Geometric Wave Engineering of Elastic Waves in Pseudohyperboloidal Solids: analytical Boundary Design, Load Encoding, Power-Normalized Multimode Scattering, and Piezoelectric Readout — A Computational Study with Stated Limits

Geometric Wave Engineering (GWE) treats an analytically prescribed outer boundary as a passive design variable: the surface is fixed by an equation, not by free-form optimization, so the response becomes differentiable in shape parameters and configurations become mutually comparable. GWE is not a theory of any particular medium: it is a medium-independent methodology — analytic geometry → boundary-value operator → organization of the physical state, formally Y = S_G[I] — in which the governing equation is never modified and only the boundary is prescribed, so that a chosen boundary alters reflection normals, propagation paths, diffraction, radiation leakage and overlap with the driven region, and nothing else. It introduces no additional interaction and no independent source of energy. To date the methodology has been applied to the control of waves in gas dynamics, acoustics, electromagnetic waves and optics. The present study applies it to elastic media — the first systematic extension to three-dimensional elastodynamics and coupled electroelasticity — under the same evidential hierarchy and the same explicit claim boundaries as the companion studies. The objects are monolithic aluminium solids of revolution whose meridional generating curves are hyperbolas, radially displaced and revolved (pseudohyperboloidal bodies, PHB). The construction is not a quadric: revolving the displaced meridian yields a pseudosurface of spatially variable negative Gaussian curvature and turns the inherited meridional focal set into external circular focal rings. The exact focal-transport contraction is retained as a geometric statement only, is not carried over to the vector elastic field, and implies no focal concentration of elastic energy. PHB are compared with equal-volume non-hyperbolic controls under identical material tensors, excitation, contact ports and receiver configuration. Because a solid carries coupled longitudinal–shear fields, all functionals are computed from three-dimensional linear elastodynamics and coupled electroelasticity, not from ray arguments; resonant operation requires explicit detuning control in units of the line half-width. The evidential structure is hierarchical — algebraic consistency, variational consistency of inertia, stiffness, coupling and bond work, and continuum convergence under joint geometric and field refinement — with physical validation of a manufactured specimen as the terminal and currently unfilled level. Results fall into four categories. Comparative: shape derivatives predicting the sign of channel change; calibrated load encoding as an inverse problem, reducing noise sensitivity and multi-channel reconstruction error within a narrow band and the calibrated source family; modal control of a selected channel; piezoelectric readout, advantageous only at nominal loss. Structural: modal localization by kinetic energy, sign persistence of ranking under mesh refinement, conversion fraction, decomposition of the power ratio into input-mobility and conversion factors, and component-resolved cycle-averaged Poynting flux with local sign reversals and outlet-region magnitude balance. Methodological: a comparison protocol and a minimal jointly reported quantity set. Verification: correctness of the numerical implementation. Geometry is thus established as a differentiable, comparable design variable and an operator-valued causal chain — not as technological superiority: the decisive comparison against an independently optimized non-hyperbolic competitor at equal optimization budget remains an unfulfilled falsification condition. Flux statements stay local and monitor-dependent, and no redirection of total time-averaged power is claimed.