Reduced-order Extrapolated Continuous Space-time Finite Element Method And Fast Airplane Crash Simulation Based On POD

The continuous space-time finite element(CSTFE)method has been rapidly developed in recent years and has been extensively used in the numerical solutions of various unsteady partial differential equations due to its many remarkable merits such as high order accuracy in space and time,computational effectiveness,geometric flexibility,and energy conservation.In the conventional CSTFE approach,however,the computational cost becomes higher as the order of the temporal interpolating polynomial increases or as we boost the resolution of the spatial discretization mesh.These problems are even more acute particularly in real-time simulations based on large-scale CSTFE systems for real-world engineering applications due to the demand of fast and accurate numerical computation,and thus significant computational issues usually occur in terms of both the computational time and memory requirements.Thus,it is important to develop the low-dimensional CSTFE model in order to decrease the computational cost and improve the computational efficiency in a manner that yields numerical solutions with high accuracy.The dynamic crash analysis of structures(DYCAST)experiments that started at NASA Langley Research Center during the late 1970s have greatly influenced the methodology and thinking of aircraft crashworthiness and survivability studies.Nevertheless,so far most of the existing work has emphasized the impact damage to the aircraft section.Issues related to potential passenger injuries have been not properly addressed in the literature,to the best of our knowledge.In addition,it could take a long time to perform aircraft crash simulations even with the use of advanced supercomputers and powerful commercial software.Therefore,it is of geat significance to study how to simulate fast the aircraft crash process so as to analyze aircraft crashworthiness and passenger survivability and optimize aircraft design.In this thesis,for the purpose of resolving the above scientific issues,we establish a reduced-order extrapolated continuous space-time finite element(ROECSTFE)method with high efficiency and accuracy based on proper orthogonal decomposition(POD).Furthermore,we verify the effectiveness and practicability for the proposed ROECSTFE method by theoretical analysis and numerical simulations.In addition,we use modal analysis and the LS-DYNA platform to study integrally impact mechanics and biomechanics of aircraft structures and passengers in dynamic crash conditions,and we use the POD non-intrusive reduced-order modeling to improve computational efficiency of the crash simulation.The research in this thesis can serve engineering problems in terms of simulation,prediction,and design.The main contents are as follows.(1)The order reduction of the classical CSTFE method for the two-dimensional Sobolev equation is studied by means of POD.The classical CSTFE model is constructed for the Sobolev equation,and the theoretical results of the existence,stability,and convergence are provided for the CSTFE solutions.Two reduced-order strategies based on POD are respectively considered.One is on the basis of reconstruction of the ROECSTFE subspace,while the other is based on the CSTFE solution coefficient vectors.The former reconstructs the POD basis and reduced-order subspace by extracting the classical CSTFE solutions at the initial few time nodes,and employs the standard CSTFE method to analyze the existence,stability,and convergence of the ROECSTFE solutions.Whereas the latter establishes the POD basis and reduced-order model by selecting the classical CSTFE solution coefficient vectors at the initial few time nodes.The existence,stability,and convergence of the ROECSTFE solutions are proved by the matrix means,resulting in significant conciseness and flexibility for theoretical analysis.The effectiveness of the developed ROECSTFE model is examined by several numerical tests.(2)The order reduction for the solution coefficient vectors of the classical CSTFE method for the two-dimensional unsteady Stokes equation is researched by POD.First,the Stokes equation is transformed into a form about vorticity-stream functions,and the classical CSTFE model is constructed,and the existence,stability,and convergence of the CCSTFE solutions are discussed.And then,the ROECSTFE model is developed by means of POD,and the existence,stability,and convergence of the ROECSTFE solutions are discussed by matrix analysis.Finally,two numerical examples of the cavity and cylinder flows are provided to confirm the correctness of the theoretical results obtained.(3)The ROECSTFE method for the non-stationary incompressible Navier-Stokes equation is developed by POD.First,the CCSTFE scheme of the Navier-Stokes equations about vorticity-stream functions is established and the existence and stability as well as convergence to the CCSTFE solutions are analyzed.And then,the ROECSTFE scheme including very few unknowns but possessing high accuracy is established by means of POD,the existence and stability as well as convergence of the ROECSTFE solutions are demonstrated by means of matrix method,and the algorithm design for solving the ROECSTFE scheme is provided.Finally,the numerical simulations for the back-step flow and the flow around airfoil problem are provided to show that the computational results agree well with the theoretical results,thereby demonstrating the effectiveness and feasibility of the proposef ROECSTFE method.(4)Impact mechanics and biomechanics of aircraft structures and passengers in dynamic crash conditions and fast aircraft crash simulation based on POD reduced-order modeling are studied.First,the geometric modeling and benchmark simulation for the single-section of an airplane are carried out in order to check the effectivess of geometric and numerical models.And then,the biomechanics by way of modal analysis of passenger dummy motions coupled with the vibration of aircraft structures are developed in order to understand their basic interactions.Next,computer simulations for general aircraft crash tests in different scenarios are carried out by using the versatile software LS-DYNA as the platform on the supercomputer clusters,and the numerical results are rendered into video animations for visualization and analysis of the mechanism in terms of impact damage to aircraft structures and passenger injuries.Finally,POD non-instrusive reduced-order modeling is used to fast simulate the aircraft crash process.In comparison with the LS-DYNA simulation results,the reliability of the POD reduced-order modeing is verified.