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Multiscale modeling using goal-oriented adaptivity and numerical homogenizationJhurani, Chetan Kumar 16 October 2009 (has links)
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. / text
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Two universality classes for random hyperbranched polymersJurjiu, A., Dockhorn, R., Mironova, O., Sommer, J.-U. 06 December 2019 (has links)
We grow AB₂ random hyperbranched polymer structures in different ways and using different simulation methods. In particular we use a method of ad hoc construction of the connectivity matrix and the bond fluctuation model on a 3D lattice. We show that hyperbranched polymers split into two universality classes depending on the growth process. For a “slow growth” (SG) process where monomers are added sequentially to an existing molecule which strictly avoids cluster–cluster aggregation the resulting structures share all characteristic features with regular dendrimers. For a “quick growth” (QG) process which allows for cluster–cluster aggregation we obtain structures which can be identified as random fractals. Without excluded volume interactions the SG model displays a logarithmic growth of the radius of gyration with respect to the degree of polymerization while the QG model displays a power law behavior with an exponent of 1/4. By analyzing the spectral properties of the connectivity matrix we confirm the behavior of dendritic structures for the SG model and the corresponding fractal properties in the QG case. A mean field model is developed which explains the extension of the hyperbranched polymers in an athermal solvent for both cases. While the radius of gyration of the QG model shows a power-law behavior with the exponent value close to 4/5, the corresponding result for the SG model is a mixed logarithmic–power-law behavior. These different behaviors are confirmed by simulations using the bond fluctuation model. Our studies indicate that random sequential growth according to our SG model can be an alternative to the synthesis of perfect dendrimers.
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Fluorescence studies of complex systems : organisation of biomoleculesMarushchak, Denys January 2007 (has links)
The homo and hetero dimerisation of two spectroscopically different chromophores were studied, namely: 4,4-difluoro-4-bora-3a,4a-diazas-indacene (g-BODIPY) and its 5-styryl-derivative (r-BODIPY). Various spectroscopic properties of the r-BODIPY in different common solvents were determined. It was shown that g- and r-BODIPY in the ground state can form homo- as well as hetero dimers. We demonstrate that the ganglioside GM1 in lipid bilayers of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) exhibits a non-uniform lateral distribution, which is an argument in favour of self-aggregation of GM1 being an intrinsic property of the GM1. This was concluded from energy transfer/migration studies of BODIPY-labelled gangliosides. An algorithm is presented that quantitatively accounts for donor–donor energy migration (DDEM) among fluorophore-labelled proteins forming regular non-covalent polymers. The DDEM algorithm is based on Monte Carlo (MC) and Brownian dynamics (BD) simulations and applies to the calculation of fluorescence depolarisation data, such as the fluorescence anisotropy. Thereby local orientations, as well as reorienting motions of the fluorescent groups are considered in the absence and presence of DDEM among them. A new method, in which a genetic algorithm (GA) was combined with BD and MC simulations, was developed to analyse fluorescence depolarisation data collected by the time-correlated single photon counting technique. It was applied to study g-BODIPY-labelled filamentous actin (F-actin). The technique registered the local order and reorienting motions of the fluorophores, which were covalently coupled to cysteine 374 (C374) in actin and interacted by means of electronic energy migration within the polymer. Analyses of F-actin samples composed of different fractions of labelled actin molecules revealed the known helical organiszation of F-actin, and demonstrated the usefulness of this technique for structure determination of complex protein polymers. The distance from the filament axis to the fluorophore was found to be considerably less than expected from the proposed position of C374 at a high filament radius. In addition, polymerisation experiments with BODIPY-actin suggest a 25-fold more efficient signal for filament formation than pyrene-actin.
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