This thesis is about the development of highly efficient semi-analytical solvers to simulate structured media in electromagnetic, acoustic and elastic settings. Mathematical approaches are developed using singular perturbation methods and matched asymptotic expansions, to approximate small scatterers in wavefields and simulate a variety of photonic and phononic crystalline media, over Bragg and deep-subwavelength scattering regimes.

The matching procedure requires ``inner'' and ``outer'' solutions, each of which has a distinct region of validity and these solutions are matched where such regions overlap. The outer solutions are expressed in terms of the free-space Green's function and their derivatives. The outer-region solutions are singular as we approach the inner region, that is as we tend towards the centre of an approximated scatterer. Interestingly, the outer solutions are conditionally convergent which leads to computational difficulties. However, upon matching the inner and outer solutions, the singular asymptotics responsible for diverging computations are shown to cancel. Subsequently, regular expressions remain which are simple to code and form the foundation of the semi-analytical solvers derived to compute: dispersion from eigensolutions associated with Floquet-Bloch waves, or scattering from finite arrays.

This methodology is applied to consider the in plane problem of arrays of Neumann inclusions placed within a Helmholtz wavefield. Once solved, this analysis is readily modified and built upon to solve more complex problems. Crucially, the characteristic lengthscale of the scatterer is considered asymptotically small relative to the lattice period, and defines the small parameter of interest for our singular perturbation problems. The Helmholtz-Neumann solution is later modified to design hybrid topologically protected photonic crystal fibres, where s-polarized obliquely propagating electromagnetic waves are guided by periodic arrays of infinitely long, perfectly conducting cylinders.

The Helmholtz-Neumann analysis is further extended to consider arrays of beams atop a thin elastic plate. The former matching procedure is replaced by approximating the beams with a simplified model, that of Euler-Bernoulli beam theory in which the arising forces and moments are well known. Approaching the centre of an approximated beam, the singular asymptotics of the elastic field of the plate is fully understood from the Helmholtz-Neumann problem. Throughout this thesis, the derived semi-analytical solutions are cross-verified against finite element computations.

In physics and engineering, there is considerable interest in the metamaterial design paradigm whereby remarkable properties are achieved by the ``effective'' response produced by the materials substructure. This thesis explores one of the dominant themes within the design of metamaterials, which is to achieve robustness by creating effects impervious to imperfections introduced during the fabrication of carefully designed substructures - such robustness is achieved by topological wavephysics. The efficient semi-analytical solvers derived within this thesis are used to investigate an important subset of topological wavephysics, that of valleytronics.

Valleytronics refers to engineering band-structures, or valleys, corresponding to extrema with locally quadratic curvature surrounding a topologically non-trivial bulk bandgap. These valleys arise from gapping symmetry induced degeneracies, creating pseudo-spin valley states characterised by opposite chirality; which localise energy by the Quantum Valley Hall Effect (QVHE). We demonstrate how symmetry induced degeneracies couple with tunable resonances, and once gapped, how the QVHE can be tuned deep subwavelength.

Within this thesis, an elegant means is established to investigate novel wavephysics within structured media, relating to both designing effects and testing the performance of crystalline media. The practicality of the highly efficient and rapid semi-analytical solvers is demonstrated for wave control over Bragg and deep subwavelength scales. Several topical examples from topological wave physics are explored, including: topological waveguides, chiral beaming, designing hybrid topologically protected photonic crystal fibres, and phononic meta-circuits from the interplay of topological interfacial and edge states.