The theory of morphoelasticity includes a family of nonlinear PDE systems and has been widely applied to understand growth and morphogenesis in living structures. For those who perceive this theory as workable, the challenge of reconciling it with the fluidic or plastic-like behaviors observed in living systems remains unresolved. Furthermore, a systematic numerical approach is needed to handle the theory in different scenarios where multiple solutions may coexist. 

In the first part of the talk, I will discuss our recent work that expands morphoelasticity to incorporate fluidic behavior and mechanical feedback on growth in the Eulerian frame. This extension involves coupling growth dynamics with elastic strains and nutrient fields based on an energy dissipation argument. I will briefly demonstrate its application in understanding tumor spheroid growth under different external forces and elastic confinements. 

In the second part of the talk, I will discuss a new finite-element method (FEM) to treat the (unextended) theory. This FEM frames the search for solutions as the optimization of the discretized energy functional. In contrast to previous methods, we utilize the analytics of the Hessian matrix to develop a gradient-descent algorithm that ensures energy stability followed by Newton’s iterations. This strategy allows us to robustly search for solutions in the evolving energy landscapes driven by growth. We showcase the effectiveness of this method through three cases involving annulus-bilayer geometry, each exhibiting distinct material parameter ratios between the interior and exterior layers. Our method can identify and characterize supercritical bifurcations leading to smooth wrinkles, capture the subcritical transitions from smooth solutions to non-smooth crease-forming solutions, and uncover smooth and non-smooth aperiodical solutions that have not been reported before. 

The first part of the talk is joint work with John Lowengrub (UCI), Nonthakorn Olaranont (UCI), and Chaozhen Wei (UESTC, China).