Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Variational ‘self-consistent’ estimates for nonlinearproblems are formulated, building on a variational formulationpreviously developed by the authors. The formulation employsa linear ‘comparison medium’ for whose propertiessome ‘self-consistent’ choice is made. In contrastto linear problems, three possible self-consistent choices presentthemselves. The results that they give are analysed for twoparticular systems (a nonlinear dielectric and a nonlinear lossycomposite) for which bounds are already available. Estimatesbased on self-consistent embedding of a single inclusion ina homogeneous matrix composed of ‘comparison material’are also developed.     

Consistent macroscopic tangent computation for FFT-based computational homogenization at finite strains

The present work addresses the efficient computation of effective properties of periodic microstructures by the use of Fast Fourier Transforms. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of an RVE, the computation of the associated moduli is not straight-forward. The contribution of the present paper is thus the derivation and implementation of an algorithmically consistent macroscopic tangent operator that comprises the effective properties of the RVE. In contrast to finite-difference based approaches, an exact solution for the macroscopic tangent is derived by means of the classical Lippmann-Schwinger equation. The problem then reduces to the solution of a system of linear equations even for nonlinear material behaviour. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of energy dissipation in visco-elastic materials at finite deformation

In this contribution, the derivation of the energy dissipation rate in generalized visco-elastic material models with internal stress-type variables and linear evolution equations is outlined. The approximated dissipation rate is computed from a positive quadratic form of the nonlinear non-equilibrium stresses and the inverse of the consistent material tangent tensor. The presented method is used to compute the energy dissipation of visco-elastic rubber material in a large scale application of a steady state rolling tire structure. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.     

Nonlinear,finite deformation,finite element analysis

The roles of the consistent Jacobian matrix and the material tangent moduli, which are used in nonlinear incremental finite deformation mechanics problems solved using the finite element method, are emphasized in this paper, and demonstrated using the commercial software ABAQUS standard. In doing so, the necessity for correctly employing user material subroutines to solve nonlinear problems involving large deformation and/or large rotation is clarified. Starting with the rate form of the principle of virtual work, the derivations of the material tangent moduli, the consistent Jacobian matrix, the stress/strain measures, and the objective stress rates are discussed and clarified. The difference between the consistent Jacobian matrix (which, in the ABAQUS UMAT user material subroutine is referred to as DDSDDE) and the material tangent moduli (C^e) needed for the stress update is pointed out and emphasized in this paper. While the former is derived based on the Jaumann rate of the Kirchhoff stress, the latter is derived using the Jaumann rate of the Cauchy stress. Understanding the difference between these two objective stress rates is crucial for correctly implementing a constitutive model, especially a rate form constitutive relation, and for ensuring fast convergence. Specifically, the implementation requires the stresses to be updated correctly. For this, the strains must be computed directly from the deformation gradient and corresponding strain measure (for a total form model). Alternatively, the material tangent moduli derived from the corresponding Jaumann rate of the Cauchy stress of the constitutive relation (for a rate form model) should be used. Given that this requirement is satisfied, the consistent Jacobian matrix only influences the rate of convergence. Its derivation should be based on the Jaumann rate of the Kirchhoff stress to ensure fast convergence; however, the use of a different objective stress rate may also be possible. The error associated with energy conservation and work-conjugacy due to the use of the Jaumann objective stress rate in ABAQUS nonlinear incremental analysis is viewed as a consequence of the implementation of a constitutive model that violates these requirements.     

Short-Time Linear Response with Reduced-Rank Tangent Map

The recently developed short-time linear response algorithm, which predicts the response of a nonlinear chaotic forced-dissipative system to small external perturbation, yields high precision of the response prediction. However, the computation of the short-time linear response formula with the full rank tangent map can be expensive. Here, a numerical method to potentially overcome the increasing numerical complexity for large scale models with many variables by using the reduced-rank tangent map in the computation is proposed. The conditions for which the short-time linear response approximation with the reduced-rank tangent map is valid are established, and two practical situations are examined, where the response to small external perturbations is predicted for nonlinear chaotic forced-dissipative systems with different dynamical properties.     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.     

A preprocessing method for parameter estimation in ordinary differential equations

Parameter estimation for nonlinear differential equations is notoriously difficult because of poor or even no convergence of the nonlinear fit algorithm due to the lack of appropriate initial parameter values. This paper presents a method to gather such initial values by a simple estimation procedure. The method first determines the tangent slope and coordinates for a given solution of the ordinary differential equation (ODE) at randomly selected points in time. With these values the ODE is transformed into a system of equations, which is linear for linear appearance of the parameters in the ODE. For numerically generated data of the Lorenz attractor good estimates are obtained even at large noise levels. The method can be generalized to nonlinear parameter dependency. This case is illustrated using numerical data for a biological example. The typical problems of the method as well as their possible mitigation are discussed. Since a rigorous failure criterion of the method is missing, its results must be checked with a nonlinear fit algorithm. Therefore the method may serve as a preprocessing algorithm for nonlinear parameter fit algorithms. It can improve the convergence of the fit by providing initial parameter estimates close to optimal ones.     

Three-point bounds for the overall properties of a nonlinear composite dielectric

The Hashin-Shtrikman methodology for nonlinear composite problemsrelies on the use of a comparison medium and in many of theexamples studied so far the comparison medium has been takento be homogeneous. A related approach originated by P. PonteCastaeda employs a comparison medium which is itself a linearcomposite with the same microgeometry as the nonlinear composite.When the method is applicable, the bounds for the nonlinearproblem then involve bounds for the energy of the linear comparisoncomposite which could include, for example, three-point informationabout the microstructure of the composite. It is, however, onlypossible to obtain at most one bound using a linear comparisonmaterial. A recent approach involves the use of a nonlinearcomparison medium and trial fields with the property of boundedmean oscillation. In this paper the approach is extended byusing a nonlinear comparison composite so that both upper andlower bounds can be obtained which incorporate three-point information.The development is in the context of bounding the propertiesof nonlinear dielectric composites, although it has wider application.     

