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

在本项目中，提出使用计算随机分析来构建一种新的非线性随机本构关系的建模和计算框架。对非线性材料本构行为中的不确定性进行建模是计算力学和材料力学的核心挑战。特别是，血管等软生物组织表现出的巨大变异性，是目前计算机辅助手术、针对患者的心血管疾病治疗和组织工程方法广泛采用的障碍。在本项目中，提出使用计算随机分析来构建一种新的非线性随机本构关系的建模和计算框架。血管构造的具体情况被有意选为结合了强各向异性和高水平随机性的典型应用。该奖项支持的研究将增强涉及动脉和脑组织等生物材料的模拟的预测能力，并将与一大类材料相关，包括受损复合材料的情况。在这一努力中促进的跨学科观点将使学生能够广泛接触到涉及不同领域的学生，如应用数学和材料科学，并将允许通过在当地高中的外联活动介绍理论和计算方面。本文主要研究非线性本构关系的计算随机分析。更具体地说，它的目标是导出概率模型，一个用于在光滑流形上采样的高性能计算环境，以及用于识别和验证空间相关的各向异性应变能函数的方法。通过解决与校准和验证密切相关的非线性本构方程的适当数学随机化问题，该奖项支持的研究将显著推进一种新的随机方法的信息论类别，其中随机性可以从潜在的多尺度实验到粗略规模的模拟来解释。该项目将涉及一系列方法学和理论发展，包括(1)建立一类多凸储能函数的基于物理的随机场模型和采样算法，以及(2)定义用于数据贫乏的随机模型的逆定标和多尺度验证的方法。新的框架将被显著地用于大规模的非线性模拟，以研究具有患者特定几何形状的随机血管结构的失败概率。

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.