Design of optimum grillages using layout optimization
利用布局优化设计最佳格栅

Abstract 摘要

Grillages are often used to form bridge decks and other constructions. However, following a period of intensive research activity in the 1970s, comparatively little attention has been paid to optimizing the layout of grillages in recent years. In this contribution a new numerical procedure is proposed which takes advantage of the adaptive solution scheme previously developed for truss layout optimization problems, enabling very large scale problems to be solved. A key benefit of the proposed numerical procedure is that it is completely general, and can therefore be applied to problems with arbitrary loading and boundary conditions. Also, unlike some previously proposed procedures, the sizes of individual beams can readily be discerned. To demonstrate its efficacy the numerical procedure is applied to a range of grillage layout design problems, including load dependent problems which could not be solved using traditional methods. It is shown that important phenomena such as “beam-weaves” can be faithfully captured and new high-precision numerical benchmark solutions are provided.
格栅通常用于桥面和其他建筑。然而，继 20 世纪 70 年代的密集研究活动之后，近年来对格栅布局优化的关注相对较少。本文提出了一种新的数值计算程序，该程序利用了之前针对桁架布局优化问题开发的自适应求解方案，能够解决非常大规模的问题。所提出的数值计算程序的一个主要优点是它具有完全的通用性，因此可应用于具有任意载荷和边界条件的问题。此外，与之前提出的一些程序不同，它可以很容易地辨别单个梁的尺寸。为了证明该数值程序的有效性，我们将其应用于一系列格栅布局设计问题，包括传统方法无法解决的与荷载相关的问题。结果表明，"梁波 "等重要现象可以被忠实地捕捉到，并提供了新的高精度数值基准解决方案。

1 Introduction 1 引言

A grillage is a planar network of intersecting beams, often used to form bridge decks and other constructions. The first significant research on beam and frame optimization dates back to the 1950s and 1960s (Heyman 1959; Morley 1966), and work on grillages gathered pace in the early 1970s with the publication of papers by Rozvany (1972a, b) and Lowe and Melchers (1972, 1973). These authors viewed the plastic design of grillages in a continuous setting, considering a notional slab comprising an infinite number of fibre-like beams. An optimum fibrous slab can be considered as analogue to an in-plane Michell structure, which is well known in the structural optimization research community; further development of the theory of Michell structures (Michell 1904) has been described by workers such as Chan (1967), Hemp (1973) and Lewinski et al. (1994a, b). Also, a numerical means of identifying Michell-type structures using the “ground structure” approach was proposed by Dorn et al. (1964) and further developed by workers such as Gilbert and Tyas (2003), Sokol (2014), and Zegard and Paulino (2014).
格栅是由相交横梁组成的平面网络，通常用于构成桥面和其他建筑。最早关于梁和框架优化的重要研究可追溯到 20 世纪 50 和 60 年代（海曼，1959 年；莫里，1966 年），而在 20 世纪 70 年代初，随着罗兹万尼（1972a, b）以及洛尔和梅尔切斯（1972 年，1973 年）的论文发表，格栅研究的步伐加快了。这些作者从连续的角度看待格栅的塑性设计，考虑了由无数纤维状梁组成的概念板。最佳纤维板可视为平面内 Michell 结构的类似物，这在结构优化研究领域是众所周知的；Chan（1967 年）、Hemp（1973 年）和 Lewinski 等人（1994a, b）等人对 Michell 结构（Michell 1904 年）理论的进一步发展进行了描述。此外，Dorn 等人（1964 年）提出了一种使用 "地面结构 "方法识别 Michell 型结构的数值方法，Gilbert 和 Tyas（2003 年）、Sokol（2014 年）以及 Zegard 和 Paulino（2014 年）等人进一步发展了这种方法。

