Direct Inversion Method For Identifying The Boundary Condition And Geometry For Transient Heat Conduction Problems Of Functionally Graded Materials

The boundary conditions and geometric shape inversions of transient heat conduction problems have been widely used in the fields of aerospace,nuclear safety protection systems,industrial production and non-destructive testing.Based on the precise integration finite element method(PIFEM),a novel direct inversion method is proposed to identify the boundary conditions and geometric shapes of two-dimensional and three-dimensional functionally graded materials(FGMs)transient heat conduction problems.The main contents of this paper are summarized as follows:(1)The forward problem of transient heat conduction in FGMs is solved based on PIFEM.The forward problem is the foundation of the research on the inversion problem.In this paper,the weak form of integral equation is established by Galerkin weighted residual method,and the Euler backward difference method(BDM)and the precise integration method(PIM)are used to deal with the time-dependent ordinary differential equations obtained by finite element discretization.Numerical examples show that the PIM is insensitive to the time step when dealing with time domain problems.(2)Based on PIFEM,a direct inversion numerical model for identifying boundary conditions of transient heat conduction problems is established.The error function is established by using matrix transformation to find the relationship between the measurement point temperature and unknown temperature and heat flux boundary conditions.The unknown boundary conditions are directly inversed by the least square method(LSM).The inversion results show that the method has high accuracy and good stability in identifying boundary conditions of transient heat conduction problems.(3)Based on PIFEM,a direct inversion numerical model for transient heat conduction geometry shape is established.The virtual boundary is introduced and new computational domain is established by the virtual boundary and the partial known boundary.The temperature boundary condition of virtual boundary is directly inversed by LSEM.The geometric shape of the unknown boundary can be obtained by searching the isothermal or the isothermal surface by using the temperature field of the new computational domain.In order to verify the validity of the method,numerical examples discuss the effects of many factors on the inversion results including different virtual boundaries,the number of measurement points,the location of measurement points,the measurement errors and the errors of measurement point position.(4)A domain propulsion and adaptive modified theory is proposed,which further improves not only the numerical precision of inversion geometric shape problems but also the ability to inverse complex geometric shapes.Based on the direct inversion geometric shape theory of heat conduction problems,a better virtual boundary position is obtained the domain propulsion process,and then an optimal virtual boundary shape is found by using the adaptive modified theory.After that the high precision estimation of geometric shape is realized.The influence of the basis function selection,the number of measurement points,the measuring errors,the selection of verification standard,the selection of adaptive convergence standard and the error of measurement point position on the inversion results are discussed in numerical examples.Numerical results show that the theory not only improves the stability of the direct inversion method to some extent,but also can be used to identify relatively complex geometric shapes.In order to reduce the ill-posed level of inverse problems,the basis function expansion method is carried out,the unknown boundary conditions will be expended as the form of undetermined coefficients and known basis functions.The inverse boundary condition problem is transformed into solving the undetermined coefficients problems,which improves the inversion efficiency to some extent.In addition,the inversion of ill-posed matrix is solved by singular value decomposition method and truncated singular value decomposition method.The direct inversion method proposed in this paper not only enriches the application field of PIFEM,but also provides a numerical method with high precision and high efficiency for inversion of boundary conditions and geometric shape problems.The research work in this paper also provides a good reference value for other inversion fields.