxy composites with optical methods

Experimental results are required to verify the simulation results obtained by the homogenisation procedure of a biaxial weftknitted fabric reinforced composite with the Binary Model [1]. Tensile, bending and shear tests were performed to investigate and validate the elastic macro‐mechanical behaviour of the composite material. Beside other methods an optical whole‐field measuring system was used to estimate the deformation in all tests. Additionally, natural frequencies and the appropriate mode shapes of plates were used to get the elements of the compliance matrix. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flexural Properties of Fiber-Reinforced Composites

To describe the macroscopic mechanical behavior of composite materials with biaxial weft-knitted fabric reinforcement, a difference between the flexural and tensile stiffness must be taken into account. Due to the periodic structure of the composite, the material behavior can be investigated on a unit cell. A homogenization technique based on the Hill-Mandel energy balance is used to replace a heterogenous Boltzmann medium by a homogenous COSSERAT continuum. To simulate the material behavior, this homogenization method is applied to the Binary Model. Tensile, shear and bending tests were performed to verify the simulation results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation and Modeling the Mechanical Behavior of Textile Reinforced Composites by Combining the Binary Model and X-FEM

This paper presents a new efficient modeling strategy based on the combination of Binary Model and X–FEM. It is applied to represent the inner architecture of textile reinforced composites where the fabric is characterized by a complex geometry. Homogenization methods are used to compute the effective elastic material properties. Thereby, the discrete formulation of periodic boundary conditions is adapted. Finally, the results in terms of effective material properties reveal a good agreement with parameters obtained by experimental tests. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental investigation of the strength of glass-reinforced plastics in biaxial compression in three planes of symmetry

Glass-reinforced plastics (GRP) with fiber (wound) and fabric (hot-molded textolites) reinforcement have been investigated in biaxial compression in the three planes of symmetry. The high strength in biaxial compression in the transverse plane, observed for all the GRP investigated, is explained. A method of investigating the tensile strength of the fiber components of the GRP from the results of testing a unidirectional composite in biaxial compression in the transverse plane is proposed. The possibility of a nondestructive estimation of the strength of GRP in biaxial compression at an arbitrary stress ratio is demonstrated.Translated from Mekhanika Polimerov, No. 1, pp. 63–72, January–February, 1976.     

On the simulation of textile reinforced composites and structures

This paper presents experimental and numerical methods to perform simulations of the mechanical behavior of textile reinforced composites and structures. The first aspect considered refers to the meso-to-macro transition in the framework of the finite element (FE) method. Regarding an effective modelling strategy the Binary Model is used to represent the discretized complex architecture of the composite. To simulate the local response and to compute the macroscopic stress and stiffness undergoing small strain a user routine is developed. The results are transfered to the macroscopic model during the solution process. The second aspect concerns the configuration of the fiber orientation and textile shear deformation in complex structural components. To take these deformations which affect the macroscopic material properties into account they are regarded in a macroscopic FE model. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of hyperelastic 2‐D and 3‐D model foams at large deformations

The influence of the modeling dimension on the determination of effective properties for hyperelastic foams is investigated by means of regular 2‐D and 3‐D model foams. For calculating the effective stress‐strain relationships of both microstructures, a strain energy based homogenization procedure is employed. The results from numerical analyses show that with a 2‐D model foam the basic deformation mechanisms of the 3‐D model can be captured. Nevertheless, due to the distinct quantitative deviations found from the homogenization analyses, 3‐D modeling approaches should be used if quantitative predictions for the effective material properties are required. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Domain decomposition preconditioners for multiscale problems in linear elasticity

We analyze two‐level overlapping Schwarz domain decomposition methods for vector‐valued piecewise linear finite element discretizations of the PDE system of linear elasticity. The focus of our study lies in the application to compressible, particle‐reinforced composites in 3D with large jumps in their material coefficients. We present coefficient‐explicit bounds for the condition number of the two‐level additive Schwarz preconditioned linear system. Thereby, we do not require that the coefficients are resolved by the coarse mesh. The bounds show a dependence of the condition number on the energy of the coarse basis functions, the coarse mesh, and the overlap parameters, as well as the coefficient variation. Similar estimates have been developed for scalar elliptic PDEs by Graham et al. 1 The coarse spaces to which they apply here are assumed to contain the rigid body modes and can be considered as generalizations of the space of piecewise linear vector‐valued functions on a coarse triangulation. The developed estimates provide a concept for the construction of coarse spaces, which can lead to preconditioners that are robust with respect to high contrasts in Young's modulus and the Poisson ratio of the underlying composite. To confirm the sharpness of the theoretical findings, we present numerical results in 3D using vector‐valued linear, multiscale finite element and energy‐minimizing coarse spaces. The theory is not restricted to the isotropic (Lamé) case, extends to the full‐tensor case, and allows applications to multiphase materials with anisotropic constituents in two and three spatial dimensions. However, the bounds will depend on the ratio of largest to smallest eigenvalue of the elasticity tensor.     

