SPH法：原理与工程应用解析

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SPH（Smoothed Particle Hydrodynamics，光滑粒子流体动力学）是一种数值模拟技术，用于解决流体力学问题，尤其在天体物理学中极为常见。这种方法的独特之处在于它不依赖于网格，而是将流体视为由大量粒子组成，每个粒子代表一小块流体，其状态通过周围邻近粒子的属性来估算。这种非结构化的方法使得SPH能够处理复杂形状和变形流动，特别是在星体演化、恒星坍缩形成黑洞以及星际物质交互等动态场景中非常适用。 SPH的基本原理基于Lagrange描述，每个粒子不仅代表局部物理量，如质量、速度和压力，还负责存储这些变量对邻近区域的加权平均值。这种方法通过计算每个粒子与其邻居之间的“重量函数”（smoothing kernel）来实现空间和时间上的平滑处理。这使得SPH能够自然地处理自由表面和界面，以及适应不同尺度的物理过程。在应用方面，"arXiv:astro-ph/0507472v120Jul2005"提到的测试程序是研究者在开发和完善SPH算法时的一个例子。作者乔治·格林斯坦通过各种极限条件和假设对程序进行了严格的测试，包括处理高质量恒星和低质量恒星、极端高温和低温环境，以及考虑超流体底层的影响。这个过程揭示了一个错误，最终导致他修复了程序，使其更加准确无误。 " FrozenStar:OfPulsars,BlackHolesandtheFateofStars"这本书可能深入探讨了SPH在研究脉冲星、黑洞和恒星命运中的具体应用，展示了如何用这种方法模拟这些天体的生命周期，从恒星的形成到死亡，包括引力塌缩、物质喷流和星风等现象。 SPH作为数值模拟的重要工具，在天体物理学领域扮演着关键角色，它的优点在于灵活性、适应性和无需预设网格的优点，使得它在处理天体系统中复杂的动态行为时表现出色。然而，与任何数值方法一样，优化和验证算法精度始终是关键，正如作者格林斯坦所强调的，确保程序的正确性和鲁棒性对于获得可靠的结果至关重要。

38 Chapter 3. Smoothed Particle Hydrodynamics

Figure 3.1: Examples of SPH smoothing kernels (solid line) together with their ﬁrst (dashed) and second

(dot-dashed) derivatives. Kernels correspond to those given in the text. The cubic spline (top left) is the

usual choice, whilst the quintic (top, middle) represents a closer approximation to the Gaussian kernel (top

right), at the cost of increased compact support. The bottom row correspond to various quintic kernels with

compact support of 2h which we derive in §3.2.6. The stability properties of all these kernels are compared

in Figure 3.2.

SPH, such as the use of Delaunay triangulations (Pelupessy et al., 2003) and normalisations of the kernel

interpolant (involving matrix inversion) which guarantee exact interpolations to arbitrary polynomial orders

(Maron and Howes, 2003; Bonet and Lok, 1999). It remains to be seen whether any such alternative proposals

are viable in terms of the gain in accuracy versus the inevitable increase in computational expense and

algorithmic complexity.

3.2 Basic formalisms 39

Finally we note that in most SPH codes, the kernel is evaluated by linear interpolation from a pre-

computed table of values, since kernel evaluations are computed frequently. The computational cost involved

in calculating the kernel function is therefore the same whatever the functional form. In the calculations given

in this thesis, the kernel is tabulated as W(q) and

∂

q, where the table is evenly spaced in q

to give a

better interpolation in the outer edges.

3.2.6 A general class of kernels

In this section we consider the possibility of constructing kernels based on smoother splines than the cubic

but which retain compact support of size 2h. A general class of such kernels may be derived by considering

kernels of the form

f(q) =











(r−q)

+ A(

−q)

+ B(

−q)

, 0 ≤q <

;

(r−q)

+ A(

−q)

≤ q <

;

(r−q)

≤ q < r;

0. q ≥ r

(3.37)

where n is the order, r is the compact support size (in this case r = 2), A and B are parameters to be de-

termined and

and

are the two matching points (with 0 <

< r), although an arbitrary number of

matching points could be added. The formulation given above guarantees that the kernel and its derivatives

are continuous at the matching points and zero at the compact support radius W(r) = dW/dq(r) = 0. To de-

termine the parameters A and B we require two further constraints on the form of the kernel. For the kernels

to resemble the Gaussian, we constrain the kernel gradient to be zero at the origin and also that the second

derivative be minimum at the origin (this also constrains n ≥ 3), ie.

′

(0) = 0, W

′′′

(0) = 0. (3.38)

For the moment we leave the matching points as free parameters. From the conditions (3.38), the parameters

A and B are given in terms of the matching points by

A =

n−3

−

)

n−3

(

−

)

, B = −

n−1

+ A

n−1

. (3.39)

In one dimension the normalisation constant is given by

n+ 1

2(A

n+1

+ B

n+1

+ r

n+1

)

. (3.40)

As an example we can construct a quintic (n = 5) kernel that closely resembles the cubic spline kernel

(3.33) in all but the continuity of the second derivative. An example of such a kernel is given by the choice

= 0.85,

= 1.87. This was chosen by constraining the second derivative to be equal to that of the cubic

spline at the origin (ie. W

′′

(0) = −2) and the turning point in the second derivative to be located as close

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