In this thesis a microdynamics theory of structured solids is formulated on the basis of probabilistic functional analysis. The theory which is developed on the principles of probabilistic micromechanics, introduces from the onset spatial and temporal scales relevant in the dynamic analysis. A general formulation of the microdynamics of a three-dimensional solid is given in terms of an abstract dynamical system; the analysis is then specialized to the kinematic subspace of the general state space, whereby the former is found to possess the topological structure of a Hilbertian-Sobolev space. / The abstract dynamical system in the microdynamics theory is developed explicitly for the wave propagation in a semi-infinite bar of a polycrystalline solid with an arbitrary cross-section. The microstructure of the solid is taken to consist of cubic grains with random physical properties. The existence of an internal and a macroscopic time is postulated, which permits the formulation of the evolution of the wave motion first in a one-dimensional solid by means of a four parametric Markovian operator having a semi-group property. This model of the cubic solid structure is shown to be asymptotically equivalent to a "generalized wave equation" of the continuum theory. A more general model of the wavefront evolution for a three-dimensional solid is then given in terms of a super-martingale (parametrized by the macrotime) on a generalized random field. / It is shown that numerical results for the wave propagation in a discrete solid in accordance with the new microdynamics theory can be obtained by the application of the Monte-Carlo simulation method. A comparison of these results with known classical and random continuum theories is given.

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Thesis (Ph.D.)--University of Iowa, 2009. / Includes separate files for thesis supplements. Thesis supervisor: David E. Stewart. Includes bibliographical references (leaves 107-109).

[pt] Esta dissertação introduz um simulador híbrido inovador que combina um resolvedor de Equações Diferenciais Parciais (EDP) numérico de Elementos Finitos (FE) com uma Rede Neural de Passagem de Mensagens (MPNN) para realizar simulações de dinâmicas de sólidos deformáveis em malhas pouco refinadas. Nosso trabalho visa fornecer simulações precisas com um erro comparável ao obtido com malhas mais refinadas em discretizações FE,mantendo a eficiência computacional ao usar um componente MPNN que corrige os erros numéricos associados ao uso de uma malha menos refinada. Avaliamos nosso modelo focando na precisão, capacidade de generalização e velocidade computacional em comparação com um solucionador numérico de referência que usa malhas 64 vezes mais refinadas. Introduzimos um novo conjunto de dados para essa comparação, abrangendo três casos de referência numéricos: (i) deformação livre após um impulso inicial, (ii) alongamento e (iii)torção de sólidos deformáveis. Baseado nos resultados de simulação, o estudo discute as forças e fraquezas do nosso método. O estudo mostra que nosso método corrige em média 95,4 por cento do erro numérico associado à discretização, sendo até 88 vezes mais rápido que o solucionador de referência. Além disso, nosso modelo é totalmente diferenciável em relaçao a funções de custo e pode ser incorporado em uma camada de rede neural, permitindo que seja facilmente estendido por trabalhos futuros. Dados e código estão disponíveis em https://github.com/Kerber31/fast_coarse_FEM para investigações futuras. / [en] This thesis introduces a novel hybrid simulator that combines a numerical Finite Element (FE) Partial Differential Equation solver with a Message Passing Neural Network (MPNN) to perform simulations of deformable solid dynamics on coarse meshes. Our work aims to provide accurate simulations with an error comparable to that obtained with more refined meshes in FE discretizations while maintaining computational efficiency by using an MPNN component that corrects the numerical errors associated with using a coarse mesh. We evaluate our model focusing on accuracy, generalization capacity, and computational speed compared to a reference numerical solver that uses 64 times more refined meshes. We introduce a new dataset for this comparison, encompassing three numerical benchmark cases: (i) free deformation after an initial impulse, (ii) stretching, and (iii) torsion of deformable solids. Based on simulation results, the study thoroughly discusses our method s strengths and weaknesses. The study shows that our method corrects an average of 95.4 percent of the numerical error associated with discretization while being up to 88 times faster than the reference solver. On top of that, our model is fully differentiable in relation to loss functions and can be embedded into a neural network layer, allowing it to be easily extended by future work. Data and code are made available on https://github.com/Kerber31/fast_coarse_FEM for further investigations.