A Numerical PDEs Approach to Evolution Equations in Shape Analysis Based on Regularized Morphoelasticity

Abstract

This work studies a variational formulation and numerical solution of a regularized morphoelasticity problem of shape evolution. The foundation of our analysis is based on the governing equations of linear elasticity, extended to account for volumetric growth. In the morphoelastic framework, the total deformation is decomposed into an elastic component and a growth component, represented by a growth tensor $G$. While the forward one-step problem -- computing displacement given a growth tensor -- is well-established, a more challenging and relevant question in biological modeling is the inverse problem in a continuous sense. While this problem is fundamentally ill-posed without additional constraints, we will explore parametrized growth models inscribed within an optimal control problem inspired by the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. By treating the growth process as a path within a shape space, we can define a physically meaningful metric and seek the most plausible, energy-efficient trajectory between configurations. In the construction, a high-order regularization term is introduced. This elevates the governing equations to a high-order elliptic system, ensuring the existence of a smooth solution. This dissertation focuses on the issue of solving this equation efficiently, as this is a key requirement for the feasibility of the overall approach. This will be achieved with the help of finite element solvers, notably from the FEniCSx library in Python. Also, we implement a Mixed Finite Element Method, which decomposes the problem into a system of coupled second-order equations as a treatment of these high-order systems that have significant computational challenges.