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The aim of this work is to provide a mathematical and numerical analysis of the static quasicontinuum (QC) method. The QC method is, in essence, a finite element method for atomistic material models. By restricting the set of admissible deformations to linear splines with respect to a finite element mesh, the computational complexity of atomistic material models is reduced considerably. We begin with a general review of atomistic material models and the QC method and, most importantly, a thorough discussion of the correct concept of static equilibrium. For example, it is shown that, in contrast to global energy minimization, a ‘dynamic’ selection procedure based on gradient flows models the physically correct behaviour. Next, an atomistic model with long-range Lennard–Jones type interactions is analyzed in one dimension. A rigorous demonstration is given for the existence and stability of elastic as well as fractured steady states, and it is shown that they can be approximated by a QC method if the mesh is sufficiently well adapted to the exact solution; this can be measured by the interpolation error. While the a priori error analysis is an important theoretical step for understanding the approximation properties of the QC method, it is in general unclear how to compute the QC deformation whose existence is guaranteed by the a priori analysis. An a posteriori analysis is therefore performed as well. It is shown that, if a computed QC deformation is stable and has a sufficiently small residual, then there exists a nearby exact solution and the error is estimated. This a posteriori existence idea is also analyzed in an abstract setting. Finally, extensions of the ideas to higher dimensions are investigated in detail.

Thanks to the advancement of additive manufacturing, periodic metallic lattice structures are gaining more and more attention. A major attraction of them is that their design can be tailored to specific applications by changing the basic repetitive pattern of the lattice, called the unit cell. This may involve the selection of optimal strut diameters and orientations, as well as the connectivity and strut lengths. Numerical simulation plays a vital role in understanding the mechanical behavior of metallic lattices and it enables the optimization of design parameters. However, conventional numerical modeling strategies in which each strut is represented by one or more beam finite elements yield prohibitively time­ consuming simulations for metallic lattices in engineering­ scale applications. The reasons are that millions of struts are involved, as well as that geometrical and material nonlinearities at the strut level need to be incorporated. The aim of this thesis is the development of multi­scale quasicontinuum (QC) frameworks to substantially reduce the simulation time of nonlinear mechanical models of metallic lattices. For this purpose, this thesis generalizes the QC method by a multi­-field interpolation enabling amongst others the representation of varying diameters in the struts’ axial directions (as a consequence of the manufacturing process). The efficiency is further increased by a new adaptive scheme that automatically adjusts the model reduction whilst controlling the (elastic or elastoplastic) model’s accuracy. The capabilities of the proposed methodology are demonstrated using numerical examples, such as indentation tests and scratch tests, in which the lattice is modeled using geometrically nonlinear elastic and elastoplastic beam finite elements. They show that the multi­scale framework combines a high accuracy with substantial model reduction that are out of reach of direct numerical simulations. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

The present thesis is on the error estimates of different energy based quasicontinuum (QC) methods, which are a class of computational methods for the coupling of atomistic and continuum models for micro- or nano-scale materials. The thesis consists of two parts. The first part considers the a priori error estimates of three energy based QC methods. The second part deals with the a posteriori error estimates of a specific energy based QC method which was recently developed. In the first part, we develop a unified framework for the a priori error estimates and present a new and simpler proof based on negative-norm estimates, which essentially extends previous results. In the second part, we establish the a posteriori error estimates for the newly developed energy based QC method for an energy norm and for the total energy. The analysis is based on a posteriori residual and stability estimates. Adaptive mesh refinement algorithms based on these error estimators are formulated. In both parts, numerical experiments are presented to illustrate the results of our analysis and indicate the optimal convergence rates. The thesis is accompanied by a thorough introduction to the development of the QC methods and its numerical analysis, as well as an outlook of the future work in the conclusion.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2018-07-26T12:29:23Z No. of bitstreams: 1 pedroandrearroyosilva.pdf: 4154699 bytes, checksum: 1875b7d54dd015591fcdd55db287ee37 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-09-03T16:20:02Z (GMT) No. of bitstreams: 1 pedroandrearroyosilva.pdf: 4154699 bytes, checksum: 1875b7d54dd015591fcdd55db287ee37 (MD5) / Made available in DSpace on 2018-09-03T16:20:02Z (GMT). No. of bitstreams: 1 pedroandrearroyosilva.pdf: 4154699 bytes, checksum: 1875b7d54dd015591fcdd55db287ee37 (MD5) Previous issue date: 2018-04-23 / A formação de padrões espaço-temporais são observados em processos químicos e bio-lógicos. Apesar dos sistemas bioquímicos serem altamente heterogêneos, aproximações homogenizadas contínuas formadas por equações diferenciais parciais são utilizadas fre-quentemente. Estas aproximações são usualmente justificadas pela diferença de escalas entre as heterogeneidades e o tamanho da característica espacial dos padrões. Em certas condições do meio, por exemplo, quando há um acoplamento fraco entre as células car-díacas, os modelos homogenizados discretos são mais adequados. Entretanto, os modelos discretos são menos manejáveis, por exemplo, na geração de malha para 2D e 3D, se comparado com os modelos contínuos. Aqui estudamos um modelo matemático homoge-nizado contínuo que se aproxima do modelo homogenizado. Este modelo é dado a partir de equações diferencias parciais com um parâmetro que depende da discretização da ma-lha. Dessa maneira nos referimos a este por um modelo matemático com parâmetros que dependem da discretização. Validamos nossa aproximação em um meio excitável genérico que simula três fenômenos em 1D: a propagação do potencial de ação transmembrânico no tecido cardíaco, a propagação do potencial de ação em filamentos de axônios cobertos por bainhas de mielina e a propagação do ativador e inibidor em microemulsões químicas. Para o caso 2D desenvolvemos uma versão da nossa aproximação que reproduz ondas espirais em um meio com acoplamento fraco. / The spatio-temporal patterns formations are observed in chemical and biological pro-cesses. Although biochemical systems are highly heterogeneous, homogenized continuum approaches formed by partial differential equations have been employed frequently. These approaches are usually justified by the difference scales between the characteristic spatial size of the patterns. Under some conditions of the medium, for instance, under weak coupling between cardiac cells, discrete models are more adequate. On the other hand discrete models may be less manageable, for instance, in terms of mesh generation, com-pared to the continuum models. Here we study a mathematical model to approach the discreteness which permits the computer implementation on non-uniform meshes. The model is cast as a partial differential equation but with a parameter that depends on the discretization mesh. Therefore we refer to it as a mathematical model with parameters dependent of discretization. We validate the approach in a generic excitable media that simulates three different phenomena in 1D: the propagation of action potential in car-diac tissue, the propation of the action potentialin filaments of axons wrapped by myelin sheaths, and the propagation of the activator/inhibitor in chemical microemulsions. For the 2D case we develop a version to this approach in microemulsions where it was possible to reproduce spiral waves with weak coupling of the medium.