Solving an inverse problem for the Euler-Bernoulli beam equation with boundary measurements.

摘要

The estimation of model parameters for real-world phenomena is the focus of much research attention. We consider an inverse problem for the steady-state Euler-Bernoulli beam equation: recover the flexural rigidity of the loaded beam, given only limited endpoint measurements. We reformulate the inverse problem as we develop two practical solution frameworks, one based on Banach's fixed point theorem and the other based on the Lax-Milgram representation theorem. In practice, each method replaces the minimization of the true approximation error by the minimization of a different objective function. Both formulations depend on the "intrinsic" parameters defining the flexural rigidity and additional parameters defining the deflection on the interior of the beam. To solve the resulting optimization problems we use Particle Swarm Ant Colony Optimization. After establishing the theoretical frameworks and their applicability to our problem, we solve numerous inverse problems for the beam. Lastly, we discuss future topics of research stemming from our work.