The computational design of embedded microchannels in actively-cooled 3D woven microvascular composite plates for high-temperature applications in hypersonic aircrafts is studied in this thesis project. Besides manufacturing constraints, the optimal network configuration in the microvascular composite depends on a variety of parameters including the dimensions of the plate, applied thermal loads, diameter and shape of the microchannels, and type of the coolant. Minimizing three objective functions are considered during the design process:​ the maximum temperature of the material, the void volume fraction of the microchannels, and the pressure drop needed to circulate the coolant in the network. One of the main challenges in evaluating the thermo-mechanical response of this system with conventional schemes such as the finite element method is the complex microstructure of the microvascular composite and thus the laborious process of creating conforming (matching) meshes. A novel Interface-enriched Generalized Finite Element Method (IGFEM) is proposed and developed to address this problem using finite element meshes that are independent of the problem morphology. To capture the temperature gradient discontinuity caused by the mismatch between the materials properties of the fluid and the solid phases, the IGFEM employs enrichment functions associated with new nodes created aSt the intersection of the microchannel surfaces with the edges of nonconforming elements of the mesh. The IGFEM formulation and its implementation for 2D and 3D conjugate heat transfer problems are presented and a detailed convergence study is provided to show that the method yields the same convergence rate and precision as those of the standard FEM with conforming meshes. A higher-order IGFEM formulation is also developed to more accurately simulate problems with complex curved interfaces without refining the finite element mesh. To validate the IGFEM solver, we compare the data collected from the thermal assessment of an actively-cooled microvascular epoxy fin with the temperature field obtained from the IGFEM solver for the same problem. The validated IGFEM solver is then employed to investigate the optimal configuration of the embedded microchannels in the actively-cooled composite. This study includes a detailed discussion of the impact of various design parameters on the thermal response of the microvascular composite.