| A periodic structure is composed of many identical structural components(unit cells)that are assembled end-to-end to form the complete system.By making full use of the special structural properties,it can be very easy and fast to accomplish the structural design,modelling analysis and product manufacturing.In addition,periodic structures usually have many excellent mechanical properties and thermal qualities,such as high specific strength,high specific stiffness,light weight,creep resistance,high temperature resistance,high energy absorption rate,low thermal conductivity and tunable material properties.Therefore,periodic structures play an extremely important role in various fields of engineering.And they have become an indispensable structural type in modern structural design and industrial production process.Since periodic structures usually serve in complex thermal environment,thermal stress problems may be induced.Especially,the transient thermal stress concentration,the increase of thermal strain and the thermal fatigue failure caused by drastic change of temperature or thermal load have a great influence on the structural strength and safety property.In order to obtain the thermal stress and thermal strain,temperature responses of the periodic structure need to be computed firstly.Hence,the research on the efficient method for transient heat conduction problems of periodic structures has not only important theoretical significance,but also great practical value.Transient heat conduction problems of periodic structures cannot be solved accurately using existing analytic methods due to the complicated configurations,boundary conditions and rapidly varying physical parameters.It is precisely this reason that we need various numerical approaches to solve the transient heat conduction in the periodic structure.Firstly,the solution domain is discretized numerically to obtain a set of ordinary differential equations in time,and then direct time integration methods are adopted to solve the above equations to obtain the responses.However,when the scale of structure is very large or the thermo-physical parameters of materials oscillate rapidly,the mesh is required to be very dense by using the finite difference method(FDM)or the finite element method(FEM),which leads that the scale of ordinary differential equations is very large.Consequently,when the direct time integration methods are used to solve the obtained equations directly,the computational efficiency is very low.Therefore,based on the periodic property of structure,the special algebraic structure of the matrix exponential,the superposition principle of linear system and the physical features of transient heat conduction,this doctoral dissertation aims at developing numerical methods with high efficiency and high precision for transient heat conduction problems of periodic structures and quasi-periodic structures.The main research works of this thesis are as follows:(1)Based on the physical features of transient heat conduction,the physical meaning of the matrix exponential and the periodic property of structure,an efficient and accurate numerical integration method is proposed for transient heat conduction in the one dimensional(1D)periodic structure.According to the fundamental solution of the transient heat conduction problem,the physical features of transient heat conduction are provided in detailed,which show that,a unit excitation applied on a point within a time step only can induce non-zero temperature responses in the domain near the point and have no effect on the other domain.Based on the above physical features,the physical meaning of the matrix exponential and the node numbering rules,it is illustrated in detail that the matrix exponential corresponding to the ID periodic structure within a reasonable time step is a sparse matrix which includes a large number of zero elements,and non-zero elements are band-shaped distributing along the diagonal of matrix.Combined with the periodic property of structure,it is observed that there are numerous repetitive elements in the matrix exponential.Then,according to the above algebraic structure of the matrix exponential and the original precise integration method(PIM),an efficient and accurate method is proposed by computing the matrix exponential corresponding to a representative periodic structure(RPS)with a few unit cells instead of computing the matrix exponential corresponding to the entire periodic structure,which can avoid computing and storing a large number of zero elements and repetitive elements.The proposed method significantly improves computational efficiency.Meanwhile,the method inherits the high accuracy and stability of the original PIM.Numerical examples show,even if the Crank-Nicholson(C-N)method with a time step which is 8 times smaller than that of the proposed method,its precision still cannot achieve that of the proposed method,which reveals that the proposed method can obtain a highly precise solution with a larger time step.In terms of computation efficiency,when the C-N method can give the reasonable result,the efficiency of the proposed method is about 20 times better than that of the C-N method.which demonstrates that the proposed method is very efficient.