# Browsing by Subject "Multiscale modeling"

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Item Multiscale modeling using goal-oriented adaptivity and numerical homogenization(2009-08) Jhurani, Chetan Kumar; Demkowicz, LeszekShow more Modeling of engineering objects with complex heterogeneous material structure at nanoscale level has emerged as an important research problem. In this research, we are interested in multiscale modeling and analysis of mechanical properties of the polymer structures created in the Step and Flash Imprint Lithography (SFIL) process. SFIL is a novel imprint lithography process designed to transfer circuit patterns for fabricating microchips in low-pressure and room-temperature environments. Since the smallest features in SFIL are only a few molecules across, approximating them as a continuum is not completely accurate. Previous research in this subject has dealt with coupling discrete models with continuum hyperelasticity models. The modeling of the post-polymerization step in SFIL involves computing solutions of large nonlinear energy minimization problems with fast spatial variation in material properties. An equilibrium configuration is found by minimizing the energy of this heterogeneous polymeric lattice. Numerical solution of such a molecular statics base model, which is assumed to describe the microstructure completely, is computationally very expensive. This is due to the problem size – on the order of millions of degrees of freedom (DOFs). Rapid variation in material properties, ill-conditioning, nonlinearity, and non-convexity make this problem even more challenging to solve. We devise a method for efficient approximation of the solution. Combining numerical homogenization, adaptive finite element meshes, and goaloriented error estimation, we develop a black-box method for efficient solution of problems with multiple spatial scales. The purpose of this homogenization method is to reduce the number of DOFs, find locally optimal effective material properties, and do goal-oriented mesh refinement. In addition, it smoothes the energy landscape. Traditionally, a finite element mesh is designed after obtaining material properties in different regions. The mesh has to resolve material discontinuities and rapid variations. In our approach, however, we generate a sequence of coarse meshes (possibly 1-irregular), and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem. This upscaling is done using Moore-Penrose pseudoinverse of the linearized fine-scale element stiffness matrices, and a material independent interpolation operator. This requires solution of a continuous-time Lyapunov equation on each element. Using the adjoint solution, we compute local error estimates in the quantity of interest. The error estimates also drive the automatic mesh adaptivity algorithm. The results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine-scale problem. Critical to the computational speed of local homogenization is computing Moore-Penrose pseudoinverse of rank-deficient matrices without using Singular Value Decomposition. To this end, we use four algorithms, each having different desirable features. The algorithms are based on Tikhonov regularization, sparse QR factorization, a priori knowledge of the null-space of the matrix, and iterative methods based on proper splittings of matrices. These algorithms can exploit sparsity and thus are fast. Although the homogenization method is designed with a specific molecular statics problem in mind, it is a general method applicable for problems with a given fine mesh that sufficiently resolves the fine-scale material properties. We verify the method using a conductivity problem in 2-D, with chessboard like thermal conductivity pattern, which has a known homogenized conductivity. We analyze other aspects of the homogenization method, for example the choice of norm in which we measure local error, optimum coarse mesh element size for homogenizing SFIL lattices, and the effect of the method chosen for computing the pseudoinverse.Show more Item Numerical methods for multiscale inverse problems(2014-05) Frederick, Christina A; Engquist, Björn, 1945-Show more This dissertation focuses on inverse problems for partial differential equations with multiscale coefficients in which the goal is to determine the coefficients in the equation using solution data. Such problems pose a huge computational challenge, in particular when the coefficients are of multiscale form. When faced with balancing computational cost with accuracy, most approaches only deal with models of large scale behavior and, for example, account for microscopic processes by using effective or empirical equations of state on the continuum scale to simplify computations. Obtaining these models often results in the loss of the desired fine scale details. In this thesis we introduce ways to overcome this issue using a multiscale approach. The first part of the thesis establishes the close relation between computational grids in multiscale modeling and sampling strategies developed in information theory. The theory developed is based on the mathematical analysis of multiscale functions of the type that are studied in averaging and homogenization theory and in multiscale modeling. Typical examples are two-scale functions f (x, x/[epsilon]), (0 < [epsilon] ≪ 1) that are periodic in the second variable. We prove that under certain band limiting conditions these multiscale functions can be uniquely and stably recovered from nonuniform samples of optimal rate. In the second part, we present a new multiscale approach for inverse homogenization problems. We prove that in certain cases where the specific form of the multiscale coefficients is known a priori, imposing an additional constraint of a microscale parametrization results in a well-posed inverse problem. The mathematical analysis is based on homogenization theory for partial differential equations and classical theory of inverse problems. The numerical analysis involves the design of multiscale methods, such as the heterogeneous multiscale method (HMM). The use of HMM solvers for the forward model has unveiled theoretical and numerical results for microscale parameter recovery, including applications to inverse problems arising in exploration seismology and medical imaging.Show more