Any structural design optimization problem can be posed in either equilibrium (primal) or kinematic (dual) form, where for a grillage the problem variables are usually moments and rotations in the equilibrium and kinematic forms respectively. However, in the paper by Rozvany (1972a) neither the equlibrium nor kinematic problem formulations are solved directly; instead a displacement-based, fully analytical method of finding the solution of the kinematic problem is proposed which essentially stems from the stress-strain optimality relation linking the solutions of the equilibrium and kinematic forms. The associated optimization problem is also confined to being applied to fully clamped slabs subject to an arbitrary, though always downward, loading. Significantly, for this class of problem the optimum layout is load-independent. This remarkable feature, combined with Rozvany’s kinematic method, provided a means of obtaining universal exact optimum grillage layouts for problems involving downward loads for both single and multiple load cases. It should however be noted that this does not furnish the optimal distribution of beam widths. For this one obviously needs to know the magnitudes of the particular loads involved, and to use the governing equilibrium equation to determine the corresponding optimal bending moment field and thus the beam widths. However, the papers by Rozvany (1972a) and by Lowe and Melchers (1972) do not describe systematic means of recovering the beam width distribution.
任何结构设计优化问题都可以用平衡（原始）或运动（对偶）形式提出，对于格栅来说，问题变量通常分别是平衡形式和运动形式下的力矩和旋转。然而，在 Rozvany（1972a）的论文中，平衡和运动问题公式都没有直接求解，而是提出了一种基于位移的、完全分析的方法来寻找运动问题的解，这种方法主要源于将平衡和运动形式的解联系起来的应力-应变优化关系。相关的优化问题也仅限于应用于承受任意（但总是向下）荷载的全夹紧板。值得注意的是，对于这类问题，优化布局与荷载无关。这一显著特点与 Rozvany 的运动学方法相结合，为涉及单载荷和多载荷情况下的向下载荷问题提供了一种获得通用精确最佳格栅布局的方法。但应注意的是，这并不能提供梁宽的最佳分布。为此，我们显然需要知道所涉及的特定荷载的大小，并使用控制平衡方程来确定相应的最佳弯矩场，从而确定梁宽。然而，Rozvany（1972a）和 Lowe 与 Melchers（1972）的论文并未描述恢复梁宽分布的系统方法。

In subsequent decades Rozvany’s analytical kinematic method was applied to slabs with a range of other boundary conditions, including simply supported edges and combinations of free and simply supported edges, and also simply supported and clamped edges (Rozvany et al. 1973; Rozvany and Hill 1976; Prager and Rozvany 1977; Rozvany and Liebermann 1994). For each of the aforementioned cases the method proved capable only of solving problems involving exclusively downward loading, as it explicitly relies on the fact that the optimum layout is load-independent. The method was then implemented by Hill and Rozvany (1985) in a computer program which allowed automatic generation of analytical optimum layouts for arbitrary polygonal slabs with partially clamped and simply supported boundaries. The authors presented exact optimum layouts for an impressive range of complex polygonal domain shapes.
在随后的几十年中，Rozvany 的分析运动学方法被应用于具有一系列其他边界条件的楼板，包括简单支撑边缘、自由边缘和简单支撑边缘的组合，以及简单支撑边缘和夹紧边缘（Rozvany 等人，1973 年；Rozvany 和 Hill，1976 年；Prager 和 Rozvany，1977 年；Rozvany 和 Liebermann，1994 年）。对于上述每一种情况，该方法都被证明只能解决只涉及向下载荷的问题，因为它明确依赖于最佳布局与载荷无关这一事实。随后，Hill 和 Rozvany（1985 年）在一个计算机程序中实施了该方法，该程序允许自动生成具有部分夹紧和简单支撑边界的任意多边形板的分析最优布局。作者提出了一系列令人印象深刻的复杂多边形域形状的精确优化布局。

It should be noted that although Rozvany’s method is capable of treating interior clamped supports, it cannot account for interior simple supports. This is because uplift may occur if such supports are present, which in turn means that there is no longer a universal kinematic solution, common for all types of downward load. This is also the case if mixed downward / upward loadings are present, or if point moment loadings are present. Less trivially, this also applies to slabs with partially clamped and free edges.
值得注意的是，尽管罗兹万尼的方法能够处理内部夹紧支撑，但却无法解释内部简支撑。这是因为如果存在此类支撑，则可能会发生上浮，这反过来又意味着不再有通用于所有类型向下荷载的通用运动学解决方案。如果存在混合向下/向上荷载，或者存在点力矩荷载，情况也是如此。不太明显的是，这也适用于部分夹紧和自由边缘的楼板。

