3-D Thermo-Acousto-Electric Diagonalized and Supplementary Equations in Fully Trianisotropic and Inhomogeneous Media Part I: Proof of Existence by Construction

Abstract

Consider a fully trianisotropic and inhomogeneous solid medium capable of supporting thermo-acousto-electric wave propagation phenomena in three spatial dimensions (3-D). Let ${\mathcal {G}}$ denote the coupled system of the governing partial differential equations. Let ${\mathcal {C}}$ denote the set of linearized constitutive equations. Let $({\mathcal {G}}, {\mathcal {C}})$ represent the collection of ${\mathcal {G}}$ and ${\mathcal {C}}$ . This paper shows that $({\mathcal {G}}, {\mathcal {C}})$ can be diagonalized $({\mathcal {D}})$ and supplemented $({\mathcal {S}})$ resulting in the collection $({\mathcal {D}}, {\mathcal {S}})$ equivalent with $({\mathcal {G}}, {\mathcal {C}})$ . The transformation $({\mathcal {G}}, {\mathcal {C}})$ $\rightarrow$ $({\mathcal {D}}, {\mathcal {S}})$ is facilitated by utilizing three ‘‘scaffolding’’ matrices and three ‘‘scaffolding’’ vectors along with their dual counterparts. The existence of $({\mathcal {D}}, {\mathcal {S}})$ has wide-ranging implications in, e.g., regularization of singularities, construction of universal functions, near- and far-field asymptotic analyses, and the design of accelerated algorithms for device modeling and simulation. This paper substantially extends author's earlier formulations for rigorously accounting for thermal phenomena in microacoustic devices in 3-D. This paper proves the existence of $({\mathcal {D}},{\mathcal {S}})$ by way of construction. In the accompanying paper (Part II) details concerning consistency have been provided.