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The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (ID) parabolic partial differential equation, TMOL engen ders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded pa rameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have con stant or variable coefficients. In this work, TMOL is applied to the unsteady ID heat equation in simple bodies (large plate, long cylinder, and sphere) with temperatureinvariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless o f time. As a consequence, usage of the more involved second adjoint quasistationary heat equation accounting for two consecutive time jumps come to be unnecessary.

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SPECIAL SESSION ON THE DIGITAL TWINSensitivity anal 1 ysis of structural systems is of great importance for structural dynamic modifications. The sensitivity analysis helps identify key input parameters that influence the dynamic response of a model. The present paper demonstrates how to accurately and efficiently obtain derivatives of linear dynamic systems using the complex step method and the generalized multicomplex step method implemented within a complex variable finite element method (ZFEM). The highly accurate derivatives are computed using a single complex variable finite element analysis of the structural system. An undamped and damped simply supported beam was modeled using ZFEM and the response derivatives were computed with respect to load amplitude, load frequency, beam cross sectional dimension, and Rayleigh damping coefficients. Comparisons of the derivatives against analytical and finite difference estimates verify the accuracy of the methodology. Nomenclatureth undamped natural frequency ! " n = n th mode shape ! K e = local stiffness matrix ! M e = local consistent mass matrix ! " = mass-normalized mode shape matrix ! " = diagonal matrix of natural frequencies squared ! " = vector of nodal degrees of freedom ! I = identity matrix ! "t = time step ! " = forcing frequency ! Q = uniform load ! " = Rayleigh damping mass proportional coefficient ! " = Rayleigh damping stiffness proportional coefficient ! * = denotes a complex variable