COLLABORATIVE PROPOSAL: Enhanced Least-Squares Methods for PIV Analysis

粒子图像测速（PIV）是一种基于粒子在图像之间的平移来获得流体速度场的方法，其中图像之间具有已知的时间跨度。 PIV的一个潜在的局限性是，只能从二维图像中获得沿单个平面沿着的二维速度场。 通常通过设计实验流动系统使第三速度分量为零或不重要来克服这一限制。 然而，在许多应用中，不可能简化完全三维的速度场（例如，在心脏的左心室中），以及与PIV分析相关的二维限制是必须克服的重要问题。 然而，是否有可能将二维PIV数据与Navier-Stokes方程的全三维数值近似结合在一起，并在心脏左心室等区域获得足够精确的三维速度场？ 这里提出了使用最小二乘有限元法（LSFEM）近似求解Navier-Stokes方程，并显著地，弱约束的解决方案沿着PIV平面匹配的实验数据。 PIV数据将基本上充当内部边界，并且数值解将与具有可变权重的数据弱匹配，该可变权重确定数据与数值解之间的耦合强度。 LSFEM非常适合以计算效率高的方式解决此类过约束问题。超声心动图专家已经开发出将微泡引入循环血液的方法，这些方法可以使用超声来解决。 微泡的位置与超声的高时间分辨率相结合，允许使用粒子图像测速法确定局部血液速度，但高时间分辨率要求也将超声扫描（以及因此的速度场数据）限制为二维。 使用FDA批准的微泡的一个目标是使用血流速度数据来计算生理学上重要的信息，例如血流的压力梯度和能量损失，但是这些计算需要完整的三维速度场。 拟议的研究的目标是开发数学技术，结合联合收割机计算流体动力学与实验速度数据，如从微泡，以获得一个完整的，三维的速度场，从而提供更深入的流动动力学。

Particle Image Velocimetry (PIV) is a method for obtaining a fluid velocity field based on the translation of particles between images with a known time span between them. A potential limitation of PIV is that only two-dimensional velocity field along a single plane can be obtained from a two-dimensional image. This limitation is normally overcome by designing the experimental flow system so that the third velocity component is either zero or unimportant. In many applications, however, it is not possible to simplify the fully three-dimensional velocity field (e.g., in the left ventricle of the heart), and the two-dimensional limitations associated with PIV analysis are a significant problem that must be overcome. Would it be possible, however, to combine the two-dimensional PIV data together with a fully three-dimensional numerical approximation to the Navier-Stokes equations and obtain a sufficiently accurate three-dimensional velocity field in a domain such as the left ventricle of the heart? The use of least-squares finite element methods (LSFEMs) is proposed here to approximately solve the Navier-Stokes equations, and, significantly, to weakly constrain the solution along the PIV plane to match the experimental data. The PIV data would basically act as an internal boundary, and the numerical solution would weakly match the data with a variable weighting that determines the strength of the coupling between the data and numerical solution. LSFEMs are uniquely well suited for solving an over-constrained problem like this in a computationally efficient manner.Echocardiologists have developed methods for introducing microbubbles into circulating blood that can be resolved using ultrasound. The location of the microbubbles combined with the high temporal resolution of ultrasound allows the local blood velocity to be determined using Particle Image Velocimetry, but the high temporal resolution requirement also limits the ultrasound scans (and, hence, the velocity field data) to two dimensions. One goal of using the FDA approved microbubbles is to use the blood velocity data to calculate physiologically important information such as pressure gradients and energy loss for the blood flow, but these calculations require a full three-dimensional velocity field. The goal of the proposed research is to develop mathematical techniques that combine computational fluid dynamics with experimental velocity data, such as that from microbubbles, to obtain a full, three-dimensional velocity field, thus providing greater insight into the dynamics of the flow.