On a Sachs bound for stress-induced phase transformations in polycrystalline shape memory alloys

In earlier work [3], a Sachs and a Taylor bound on the transformation yield stress in shape memory polycrystals were derived in the context of a variational model. The aim of this article is to compare the Sachs with the Taylor bound for cubic-toorthorhombic phase transformations under biaxial loading, where the material parameters are chosen explicitly (CuAlNi). It turns out that the gap between the two bounds can be quite large depending on the underlying texture and the loading direction. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibration of submerged floating tunnels due to moving loads

The concept of submerged floating tunnel (SFT) has become an increasingly attractive idea to cross the straits. The structural solution in this scheme includes buoyancy force on tunnel body plus tension in mooring tethers. This paper investigates the effect of submergence on the dynamic response of SFT due to moving load. The inertial effect of the fluid is accounted for by evaluating the added mass of tunnel using two and three dimensional models. It is found that fluid–structure interaction increases dynamic amplification of the tunnel deflection (in some cases very significantly). The results show that although the 3D model predicts lesser inertial contribution for surrounding fluid, it is not always possible to associate the larger response with the 2D or 3D models. The discrepancy between the results of the two models decreases as the tether stiffness increases. This indicates that the adoption of Morison’s equation for evaluating the fluid loading on the tunnel is a reasonable assumption when the tether stiffness is high. It is also found that by increasing the tether stiffness, it is possible to introduce a major reduction in the dynamic amplification of the response and by this way control the dynamic response of the SFT.     

Examination and prediction of fog and haze pollution using a Multi-variable Grey Model based on interval number sequences

In this study, a new Multivariable Grey Model (1,m) aimed at interval grey number sequences with known possibility functions is built using the kernel and degree of greyness under new definitions. Based on the new model, formulae are deduced to calculate and predict the upper and lower bounds of interval grey numbers. Since the grey system model and fog- and haze-prone weather have the same characteristics of uncertainty, this model was applied to simulate and predict the measurable indicators of fog and haze in Nanjing, China. We selected visibility data and particulate matter data with a diameter of 2.5 µm to build a new Multivariable Grey Model (1,2) with a new kernel and degree of greyness sequence. In addition, we established the traditional Multivariable Grey Model (1,2) with the original kernel and degree of greyness and the Auto-Regressive Integrated Moving Average Model (1,1,0). The results show that the new Multivariable Grey Model (1,2) has the best simulation and prediction effects among the three models, with average relative errors of simulation and prediction at 1.32% and 0.32%, respectively. To further verify the validity and feasibility of the proposed model, we added another real-world example to establish the three models mentioned above. The results prove that the proposed model has evidently superior performance to another two models.     

An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

An analytical spectral stiffness method is proposed for the efficient and accurate buckling analysis of rectangular plates on Winkler foundation subject to general boundary conditions (BCs). The method combines the advantages of superposition method, stiffness-based method and the Wittrick–Williams algorithm. First, exact general solutions of the governing differential equation (GDE) of plate buckling considering both elastic foundation and biaxial loading is derived by using a modified Fourier series. The superposition of such general solutions satisfy the GDE exactly and BCs approximately, which guarantees the rapid convergence and high accuracy. Then, based on the exact general solution, the spectral stiffness matrix which relates the coefficients of plate generalized displacement BCs and force BCs is symbolically developed. As a result, arbitrary BCs can be prescribed straightforwardly in the stiffness-based model. As an efficient and reliable solution technique, the Wittrick–Williams algorithm with the J₀ problem resolved is applied to obtain the critical buckling solutions. The accuracy and efficiency of the method are verified by comparing with other methods. Benchmark buckling solutions are provided for plates with all possible boundary conditions. Also, dependence of various factors such as foundation stiffness, load combinations and aspect ratio on the buckling behaviors are investigated.     

A drift‐free simulation method for pricing commodity derivatives

Having in view the pricing of commodity derivatives in Libor Market Model (LMM) setting, we first analyze the set of basic rates we need to formulate the model by using the spanning tree concept taken from graph theory. Next, we present an efficient procedure for Monte Carlo simulation of the dynamics of the rates associated to LMM, avoiding the presence of the rates dependent drifts (drift‐free simulation) and the presence of negative deflated bond prices and negative forward rates. The method is based upon a new parameterization of the martingales introduced by Glasserman and Zhao and it is extended to a Cross‐Market Model for commodities. Finally, a particular example of commodity derivative (spread option) pricing problem is considered. Copyright © 2014 John Wiley & Sons, Ltd.     