The load-sensitivity of many real-world problems encouraged researchers to seek general numerical methods. In the paper by Sigmund et al. (1993) the ground structure method was, apparently for the first time, used to obtain solutions to the grillage compliance minimization problem. (A “ground structure” comprises a network of structural members interconnecting nodes laid out on a grid from which the subset of members defining the optimum structure is sought, after Dorn et al. 1964.) By using the DCOC method in combination with linearly tapering beam finite elements the authors found a number of new grillage layouts, showing solutions for problems involving clamped and free edges; later the method was applied to problems involving mixed downward and upward loadings (Rozvany 1997). Low resolution ground structures were used, in part because of the available computing capabilities of the time. However because the method does not take advantage of modern adaptive solution schemes (e.g. Gilbert and Tyas 2003), the scale of problems that can be tackled even now appears to be limited. More recently Zhou (2009) proposed a method which involved recovering principal moment trajectories, but this is likely to be rather cumbersome in practice.
许多实际问题对荷载的敏感性促使研究人员寻求通用的数值方法。在 Sigmund 等人的论文（1993 年）中，地面结构法显然是第一次被用来求解格栅顺应性最小化问题。(地面结构 "包括一个由结构构件组成的网络，这些结构构件相互连接，节点布置在网格上，根据 1964 年 Dorn 等人的研究，从网格中寻找确定最佳结构的构件子集）。通过将 DCOC 方法与线性锥形梁有限元相结合，作者发现了许多新的格栅布局，显示了涉及夹紧和自由边缘问题的解决方案；后来，该方法被应用于涉及混合向下和向上荷载的问题（Rozvany，1997 年）。使用低分辨率地面结构的部分原因是当时可用的计算能力。然而，由于该方法没有利用现代自适应求解方案（如 Gilbert 和 Tyas，2003 年），即使是现在，能解决的问题规模似乎也很有限。最近，Zhou（2009 年）提出了一种涉及恢复主矩轨迹的方法，但这在实践中可能相当麻烦。

In summary, although the analytical approach initiated by Rozvany and co-workers had reasonably broad applicability, and allowed new insights to be drawn, it left a wide range of grillage optimization problems unsolvable, due to their inherent load sensitivity. Moreover, even for grillage problems which could be solved, the optimum beam width distribution was not identified. In the present paper the authors propose that a ground structure approach is adopted, and that techniques now well established in the field of truss layout optimization (e.g. see Gilbert and Tyas 2003; Sokol 2014) or limit analysis via discontinuity layout optimization (see Smith and Gilbert 2007; Gilbert et al. 2014) are applied. Here a plastic design formulation is used by posing two mutually dual linear programming forms: equilibrium and kinematic. The goal is to minimize the volume of material for specified applied loading. This leads to a simple linear formulation which can be used in conjunction with an adaptive solution scheme to solve problems involving ground structures consisting of many million beams, thus generating optimum layouts closely approximating the analytical solutions found by Rozvany et al., though with a far greater range of applicability.
总之，尽管罗兹万尼及其合作者提出的分析方法具有相当广泛的适用性，并能得出新的见解，但由于其固有的荷载敏感性，使得大量格栅优化问题无法解决。此外，即使是可以解决的格栅问题，也无法确定最佳梁宽分布。在本文中，作者建议采用地面结构方法，并应用目前在桁架布局优化（例如，见 Gilbert 和 Tyas，2003 年；Sokol，2014 年）或通过不连续布局优化进行极限分析（见 Smith 和 Gilbert，2007 年；Gilbert 等，2014 年）领域成熟的技术。这里采用的是塑性设计方案，提出两个相互对偶的线性编程形式：平衡编程和运动编程。目标是在指定的应用载荷下使材料体积最小化。这就产生了一个简单的线性公式，可与自适应求解方案结合使用，以解决涉及由数百万根梁组成的地面结构的问题，从而产生近似于 Rozvany 等人所发现的分析解决方案的最佳布局，但适用范围要大得多。

