A second-order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

An efficient second-order reduced asymptotic homogenization approach is developed for nonlinear heterogeneous media with large periodic microstructure. The two salient features of the proposed approach are (i) an asymptotic higher-order nonlinear homogenization that does not require higher-order continuity of the coarse-scale solution and (ii) an efficient model reduction scheme for solving higher-order nonlinear unit cell problems at a fraction of computational cost in comparison to the direct computational homogenization. The former is a consequence of a sequential solution of increasing order solutions, which permits evaluation of higher-order coarse-scale derivatives by postprocessing from the zeroth-order solution. The efficiency and accuracy of the formulation in comparison to the classical zeroth-order homogenization and direct numerical simulations are assessed on hyperelastic and elastoplastic periodic structures.     

A staggered nonlocal multiscale model for a heterogeneous medium

A staggered nonlocal multiscale model for a heterogeneous medium is developed and validated. The model is termed as staggered nonlocal in the sense that it employs current information for the point under consideration and past information from its local neighborhood. For heterogeneous materials, the concept of phase nonlocality is introduced by which nonlocal phase eigenstrains are computed using different nonlocal phase kernels. The staggered nonlocal multiscale model has been found to be insensitive to finite element mesh size and load increment size. Furthermore, the computational overhead in dealing with nonlocal information is mitigated by superior convergence of the Newton method. Copyright © 2012 John Wiley & Sons, Ltd.     

A Koiter reduction technique for the nonlinear thermoelastic analysis of shell structures prone to buckling

The multi-modal Koiter method is a reduction technique for estimating quickly the nonlinear buckling response of structures under mechanical loads requiring a fine discretization. The reduced model is based on a quadratic approximation of the full model using a few linear buckling modes and their second order corrections, followed by the projection of the equilibrium equations onto the modal subspace. In this article, the method is reformulated for geometrically nonlinear thermoelastic analyses of shell structures. The starting point is an isogeometric solid-shell discretization with an accurate modeling of thermal strains, temperature-dependent materials and general temperature distributions. The equilibrium path is defined as a displacement versus temperature amplifier curve, while mechanical loads are kept constant. The strain energy nonlinearity with respect to the temperature amplifier is the main obstacle in the definition of an accurate reduced model. This task is achieved by coherent asymptotic expansions in mixed form using independent stress variables at the integration points and accurate linear buckling modes obtained by a two-point mixed eigenvalue problem. Structures made of isotropic and multi-layered composite materials, including variable angle tow laminates, are considered to show the accuracy of the proposed method.     

A bispace parameterization method for shape optimization of thin‐walled curved shell structures with openings

Simultaneous shape optimization of thin‐walled curved shell structures and involved hole boundaries is studied in this paper. A novel bispace parameterization method is proposed for the first time to define global and local shape design variables both in the Cartesian coordinate system and the intrinsic coordinate system. This method has the advantage of achieving a simultaneous optimization of the global shape of the shell surface and the local shape of the openings attached automatically on the former. Inherent problems, for example, the effective parameterization of shape design variables, mapping operation between two spaces, and sensitivity analysis with respect to both kinds of design variables are highlighted. A design procedure is given to show how both kinds of design variables are managed together and how the whole design flowchart is carried out with relevant formulations. Numerical examples are presented and the effects of both kinds of design variables upon the optimal solutions are discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

An algorithm for optimization of non-linear shell structures

An algorithm for optimal design of non-linear shell structures is presented. The algorithm uses numerical optimization techniques and nonlinear finite element analysis to find a minimum weight structure subject to equilibrium conditions, stability constraints and displacement constraints. A barrier transformation is used to treat an apparent non-smoothness arising from posing the stability constraints in terms of the eigenvalues of the Hessian of the potential energy of the structure. A sequential quadratic programming strategy is used to solve the resulting non-linear optimization problem. Matrix sparsity in the constraint Jacobian is exploited because of the large number of variables. The usefulness of the proposed algorithm is demonstrated by minimizing the weight of a number of stiffened thin shell structures.     

Stress recovery and error estimation for shell structures

The Penalized Discrete Least‐Squares (PDLS) stress recovery (smoothing) technique developed for two‐dimensional linear elliptic problems [1–3] is adapted here to three‐dimensional shell structures. The surfaces are restricted to those which have a 2‐D parametric representation, or which can be built‐up of such surfaces. The proposed strategy involves mapping the finite element results to the 2‐D parametric space whichdescribes the geometry, and smoothing is carried out in the parametric space using the PDLS‐based Smoothing Element Analysis (SEA). Numerical results for two well‐known shell problems are presented to illustrate the performance of SEA/PDLS for these problems. The recovered stresses are used in the Zienkiewicz–Zhu a posteriori error estimator. The estimated errors are used to demonstrate the performance of SEA‐recovered stresses in automated adaptive mesh refinement of shell structures. The numerical results are encouraging. Further testing involving more complex, practical structures is necessary. Copyright © 2000 John Wiley & Sons, Ltd.     

