An adaptive order finite element method for poroelastic materials described through the Biot equations

Poroelastic materials are essential to engineers to improve the sound and vibration properties of a product. The most common mathematical model that describes the vibro-acoustics of these materials is the Biot theory, which considers the fully coupled behavior of a homogenized solid phase, based on the structural skeleton, and a homogenized fluid phase, describing the inter-penetrating fluid. This article focuses on an adaptive higher order finite element implementation to solve poroelastic models. Owing to the presence of several wave solutions, namely, two compressional wave types and one shear wave type, it is difficult to define clear meshing guidelines a priori for poroelastic material modeling and trial and error is required to ensure reliable results. Multiple scales may be present, and local resonance phenomena may significantly modify the resolution requirements in certain frequency bands. In order to remedy this, and to be able to deliver a (nearly) mesh-independent solution, a physics based adaptivity mechanism is developed which determines the distribution of (directional) element orders at each frequency prior to the calculation, to reach a user-defined target error. The resulting approach is verified for different element topologies and is further applied to relevant applications of poroelastic materials in vibro-acoustics.