This thesis develops a novel meshfree numerical method for simulating general fluid flows. Drawing from concepts in optimal mass transport theory and in combination with the notion of material point sampling and meshfree interpolation, the optimal transport meshfree (OTM) method provides a rigorous mathematical framework for numerically simulating three-dimensional general fluid flows with general, and possibly moving boundaries (as in fluid-structure interaction simulations). Specifically, the proposed OTM method generalizes the Benamou-Brenier differential formulation of optimal mass transportation problems which leads to a multi-field variational characterization of general fluid flows including viscosity, equations of state and general geometries and boundary conditions. With the use of material point sampling in conjunction with local max-entropy shape functions, the OTM method leads to a meshfree formulation bearing a number of salient features. Compared with other meshfree methods that face significant challenges to enforce essential boundary conditions as well as couple to other methods, such as the finite element method, the OTM method provides a new paradigm in meshfree methods. The OTM method is numerically validated by simulating the classical Riemann benchmark example for Euler flow. Furthermore, in order to highlight the ability of the OTM to simulate Navier-Stokes flows within general, moving three-dimensional domains, and naturally couple with finite elements, an illustrative strongly coupled FSI example is simulated. This illustrative FSI example, consisting of a gas-inflated sphere impacting the ground, is simulated as a toy model of the final phase of NASA's landing scheme devised for Mars missions, where a network of airbags are deployed to dissipate the energy of impact.udud
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机译:本文提出了一种新型的无网格数值模拟方法来模拟一般的流体流动。借鉴最佳质量传输理论中的概念,并结合材料点采样和无网格插值的概念,最佳传输无网格(OTM)方法提供了一个严格的数学框架,用于数值模拟具有一般运动和可能运动的三维一般流体流动边界(如在流固耦合模拟中)。具体而言,提出的OTM方法概括了最佳质量运输问题的Benamou-Brenier微分公式,这导致了一般流体的多场变化特征,包括粘度,状态方程和一般几何形状以及边界条件。通过将材料点采样与局部最大熵形状函数结合使用,OTM方法可生成无网格配方,并具有许多显着特征。与强制执行基本边界条件以及与其他方法(例如有限元方法)面临巨大挑战的其他无网格方法相比,OTM方法在无网格方法中提供了新的范例。通过模拟欧拉流的经典Riemann基准示例对OTM方法进行了数值验证。此外,为了突出OTM在一般的,移动的三维域内自然地与有限元耦合的Navier-Stokes流进行仿真的能力,模拟了一个示例性的强耦合FSI示例。这个说明性的FSI示例由一个撞击地面的充气球组成,被模拟为NASA为火星任务设计的着陆方案最后阶段的玩具模型,其中部署了安全气囊网络以消散冲击能量。 ud ud
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