Dispersion analysis of elastic waves in isotropic media ... - WCCM 2016
Dispersion analysis of elastic waves in isotropic media discretized by the energy-orthogonal twenty-node hexahedral finite element Francisco José Brito Castro Departamento de Ingeniería Industrial, Universidad de La Laguna Calle Mendez Nuñez 67-2C Santa Cruz de Tenerife 38001, Spain This contribution studies the propagation of plane harmonic waves in homogeneous and isotropic elastic media discretized by the twenty-node hexahedral finite element formulated in energy-orthogonal form. In this formulation the element stiffness matrix is split into basic and higher order components which are respectively related to the mean and deviatoric components of the strain field [1]. This decomposition is applied to the element elastic energy and holds for the finite element assemblage. The dispersion properties and the period-averaged elastic energy density are computed for plane harmonic waves in unbounded media discretized by a regular mesh of finite elements which can be distorted preserving the element volume. Given the mesh, in the limit of long wavelength, although the elastic energy density does not vanish, its higher order component does vanish. Similarly, given the wavelength, as the solution converges on account of mesh refinement, the elastic energy density is increasingly dominated by its basic component. The above heuristic argument motivates to research the relationship between the percentage of higher order elastic energy eh = Eh/E and the elastic energy error ε = E/E0 - 1 in order to explore the behaviour of this energy component as an error indicator, where: E0 (E), exact (approximate) period-averaged elastic energy density; Eh, higher order component of E. Given the mesh, both for longitudinal waves P and transverse waves S, the mapping ε versus eh is depending on the direction of wave polarization and the Poisson’s ratio of the elastic medium ν. To be precise, by the dispersion analysis, both for P and S waves, the root-mean-square elastic energy error is computed versus the Poisson’s ratio for two reference values of the percentage of higher order elastic energy. Reference averaged values of the root-mean-square elastic energy error are then computed by considering different meshes, Table 1. The above reference values roughly correspond, in an averaged sense, to six and four cubic elements per wavelength, respectively. Table 1. Reference averaged values of elastic energy error versus Poisson’s ratio.
eh1 0.10 eh 2 0.20
P S P S
= 0.45 0.006190 0.003018 0.033621 0.012421
= 0.33 0.004928 0.002781 0.024421 0.011963
= 0.25 0.004557 0.002785 0.022161 0.012115
= 0.05 0.004075 0.002844 0.019373 0.012544
Finally, both for P and S waves, by using the reference averaged values listed in Table 1, the coefficients for a standard cubic correlation between the elastic energy error and the percentage of higher order elastic energy are computed versus the Poisson’s ratio, 2
P ( S ) [ AP ( S ) ( ) * eh B P ( S ) ( )] * eh ,
0 eh 0.20
The use of both the P and S standard correlations as a reference to apply the higher order elastic energy as an error indicator for the elastic vibration modes computed by the finite element method is explored. The numerical research reveals that the modal elastic energy error computed by both the P-correlation and the more demanding S-correlation generally overestimate the modal elastic energy error computed by mesh halving. As conclusion, by the proposed standard correlations, the accuracy of a finite element model can be confidently verified in order to properly select a cutoff modal order. [1] C.A. Felippa, B. Haugen and C. Militello, From the Individual Element Test to Finite Element Templates: Evolution of the Patch Test, Int. J. Meth. Eng., vol. 38, pp. 199-229, 1995.
eh1 0.10 eh 2 0.20
P S P S
= 0.45 0.006190 0.003018 0.033621 0.012421
= 0.33 0.004928 0.002781 0.024421 0.011963
= 0.25 0.004557 0.002785 0.022161 0.012115
= 0.05 0.004075 0.002844 0.019373 0.012544
Finally, both for P and S waves, by using the reference averaged values listed in Table 1, the coefficients for a standard cubic correlation between the elastic energy error and the percentage of higher order elastic energy are computed versus the Poisson’s ratio, 2
P ( S ) [ AP ( S ) ( ) * eh B P ( S ) ( )] * eh ,
0 eh 0.20
The use of both the P and S standard correlations as a reference to apply the higher order elastic energy as an error indicator for the elastic vibration modes computed by the finite element method is explored. The numerical research reveals that the modal elastic energy error computed by both the P-correlation and the more demanding S-correlation generally overestimate the modal elastic energy error computed by mesh halving. As conclusion, by the proposed standard correlations, the accuracy of a finite element model can be confidently verified in order to properly select a cutoff modal order. [1] C.A. Felippa, B. Haugen and C. Militello, From the Individual Element Test to Finite Element Templates: Evolution of the Patch Test, Int. J. Meth. Eng., vol. 38, pp. 199-229, 1995.
