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    Area of Science:

    • Multidisciplinary applications of tensor data analysis.
    • Physics, engineering, and earth sciences utilize tensorial data for fundamental equations.

    Background:

    • Existing tensor visualization techniques are limited, especially for higher-order tensors.
    • Current tensor decomposition methods hinder the visualization of complex, arbitrary-order tensors.

    Purpose of the Study:

    • To propose a novel line-based visualization technique for three-dimensional tensors of arbitrary order.
    • To leverage deviatoric decomposition and multipoles for enhanced tensor data representation.

    Main Methods:

    • Deviatoric decomposition to split tensors into totally symmetric, traceless deviators.
    • Definition of multipoles and multipole lines for tensor description.
    • Computation of multipole lines analogous to tensor lines.

    Main Results:

    • Successful line-based visualization of symmetric second-order and fourth-order tensor fields.
    • Established connections between multipoles and eigenvectors/eigenvalues for second-order tensors.
    • Derived relations between multipoles and physical quantities (e.g., shear moduli) for fourth-order stiffness tensors.

    Conclusions:

    • Deviatoric decomposition offers a powerful approach for visualizing high-order tensors.
    • Multipole lines provide an interpretable and effective method for 3D tensor visualization.
    • The technique has broad applicability in scientific fields dealing with complex tensor data.