As applications of materials continue to increase in complexity, there is a clear need to quantitatively assess and optimize material performance in the presence of uncertainties. Insufficient knowledge of the physical phenomena at different length scales, the lack of understanding of the way information propagates from one length scale to another and the presence of inherent uncertainties leads to material response that cannot be accurately predicted using deterministic models. In this work, a novel computational framework for uncertainty modeling and design of complex systems is developed.

In the first part of the thesis, computational tools for stochastic modeling of material systems is discussed. Probability distribution functions (PDFs) providing a complete representation of microstructural variability is discussed. We use the maximum entropy (MaxEnt) principle to compute a PDF of microstructures based on given information. Microstructural features are incorporated into the maximum entropy framework using data obtained from experiments or simulations. Microstructures are sampled from the computed MaxEnt PDF using concepts from Gibbs sampling, computational geometry and voronoi-cell tessellations. The MaxEnt technique is applied on a wide range of materials including multi-phase and polycrystalline structures. These microstructures are then interrogated in virtual deformation tests to compute the variability of non-linear stress-strain curve, elastic moduli as well as fracture-initiation stress.

In the second half, we explore the design of material systems in the presence of uncertainties - both in input variables as well as design variables. The robust design problem is posed as a stochastic optimization problem. The concept of stochastic sensitivities is introduced and a stochastic gradient descent approach is proposed to compute the optimal solutions. The sparse grid stochastic collocation technique is utilized to accelerate computing the optimal stochastic solution. These techniques are used in conjunction with Finite Element techniques for the simulation of physical phenomenon in material systems. The technique is validated on stochastic inverse problems in thermal-diffusive systems and problems involving flow in porous media. Finally, examples on robust design for large-deformation processes is discussed and scope for future work are discussed.