Distributed material density and anisotropy for optimized eigenfrequency of 2D continua

A practical approach to optimize a continuum/structural eigenfrequency is presented, including design of the distribution of material anisotropy. This is often termed free material optimization (FMO). An important aspect is the separation of the overall material distribution from the local design of constitutive matrices, i.e., the design of the local anisotropy. For a finite element (FE) model the amount of element material is determined by a traditional optimality criterion (OC) approach. In this respect the major value of the present formulation is the derivation of simple eigenfrequency gradients with respect to material density and from this values of the element OC. Each factor of this expression has a physical interpretation. Stated alternatively, the optimization problem of material distribution is converted into a problem of determining a design of uniform OC values. The constitutive matrices are described by non-dimensional matrices with unity norms of trace and Frobenius, and thus this part of the optimized design has no influence on the mass distribution. Gradients of eigenfrequency with respect to the components of these non-dimensional constitutive matrices are therefore simplified, and an additional optimization criterion shows that the optimized redesign of anisotropy are described directly by the element strains. The fact that all components of an optimal constitutive matrix are expressed by the components of a strain state, imply a reduced number of independent components of an optimal constitutive matrix. For 3D problems from 21 to 6 parameters, for 2D from 6 to 3 parameters, and for axisymmetric problems from 10 to 4 parameters.