Modeling and homogenization of textile reinforced composites based on the isogeometric analysis

In this contribution, the isogeometric analysis is used to compute the effective material properties of textile reinforced composites. The isogeometric analysis based on non-uniform rational B-splines (NURBS) provides an efficient approach for numerical modeling because there is no need for a mesh generation. There are further advantages such as the availability of a geometry representation based on NURBS in computer-aided design software and the possibility to apply different refinement methods which do not change the geometry of the numerical model. These properties motivate the combination of the isogeometric analysis with the homogenization method. Therefor, the unit cell model representing the inner architecture of a textile reinforced composite is defined using NURBS. In order to compute the effective mechanical properties of the heterogeneous material, the homogenization method with periodic boundary conditions is applied. Finally, two examples demonstrate the advantages of this approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective transport properties for flashing ratchets using homogenization theory

We study effective transport properties of Brownian motor models of molecular motors. The effective drift and diffusivity can be calculated by solving cell problems, given explicitly by homogenization theory. We briefly describe how this approach is equivalent to theWang-Peskin-Elston (WPE) [3] numerical algorithm for calculating effective transport properties of flashing ratchets. For an on-off flashing ratchet we examine the optimization of the Peclet number as a function of the free parameters of the system. We also present a numerical method for solving the cell equations for a flashing ratchet with Gaussian multiplicative noise. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of an elastic-plastic constitutive model for highly anisotropic materials

Homogenization methods are used to obtain the effective properties of highly anisotropic materials such like textile reinforced composites. The state of the art is the utilization of this method to compute elastic properties. But the consideration of inelastic and anisotropic properties requires the application of more advanced techniques such as the FE²-method. Due to the high numerical effort induced by this approach, this paper presents a new method to evaluate inelastic properties of an heterogeneous elastic-plastic material. The parameters describing the inelastic properties require a modification of the return mapping algorithm which is used for the numerical implementation. Finally, the verification shows the accuracy of the results obtained with this new homogenization method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization

Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL ^T factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraints.     

Calibration of estimator-weights via semismooth Newton method

Weighting is a common methodology in survey statistics to increase accuracy of estimates or to compensate for non-response. One standard approach for weighting is calibration estimation which represents a common numerical problem. There are various approaches in the literature available, but quite a number of distance-based approaches lack a mathematical justification or are numerically unstable. In this paper we reformulate the calibration problem as a system of nonlinear equations. Although the equations are lacking differentiability properties, one can show that they are semismooth and the corresponding extension of Newton’s method is applicable. This is a mathematically rigorous approach and the numerical results show the applicability of this method.     

Numerical assessment of disorder effects in metal foam core sandwich beams based on a local homogenization procedure

The present study is concerned with a numerical procedure for prediction of uncertainty effects in sandwich structures with disordered cores. The approach is based on probability distributions of different material properties and their spatial correlation which are the results of the multiple homogenization analysis of testing volume elements. In order to illustrate the essential difference in the results of material uncertainties between computations using random fields and a deterministic approach both methods are applied to a single edge clamped sandwich beam with a metal foam core which is loaded by a force at the free end. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Design of a new dynamical core for global atmospheric models based on some efficient numerical methods

A careful study on the integral properties of the primitive hydrostatic balance equations for baroclinic atmosphere is carried out, and a new scheme to design the global adiabatic model of atmospheric dynamics is presented. This scheme includes a method of weighted equal-area mesh and a fully discrete finite difference method with quadratic and linear conservations for solving the primitive equation system. Using this scheme, we established a new dynamical core with adjustable high resolution acceptable to the available computer capability, which can be very stable without any filtering and smoothing. Especially, some important integral properties are kept unchanged, such as the anti-symmetries of the horizontal advection operators and the vertical convection operator, the mass conservation, the effective energy conservation under the standard stratification approximation, and so on. Some numerical tests on the new dynamical core, respectively regarding its global conservations and its integrated performances in climatic modeling, incorporated with the physical packages from the Community Atmospheric Model Version 2 (CAM2) of National Center for Atmospheric Research (NCAR), are included.     