2 Formulation 2 配方

2.1 Equilibrium formulation
2.1 平衡公式

Consider a ground structure (Dorn et al. 1964) consisting of a design domain discretized using n nodes and b beams, as shown on Fig. 1. For beam i, assume its cross sectional area varies linearly from a_i1 to a_i2. The total volume V of the structure can be written as:
考虑一个地面结构（Dorn 等人，1964 年），如图 1 所示，该结构由一个设计域（使用 n 个节点和 b 根梁离散）组成。对于横梁 i，假设其横截面积从 a _i1 到 a _i2 呈线性变化。结构的总体积 V 可以写成

$$V={\mathbf{l}}^{\mathrm{T}}\mathbf{a},$$

(1)

where $\mathbf{l}=[{\displaystyle \frac{l_1}{2}},{\displaystyle \frac{l_1}{2}},{\displaystyle \frac{l_2}{2}},{\displaystyle \frac{l_2}{2}},...,{\displaystyle \frac{l_b}{2}},{\displaystyle \frac{l_b}{2}}{]}^{\mathrm{T}}$ and a = [a₁₁, a₁₂, a₂₁, a₂₂,...,a_b1, a_b2]^T are, respectively, vectors of beam lengths and areas.
其中 $\mathbf{l}=[{\displaystyle \frac{l_1}{2}},{\displaystyle \frac{l_1}{2}},{\displaystyle \frac{l_2}{2}},{\displaystyle \frac{l_2}{2}},...,{\displaystyle \frac{l_b}{2}},{\displaystyle \frac{l_b}{2}}{]}^{\mathrm{T}}$ 和 a = [a ₁₁ , a ₁₂ , a ₂₁ , a ₂₂ ,...,a _b1 , a _b2 ]。 ^T 分别是光束长度和面积的矢量。

Fig. 1 图 1

Ground structure for a design domain, in this case a simple square domain discretized using n = 9 nodes and b = 36 interconnecting beams (including overlaps)
设计域的地面结构，在本例中是一个简单的正方形域，使用 n = 9 个节点和 b = 36 个相互连接的梁（包括重叠部分）进行离散处理

For each node, moment equilibrium needs to be enforced in the x and y directions and force equilibrium in the z direction. Denoting m_i1 and m_i2 as the moments at the two ends of beam i, the local equilibrium matrix can be expressed as:
对于每个节点，需要在 x 和 y 方向上实现力矩平衡，在 z 方向上实现力平衡。将 m _i1 和 m _i2 表示梁 i 两端的力矩，局部平衡矩阵可表示为

$${\mathbf{B}}_i=\left[\begin{array}{cc}-\sin {\theta }_i& 0\\ \cos {\theta }_i& 0\\ {\displaystyle \frac{1}{l_i}}& -{\displaystyle \frac{1}{l_i}}\\ 0& -\sin {\theta }_i\\ 0& \cos {\theta }_i\\ -{\displaystyle \frac{1}{l_i}}& {\displaystyle \frac{1}{l_i}}\end{array}\right],$$

(2)

where 𝜃_i is the angle of beam i to the x⁺ axis, and l_i is its length.
其中，𝜃 _i 是光束 i 与 x ⁺ 轴的夹角，l _i 是其长度。

Also, assuming that the beam cross-sections are of uniform depth, let $m_p^{+}$ and $m_p^{-}$ denote the limiting moments per unit area. The yield condition of beam i can thus be written as:
另外，假设梁的横截面深度均匀，让 $m_p^{+}$ 和 $m_p^{-}$ 表示单位面积的极限力矩。因此，梁 i 的屈服条件可写成

$$-m_p^{-}{a}_i\le m_{i1},m_{i2}\le m_p^{+}{a}_i.$$

(3)

The grillage layout optimization problem can therefore be written as:
因此，格栅布局优化问题可以写为

$$\begin{array}{rcl}\underset{\mathbf{a},\mathbf{q}}{min}& & V={\mathbf{l}}^{\mathrm{T}}\mathbf{a}\end{array}$$

