Abstract
Solving transient heat transfer equations is required to understand the evolution of temperature and heat flux. This physics is highly dependent on the materials and environmental conditions. If these factors change with time and temperature, the process becomes nonlinear and numerical methods are required to predict the thermal response. Numerical tools are even more relevant when the number of parameters influencing the model is large, and it is necessary to isolate the most influential variables. In this regard, sensitivity analysis can be conducted to increase the process understanding and identify those variables. Here, we combine the complex-variable differentiation theory with the finite element formulation for transient heat transfer, allowing one to compute efficient and accurate first-order sensitivities. Although this approach takes advantage of complex algebra to calculate sensitivities, the method is implemented with real-variable solvers, facilitating the application within commercial software. We present this new methodology in a numerical example using the commercial software Abaqus. The calculation of sensitivities for the temperature and heat flux with respect to temperature-dependent material properties, boundary conditions, geometric parameters, and time are demonstrated. To highlight, the new sensitivity method showed step-size independence, mesh perturbation independence, and reduced computational time contrasting traditional sensitivity analysis methods such as finite differentiation.
Highlights
Introduction published maps and institutional affilThe physics of transient heat transfer allows one to evaluate the dynamics of energy movements for bodies subjected to changes in temperature, enabling the analysis of the heat flux, temperature history, and temperature rate of different processes
To obtain sensitivity information for transient heat transfer problems, we propose a methodology that combines the solution of the transient heat transfer equation using the finite element method with the Complex Taylor Series Expansion
Time to solve the Transient Thermal ZFEM model and obtain sensitivities compared to using built-in Abaqus functions and finite differentiation (FD)
Summary
The forfor transient heat transfer for for a solid body withwith unit unit length, Thegoverning governingequation equation transient heat transfer a solid body length, isotropic thermal conductivity k [W/mK], is presented in Equation (1), where ρ [kg/m. Combining Equations (1)–(6), we obtain Equation (7), which represents the transient heat transfer FEM formulation for any time step (ts) In this equation, Me [J/K] is the element capacitance matrix, Kce [W/K] is the symmetric element conductivity matrix, Khe conv [W/K] and Khe [W/K] are the rad asymmetric convective and radiative flux matrixes, respectively. Setting β = 1 in Equation (8) minimizes numerical integration problems, prevents spurious oscillations, and ensures stability for the solution [82,83] With this condition, the system of equations becomes the fully implicit system Ae Ttse +1 = RHSe , where the coefficient matrix for the time increment (Ae [W/K]) and the right-hand side of the equation (RHSe [W]) are defined in Equations (9) and (10), respectively. UV should be tuned to minimize the error of the temperature response of the FEM and a reference solution
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