Non-iterative Algorithm For Identifying Boundary Conditions And Geometry Shapes In Heat Conduction Problems

Based on the finite element method(FEM),a new non-iterative inverse method is proposed to identify the boundary conditions and boundary geometry shapes in multi-dimensional steady heat conduction problems.The inverse process for estimating unknown boundary temperatures includes two steps.Step 1,on the basis of exact boundary conditions,the measured temperatures can be obtained by calculating the direct heat conduction problem.Step 2,by minimizing the difference between the exact measured temperatures and the estimated temperatures,a system of equations containing the unknown temperatures is formed.The inversion results can be acquired by solving the equations.For boundary heat flux identification problems,the boundary temperatures can be identified firstly.Then,a system of equations containing the unknown heat fluxes is formed by matrix transformation.The unknown boundary heat fluxes can be acquired by solving the equations.The inverse process for identifying unknown boundary geometry shapes can be divided into three steps.Step 1,the direct heat conduction problem with the exact domain is solved by the FEM and the temperatures of measurement points are obtained.Step 2,by introducing a virtual boundary,a virtual domain is formed.By minimizing the difference between the temperatures of measurement points in the exact domain and those in the virtual domain,the temperatures of the points on the virtual boundary are calculated based on the least square error method.Step 3,the objective geometry shapes can be estimated by the method of searching the isothermal curve or isothermal surface for two-dimensional(2-D)or three-dimensional(3-D)problems,respectively.In the process,no iterative calculation is needed.The mesh reconstruction problem is avoided in boundary geometry identification problems.Compared with iterative inverse algorithms,the proposed method has a tremendous advantage in reducing the computational time for the inverse problems.Additionally,the inverse process is more concise.The ill-posed problem is solved by the singular value decomposition(SVD)method and Tikhonov regularization algorithm.The SVD is employed to decompose the ill-conditioned matrix.Thus,a matrix inversion of the ill-conditioned matrix can be performed.Tikhonov regularization has advantages in dealing with the ill-conditioned equations.The Tikhonov regularization method can solve the ill-conditioned equations very well.It can guarantee the accuracy and stability of the inversion results as well as the anti-noise ability of the algorithm.Numerical examples are presented to test the validity of proposed method.In 2-D examples,the influence of measurement noise,measurement point number and measurement point position on the inversion results are investigated.Meanwhile,for boundary geometry identification problems,the inversion results with different virtual boundaries are compared.In 3-D problems,the influence of the measurement noise on the inversion results is considered.The solutions show that the method is accurate and efficient to identify the unknown boundary conditions and boundary geometry configurations for 2-D and 3-D heat conduction problems.