Lumped mass acoustic and membrane eigenanalysis using the global collocation method

Christopher Provatidis,Duc Pham

doi:10.1080/23311916.2014.981366

Abstract

AbstractThe paper proposes a direct way to build lumped masses for performing eigenvalue analysis using the global collocation method in conjunction with tensor product Lagrange polynomials. Although the computational mesh is structured, it has a non-uniform density, in such a way that the internal nodes are located at the position of Gaussian points or the images of the roots of Chebyshev polynomials of second kind. As a result, the mass matrix degenerates to the identity matrix. In this particular nodal collocation procedure, no complex eigenvalue appears. The theory is successfully applied to rectangular and circular acoustic cavities and membranes.

Highlights

Transfinite elements were inspired in early 1970s (Gordon & Hall, 1973) for the purposes of CAD/CAE integration, but only much later were applied for the numerical solution of static and dynamic engineering problems using natural cubic B-splines
Christopher Provatidis is a full professor of Mechanical Engineering at National Technical University of Athens (NTUA), Greece
He has performed computer modeling in biomechanics, acoustics, and elastodynamics using novel boundary element methods, static-eigenvalue-transient analysis using novel CAD-based isoparametric macroelements, fracture mechanics, textile micromechanics, active noise control, inverse problems for NDT using a variety of optimization algorithms and neural networks, shape and motion reconstruction, structural size–shape– topology optimization, and wave propagation in thin silica films and laser cleaning of paintings

Summary

Introduction

Transfinite elements were inspired in early 1970s (Gordon & Hall, 1973) for the purposes of CAD/CAE integration, but only much later were applied for the numerical solution of static and dynamic engineering problems using natural cubic B-splines The second major problem is that when applying the so-called “nodal collocation” technique, according to which the collocation points coincide with the internal nodes (no actual computation of the mass matrix is required), some of the obtained numerical eigenvalues are found to be complex This numerical shortcoming has been resolved by taking the collocation points at the position of Gaussian points (roots of Legendre polynomials) or at the images of the roots of Chebyshev polynomials ( of second kind) (Provatidis, 2008b); these choices constitute what is usually called “orthogonal collocation.”. The theory is sustained by four examples in simple rectangular and circular shapes, which concern either acoustic cavities or membranes

The usual global collocation method

Matrix formulation

Basis change

Discussion and conclusions