(4a)

$$\begin{array}{rcl}\text{s.t.}& & \mathbf{B}\mathbf{q}=\mathbf{f}\end{array}$$

(4b)

$$\begin{array}{rcl}& & -m_{\mathrm{p}}^{-}\mathbf{a}\le \mathbf{q}\le m_{\mathrm{p}}^{+}\mathbf{a}\end{array}$$

(4c)

$$\begin{array}{rcl}& & \mathbf{a}\ge \mathbf{0},\end{array}$$

(4d)

where q = [m₁₁, m₁₂, m₂₁, m₂₂,...,m_b1, m_b2]^T is a vector containing the moments at the two ends of each beam. B is the global equilibrium matrix, assembled from the local matrices B_i given in (2) and $\mathbf{f}=[m_1^x,m_1^y,f_1^z,m_2^x,m_2^y,f_2^z,...,m_n^x,m_n^y,f_n^z{]}^{\mathrm{T}}$ is the external loading applied at each node. Problem (4) is a linear programming (LP) problem which can be solved using well-developed algorithms.
其中 q = [m ₁₁ , m ₁₂ , m ₂₁ , m ₂₂ ,... , m _b1 , m _b2 ]。 ^T 是包含每根梁两端力矩的向量。B 是全局平衡矩阵，由 ( 2) 中给出的局部矩阵 B _i 组合而成， $\mathbf{f}=[m_1^x,m_1^y,f_1^z,m_2^x,m_2^y,f_2^z,...,m_n^x,m_n^y,f_n^z{]}^{\mathrm{T}}$ 是施加在每个节点上的外部荷载。问题 ( 4) 是一个线性规划（LP）问题，可以用成熟的算法解决。

2.2 Adaptive solution scheme
2.2 自适应解决方案

When a fully-connected ground structure is used the number of beams b grows rapidly with the number of nodes n, limiting the size of problem that can be solved (since in this case b = n(n − 1)/2). This issue was addressed for truss layout optimization problems by Gilbert and Tyas (2003) who proposed an adaptive solution scheme, later further developed by Sokol (2014). This scheme employs an initial sparsely connected ground structure and uses constraints from the dual problem to check whether the solution could potentially be improved by adding additional members, as part of an iterative process. As problem formulation (4) is very similar to the truss formulation used by e.g. Gilbert and Tyas (2003), the same basic technique can be applied in this case, where the dual formulation of (4) involves maximizing virtual work:
当使用全连接地面结构时，梁的数量 b 会随着节点数 n 的增加而快速增长，从而限制了可求解问题的规模（因为在这种情况下，b = n(n-1)/2）。Gilbert 和 Tyas（2003 年）针对桁架布局优化问题提出了一种自适应求解方案，后来 Sokol（2014 年）进一步发展了这一方案。该方案采用初始稀疏连接的地面结构，并利用对偶问题中的约束条件来检查是否可以通过增加额外的构件来改进求解，这是迭代过程的一部分。由于问题表述 ( 4) 与 Gilbert 和 Tyas（2003 年）等人使用的桁架表述非常相似，因此相同的基本技术也可应用于本案例，其中 ( 4) 的对偶表述涉及最大化虚拟功：

$$maxW={\mathbf{f}}^{\mathrm{T}}\mathbf{u},$$

(5)

where, W is the virtual work and u collects the virtual rotations in the x and y directions and out-of-plane displace- ment in the z direction. Also the following constraint must be satisfied:
其中，W 是虚功，u 收集了 x 和 y 方向上的虚旋转和 z 方向上的平面外位移。此外，还必须满足以下约束条件：

$$-{\displaystyle \frac{\mathbf{l}}{m_p^{-}}}\le {\mathbf{B}}^{\mathrm{T}}\mathbf{u}\le {\displaystyle \frac{\mathbf{l}}{m_p^{+}}},$$

