Various engineering applications rely on efficient, high performance materials to overcome design challenges. This high performance can be achieved by engineering micro-heterogenous materials also known as composites. Since the behavior of composites relies heavily on micro-scale interactions between different components, modeling macrostructures with fully-represented microscopic geometry is needed. Thus, the standard finite element modeling approach becomes impractical. Computational homogenization, also known as concurrent finite element analysis (FE$^2$), is a method that is employed to model materials with distinct multi-scaled structure. FE$^2$ employs the concept of embedding a representative volume element (RVE), at each integration point of the macro-scale problem and obtaining the macroscopic constitutive behavior through homogenization, thus bypassing the need to develop a macro-scale constitutive model. Although it succeeds in upscaling the microscopic material behavior accurately, this method comes with the major drawback of being computationally expensive due to its nested structure. Developing methods to bypass the aforementioned computational bottleneck of FE$^2$ is an ongoing research endeavor. Employing machine learning algorithms to create surrogate constitutive models for microscopic behavior is one possible approach. However, creating surrogate models is not an easy task. A thoroughly collected training set is needed for the surrogate model to be representative. Thus, investigation of surrogate model creation strategies with machine learning while trying to reduce the computational burden of this extit{offline} training process is a compelling area of study. Gaussian Process Regression (GPR) is a probabilistic machine learning model. It can be utilized to create surrogate constitutive models effectively in the aforementioned context. Moreover, the computational burden of the training procedure can be decreased by extending the conventional GPR technique into a co-kriging regression with multi-fidelity information (multiGPR). This surrogate modeling strategy reduces the need to collect high-fidelity information by collecting information from a low-fidelity model thoroughly. Thus, enabling accurate training datasets to be created from a less representative, but computationally less taxing models. In this work, the multiGPR approach is used to construct accurate and efficient surrogates for the behavior of fiber-reinforced composite materials. Training data is obtained from selected RVE configurations that consist of linear-elastic fibers randomly embedded in a matrix that has pressure-dependent plasticity and used to train single and multi-fidelity GPR constitutive models. Both approaches are trained with various combinations of loading scenarios and their prediction capabilities are investigated to represent the training cases in addition to their prediction capabilities under unseen load cases.

This research considers the offline training stage of the Reduced Order Models (ROM), that has been getting attention recently on the endeavor to come up with efficient solutions for the highly complex numerical models. In this work, a simply supported beam problem has been considered, for which a reduced basis creation has been investigated. Reduced basis creation is in utmost importance for the accuracy and reliability of the ROM. Main focus is on the efficient parameter sampling strategies to enrich the reduced basis, which brings forth computational burden. To decrease this burden, a statistical tool Gaussian Processes Regression (GPR) based Bayesian Optimization (BO) is utilized. These tools are used to create a surrogate function of error indicator that is used to select additional training points for ROM. Results of this work show that randomness in the proposed procedure influences parameter sampling but does not have an impact on the overall accuracy. Finally, this work suggests further work on creation of a stopping criteria and finding a method of storing previous information and combining it with current information regarding training points without losing information. With the help of proposed further research topics, this work intends to be used as a foundation for efficient reduced basis construction.