The direct method of lines for forward and inverse problems of nearly-incompressible composite materials in star-shaped domains

In this paper, the direct method of lines is extended to (nearly) incompressible composite materials problems in star-shaped domains. We also study its application in inverse elasticity problems. By introducing a suitable curvilinear coordinate, the irregular star-shaped domains are transformed into regular semi-infinite stripes. The (nearly) incompressible elastic equations are discretized in the angular variable and we resolve the resulting semi-discrete approximation using the direct method. Then we present an optimal error estimate for the nearly-incompressible composite materials problem. There is no rise to the “locking effect" and the convergence order will not reduce. For the inverse problems, we determine the Lamé coefficient from the measured data by minimizing the regularized energy functionals. We apply the direct method of lines as a forward solver to deal with the irregularity of the domain and the possible singularity in the forward problem. The effectiveness and accuracy of this method for forward and inverse problems are verified by abundant numerical examples.