(2)Based on the superposition principle of linear system,the physical features of the transient heat conduction and the periodic property of structure,an accurate,efficient and stable integration method is developed to compute temperature responses of the two dimensional(2D)periodic structure.Firstly,based on the superposition principle of linear system,the solution of the original finite element model can be obtained by the superposition of solutions of a series of basic finite element models.Secondly,according to the physical feature of transient heat conduction,computation of responses of basic finite element models can be transformed into computation of responses of small-scale models.That can reduce the computational scale and improve the computational efficiency.Finally,according to the periodic property of structure.matrix exponentials of many small-scale models are identical,so only the matrix exponentials of a few small-scale models need to be computed by the PIM.That can reduce the computational numbers of matrix exponentials to improve the computational efficiency further.The proposed method not only inherits the accuracy and stability of the PIM but also significantly improves computational efficiency and optimizes the memory usage.Numerical examples show that if the time step of the C-N method is the same as that of the proposed method,its relative error is very large.In order to obtain the reasonable result,the smaller time step must be adopted for the C-N method.However,even when the time step of the C-N method is four times smaller than that of the proposed method,its precision is still less than one-thousandth of that of the proposed method.It means that the proposed method can obtain highly accurate results with a larger time step.For the 2D periodic structure with approximately 5.8 million DOFs,when the calculation result is reasonable,the efficiency of the proposed method is about 27 times better than that of the C-N method with the Preconditioning Conjugate Gradient(PCG)solver,while the direct method for the C-N method cannot be performed due to computer memory limitations.Thus,the efficiency of the proposed method is very high and the memory usage is very small.(3)According to the spatial and temporal distribution of the Gaussian moving heat source,the periodic property of structure and the physical features of transient heat conduction,an efficient numerical integration method is established to analyse the transient heat conduction in a periodic structure with moving heat sources.The moving heat source is modelled as a localized Gaussian distribution in space.Although the central position of the heat source is time-varying,the spatial distribution of heat flux is relatively time-invariant and localized for every moment.Based on the spatial and temporal distribution features,the periodic property of structure and the physical feature of transient heat conduction,within a time step,when the heat source moves in unit cells which are far away from the boundaries of structure,the contribution caused by the heat source is repetitive.Based on this feature,an efficient numerical method is proposed to reduce the computation of repetitive contribution.Then,combined with the superposition principle of linear system,within a small time-step,computation of results corresponding to the whole structure excited by the Gaussian heat source is transformed into that of some small-scale structures excited by several basic equivalent heat loads individually,which can improve the efficiency further by reducing the computational scale.The numerical examples demonstrate that the relative error of the proposed method is about 10-4,which means the precision of the proposed method is very high.For the single moving heat source problem with about 0.21 million DOFs and the multiple moving heat sources problem with about 1.1 million DOFs,the efficiency of the proposed method is about 4 and 5 times better than that of the C-N method with PCG solver,respectively.(4)Based on the local features of temperature responses caused by the external excitation within a time step,a quasi-superposition principle for transient heat conduction in a quasi-periodic structure with nonlinear defects is presented,and then a novel numerical method with high precision and high efficiency is proposed for analyzing transient heat conduction in a quasi-periodic structure.According to the physical features of transient heat conduction and the positions of defects,the whole external excitation of structure can be properly divided into two groups,and within a time step,the temperature response of the quasi-periodic structure with defects can be obtained by the superposition of responses induced by both groups of external excitations individually.Based on this fact,a new method is developed to transform the transient problem of the quasi-periodic structure into a linear problem of a perfect periodic structure and nonlinear transient problems of some small-scale substructures with defects.Since the scales of substructures with defects are far smaller than that of the original structure,thus the temperature responses for the substructures with defects can be efficiently computed by traditional numerical methods,while the temperature responses of the perfect periodic structure can be efficiently computed by using the numerical method proposed in this PHD paper.The proposed method can not only avoid solving large-scale nonlinear equations iteratively,but also avoid updating the heat conduction and heat capacity matrices of the whole structure.Thus,compared with the traditional method,the proposed method needs less memory and has higher efficiency.Numerical examples show that the relative error of the proposed method with a larger time step is about 10-3.In terms of the computational efficiency,for a quasi-periodic structure with nonlinear defects having about 5 million DOFs,when the results obtained are reasonable,the efficiency of the proposed method is more than 10 even 60 times better than that of the traditional method. |
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