(6)

which imposes limits on the maximum and minimum virtual rotation that can occur in each beam. Note that u is obtained automatically after solving (4), and (6) is only guaranteed to be satisfied in beams that are present in (4). This means that potential beams, not currently represented in the problem, may violate (6); in this case the beams most in violation should be added to problem (4) to prevent this violation in the next iteration. The process repeats until no violation is found in (6); for further details of the algorithm readers are referred to Gilbert and Tyas (2003) and Sokol (2014).
这对每个梁中可能出现的最大和最小虚拟旋转施加了限制。需要注意的是，u 是在求解 ( 4) 后自动获得的，而 ( 6) 只保证满足 ( 4) 中存在的梁。这意味着当前问题中没有体现的潜在横梁可能会违反 ( 6)；在这种情况下，应将违反情况最严重的横梁添加到问题 ( 4) 中，以防止下一次迭代中出现这种违反情况。该过程重复进行，直到没有发现违反 ( 6) 的情况；关于该算法的更多细节，读者可参阅 Gilbert 和 Tyas ( 2003) 以及 Sokol ( 2014)。

2.3 Commentary 2.3 评注

The grillages considered herein are assumed to be rigid-jointed, but with torsional resistance neglected. This assumption, also made by Sigmund et al. (1993), is justified by the fact that for most cross-sections used in practice the torsional resistance is low compared with the bending resistance. This is particularly true for open cross-sections, such as I-beams. In the latter case by varying only the flange width the linearity of the formulation is preserved.
此处考虑的格栅假定为刚性连接，但忽略了抗扭性。Sigmund 等人（1993 年）也做出了这一假设，其理由是，与弯曲阻力相比，实际中使用的大多数横截面的扭转阻力较小。这一点对于工字钢等开口截面尤为明显。在后一种情况下，只需改变翼缘宽度，就能保持公式的线性。

Now consider beam i such that m_i1 ⋅ m_i2 < 0, i.e. the bending moment changes sign across the length of the beam. Notice that restricting the cross sectional area function to vary linearly from a_i1 = |m_i1| /m_p to a_i2 = |m_i2| /m_p will not lead to an optimal beam being generated in this case, since each intermediate point along the length of the beam is overdesigned (e.g. consider the intermediate point where the bending moment vanishes). This can potentially be addressed in two ways: (i) via use of a non-linear relation for the grillage volume (1) (see Bolbotowski 2018); (ii) ensuring that a large number of nodes and interconnecting beams are employed in the problem, such that any inaccuracy is small. A drawback with (i) is that it requires the use of computationally expensive non-linear optimizers and thus (ii) is adopted here. However, note that the single load case plastic design problem considered here is only equivalent to the corresponding compliance minimization problem when (i) is adopted; the same holds for the grillage-like continuum addressed by Rozvany et al.
现在考虑横梁 i，使 m _i1 ⋅ m _i2 < 0，即弯矩在横梁长度上改变符号。请注意，限制横截面积函数从 a _i1 = |m _i1 | /m _p 到 a _i2 = |m _i2 | /m _p 的线性变化将不会导致在这种情况下生成最佳横梁，因为横梁长度上的每个中间点都是过度设计的（例如，考虑弯矩消失的中间点）。解决这一问题的潜在方法有两种：(i) 使用格栅体积的非线性关系 ( 1)（见 Bolbotowski 2018）；(ii) 确保在问题中使用大量节点和相互连接的梁，这样任何误差都会很小。(i)的缺点是需要使用计算昂贵的非线性优化器，因此这里采用(ii)。不过，请注意，此处考虑的单载荷情况下的塑性设计问题仅等同于采用(i)时相应的顺应性最小化问题；Rozvany 等人研究的格栅状连续体问题也是如此。

The above argument suggests that, when $m_p^{+}=m_p^{-}=m_p$, the volume V in (4a) approximates to the scaled (by m_p) integral of the absolute value of the bending moment diagram taken over the grillage. Thus when a high resolution ground structure is adopted the equilibrium form (4) can be viewed as a discrete version of the continuous problem addressed in Rozvany (1972a), Save and Prager (1985) and others.
上述论证表明，当 $m_p^{+}=m_p^{-}=m_p$ 时，( 4a) 中的体积 V 近似于格栅上弯矩图绝对值的按比例（按 m _p ）积分。因此，当采用高分辨率地面结构时，可将 ( 4) 中的平衡形式视为 Rozvany ( 1972a)、Save 和 Prager ( 1985) 等人研究的连续问题的离散版本。

