Finite Element Method For Boundary Condition Inverse Problems In Elasticity

This paper introduces the concept, characteristic and difficulty of solvingmathematical physical inverse problems, and focuses on introducing an iteration method–the conjugate gradient method(CGM). This paper expounds the basic principle of thefinite element method, and the solving thought of the finite element method(FEM) forsolving the inverse problem. The boundary condition inverse problems in2-D isotropicelasticity are solved.The displacement and force boundary condition inverse problems for truss, beam andplate are discussed. The original data of the structure models are extracted by ANSYSfinite element software. The original data are reconstructed to form the systemequations with the unknown boundary conditions by MATLAB software. In general, thesystem equations are ill-posed, so the problems are treated in combination with theCGM to obtain all unknown displacement and force boundary conditions.The comparison of inversion numerical solutions and the solutions of finite elementanalysis(as exact solutions) for examples shows that the inversion results converge toexact solutions, if there are sufficient original data and the number of the constraints isappropriately reduced. When the original data have the random noise, the inversionsolutions still have the good precision, and converge to the exact solution with reducingthe random noise added into the known data. The numerical examples demonstrate thatthe FEM in conjunction with the CGM method is feasible, accurate and stable to solvethe inverse problem.