On the use of three-dimensional h- and p-version finite elements in solving vibration response problems

The potential for reliably solving vibration response problems by using h- and p-versions of the finite element (FE) method is investigated. Each FE model is based on full three-dimensional (3-D) displacement-based continuum theory. Special attention is given to the ability to handle thin and/or nearly incompressible elastic and viscoelastic materials. Steady-state time-harmonic problems in the low- and mid-frequency ranges are treated. Convergence studies are performed on plate-like model problems with simply supported boundary conditions, using a series elasto-dynamic Navier solution extended to the viscoelasto-dynamic case by means of the elasto-viscoelastic correspondence principle. The importance of eliminating the discretisation error when using numerical solutions to estimate the frequency-dependent viscoelastic material parameters from experiments is stressed. perior efficiency of p-enrichments compared to h-refinements for resolving regular elastostatic solutions is observed in elasto/viscoelasto-dynamics as well. Requirements on p-enrichments to avoid shear- and volumetric locking are given. The frequently used polynomial interpolation functions employed are shown to produce a mass contribution to the dynamic stiffness, whose condition number deteriorates with increasing polynomial order. The practical implications and limitations of this observation are outlined. The recommended approach is corroborated using the measured frequency response on a laminate consisting of two aluminium plates and a constrained viscoelastic polymer damping treatment.