A Spectral Finite Element Approach to Modeling Soft Solids Excited with High-Frequency Harmonic Loads

An approach for efficient and accurate finite element analysis of harmonically excited soft solids using high-order spectral finite elements is presented and evaluated. The Helmholtz-type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness (i.e. high wave number), which is the case, for example, for soft tissues subject to ultrasound excitations. The use of high-order polynomial elements allows for a reduction in this pollution error, but requires additional consideration to counteract Runge's phenomenon and/or poor linear system conditioning, which has led to the use of spectral element approaches. This work examines in detail the computational benefits and practical applicability of high-order spectral elements for such problems. The spectral elements examined are tensor product elements (i.e. quad or brick elements) of high-order Lagrangian polynomials with non-uniformly distributed Gauss-Lobatto-Legendre nodal points. A shear plane wave example is presented to show the dependence of the accuracy and computational expense of high-order elements on wave number. Then, a convergence study for a viscoelastic acoustic-structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown. The number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order. However, the computationally optimal element order was found to strongly depend on the wave number.