the thermal efficiency of the heater was improved. [PERSON] and [PERSON] [4] reviewed two methods of calculating the overall heat transfer coefficient (U) in steady state modelling of a steam cracking furnace radiant section using Aspen PLUS. They evaluated and compared some U values to find the optimal U for their proposed model. They found that the U value of 165 W/m\({}^{2}\cdot\)K gave the best estimate for the industrial data. [PERSON] [5] combined the experimental study and finite element simulation to design different tube structures for the requirements of different high temperatures, including Co-C 1324 \({}^{\circ}\)C, Pt-C 1738 \({}^{\circ}\)C, and Re-C 2474 \({}^{\circ}\)C. The main research topics included (1) the study of the precise measurement method for the axial temperature profiles of a furnace tube and (2) the methods of tube structure improvement using software simulation and experimental methods. The author compared different curves of axial temperature distribution to determine the optimal structure of the tube. Experiments were used to verify the simulation results. [PERSON], et al. [6], used stick multi-degree-of-freedom models to numerically analyze the response of tall, reinforced concrete (RC) chimneys under an earthquake. The effects of eccentricity for two RC chimneys (150 m and 210 m tall) were studied. They also studied the seismic stresses and influences of the vertical ground motion for the two chimneys. Numerical results revealed that by considering existing eccentricities, the tensile and compressive stresses of the two chimneys might increase under the actions of some specific earthquake waves. [PERSON] and [PERSON] [7] numerically analyzed the flow within and outside of a chimney that had a low length-to-diameter ratio and a lateral inlet for combustion gases. They discussed the dispersion of gases in the atmosphere and the problematic flow configuration detected. They also proposed a flow straightener to reduce the vorticity within the chimney. Numerical results revealed that the flow straightener proposed could reduce the streamlines' deviation from 35-40\({}^{\circ}\) to less than 5\({}^{\circ}\). From the above literature related to heaters, it can be seen that in addition to experimental measurements, the numerical simulation (e.g., finite element method (FEM)) has also been used successfully to analyze vibrations and stresses of the structure. Several relevant studies using numerical simulation to analyze vibrations and stresses of the structure are reviewed below. [PERSON], et al. [8], proposed a unified and generic numerical modeling approach for stiffened carbon fiber-reinforced polymer (CFRP) panels considering both the intra- and inter-laminar damage modes. A progressive damage model (PDM) based on the 3D finite element method was proposed to simulate the collapse behavior of a single blade-stiffened composite CFRP panel under uniaxial compression loading. The authors validated the developed PDM by comparing the stability response, collapse load results, damage evolution, failure mechanisms, ultimate load, and the displacement data with the experimental observations and estimates. [PERSON], et al. [9], applied a multibody calculation method to analyze the vibration of a turbocharged diesel engine. They considered different modeling assumptions and simulated the angular speed variation of a crankshaft. Moreover, they performed time dependent simulation of the engine supports and compared with experimental results. They also used a multibody model created with the AVL/EXCITE software to do the modal analysis, considering the damping aspects. Finally, they evaluated the influence of crankshaft torsional frequencies on the rotational speed behavior to reduce the vibration phenomena. In a later study [10], [PERSON], et al., adopted a reduced modelling strategy to reduce the full FEM model size of the engine. The strategy was based on the component mode synthesis and was found to reduce the required runtime significantly.