In the field of truss optimization it is well-established that when a single load case is involved there must always exist a statically determinate optimum truss solution; see for example Achtziger (1997). The simplest proof of this statement revolves around existence of a basic solution to the underlying LP problem. As this also applies to the grillage optimization problem (4), it follows that there must always exist a statically determinate optimum grillage.
在桁架优化领域，当涉及单一载荷情况时，必须始终存在一个静态确定的最佳桁架解决方案，这一点已得到公认；例如，请参见 Achtziger ( 1997)。对这一论断的最简单证明就是基本 LP 问题存在基本解。由于这也适用于格栅优化问题 ( 4)，因此必然存在一个静态确定的最优格栅。

Finally, although thus far attention has focussed on single load case problems, it is well known that the plastic truss layout optimization can be extended to treat multiple load case problems (e.g. see Hemp 1973), and, though beyond the scope of the present contribution, it is worth pointing out that the grillage layout optimization formulation described herein can be similarly extended.
最后，尽管到目前为止，人们的注意力主要集中在单一荷载情况下的问题上，但众所周知，塑性桁架布局优化可以扩展到处理多重荷载情况下的问题（例如，见 Hemp 1973），尽管超出了本文的研究范围，但值得指出的是，本文所述的格栅布局优化公式也可以进行类似的扩展。

3 Numerical examples 3 数字示例

The proposed numerical method was programmed independently in both Mathematica 11.1.0.0 and Matlab 2015a, respectively using the default Mathematica solver and Mosek 7 to obtain solutions to the LP problems involved. In all cases tried the results obtained from the two programs were identical for all quoted significant figures, though for the larger problems the Matlab / Mosek combination was favoured due to lower associated run times. All quoted CPU times are single core values obtained using a workstation equipped with Intel Xeon E5-2680v2 processors running 64-bit CENTOS Linux.
建议的数值方法分别在 Mathematica 11.1.0.0 和 Matlab 2015a 中独立编程，使用默认的 Mathematica 求解器和 Mosek 7 求解相关 LP 问题。在所有情况下，两个程序得到的结果在所有引用的有效数字上都是相同的，但对于较大的问题，由于相关运行时间较短，Matlab 和 Mosek 的组合更受青睐。所有引用的 CPU 时间都是使用配备英特尔至强 E5-2680v2 处理器、运行 64 位 CENTOS Linux 的工作站获得的单核心数值。

The efficacy of the method is demonstrated through application to a range of numerical example problems. For sake of simplicity beam moment capacities were in all cases taken to be equal for sagging and hogging, i.e. $m_p^{+}=m_p^{-}=m_p$, and nodes were evenly distributed over each problem domain, with pressure loads approximated using point loads applied at these nodes. In this case the magnitude of each point load was calculated by taking into account the area of the surrounding domain, taking the load applied at an intermediate node along an edge to be half that applied at an interior node, and the load applied at a corner node as one quarter. For example, considering the domain shown in Fig. 1, and assuming a uniform pressure load of total magnitude pL² is applied, the loads applied to nodes A, B and I would be pL²/16, pL²/8 and pL²/4 respectively. Note that because of the presence of supports the grillage designed would in this case only need to carry a load of pL²/4, leading to an underestimate in the volume of material required. To address this load discretization error, and also the nodal discretization error that limits the range of layouts that can be identified, and hence tends to overestimate the required volume of material, most problems described were solved using a sequence of increasing nodal divisions, enabling approximations of the exact values to be obtained via extrapolation (see Appendix A for details); these latter approximations are quoted in the main text, whilst tabulated results are presented in Tables 1–3 of Appendix A. However, in the interests of visual clarity the graphical results presented correspond to problems with a moderate number of nodal divisions. Beams are drawn in blue and red to indicate sagging and hogging respectively in the graphical solutions, with line widths proportional to beam cross-sectional areas. In the interests of visual clarity, beams with very small cross-sectional areas have been filtered out.
通过对一系列数值示例问题的应用，证明了该方法的有效性。为简便起见，在所有情况下，梁的弯矩承载力都相等，即 $m_p^{+}=m_p^{-}=m_p$ ，节点均匀分布在每个问题域中，压力荷载近似使用施加在这些节点上的点荷载。在这种情况下，每个点载荷的大小是通过考虑周围域的面积来计算的，沿边缘施加在中间节点上的载荷是施加在内部节点上的载荷的一半，而施加在角落节点上的载荷是四分之一。例如，考虑到图 1 所示的域，并假设施加总大小为 pL ² 的均匀压力载荷，则节点 A、B 和 I 所受的载荷分别为 pL ² /16、pL ² /8 和 pL ² /4。需要注意的是，由于支撑物的存在，在这种情况下设计的格栅只需承受 pL ² /4 的荷载，从而导致所需材料的体积被低估。为了解决这种载荷离散化误差，以及节点离散化误差（节点离散化误差限制了可确定的布局范围，因此往往会高估所需的材料体积），所述的大多数问题都采用了节点划分递增序列来解决，从而能够通过外推法获得精确值的近似值（详见附录 A）；正文中引用了后一种近似值，附录 A 表 1-3 中列出了表列结果。不过，为了直观清晰起见，所展示的图形结果与节点划分数量适中的问题相对应。 在图形解决方案中，横梁用蓝色和红色分别表示下垂和阻塞，线宽与横梁横截面积成正比。为了视觉清晰，横截面积非常小的横梁已被过滤掉。