A preconditioner for generalized saddle point problems: Application to 3D stationary Navier‐Stokes equations

In this article we consider the stationary Navier‐Stokes system discretized by finite element methods which do not satisfy the inf‐sup condition. These discretizations typically take the form of a variational problem with stabilization terms. Such a problem may be transformed by iteration methods into a sequence of linear, Oseen‐type variational problems. On the algebraic level, these problems belong to a certain class of linear systems with nonsymmetric system matrices (“generalized saddle point problems”). We show that if the underlying finite element spaces satisfy a generalized inf‐sup condition, these problems have a unique solution. Moreover, we introduce a block triangular preconditioner and we show how the eigenvalue bounds of the preconditioned system matrix depend on the coercivity constant and continuity bounds of the bilinear forms arising in the variational problem. Finally we prove that the stabilized P1‐P1 finite element method proposed by Rebollo is covered by our theory and we show that the condition number of the preconditioned system matrix is independent of the mesh size. Numerical tests with 3D stationary Navier‐Stokes flows confirm our results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Ill‐conditioning in the virtual element method: Stabilizations and bases

In this article, we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the virtual element method. It turns out that ill‐conditioning appears when considering high‐order methods or in presence of “bad‐shaped” (for instance nonuniformly star‐shaped, with small edges…) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number.     

Mathematical modelling of tongue deformation during swallow in patients with head and neck cancer

ABSTRACT

Cancer localized to the tongue is often characterized by increased stiffness in the affected region. This stiffness affects swallow in a manner that is difficult to quantify in patients. A biomechanical model was developed to simulate the spatio-temporal deformation of the tongue during the pharyngeal phase of swallow in patients with cancer of the tongue base. The model involves finite element analysis (FEA) of a three-dimensional (3D) model of the tongue reconstructed from magnetic resonance images (MRI). The tongue tissue is assumed to be hyper-elastic. In order to examine the effects of tissue change (increased stiffness) due to the presence of cancer localized to the tongue base, various sections of the 3D geometry are modified to exhibit different elastic properties. Three cases are considered, representing the normal tongue, a tongue with early-stage cancer, and tongue with late-stage cancer. Early- and late-stage cancers are differentiated by the degree of stiffness within the base of tongue tissue. Analysis of the model suggests that healthy tongue has a maximum deformation of 9.38 mm, whereas tongues having mild cancer and severe cancer have a maximum deformation of 8.65 and 6.17 mm, respectively. Biomechanical modelling is a useful tool to explain and estimate swallowing abnormalities associated with tongue cancer and post-treatment characteristics.     

Optimal harvesting policy of an inland fishery resource under incomplete information

A new mathematical model for finding the optimal harvesting policy of an inland fishery resource under incomplete information is proposed in this paper. The model is based on a stochastic control formalism in a regime‐switching environment. The incompleteness of information is due to uncertainties involved in the body growth rate of the fishery resource: a key biological parameter. Finding the most cost‐effective harvesting policy of the fishery resource ultimately reduces to solving a terminal and boundary value problem of a Hamilton‐Jacobi‐Bellman equation: a nonlinear and degenerate parabolic partial differential equation. A simple finite difference scheme for solving the equation is then presented, which turns out to be convergent and generates numerical solutions that comply with certain theoretical upper and lower bounds. The model is finally applied to the management of Plecoglossus altivelis, a major inland fishery resource in Japan. The regime switching in this case is due to the temporal dynamics of benthic algae, the main food of the fish. Model parameter values are identified from field measurement results in 2017. Our computational results clearly show the dependence of the optimal harvesting policy on the river environmental and biological conditions. The proposed model would serve as a mathematical tool for fishery resource management under uncertainties.     

Justifying the small‐world phenomenon via random recursive trees

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three‐fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank‐based selection model, the Aiello‐Chung‐Lu models, the generalized linear preference model, directed scale‐free graphs, the Cooper‐Frieze model, and random unordered increasing k‐trees. Our results shed light on why the small‐world phenomenon is observed in so many real‐world graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 201–224, 2017     

A ritz method for an optimal control problem

We generalize and simplify the proofs of the basic papers of Bosarge and Johnson (Refs. 1-3) on a variational procedure for approximating the solution of thestate regular problem. We derive generala priori error bounds for this procedure and apply these results to obtain asymptotic error bounds for the special case of spline-type approximations.     

Convex upper and lower bounds for present value functions

In this paper we present an efficient methodology for approximating the distribution function of the net present value of a series of cash‐flows, when discounting is presented by a stochastic differential equation as in the Vasicek model and in the Ho–Lee model. Upper and lower bounds in convexity order are obtained. The high accuracy of the method is illustrated for cash‐flows for which no analytical results are available. Copyright © 2001 John Wiley & Sons, Ltd.     

A simplified description of the stress-strain state of a warp-knitted fabric

A new technique for describing the deformation and failure of a warp-knitted fabric is proposed. The stress-strain state of the material is specified not by the principal stretch ratios, but by an effective strain, introduced as an analogue of the radial strain, and an angle characterizing the deformation regime. Within the framework of the approach suggested, the flexural stiffness of threads is ignored, which allows us to linearize the relation between the specific loads in the fabric and the effective strains, which is approximated by trigonometric functions (squares of sines and cosines). As a result, the mechanical properties of the fabric are characterized by ten parameters. They are determined by employing the method of separating a structural element, which has been elaborated previously. The results obtained are compared with those given by a more complex mathematical model without approximations and with experimental data found by the ball penetration method.