Acceleration-Energy Filter and Bias Compensation for

    Stabilizing Equation Error Estimator

    in Inverse Analysis Using Dynamic Dispalcement

    Kwang Youn Park



New stabilization schemes which correct the equation error estimator (EEE) in the inverse analysis using dynamic responses of linear elastic continua are presented. The goal of the inverse analysis run in these cases is the proper identification of material properties. Stabilization schemes consist of the acceleration-energy filter and the bias compensation.

The acceleration-energy filter stabilizes the ill-posedness of the inverse analysis. The acceleration-energy filter replaces the techniques known as truncated singular value decomposition (TSVD), L1-norm regularization and L2-norm regularization (or Tikhonov regularization). Existing regularization techniques do not work properly for cases involving hard inclusions, i.e., tumors of organ and suspensions of vehicles. The Acceleration-energy filter, however, work properly for cases involving both hard and soft inclusions. The acceleration-energy filter is separated into the acceleration filter and the energy filter. Dividing them in this manner simplifies a filtering process.

The acceleration filter imposes finiteness condition of accelerations, the second derivatives of the measured displacements. Accelerations can be considered as finite functions when impact loads do not exist. The acceleration filter requires two initial and two final values, but the overlapping moving time window technique is employed so that the initial and final values can be ignored. The final form of the acceleration filter is a low-pass finite impulse response (FIR) filter. However, the acceleration filter differs from typical low-pass FIR filters because it has physical meaning which guarantees consistency with the energy regularization.

The energy filter imposes finiteness of strain energy, which is internal energy of linear elastic continua. The final form of the energy filter is very similar to low-pass spatial filters used with image processing, but the energy filter has three advantages. The first of these are the boundary conditions. The boundary conditions of the energy filter are identical to these of an equilibrium equation for the continuum, and are always satisfied by all continuum examples. The second is the available meshes. The energy filter involves the connectivity information of nodes and can handle complicatedly meshed FEM models, whereas typical low-pass spatial filters can handle only rectangular meshes. The third advantage is the physical meaning which guarantees consistency with the acceleration filter.

The acceleration filter and the energy filter must satisfy consistency of the elastic waves and the temporal wave. The solution of the inverse analysis without the consistency is not trustable because the strain and the acceleration do not have equivalent information. The physical meaning of two filters gives consistency between two filter.

The biases of the solutions are ignored in existing studies. However, the inverse analysis using EEE for linear elastic continua must consider the biases. If the noise variances are known, the biases of the solution could be perfectly eliminated by means of bias compensation.

Aluminum plate and medical imaging examples are demonstrated to show the effectiveness of the schemes described above.


Key Word

Wide steel box-girder, Ultimate compressive strength, Stiffened flange, U-rib, Banding stiffness of diaphragm, Force control, Displacement control, FHWA provision


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