Modeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield's seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might have no solution even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier's iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, where a one-level additive Schwarz and generalized Dryja-Smith-Widlund provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios.

翻译：建模生物组织与器官行为通常需要掌握其在无外部载荷时的形态。然而，当通过成像技术活体获取其几何结构时，组织通常因外力作用发生机械形变，因此需要重构无载荷构型。本文针对这一关键但常被忽视的逆弹性问题（IEP），从理论与数值两方面展开深入探讨，并特别关注心脏力学领域。在本研究中，我们将Shield的开创性工作拓展至具有任意材料非均匀性且存在体力和主动力的情况，这些要素在计算心脏病学中具有基础性意义，同时证明它们可能破坏逆问题的变分结构。此外，我们进一步论证了即便在恒定诺伊曼边界条件与多凸应变能函数条件下，逆问题仍可能无解。随后，我们通过大量数值实验验证理论框架，并刻画直接数值逼近IEP所面临的计算挑战。具体而言，研究表明本框架在鲁棒性和最优性上均优于现有方法（包括经加速技术改进的Sellier迭代法）。值得关注的是，多网格预处理器在标准弹性问题中有效，但在本场景中效率较低，而单层加性施瓦茨方法及广义Dryja-Smith-Widlünd格式则是更可靠的替代方案。最终，我们成功解决了全心脏几何结构的IEP，证明该公式可计算实际场景中的无应力构型。