Computational approach for composite materials with coupled constitutive laws

A numerical scheme based on fast Fourier transform is presented to compute the effective response and the local fields within a heterogeneous material which exhibits a coupled constitutive law. It consists in the iterative resolution of periodic coupled Lippmann–Schwinger equations. This approach is illustrated in the case of electroelastic composite materials. By using an augmented Lagrangian formulation, a simple iterative scheme relying on the uncoupled Green operators for the elastic and electrostatics problems is proposed. This computational framework, which allows to consider composite materials with an infinite contrast on the local properties, is assessed in the case of porous and fiber-reinforced piezoelectric materials.     

Composite Grid Finite Element Method: Implementation and Iterative Solution with Inexact Subproblems

This paper concerns the composite grid finite element (FE) method for solving boundary value problems in the cases which require local grid refinement for enhancing the approximating properties of the corresponding FE space. A special interest is given to iterative methods based on natural decomposition of the space of unknowns and to the implementation of both the composite grid FEM and the iterative procedures for its solution. The implementation is important for gaining all benefits of the described methods. We also discuss the case of inexact subproblems, which can frequently arise in the course of hierarchical modelling.     

A local iteration scheme for nonlinear two-dimensional steady-state heat conduction: a BEM approach

An efficient algorithm is proposed to solve the steady-state nonlinear heat conduction equation using the boundary element method (BEM). Nonlinearity of the heat conduction equation arises from nonlinear boundary conditions and temperature dependence of thermal conductivity. Using Kirchhoff's transformation, the case of temperature dependence of thermal conductivity can be transformed to the nonlinear boundary conditions case. Applying the BEM technique, the resulting matrix equation becomes nonlinear. The nonlinearity, however, only involves the boundary nodes that have nonlinearboundary conditions. The proposed local iterative scheme reduces the entire BEM matrix equation to a smaller matrix equation whose rank is the same as the number of boundary nodes with nonlinear boundary conditions. The Newton-Raphson iteration scheme is used to solve the reduced nonlinear matrix equation. The local iterative scheme is first applied to two one-dimensional problems (analytical solutions are possible) with different nonlinear boundary conditions. It is then applied to a two-region problem. Finally, the local iterative scheme is applied to two cavity problems in which radiation plays a role in the heat transfer.     

Numerical solutions for some coupled systems of nonlinear boundary value problems

Summary In the well-known Volterra-Lotka model concerning two competing species with diffusion, the densities of the species are governed by a coupled system of reaction diffusion equations. The aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems. This iterative scheme is based on the method of upper-lower solutions which leads to two monotone sequences from some uncoupled linear systems. It is shown that each of the two sequences converges to a nontrivial solution of the discrete equations. The model under consideration may have one, two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration. Numerical results are given for these solutions under both the Dirichlet boundary condition and the mixed type boundary condition.     

An iterative solver for a coupled system of Helmholtz equations

An iterative solver for a pair of coupled partial differential equations that are related to the Maxwell equations is discussed. The convergence of the scheme depends on the choice of two parameters. When the first parameter is fixed, the scheme is seen to be a successive under-relaxation scheme in the other parameter. A theory for convergence of the scheme is discussed for a special case of the equations, and several numerical examples are presented. © 1996 John Wiley & Sons, Inc.     

关于m-增生算子零点的粘性复合迭代序列的强收敛性

在一致光滑的Banach空间的框架下,引入关于m-增生算子的一种新粘性复合迭代序列{xn},并证明了在适当的控制条件下,此迭代序列强收敛于m-增生算子的一个零点,用不同方法推广了相关文献的近代结果.     

Coupled solutions and monotone iterative techniques for some nonlinear initial value problems on time scales

A unified approach to monotone iterative technique concerning the existence of coupled maximal and minimal solutions for dynamic IVP's on time scales is developed in the case of the nonlinear term involved admitting a splitting of a difference of two monotone functions. Besides pointing out the relevance of such problems in mathematical biology involving both differential and difference equations, the results involved prove to be useful in the sense of including several known results as well as some interesting new results in the monotone iterative scheme theory, especially in the case of the corresponding difference equations.     

