P. Helnwein:
"Some Remarks on the Compressed Matrix Representation of Symmetric Second-Order and Fourth-Order Tensors";
Computer Methods in Applied Mechanics and Engineering, 190 (2001), 22-23; 2753 - 2770.
The mathematical description of physical phenomena requires the use of scalars and tensors of various order. Since many of the second-order and fourth-order tensors used in continuum mechanics possess certain symmetries, a compressed vector or matrix representation, respectively, is frequently used in computational applications such as the FEM and BEM.Use of different storage schemes for different tensors leads to an hypothetical non-uniqueness of such matrix representations. The present paper offers a clarification by investigation of the structure of the underlying 6-dimensional vector space. It identifies various types of matrix representations as covariant, contravariant or mixed-variant coordinates in that vector space and thus proves consistency of the matrix representation with classical tensor analysis in R3. Furthermore, it is shown that an ortho-normal basis for the underlying tensor representation in R3 does not automatically lead to a normalized space for the compressed matrix representation in R6. Thus, distinction of covariant and contravariant coordinates is necessary even in that case.
Theoretical findings are worked out in detail for symmetric second-order and fourth-order tensors in R3 Example applications on commonly used fourth-order tensors as well as a comparison of possible computational implementations close the paper.
Keywords: matrix representation, metric, tensor, vector space
http://dx.doi.org/10.1016/S0045-7825(00)00263-2