Stochastic Constitutive Laws in Nonlinear Mechanics: Application to the Multiscale Modeling of Arterial Walls for Robust Vascular Grafting

本研究提出利用计算随机分析的方法来建立一个新的非线性随机本构关系的建模和计算框架。非线性材料本构行为的不确定性建模是计算力学和材料力学的核心挑战。特别地，由软生物组织（诸如血管）表现出的大的可变性是当前计算辅助手术、针对心血管疾病的患者特异性治疗和组织工程方法的广泛采用的障碍。本研究提出利用计算随机分析的方法来建立一个新的非线性随机本构关系的建模和计算框架。血管结构的特定情况被故意选择为结合强各向异性和高水平随机性的原型应用。该奖项支持的研究将增强涉及生物材料（如动脉和脑组织）的模拟的预测能力，并将与一大类材料相关，包括受损复合材料的情况。在这一努力中提倡的跨学科观点将使涉及应用数学和材料科学等各个领域的学生能够广泛接触，并将通过在当地高中的外联活动介绍理论和计算方面。本研究的重点是非线性本构关系的计算随机分析。更具体地说，它的目的是推导概率模型，一个高性能的计算环境，采样光滑流形和方法的空间依赖性各向异性应变能函数的识别和验证。通过解决与校准和验证问题密切相关的非线性本构方程的适当数学随机化，该奖项支持的研究将显着推进一种新的信息理论随机方法，其中随机性可以从潜在的多尺度实验到粗尺度模拟。该项目将涉及一系列方法和理论发展，包括（1）构建基于物理的随机场模型和一类多凸储能函数的采样算法，以及（2）定义用于随机模型的数据贫乏逆校准和多尺度验证的方法。该新框架将特别用于大规模非线性模拟，以研究具有患者特定几何形状的随机血管结构的失效概率。

In this project, the use of computational stochastic analysis is put forward in order to construct a new modeling and computational framework for nonlinear stochastic constitutive laws. Modeling the uncertainties in the constitutive behavior of nonlinear materials is a central challenge in computational mechanics and mechanics of materials. In particular, the large variability exhibited by soft biological tissues, such as vascular vessels, is a current roadblock to computational assisted surgeries, patient-specific treatments for cardiovascular diseases and wide adoption of tissue engineering approaches. In this project, the use of computational stochastic analysis is put forward in order to construct a new modeling and computational framework for nonlinear stochastic constitutive laws. The specific case of vascular constructs is purposely chosen as a prototypical application combining strong anisotropy and a high level of stochasticity. The research supported by this award will enhance the predictive capabilities of simulations involving biological materials, such as arterial and brain tissues, and will be relevant to a large class of materials, including the case of damaged composites. The interdisciplinary standpoint promoted in this effort will enable a broad exposure to students involved in various fields, such as applied mathematics and materials science, and will allow theoretical and computational aspects to be introduced through outreach activities in local high schools. This research is focused on computational stochastic analysis for nonlinear constitutive laws. More specifically, it aims at deriving probabilistic models, a high-performance-computing environment for sampling on smooth manifolds and methodologies for the identification and validation of spatially dependent anisotropic strain energy functions. By addressing the proper mathematical randomization of nonlinear constitutive equations in close relation with calibration and validation concerns, the research supported by this award will notably advance a new information-theoretic class of stochastic methods where randomness can be accounted for from potentially multiscale experiments to coarse-scale simulations. The project will involve a set of methodological and theoretical developments, including (1) the construction of physics-based random field models and sampling algorithms for a class of polyconvex stored energy functions, and (2) the definition of methodologies for the data-poor inverse calibration and multiscale validation of the stochastic models. The novel framework will notably be used within large-scale nonlinear simulations to investigate the probability of failure of stochastic vascular constructs with patient-specific geometries.