An alternative technique for solving a coupled PDE system in interphase heat transfer

Recently an accelerated iterative procedure was studied for solving a coupled partial differential equation system in interphase heat transfer to improve some existing iterative procedures in the literature. In that procedure, at each step of the iteration one has to evaluate the derivative of a well-known function at a new point. In this paper, an alternative approach is proposed in which one has to evaluate the derivative only once throughout the procedure. The proposed new iterative scheme also has the same order of convergence and takes lesser number of iterations for certain benchmark problems. An interesting theoretical study on the monotone convergence as well as error estimate of the proposed iterative procedure are provided for continuous as well as discretized problems. The proposed iterative procedure also supplements the existence and uniqueness of the solution in both the cases. A comparative numerical study is also done to demonstrate the efficacy of the proposed scheme.     

An efficient parallel processing optimal control scheme for a class of nonlinear composite systems

This article presents an efficient parallel processing approach for solving the optimal control problem of nonlinear composite systems. In this approach, the original high-order coupled nonlinear two-point boundary value problem (TPBVP) derived from the Pontryagin's maximum principle is first transformed into a sequence of lower-order decoupled linear time-invariant TPBVPs. Then, an optimal control law which consists of both feedback and forward terms is achieved by using the modal series method for the derived sequence. The feedback term specified by local states of each subsystem is determined by solving a matrix Riccati differential equation. The forward term for each subsystem derived from its local information is an infinite sum of adjoint vectors. The convergence analysis and parallel processing capability of the proposed approach are also provided. To achieve an accurate feedforward-feedback suboptimal control, we apply a fast iterative algorithm with low computational effort. Finally, some comparative results are included to illustrate the effectiveness of the proposed approach.     

A Robin-type non-overlapping domain decomposition procedure for second order elliptic problems

This article deals with the analysis of an iterative non-overlapping domain decomposition (DD) method for elliptic problems, using Robin-type boundary condition on the inter-subdomain boundaries, which can be solved in parallel with local communications. The proposed iterative method allows us to relax the continuity condition for Lagrange multipliers on the inter-subdomain boundaries. In order to derive the corresponding discrete problem, we apply a non-conforming Galerkin method using lowest order Crouzeix–Raviart elements. The convergence of the iterative scheme is obtained by proving that the spectral radius of the matrix associated with the fixed point iterations is less than 1. Parallel computations have been carried out and the numerical experiments confirm the theoretical results established in this paper.     

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multigrid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical two-grid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.     

Numerical analysis for the quadratic matrix equations from a modification of fixed‐point type

In this paper, we study the quadratic matrix equations. To improve the application of iterative schemes, we use a transform of the quadratic matrix equation into an equivalent fixed‐point equation. Then, we consider an iterative process of Chebyshev‐type to solve this equation. We prove that this iterative scheme is more efficient than Newton's method. Moreover, we obtain a local convergence result for this iterative scheme. We finish showing, by an application to noisy Wiener‐Hopf problems, that the iterative process considered is computationally more efficient than Newton's method.     

Variational Structural and Material Stability Analysis in Finite Electro-Magneto-Mechanics of Active Materials

Soft matter electro-elastic, magneto-elastic and magneto-electro-elastic composites exhibit coupled material behavior at large strains. Examples are electro-active polymers and magneto-rheological elastomers, which respond by a deformation to applied electric or magnetic fields, and are used in advanced industrial environments as sensors and actuators. Polymer-based magneto-electro-elastic composites are a new class of tailor-made materials with promising future applications. Here, a magneto-electric coupling effect is achieved as a homogenized macro-response of the composite with electro-active and magneto-active constituents. These soft composite materials show different types of instability phenomena, which even might be exploited for future enhancement of their performance. This covers micro-structural instabilities, such as buckling of micro-fibers or particles, as well as material instabilities in the form of limit-points in the local constitutive response. Here, the homogenization-based scale bridging links long wavelength micro-structural instabilities to material instabilities at the macro-scale. This work outlines a comprehensive framework of an energy-based homogenization in electro-magneto-mechanics, which allows a tracking of postcritical solution paths such as those related to pull-in instabilities. Representative simulations demonstrate a tracking of inhomogenous composites, showing the development of postcritical zones in the microstructure and a possible instable homogenized material response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear statics and dynamics of antisymmetric composite laminated square plates supported on nonlinear elastic subgrade