A multiscale projection method for macro/microcrack simulations

We present a new multiscale method for crack simulations. This approach is based on a two‐scale decomposition of the displacements and a projection to the coarse scale by using coarse scale test functions. The extended finite element method (XFEM) is used to take into account macrocracks as well as microcracks accurately. The transition of the field variables between the different scales and the role of the microfield in the coarse scale formulation are emphasized. The method is designed so that the fine scale computation can be done independently of the coarse scale computation, which is very efficient and ideal for parallelization. Several examples involving microcracks and macrocracks are given. It is shown that the effect of crack shielding and amplification for crack growth analyses can be captured efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

A multiscale mass scaling approach for explicit time integration using proper orthogonal decomposition

One of the main computational issues with explicit dynamics simulations is the significant reduction of the critical time step as the spatial resolution of the finite element mesh increases. In this work, a selective mass scaling approach is presented that can significantly reduce the computational cost in explicit dynamic simulations, while maintaining accuracy. The proposed method is based on a multiscale decomposition approach that separates the dynamics of the system into low (coarse scales) and high frequencies (fine scales). Here, the critical time step is increased by selectively applying mass scaling on the fine scale component only. In problems where the response is dominated by the coarse (low frequency) scales, significant increases in the stable time step can be realized. In this work, we use the proper orthogonal decomposition (POD) method to build the coarse scale space. The main idea behind POD is to obtain an optimal low‐dimensional orthogonal basis for representing an ensemble of high‐dimensional data. In our proposed method, the POD space is generated with snapshots of the solution obtained from early times of the full‐scale simulation. The example problems addressed in this work show significant improvements in computational time, without heavily compromising the accuracy of the results. Copyright © 2013 John Wiley & Sons, Ltd.     

A coupled thermo-chemo-mechanical reduced-order multiscale model for predicting process-induced distortions,residual stresses,and strength

We study residual stresses and part distortion induced by a manufacturing process of a polymer matrix composite and its effect on the component strength. Unlike most of the thermo-chemo-mechanical models in the literature where governing multiphysics equations are directly formulated on the macroscale, we present a multiscale-multiphysics approach. To address the enormous computational complexity involved, a reduced-order homogenization was originally developed for a single physics problem is employed. The proposed reduced-order two-scale thermo-chemo-mechanical model has been validated for predicting part distortion beam strength in three-point bending test. It is shown that while macroscopic stresses are relatively low, and therefore often ignored in practice, stresses at the scale of microconstituents are significant and may have an effect on the overall composite component strength.     

A multiscale virtual element method for the analysis of heterogeneous media

We introduce a novel heterogeneous multiscale method for the elastic analysis of two-dimensional domains with a complex microstructure. To this end, the multiscale finite element method is revisited and originally upgraded by introducing virtual element discretizations at the microscale, hence allowing for generalized polygonal and nonconvex elements. The microscale is upscaled through the numerical evaluation of a set of multiscale basis functions. The solution of the equilibrium equations is performed at the coarse scale at a reduced computational cost. We discuss the computation of the multiscale basis functions and corresponding virtual projection operators. The performance of the method in terms of accuracy and computational efficiency is evaluated through a set of numerical examples.     

A parallel noninvasive multiscale strategy for a mixed domain decomposition method with frictional contact

A multiscale extension for a parallel noninvasive mixed domain decomposition method is presented. After briefly exposing our noninvasive implementation of the Latin method, we present how the scalability of the algorithm is obtained by the partial verification of the constitutive law of the interfaces. We propose a new interpretation of the classical macrostrategy that imposes the overall balance of interface's forces. We also propose a new study of the macrostrategy that consists in enforcing the coarse continuity of the interfaces' displacement.     

A reduced SAND method for optimal design of non-linear structures

An SQP-based reduced Hessian method for simultaneous analysis and design (SAND) of non-linearly behaving structures is presented and compared with conventional nested analysis and design (NAND) methods. It is shown that it is possible to decompose the SAND formulation to take advantage of the particular structure of the problem at hand. The resulting reduced SAND method is of the same size as the conventional NAND method but it is computationally more efficient. The presentation here builds on previous research on SAND methods generalizing the solution approach to cases with both equality and inequality constraints. The new version of the reduced SAND method is tested in the context of weight minimization of 3-D truss structures with geometrically non-linear behaviour. © 1997 John Wiley & Sons, Ltd.     

Influence functions and goal‐oriented error estimation for finite element analysis of shell structures

In this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal‐oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over‐estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal‐oriented error estimation procedure shows good effectivity indices. Copyright © 2005 John Wiley & Sons, Ltd.     

A FE2 shell model with periodic boundary conditions for thin and thick shells

In this article a FE2 shell model for thin and thick shells within a first order homogenization scheme is presented. A variational formulation for the two-scale boundary value problem and the associated finite element formulation is developed. Constraints with 5 or 9 Lagrange parameters are derived which eliminate both rigid body movements and dependencies of the shear stiffness on the size of the representative volume elements (RVEs). At the bottom and top surface of the RVEs which extend through the total thickness of the shell stress boundary conditions are present. The periodic boundary conditions at the lateral surfaces are applied in such a way that particular membrane, bending and shear modes are not restrained. This is shown by means of a homogeneous RVE. The first of all linear formulation is extended to finite strain problems introducing transformation relations for the stress resultants and the material matrix. The transformations are performed at the Gauss points on macro level. Several boundary value problems including large deformations, stability and inelasticity are computed and compared with 3D reference solutions.     

A variational multiscale method with subscales on the element boundaries for the Helmholtz equation

In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.     

A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

In this paper, we present a non‐linear finite element formulation for piezoelectric shell structures. Based on a mixed multi‐field variational formulation, an electro‐mechanical coupled shell element is developed considering geometrically and materially non‐linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non‐linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three‐dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.     

A generalized phase field multiscale finite element method for brittle fracture

A generalized multiscale finite element method is introduced to address the computationally taxing problem of elastic fracture across scales. Crack propagation is accounted for at the microscale utilizing phase field theory. Both the displacement-based equilibrium equations and phase field state equations at the microscale are mapped on a coarser scale. The latter is defined by a set of multinode coarse elements, where solution of the governing equations is performed. Mapping is achieved by employing a set of numerically derived multiscale shape functions. A set of representative benchmark tests is used to verify the proposed procedure and assess its performance in terms of accuracy and efficiency compared with the standard phase field finite element implementation.     

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

We present a computational framework for the simulation of J₂‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.