Algorithms and modeling for nonlocal models of diffusion and mechanics and for plasmas

该项目涉及三个物理系统，即扩散，固体力学和等离子体的新模型的建模，分析，算法发明和数值分析。在第一个系统中，PI考虑非Fickian（或异常）扩散的非局部积分模型。该模型具有分数拉普拉斯和分数阶导数模型的特殊情况，但在几个方面概括了这些模型，例如允许空间异质性和各向异性，不太光滑的解决方案的行为，以及一个简单的方法来处理有界域上的问题。该项目解决了几个问题，是至关重要的非局部扩散模型的高效，鲁棒性和准确的近似。特别是，截断内核和数据，使计算可以在有限域上进行的影响进行了分析，结合有限元/域和数据截断误差估计，并以目标为导向，伴随方程为基础的网格自适应方法的开发，分析和测试。在第二个系统中，PI考虑固体力学的非局部、空间导数自由的周期性模型。该模型允许不连续的解决方案，因此它特别适合于建模缺陷。因此，它的功效已被证明在几个复杂的应用，包括复合材料的断裂和失效，裂纹不稳定性，多晶体的断裂，和断裂网络。该项目涉及的计算方法的发展，充分利用固有的多尺度特性，在peridendics和结果从现象学的地平线参数，限制了相互作用的程度。在第三个系统中，被认为是用于模拟从高到低密度区域的离子膨胀的Vlasov-Poisson和冷离子系统。该项目包括开发一种新的冷离子模型，该模型在奇点产生之后仍然有效，表明新模型的解决方案仍然是极限（当离子温度趋于零）的解决方案的弗拉索夫泊松系统，研究分析和计算的过渡发生的密度比下降，从解决方案没有奇异性的，处理多个种类的离子的问题，并开发有效的数值方法，为二维和三维设置。该项目解决的基本问题，出现在数学和计算处理的三个物理系统是非常重要的，在各种各样的应用。这些包括但不限于缺陷的成核和传播（裂纹、分层等）。固体（飞机机翼、核反应堆安全壳等）;在地下水、油和气流、动物觅食行为、聚合物流、外来材料等中观察到的异常扩散行为;获得有关这些复杂系统的有用信息需要大量的计算工作，这可以大大有助于改进数学模型，定性理论信息的解决方案，这些模型，最重要的是，通过更好的，更有效的和更准确的计算算法。由于该项目影响的环境范围广泛，所获得的结果将引起参与物理现象理论研究、新设备设计和制造以及其他领域的科学家、工程师和政策制定者的极大兴趣。风险评估和补救设计。

The project involves the modeling, analysis, algorithmic invention, and numerical analysis of novel models for three physical systems, namely diffusion, solid mechanics, and plasmas. In the first system, the PI considers a nonlocal, integral model for non-Fickian (or anomalous) diffusion. The model has as special cases fractional Laplacian and fractional derivative models but generalizes these models in several ways such as allowing for spatial heterogeneity and anisotropy, less smooth solution behavior, and a simple means for treating problems posed on bounded domains. The project addresses several issues that are crucial to the efficient, robust, and accurate approximation of the nonlocal diffusion model. In particular, the effects of truncating kernels and data so that computations can be done on finite domains are analyzed, combined finite element/domain and data truncation error estimates are obtained, and a goal oriented, adjoint equation-based grid adaptation methodology is developed, analyzed, and tested. In the second system, the PI considers the nonlocal, spatial derivative free peridynamics model for solid mechanics. The model allows for discontinuous solutions so it is especially well suited for modeling defects. As such, its efficacy has been demonstrated in several sophisticated applications, including the fracture and failure of composites, crack instability, fracture of polycrystals, and nanofiber networks. The project involves the development of a computational methodology that takes full advantage of the multiscale properties inherent in peridynamics and which results from the phenomenological horizon parameter that limits the extent of interactions. In the third system, both the Vlasov-Poisson and cold-ion systems for modeling the expansion of ions from high to low-density regions are considered. The project includes the development of a new cold-ion model that remains valid beyond the time of singularity creation, showing that solutions of the new model remain the limit (as the ion temperature tends to zero) of solutions of the Vlasov-Poisson system, studying analytically and computationally the transition that occurs as the density ratio decreases from solutions having no singularities to ones that do, treating problems with multiple species of ions, and developing efficient numerical methods for two and three-dimensional settings.The project addresses fundamental issues that arise in the mathematical and computational treatment of three physical systems that are of huge importance in a wide variety of applications. These include but are not limited to the nucleation and propagation of defects, (cracks, delaminations, etc.) in solid bodies (airplane wings, nuclear reactor containment vessels, etc.); anomalous diffusive behavior observed in subsurface water, oil, and gas flows, in animal foraging behaviors, in polymeric flows, in exotic materials, etc.; and in ionized flows in lasers, space propulsion systems, supernovae, etc. Obtaining useful information about such complex systems requires massive computational efforts which can be greatly aided by improvements in mathematical models, qualitative theoretical information about solutions of those models, and, most of all, by better, more efficient and more accurate computational algorithms. Because of the wide-ranging settings that the project impacts, the results obtained will be of great interest to scientists, engineers, and policy makers involved in, among other things, the theoretical study of physical phenomena, in the design and manufacture of new devices, and in the assessment of risks and design of remediations.