Investigation On Domain Decomposition Preconditioner In Heat Conduction Problem And Research Of Cooling Method In Transition Piece

In the related fields, facing the trial costly, lengthy test period, a huge cost ofcomputation. Computer simulation technology has irreplaceable advantages. But when usingsuch methods, people often encounter many problems.The finite element method is commonly used to solve engineering problems. For thedemand point of view, the calculation model is actually facing an engineering or a physicalproblems. Because of the different model, structures, equations, parameters, functions andboundary conditions will change. When using finite element method, people usually choosethe iteration method to deal with the problem which the scale of FEM calculation is too large.But the application of the finite element method often obtain an ill-condition stiffness matrix,which result in the convergence difficulties. This naturally requires the calculation method canadapt to this. Furthermore, When design a structure with special features. Such as the designissue of the cooling performance of the gas turbine transition section. Need to be consideredthe combination of factors, which requires the introduction of a large number of designelements and design concepts. How to design a specified structure which have a goodperformance when coupling factors exsit then get the optimization between schemes. It havebeen research value to use numerical simulation method to reduce costs.The main work of this paper as follow:(1), This article gave a brief describes of the Krylov subspace method, conjugategradient method, and preconditioned conjugate gradient method. Analyzed the disadvantagesof the traditional conjugate gradient method then investigate the advantage of usingpreconditioned conjugate gradient method. Analysis the transient heat conduction problems.Conclution that, the original problem can be discreted by the mixed finite elemen after thedivision by the time step, then get mixed variational form.(2), From the perspective of the transient heat conduction problem, consider the problem of partial differential equation with variable coefficients. Preprocessor is built with Domain decomposition method (DDM) in this paper, then combian it with the conjugate gradient method. First, analysis the two-level overlapping Schwarz algorithm which based on the overlapping domain decomposition in the case which coefficients have jumps between subdomains. Use the additive Schwarz operators Spa:=∑i=0NSpi, define the coarse space with the minimize energy norm by the discrete harmonic extension. Decompose the original domain into some subregions, set the coefficients constant in individual subdomain. Because of many reasons such as the material parameters, the relationship of size between coefficients maybe different, investigate the relationship one by one. Finally find the condition number is independent of the numbers of subdomains, only get lager with max1≤i≤N(1+log(Di/di)) and max1≤i≤N(1+log (Di/δi)). So people can divided the original domain into multiple subdomains according to the actual without consider the convergence property when using the domain decomposition method.(3), There are two main family of domain decomposition method. One is overlapping Schwarz method, the other one is iterative substructive method. The other work of this paper is analysis the BDDC method which based on the nonoverlapping domain decomposition. The BDDC method is one kind of iterative substructive method. In this article, use an intermediate space GΓ, which defined on the interface of subdomains, the GΓ space can be decomposed into primal subspace GΠ and dual subspace GΔ two subspace. Consider the jumps of coefficients between subdomains, take the weighted value on the interface by the scaled operator δi(?)(x). Then set the BDDC operator as PB-1=RD,Γ～TΓ～-1～RD,Γ by using de restriction operatorRD,Γ～and RΓ～, Estimate the upper bound of the condition number is Cmax1≤i≤N(1+log(Di/di))2. It’s easy to see that, this condition number bound is independent of the values and jumps of the coefficients across the interface and has a polylogarithmiccondition number bound in terms of the number of degrees of freedom of the individual subdomains.(4), A one-fourth cylinder double chamber model with a single row of impinging holes in the outer-wall was simplified from the gas turbine transition piece, and its operatingconditions selected in this paper are featured by high temperature, high pressure, and highvelocity of coolant. A complete three-dimensional numerical simulation is employed withdifferent pin-fin structures adopted in the impinging cooling. Mist cooling technology takeninto the impinging cooling is tested and investigated. And it has been proven that the pin finscould enhance the cooling effectiveness without excess pressure loss. The temperature couldobtain a lower value when injecting mist into the cooling chamber. The optimal combinationof the pin-fin parameters and mist imported brings the optimum impinging cooling efficiencywithout excess pressure drop.