Automated parameter tuning based on RMS errors for nonequispaced FFTs

In this paper we study the error behavior of the well known fast Fourier transform for nonequispaced data (NFFT) with respect to the \(\mathcal {L}_{2}\)-norm. We compare the arising errors for different window functions and show that the accuracy of the algorithm can be significantly improved by modifying the shape of the window function. Based on the considered error estimates for different window functions we are able to state an easy and efficient method to tune the involved parameters automatically. The numerical examples show that the optimal parameters depend on the given Fourier coefficients, which are assumed not to be of a random structure or roughly of the same magnitude but rather subject to a certain decrease.     

Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems

This paper is concerned with the numerical solution of a symmetric indefinite system which is a generalization of the Karush–Kuhn–Tucker system. Following the recent approach of Luk?an and Vl?ek, we propose to solve this system by a preconditioned conjugate gradient (PCG) algorithm and we devise two indefinite preconditioners with good theoretical properties. In particular, for one of these preconditioners, the finite termination property of the PCG method is stated. The PCG method combined with a parallel version of these preconditioners is used as inner solver within an inexact Interior‐Point (IP) method for the solution of large and sparse quadratic programs. The numerical results obtained by a parallel code implementing the IP method on distributed memory multiprocessor systems enable us to confirm the effectiveness of the proposed approach for problems with special structure in the constraint matrix and in the objective function. Copyright © 2002 John Wiley & Sons, Ltd.     

Design of a new dynamical core for global atmospheric models based on some efficient numerical methods

A careful study on the integral properties of the primitive hydrostatic balance equations for baroclinic atmosphere is carried out, and a new scheme to design the global adiabatic model of atmospheric dynamics is presented. This scheme includes a method of weighted equal-area mesh and a fully discrete finite difference method with quadratic and linear conservations for solving the primitive equation system. Using this scheme, we established a new dynamical core with adjustable high resolution acceptable to the available     

A delay differential equation solver based on a continuous Runge–Kutta method with defect control

We have recently developed a generic approach for solving neutral delay differential equations based on the use of a continuous Runge–Kutta formula with defect control and investigated its convergence properties. In this paper, we describe a method, DDVERK, which implements this approach and justify the strategies and heuristics that have been adopted. In particular we show how the assumptions related to error control, stepsize control, and discontinuity detection (required for convergence) can be efficiently realized for a particular sixth-order numerical method. Summaries of extensive testing are also reported. This revised version was published online in August 2006 with corrections to the Cover Date.     

Lagrange-projection scheme for two-phase flows 1. Applications to realistic test cases

We are concerned with the numerical simulation of two-phase flows representing oil transportation along a 1-D pipeline. More specifically, we wish to show that, for this kind of problems, it is highly recommended to make use of the Lagrange-Projection formalism. Indeed, this approach naturally splits the effects of fast (acoustic) and slow (kinematic) waves, thus enabling us to design numerical schemes with many desirable properties: explicit with respect to slow waves and implicit with respect to fast waves. The design of the overall explicit-implicit Lagrange-Projection method involves many interesting features, among which the positivity of the partial densities is of utmost interest to us. In this paper, the emphasis will be laid on the motivation and the general philosophy of the Lagrange-Projection method we propose. Extensive numerical results for realistic test cases are then presented in order to illustrate the capabilities of the method     

Symmetric Projection Methods for Differential Equations on Manifolds

Projection methods are a standard approach for the numerical solution of differential equations on manifolds. It is known that geometric properties (such as symplecticity or reversibility) are usually destroyed by such a discretization, even when the basic method is symplectic or symmetric. In this article, we introduce a new kind of projection methods, which allows us to recover the time-reversibility, an important property for long-time integrations.     

Effective elastic properties of particulate-reinforced compossite materials

A numerical approach for determination of the effective properties of particulate composite materials has been developed. A representative volume element (RVE) of the composite material is analyzed with help of the finite-element method. Uniform boundary displacements or tractions are applied on the boundaries of the RVE for introducing the known average strain in the RVE. Local stress and strain distributions in the RVE are calculated using the finite-element method. Different effective elastic constants can be calculated by averaging the local fields corresponding to different sets of boundary conditions. The present approach allows us to determine the effective properties of particle-reinforced composites with acceptable accuracy. The calculated effective properties of the composite are between the upper and lower Hashin—Shtrikman bounds. The results based on the present approach lead to higher stiffness of composites in comparison with analytical approaches.Institute fur Werkstoffwissenschaften, Fachberech Werkseoffwissenschaften, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 450–459, July–August, 1997.     