Non-linear static and dynamic analysis is presented for composite laminated anti-symmetric square plates supported on non-linear elastic foundation subjected to uniformly distributed transverse and step loading, respectively. The formulation is based on first order shear deformation theory (FSDT) and Von-Karman non-linearity, subgrade interaction is modeled as shear deformable with cubic nonlinearity. The methodology of solution is based on the Chebyshev series technique. The coupled non-linear partial differential equations are linearized using quadratic extrapolation technique. Houbolt time marching scheme is employed for temporal discretisation. An incremental iterative approach is employed for the solution. The effects of foundation stiffness parameters and boundary conditions on the non-linear static and dynamic analysis on the central response are studied.     

A numerical method for mass conservative coupling between fluid flow and solute transport

We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier-Stokes equations for the flow are discretized by the divergence-free Scott-Vogelius element on barycentrically refined meshes guaranteeing LBB stability. The convection-diffusion equation for species transport is discretized by the Voronoi finite volume method. In accordance to the continuous setting, due to the exact integration of the normal component of the flow through the Voronoi surfaces, the species concentration fulfills discrete global and local maximum principles. Besides of the numerical scheme itself, we present important aspects of its implementation. Further, for the case of homogeneous Dirichlet boundary conditions, we give a convergence proof for the coupled scheme. We report results of the application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell with cylindrical shape.     

Conjugate gradient-boundary element solution for distributed elliptic optimal control problems

An optimality system of equations for the optimal control problem governed by Helmholtz-type equations is derived. By the associated first-order necessary optimality condition, we obtain the conjugate gradient method (CGM) in the continuous case. Introducing the sequence of higher-order fundamental solutions, we propose an iterative algorithm based on the conjugate gradient-boundary element method using the multiple reciprocity method (CGM+MRBEM) for solving the discrete control input. This algorithm has an advantage over that of the existing literatures because the main attribute (the reduced dimensionality) of the boundary element method is fully utilized. Finally, the local error estimates for this scheme are obtained, and a test problem is given to illustrate the efficiency of the proposed method.     

A class of generalized Newton iterative schemes

We study an N-step iterative scheme which generalizes several Newton-type schemes that have appeared in the literature. We show that, under generalized Zabrejko-Nguen conditions, the iterative scheme converges whenever 1 ≤ N ≤ ∞. This proves in a unified context the convergence of an infinite number of iterative schemes which include as special cases the classical Newton scheme, the classical chord scheme, and the generalized Newton scheme.     

Perturbed iterative solution of coupled nonlinear systems with application to fluid dynamics

The perturbed iterative scheme developed in [3] is extended in this work to solve coupled systems of nonlinear equations. The algorithm consists of computing distinct perturbation parameters for each system at each iteration and adding these to corresponding nonlinear Gauss-Seidel iterates. It has been found computationally that such an algorithm significantly improves the convergence properties of Gauss-Seidel iterations. The method has been successfully applied to several coupled nonlinear systems of equations some of which are discussed in this work.     

An iterative scheme for a class of quasi variational inequalities

An iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities. It is shown that the approximate solution obtained by the iterative scheme converges strongly in the Hilbert space to the exact solution. As a special case, we obtain the corresponding iterative scheme for variational inequalities.     

Numerical methods for a coupled system of differential equations arising from a thermal ignition problem

This article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first‐order partial differential equation and an ordinary differential equation which arises from fast‐igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite‐difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite‐difference solution. Particular attention is given to the “finite‐time” blow‐up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow‐up solution are obtained. Also given is the convergence of the finite‐difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow‐up time and the critical value of a physical parameter which determines the global existence and the blow‐up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013