Inverse analysis for the computation of growth and boundary conditions in elastic bodies

Abstract

AbstractA common approach for modelling growth processes in tissues and organs is the decomposition of the deformation gradient into elastic and growth contributions. The latter is usually deduced from empirically motivated laws, and validated by comparing the simulated and experimental final shapes. We here solve the associated inverse problem: deduce the growth distribution that best fits the measured deformation of an elastic body. Since in practical problems, boundary conditions are also in general unknown, we extend our methodology and optimality conditions to the inference of the boundary reaction forces. The system of equations is guaranteed to have a solution by resorting to an iterative regularisation process that we numerically analyse in order to deduce suitable parameters. Uniqueness of the solution is analysed as a function of the given measured positions, and ensured when the measured data encompass all the nodes of the mesh. In other situations, uniqueness can be guaranteed under some conditions on the data and elasticity stiffness matrix. We test and demonstrate the effectiveness of our methodology to capture arbitrary deformed shapes with a set of three-dimensional synthetic problems.