A Numerical Approach of the sentinel method for distributed parameter systems

In this paper we consider the problem of detecting pollution in some non linear parabolic systems using the sentinel method. For this purpose we develop and analyze a new approach to the discretization which pays careful attention to the stability of the solution. To illustrate convergence properties we give some numerical results that present good properties and show new ways for building discrete sentinels.      

Comparison of Taylor finite difference and window finite difference and their application in FDTD

The finite difference time domain (FDTD) method is an important tool in numerical electromagnetic simulation. There are many ways to construct a finite difference approximation such as the Taylor series expansion theorem, the filtering theory, etc. This paper aims to provide the comparison between the Taylor finite difference (TFD) scheme based on the Taylor series expansion theorem and the window finite difference (WFD) scheme based on the filtering theory. Their properties have been examined in detail, separately. In addition, the formula of the generalized finite difference (GFD) scheme is presented, which can include both the TFD scheme and the WFD scheme. Furthermore, their application in the numerical solution of Maxwell's equations is presented. The formulas for the stability criterion and the numerical dispersion relation are derived and analyzed. In order to evaluate their performance more accurately, a new definition of error is presented. Upon it, the effect of several factors including the grid resolution, the Courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined.     

A hybrid prediction for wind buffeting noises of vehicle rear window based on LES-LAA method

Based on the theories of computational fluid dynamics (CFD) and aeroacoustics, a hybrid simulation technique, the so-called LES-LAA method is proposed in this paper, for predicting the wind buffeting noises generated by opening rear windows of a running vehicle. The LES-LAA is developed by combining the large eddy simulation (LES) and the Lighthill acoustic analogy (LAA) methods. Based on the established vehicle and wind tunnel models, the wind buffeting noises from rear windows are predicted by using the proposed LES-LAA method and the obtained results are compared with experimental data. The results show that the calculation error of sound pressure level (SPL) from the LES-LAA method is less than 2%, which suggests that the proposed method has good accuracy in predicting the wind noise of the rear window of a vehicle. The wind noise when both sides of the rear window are opened at the same time is much lower than the case when just one window is opened. In conclusion, the hybrid LES-LAA technique presented in this paper is effective and feasible for predicting the wind buffeting noise, which can be applied to other types of vehicle and is a promising approach for solving other aero-acoustical engineering problems.     

Substructured two-grid and multi-grid domain decomposition methods

Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

Conservative finite-difference scheme for computer simulation of contrast 3D spatial-temporal structures induced by a laser pulse in a semiconductor

The numerical approach for computer simulation of femtosecond laser pulse interaction with a semiconductor is considered under the formation of 3D contrast time-dependent spatiotemporal structures. The problem is governed by the set of nonlinear partial differential equations describing a semiconductor characteristic evolution and a laser pulse propagation. One of the equations is a Poisson equation concerning electric field potential with Neumann boundary conditions that requires fulfillment of the well-known condition for Neumann problem solvability. The Poisson equation right part depends on free-charged particle concentrations that are governed by nonlinear equations. Therefore, the charge conservation law plays a key role for a finite-difference scheme construction as well as for solvability of the Neumann difference problem. In this connection, the iteration methods for the Poisson equation solution become preferable than using direct methods like the fast Fourier transform. We demonstrate the following: if the finite-difference scheme does not possess the conservatism property, then the problem solvability could be broken, and the numerical solution does not correspond to the differential problem solution. It should be stressed that for providing the computation in a long-time interval, it is crucial to use a numerical method that possessing asymptotic stability property. In this regard, we develop an effective numerical approach—the three-stage iteration process. It has the same economic computing expenses as a widely used split-step method, but, in contrast to the split-step method, our method possesses conservatism and asymptotic stability properties. Computer simulation results are presented.