An adaptive approach for modeling a fiber-matrix composite with the FE2 method

In materials with a complicated microstructre [1], the macroscopic material behaviour is unknown. In this work a Fiber-Matrix composite is considered with elasto-plastic fibers. A homogenization of the microscale leads to the macroscopic material properties. In the present work, this is realized in the frame of a FE² formulation. It combines two nested finite element simulations. On the macroscale, the boundary value problem is modelled by finite elements, at each integration point a second finite element simulation on the microscale is employed to calculate the stress response and the material tangent modulus. One huge disadvantage of the approach is the high computational effort. Certainly, an accompanying homogenization is not necessary if the material behaves linear elastic. This motivates the present approach to deal with an adaptive scheme. An indicator, which makes use of the different boundary conditions (BC) of the BVP on microscale, is suggested. The homogenization with the Dirichlet BC overestimates the material tangent modulus whereas the Neumann BC underestimates the modulus [2]. The idea for an adaptive modeling is to use both of the BCs during the loading process of the macrostructure. Starting initially with the Neumann BC leads to an overestimation of the displacement response and thus the strain state of the boundary value problem on the macroscale. An accompanying homogenization is performed after the strain reaches a limit strain. Dirichlet BCs are employed for the accompanying homogenization. Some numerical examples demonstrate the capability of the presented method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bounds and an isotropically self-consistent singular approximation of the linear elastic properties of cubic crystal aggregates for application in materials design

For crystal aggregates, the orientation distribution of single crystals affects the anisotropic linear elastic properties. In the singular approximation for cubic materials, this influence is reflected by a fourth-order texture coefficient. From this approximation, the statistical bounds of Voigt, Reuss and Hashin-Shtrikman, and an isotropically self-consistent singular approximation can be obtained. Here, an approximation is called isotropically self-consistent, if, for a vanishing texture, it results in the isotropic self-consistent approximation. The isotropically self-consistent singular approximation has the following advantages: i) it lies between the bounds of Voigt, Reuss and Hashin-Shtrikman, ii) it offers a useful approximation of the effective material behavior of textured anisotropic polycrystals, and iii) it can be used for material design purposes tailoring anisotropic properties mainly depending on the crystallographic texture. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the best parametrization

The numerical solution to a system of nonlinear algebraic or transcendental equations with several parameters is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are proved for choosing the best parameters, which provide the best condition number for the system of linear continuation equations. Such parameters have to be sought in the subspace tangent to the solution space of the system of nonlinear equations. This subspace is obtained if the original system of nonlinear equations is solved at the various parameter values from a given set. The parametric approximation of curves and surfaces is considered.     

Computational homogenization and nonlinear effects in heterogeneous fluid saturated porous media

For the two-scale modelling of deforming fluid-saturated porous structure we apply the asymptotic homogenization approach to the fluid-structure interaction problem involving linear elastic porous skeleton and the Newtonian compressible fluid. The sensitivity analysis of effective coefficients depending on the geometrical configuration is used to introduce a weakly nonlinear formulation which enables to capture influences of the deformation on the material properties. The paper is devoted to the comparison of the linear and the weakly nonlinear models (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local strong uniqueness for nonlinear approximation

The local constant of strong uniqueness for nonlinear approximation with respect to a norm is the local constant of strong uniqueness for approximation in the associated problem of linear approximation by the tangent space.     

Numerical calculation of the tangent stiffness matrix in materials modeling

The Finite Element Method in the field of materials modeling is closely connected to the tangent stiffness matrix of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness matrix can be derived analytically only with great effort, if at all. Numerical methods are therefore widely used for its calculation. We present our method to calculate the tangent stiffness matrix for the logarithmic strain measure. The approach is compared with other commonly used procedures. A significant increase in accuracy can be achieved with the proposed method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A class of differential descent methods for constrained optimization

In this paper a class of algorithms is presented for minimizing a nonlinear function subject to nonlinear equality constraints along curvilinear search paths obtained by solving a linear approximation to an initial-value system of differential equations. The system of differential equations is derived by introducing a continuously differentiable matrix whose columns span the subspace tangent to the feasible region. The new approach provides a convenient way for working with the constraint set itself, rather than with the subspace tangent to it. The algorithms obtained in this paper may be viewed as curvilinear extensions of two known and successful minimization techniques. Under certain conditions, the algorithms converge to a point satisfying the first-order Kuhn-Tucker optimality conditions at a rate that is asymptotically at least quadratic.     

Locality of Interatomic Interactions in Self-Consistent Tight Binding Models

A key starting assumption in many classical interatomic potential models for materials is a site energy decomposition of the potential energy surface into contributions that only depend on a small neighbourhood. Under a natural stability condition, we construct such a spatial decomposition for self-consistent tight binding models, extending recent results for linear tight binding models to the nonlinear setting.

     

Nonlinear homogenization of metal-ceramic composites with lamellar microstructure

Using nonlinear homogenization methods for the computation of the material response of metal-ceramic composites with lamellar microstructure is a power approach to do computation less costly in comparison to finite elements modeling. A modified secant homogenization method is utilized in this study for simulation of inelastic behaviors of the composite micro-constituents. A nonlinear homogenization method is based on a linear homogenization scheme for multilayer composites. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)