We develop a novel patterning technique to create 3D patterns of micro and nanoparticle assembly via evaporative self-assembly based on the coffee-ring effect of an evaporating suspension. The principle of the technique is analyzed theoretically by the scaling analysis of main parameters of the process and the scaling effect, the effect of the volume, the concentration of the suspension, and the effect of surface treatment on the patterning are studied. On the basis of the presented technique, we demonstrate that the patterns of 3D assembly of various sizes of microparticles (Silica), metal oxide nanoparticles (TiO(2), ZnO), and metallic nanoparticles (Ag) can be successfully generated by low-concentrated particle suspension (1.25-5 wt %) without additional sintering steps, and we also show the geometries of the patterns can be finely controlled by adjusting the parameters of the process.

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In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress-strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.

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Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers' equations. A speedup of up to 2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Finally, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.