Computational mechanics tools make up a crucial part of modern engineering analysis.The most widely used softwares typically employ finite element analysis (FEA) to approximate the solutions to partial differential equations (PDEs). FEA relies on the construction of high- quality boundary fitted meshes which can be costly to construct. More specialized numerical methods are widely used within the academic community to not only avoid mesh generation and remeshing, but also to tackle problems with unique requirements not well suited to FEA. This work expands upon the theory of approximate extraction and interpolation-based methods, enabling more advanced computational tools to be implemented within existing finite element softwares. The first method discussed in this work is the interpolation-based immersed bound- ary method. Immersed boundary methods avoid mesh generation by embedding a problem’s computational domain in a structured background grid. With non-conforming meshes comes additional difficulty in integration steps, limiting the execution of most immersed boundary methods to custom research codes. Interpolation-based immersed boundary methods augment existing FEA software to non-invasively implement immersed-boundary capabilities through extraction. Extraction, which has previously been applied to implement isogeometric analysis in finite element codes, interpolates the structured background basis as a linear combination of Lagrange polynomials which can be easily integrated with existing software. In addition to classic immersed boundary methods, immersed-isogeometric methods are also implemented, using B-splines as background basis functions. B-splines offer higher levels of continuity than classic finite element basis functions, enabling the method to model high-order derivative PDEs, such as shell problems. Heaviside enrichment of the background basis as well as local refine- ment through truncated hierarchically refined B-splines extends the capability of this method to multi-material problems, including image-based analysis of composite materials. Extending the theory of interpolation, this work also introduces interpolation-based meshfree methods, specifically interpolation-based reproducing kernel particle method (RKPM) . Developed for similar reasons as immersed boundary methods, meshfree methods do not require a mesh connectivity data structure and are optimal for modeling large deformation, fracture, or extreme events. Also like immersed boundary methods, meshfree methods require sophisticated integration techniques that make them challenging to implement outside a research setting. Interpolation-based RKPM is implemented within existing FEA software, and applied in this work to a variety of problems, including high order derivative PDEs and multi-material problems