Research On The Efficient Solution Method Driven By Application Features For Linear Equation Systems

The cost of solving linear equation systems is often the main cost of the numeri-cal simulations of real complex applications.The preconditioned iterative method is the most widely used method to solve large-scale sparse linear equation systems.Both the preconditioner and the iterative method have been focused on their general performance and lack of consideration for the characteristics in numerical simulations of real appli-cations.Inertial confinement fusion(ICF)is a kind of radiation fluid application with strong nonlinearity,strong discontinuity,large deformation,and multi-media? it shows various characteristics in numerical simulation: on the one hand,during the early period of simulation,some physical variables change wildly in some computational areas,but little in other areas? on the other hand,the characteristics of strong discontinuity and large deformation make the coefficient matrices have orders of magnitude difference among its entries.Underwater vehicle is a kind of application of incompressible fluid.In the numerical simulation,the segregated iterative algorithm is usually used to solve the in-compressible Navier-Stokes equation.In the segregated iterative algorithm,the variables on each grid appear to be oscillatory.In this thesis,the characteristics in two kinds of applications,inertial confinement fusion(ICF)and underwater vehicle,are used to drive the efficient solution of the linear equations generated in their simulations,respectively.The main contributions and innovations of this thesis are as follows:(1)Considering the local character in the numerical simulations,this thesis proposes a local character-based method to solve the linear systems.The proposed method,which is an algebraic method,consists of three steps: firstly,a local domain(algebraic domain)where the variables change wildly is constructed? secondly,the subsystem on the local domain is solved? and lastly,the whole system will be solved.In this thesis,the local character is abstracted mathematically,and the definition and related properties of the local character linear equations are given.Two methods are given to construct the lo-cal domain.One is based on the spatial gradient,and the other is based on the residual.Numerical tests for a two-dimensional heat conduction model problem,and two real ap-plication models,the multi-group radiation diffusion equations and the three temperature energy equations,are conducted.The test results show that the solution time for solving the linear system can be reduced dramatically by using the local character-based method.The speedups of this method reach up to 1.61 x and 1.65 x,respectively,in the test set of linear equations of multi group radiation diffusion equations and three temperature energy equations.(2)Considering the multi-scale character in the numerical simulations,this thesis proposes a filter to remove small entries in the coefficient matrix while constructing the preconditioner.The small entries are filtered first according to a so-called weak depen-dence matrix,which is constructed based on the conception of the strength of connections in AMG.The preconditioner is then built based on the filtered matrix instead of the orig-inal matrix.Four strategies of filtering out entries are designed and investigated.Numer-ical results for various model-type problems and two real application problems,i.e.,the multi-group radiation diffusion equations and the three temperature energy equations,are provided to show the effectiveness of the proposed method.The speedups of this method reach up to 1.47 x and 1.55 x,respectively,in the test set of linear equations of multi group radiation diffusion equations and three temperature energy equations.(3)Considering the discontinuous character in the numerical simulations,this the-sis proposes a combined coarsening strategy for the classical algebraic multigrid method.In this method,the combined coarsening algorithm constructs the coarse grid within two stages.In the first stage,a basic set of coarse grid is built by the classical coarsening algo-rithm? in the second stage,the quality of the coarse grid is measured based on compatible relaxation,and the points out of the coarse grid that converge slowly in the relaxation are added into the previous set.We test various discontinuous coefficients Poisson-like problems,the three temperature energy equations,as well as some linear equation sys-tems arising from 3D blended-wing-body airfoil case,to verify the effectiveness of our method.(4)Considering the numerical oscillatory character in the segregated iterative algo-rithm,this thesis proposes a weighted group extrapolation method to obtain a superior initial guess instead of the general one,the solution of the previous linear equation sys-tem.The proposed method consists of three steps: firstly,all the linear equation systems are partitioned into several lanes according to the settings of the segregated iterative al-gorithm,and the previous solutions that used to extrapolate the predicted solutions are carefully organized by groups to address the oscillatory solution on each grid? secondly,each group will predict a solution for the current linear system? lastly,the proposed method uses a weighted average of the predicted solutions as the new initial guess to avoid over extrapolating.Three numerical test results of pitz Daily,2D NACA0012 and 3D blended-wing-body airfoil show that the proposed method can accelerate the iterative solution of most linear equation systems and reduce the simulation time.The speedups of this method reach up to 2.58 x and 1.87 x,respectively,in the simulations of 2D NACA0012 and 3D blended-wing-body airfoil cases.