High performance reduced order modeling techniques based on optimal energy quadrature: application to geometrically non-linear multiscale inelastic material modeling

A High-Performance Reduced-Order Model (HPROM) technique, previously presented by the authors in the context of hierarchical multiscale models for non linear-materials undergoing infinitesimal strains, is generalized to deal with large deformation elasto-plastic problems. The proposed HPROM technique uses a Proper Orthogonal Decomposition procedure to build a reduced basis of the primary kinematical variable of the micro-scale problem, defined in terms of the micro-deformation gradient fluctuations. Then a Galerkin-projection, onto this reduced basis, is utilized to reduce the dimensionality of the micro-force balance equation, the stress homogenization equation and the effective macro-constitutive tangent tensor equation. Finally, a reduced goal-oriented quadrature rule is introduced to compute the non-affine terms of these equations. Main importance in this paper is given to the numerical assessment of the developed HPROM technique. The numerical experiments are performed on a micro-cell simulating a randomly distributed set of elastic inclusions embedded into an elasto-plastic matrix. This micro-structure is representative of a typical ductile metallic alloy. The HPROM technique applied to this type of problem displays high computational speed-ups, increasing with the complexity of the finite element model. From these results, we conclude that the proposed HPROM technique is an effective computational tool for modeling, with very large speed-ups and acceptable accuracy levels with respect to the high-fidelity case, the multiscale behavior of heterogeneous materials subjected to large deformations involving two well-separated scales of length.Peer ReviewedPostprint (author's final draft

Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers

We first consider an elastic thin heterogeneous cylinder of radius of order epsilon: the interior of the cylinder is occupied by a stiff material (fiber) that is surrounded by a soft material (matrix). By assuming that the elasticity tensor of the fiber does not scale with epsilon and that of the matrix scales with epsilon square, we prove that the one dimensional model is a nonlocal system. We then consider a reference configuration domain filled out by periodically distributed rods similar to those described above. We prove that the homogenized model is a second order nonlocal problem. In particular, we show that the homogenization problem is directly connected to the 3D-1D dimensional reduction problem

Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT

We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients $\sigma(x)$ in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions $s(x)$ over the domain $\Omega$. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of $\Omega$ when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we obtain an optimally robust homogenization algorithm for arbitrary rough coefficients. Next, we consider inverse homogenization and show how to decompose it into a linear ill-posed problem and a well-posed non-linear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT). It is known that the EIT problem admits at most one isotropic solution. If an isotropic solution exists, we show how to compute it from any conductivity having the same boundary Dirichlet-to-Neumann map. It is known that the EIT problem admits a unique (stable with respect to $G$-convergence) solution in the space of divergence-free matrices. As such we suggest that the space of convex functions is the natural space in which to parameterize solutions of the EIT problem

Tail estimates for homogenization theorems in random media

It is known that a random walk on $\Z^d$ among i.i.d. uniformly elliptic random bond conductances verifies a central limit theorem. It is also known that approximations of the covariance matrix can be obtained by considering periodic environments. Here we estimate the speed of convergence of this homogenization result. We obtain similar estimates for finite volume approximations of the effective conductance and of the lowest Dirichlet eigenvalue. A lower bound is also given for the variance of the Green function of a random walk in a random non-negative potential.Comment: 26 page

Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Energy-based comparison between the Fourier--Galerkin method and the finite element method

The Fourier-Galerkin method (in short FFTH) has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom. Because the method incorporates the fast Fourier transform (FFT) in the linear solver, it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method (FEM). Here, we systematically compare these two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenised properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods' effectiveness relies on the smoothness (regularity) of the solution and thus on the material coefficients. Thanks to its approximation properties, FEM outperforms FFTH for problems with jumps in material coefficients, while ambivalent results are observed for the case that the material coefficients vary continuously in space. FFTH profits from a good conditioning of the linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom to reach the same approximation accuracy. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM but not in FFTH).Comment: 24 pages, 10 figures, 2 table