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.Army Research Office
Air Force Office of Scientific Researc
