The numerical simulation of nonlinear fluid-structure interaction problems is significant in many engineering and scientific applications. Examples include aircraft flutter in transonic flows, underwater implosions, pipeline explosions, flapping wings for micro aerial vehicles, and biomedical flows in heart and blood vessels, reactor fuel performance and so on. These problems are generally highly nonlinear, feature multiple scales and strong coupling effects, and might require heterogeneous discretizations for the various physics subsystems. Developing high-fidelity multidisciplinary models for such problem, with an emphasis on real world applications, is the motivation of the thesis. More specifically, the goal is to understand the supersonic parachute inflation dynamics for Mars landing. In 1972, NASA recorded the 1st known instance of a successful parachute operation in the wake of a blunt object at supersonic speeds, during the preparation for the Viking missions to Mars. However, the problem still remains open. "The stress analysis of the canopy cloth is extremely complicated, since maximum stresses occur during opening, which is when the shape and load change rapidly." (Houmard, the Goodyear Aerospace Corporation, 1972) "Technology in parachute structural analysis is far from desirable and no authoritative individual and/or company can really be said to exist." (Alley, NASA, 1972) "No information exists fully explaining parachute behavior in supersonic flow." (Lingard, UK-based Vorticity Ltd, 2010) This thesis first describes an embedded boundary framework for highly nonlinear fluid-structure interaction problems. This is based on the Finite Volume method with Exact two-material Riemann Problems (FIVER), developed in Farhat Research Group [1], and capable of handling evolving material interfaces, including structural fracture. The framework further incorporates a parallel adaptive mesh refinement based on newest vertex bisection, therefore, becomes more efficient for problems with large structure deformations and evolving shock waves. And several modifications are proposed to relieve the ill-conditioning near the fluid-structure interface due to extrapolations, hence, the framework becomes mesh position and orientation independent and delivers smooth pressure coefficient and skin-friction coefficient. Furthermore, this thesis makes a variety of contributions to make the framework more suitable for the parachute inflation simulations. These contributions include a homogeneous porous wall model to capture the porous effect of the canopy on the fluid, and a suspension line treatment that allows for the interactions of the sub-grid scale suspension lines with the flow. Finally, several full-size parachute inflation simulations in the low-density, low-pressure Mars atmosphere are conducted. The computed drag performance is in good agreement with the data collected by the NASA Curiosity Rover during its Mars atmospheric entry. Besides, basic material failure analysis is conducted, which indicates the parachute decelerator system of Curiosity survives with a safety factor about 5.0. This framework demonstrates the potential of using CFD and FSI based simulation tools for the future supersonic parachute design. In addition to the parachute project, several other related projects are reported in this thesis. The first one is a new approach to learning constitutive relations from indirect observation data and its uncertainty quantification for material coupons, such as parachute fabric. In contrast to direct input-output data curve-fitting, this approach focuses on problems, where obtaining comprehensive model input-output data is difficult. But indirect data, like deflections of the structure coupon under different load conditions, are available. Traditional approaches, such as the finite element method, bridge the indirect data with neural network-based~(or its counterparts) constitutive relation models. Its mathematical properties, like accuracy and applicability properties of the approach and the strength of neural networks, are thoroughly studied. Another project aims to extend the multiphysics framework to higher order. Arbitrarily high-order, stable, partitioned solvers are constructed, where different subsystems are modeled and discretized separately, and the resulting equations are solved independently. These solvers are used with the discontinuous Galerkin method for fluid-structure interaction problems with moderate structure deformations