| Single or multiple medium transient heat conduction in complex structures needs to be predicted and calculated rapidly, accurately, and real-timely in the field of aerodynamics and thermodaynamics, as well as thermal engineering which is related to industrial applications, such as aerospace, nuclear power, metallurgy and casting. At present, some common numerical methods, such as finite element method (FEM), finite volume method (FVM), finite difference method(FDM), boundary element method(BEM), can not meet the requirements of real-time control and rapid prediction when they are used to solve the complex partial differential equations of a given initial boundary value, because of the problem of long computing time and large storage space caused by large degrees of freedom in the discrete format. At this point, when such complex and multiple medium transient heat conduction problems are analyzed by numerical methods such as FEM or BEM, with a very small workload to obtain an low order calculation model which can approximatively predict the characteristics of the original physical model is very valuable. This paper is devoted to the research in this field, namely, based on FEM/BEM, using the idea of proper orthogonal decomposition (POD) to obtain the reduced order models of unsteady heat conduction problems. The main work includes:(1) A reduced order analysis method for transient heat conduction problems with time-varying boundary conditions is proposed by using POD modes obtained from the results of using constant boundary conditions based on FEM. This method can perform interpolation and extrapolation analysis for temperature field at any times. First, POD modes are formed by calculating eigenvectors of an autocorrelation matrix composed of snapshots which are clustered by the given results obtained from experiments, FEM or other numerical methods for transient heat transfer problem with constant boundary conditions. Then, the reduced order model (ROM) for problems with time-varying boundary conditions is obtained by projecting the finite element discrete format on reduced POD modes determined based on eigen-value error analysis of the original POD modes. One feature is that the POD modes need not to be formed when the boundary conditions are time-varying, and one can use a few modes to capture as much as 99.99% of energy of the whole order model. Examples show that this method is correct and effective. The same POD modes can accurately analyze transient heat conduction problems with the same geometric domain but variety of smooth and time-varying boundary conditions.(2) A reduced order model of transient heat conduction problems consisting of multiple media is established, and a multi-domain POD (MDPOD) analysis method is proposed based on FEM results. Fist, according to the specific property of multiple media, the computational domain of the problem is decomposed into some subdomains. For each subdomain, the snapshots are obtained and the POD modes matrix for oder reduction is established. Secondly, through the finite element discrete format, obtain the original heat capacity matrix, stiffness matrix, and the right side load vector for each subdomain, and establish a reduced order model for each subdomain after dealling with the first kind of boundary conditions. Finally, the overall coupling reduced order model is established according to the continuity conditions of the temperature and heat flux on the interfaces of media. Numerical examples are given to prove the correctness and effectiveness of the method, indicating that 1) the low order POD modal matrix obtained from the temperature results under constant boundary conditions of the problem, can be used to accurately predict the temperature of the problem under complex time-varying boundary conditions; 2) the POD modal matrix obtained from the subdomain, can used to forecast the whole computational domain temperature field, which enables us to deal with multiple media heat conduction problems by using the results of single medium heat conduction; 3) the computation time and the storage space costed for computing temperature field of single subdomain is much smaller than that of the entire domain. Therefore, this MDPOD analysis method can be used to solve large-scale multiple media heat conduction problems efficiently.(3) Order reduction technique of POD is applied to the BEM for solving variable coefficient transient heat conduction problems and the reduced order models of these problems are established based on BEM. The low order POD modal matrix obtained from the temperature results under constant boundary conditions of the unsteady heat conduction problem, can be used to accurately predict the temperature results of the problem under complicated time-varying boundary conditions. First, for a variable coefficient transient heat conduction problem, the boundary integral equation is established and the domain integrals are transformed into boundary integrals. Secondly, the time differential equations whose order will be reduced later are obtained after discreting and reorganizing the integral equation. Finally, the reduced order model is established and solved by projecting the time differential equations on reduced POD modes. Numerical examples demonstrate the validity and effectiveness of the proposed method. Usually, in BEM the time differential equations are solved by using the time difference marching technology, which makes the stability of the algorithm is closely related to the time step size and the solution speed is slow when the order of equations is big. When using the method, all the above problems are solved due to the degrees of freedom for a reduced order model is very small. That is, the solution speed of that time differential equations is fast and the results are accurate.(4) Time difference marching technology is generally adopted to solve differential equations when BEM is used to solve the transient heat conduction problems. And a system of linear equations needs to be solved at each time step. Therefore, a direct method for solving large scale linear algebraic equations is developed, which is named the simultaneous elimination and back substitution method (SEBSM). SEBSM performs both the elimination and back-substitution procedures when each row of the system is formed. When the last row is finished for assembling, the solutions of the system are obtained at the same time, without the need of the last back-substitution procedure. Therefore, SEBSM need not to load the coefficient matrix into the computer's memory in advance as other direct methods do, which results in memory pressure or insufficient to calculate the problem. In addition, the maximum storage space required in the process of SEBSM is about a quarter of the size of the coefficient matrix for a full matrix equation. And it is much smaller for a sparse matrix equation. Examples show that, SEBSM is very effective in solving large asymmetric and non positive definite sparse matrix equations, with the characteristics of less data storage and faster computing speed. Overall, SEBSM can provide BEM a robust equation solver for solving large engineering problems.(5) Singular integrals may appear when forming the coefficient matrix of discrete integral equations by BEM. Therefore, a method for the numerical evaluation of high order singular boundary and domain integrals is developed. The main idea of this method is that the multiple integral can be converted into a radial integral and a low order boundary integral by using the radial integration formula. At this point, when the source point is within the integration region, the singularity is concentrated in the radial integral.Then, the analytical elimination of singularities condensed to the radial integral formulas can be accomplished by expressing the nonsingular part of the integration kernels as a series of cubic B-spline basis functions and by considering all contributions to free terms of adjacent elements around the source point. Numerical examples show that the method has good numerical stability compared with the power series expansion method. |
PDF Full Text Request