Galerkin-projected evolution equations for reduced 1D thermal stress resultant model derived from transient 2D thermoelasticity

Abstract

A framework for the dimensional reduction of transient thermo-structural problems is presented in which the cross-sectional heat diffusion is encoded as an autonomous evolution system for the thermal stress resultants. A through-the-thickness temperature ansatz satisfying the essential boundary conditions is postulated and the heat equation is projected via a weighted-residual Galerkin method, yielding coupled first-order evolution equations for the thermally induced membrane force and bending moment. The resulting 1D model requires no resolved cross-sectional temperature computation at runtime, yet preserves the diffusive time scales of the parent 2D problem; the structural response follows from Euler–Bernoulli beam theory driven by the evolved thermal resultants. The projection procedure is methodologically general; its accuracy is demonstrated here for a homogeneous rectangular beam under mixed Dirichlet–Neumann thermal conditions, for which an exact 2D Fourier series solution is derived independently. A three-way numerical validation against the exact solution and a high-fidelity 2D finite element reference confirms that the reduced 1D model reproduces deformationfields to within a few percent, although the reconstructed cross-sectional stress underestimates the peak thermal eigenstress by approximately 43% due to the quadratic temperature approximation. For the present constant-coefficient problem, the reduced evolution system admits a closed-form eigenfunction expansion, rendering the model entirely mesh-free and yielding wall-clock speedups exceeding three orders of magnitude; for variable-coefficient or nonlinear problems, the dimensional reduction still provides substantial computational savings.