Typical pipelines for model geometry generation in computational biomedicine stem from images, which are usually considered to be at rest, despite the object being in mechanical equilibrium under several forces. We refer to the stress-free geometry computation as the reference configuration problem, and in this work we extend such a formulation to the theory of fully nonlinear poroelastic media. The main steps are (i) writing the equations in terms of the reference porosity and (ii) defining a time dependent problem whose steady state solution is the reference porosity. This problem can be computationally challenging as it can require several hundreds of iterations to converge, so we propose the use of Anderson acceleration to speed up this procedure. Our evidence shows that this strategy can reduce the number of iterations up to 80\%. In addition, we note that a primal formulation of the nonlinear mass conservation equations is not consistent due to the presence of second order derivatives of the displacement, which we alleviate through adequate mixed formulations. All claims are validated through numerical simulations in both idealized and realistic scenarios.

翻译：计算生物医学中模型几何生成的典型流程源于图像，这些图像通常被认为处于静止状态，尽管物体在多种力作用下处于机械平衡。我们将无应力几何计算称为参考构型问题，并在本工作中将该公式扩展到完全非线性孔隙弹性介质理论。主要步骤包括：(i) 以参考孔隙度形式改写方程；(ii) 定义一个时间依赖问题，其稳态解即为参考孔隙度。该问题在计算上具有挑战性，可能需要数百次迭代才能收敛，因此我们提出使用安德森加速来加快这一过程。实验证据表明，该策略可将迭代次数减少高达80%。此外，我们注意到非线性质量守恒方程的原始公式因存在位移的二阶导数而不一致，我们通过适当的混合公式缓解了这一问题。所有结论均在理想化与真实场景中通过数值模拟验证。