Symbols used in the paper are illustrated in Fig. 2. The symbols used by e.g. Rozvany (1972a) for region type are used herein to describe the analytical optimum layouts; see for example Fig. 3a. Specifically, a design domain can be divided into regions where each region is labelled to indicate the optimum directions of beams of possibly non-zero cross section, whether sagging (“+” symbol) or hogging (“−” symbol) moments are involved. The circle symbols denote so called “indeterminate regions”, where the optimum beam direction is arbitrary.
本文所用符号如图 2 所示。本文使用罗兹万尼（1972a）等人使用的区域类型符号来描述分析最优布局；例如，请参见图 3a。具体来说，设计域可划分为多个区域，每个区域都标有横截面可能不为零的梁的最佳方向，无论涉及下垂力矩（"+"符号）还是曳引力矩（"-"符号）。圆圈符号表示所谓的 "不确定区域"，即最佳横梁方向是任意的。

Fig. 2 图 2

Symbols used in the paper
文件中使用的符号

Fig. 3 图 3

Square domain with simple supports: a optimum layout derived analytically by e.g. Morley (1966), $V_{\text{exact}}=\frac{5}{96}p{L}^4/m_{\mathrm{p}}\approx 0.05208p{L}^4/m_{\mathrm{p}}$; b result obtained by numerical layout optimization, $V_{\text{num}}=\hspace{0.17em}0.05208p{L}^4/m_{\mathrm{p}}$
具有简单支撑的方形域：a 由莫里（1966 年）等人通过分析得出的最佳布局， $V_{\text{exact}}=\frac{5}{96}p{L}^4/m_{\mathrm{p}}\approx 0.05208p{L}^4/m_{\mathrm{p}}$ ；b 通过数值布局优化得出的结果， $V_{\text{num}}=\hspace{0.17em}0.05208p{L}^4/m_{\mathrm{p}}$ 。

Due to the presence of indeterminate regions it was found that the numerical layouts obtained often became complex in form. This is because the interior point method used to solve the underlying LP problem (4) will normally identify a solution that combines all possible designs in such regions, e.g. see Fig. 4b. To address this, the length vector l can be modified by adding a constant joint cost / length (j = ± 10^− 6 unit length), i.e. ${\tilde{l}}_i=l_i+j$. Joint costs were first introduced by Parkes (1978) as a simple means of rationalizing optimum trusses. Here a very small joint cost is used to ensure the numerical layout is pushed towards a basic LP solution to increase visual clarity. Furthermore, numerical tests showed that the clearest visual results could be obtained when j is taken as a small positive value for hogging beams and a small negative value for sagging beams. Notwithstanding this, all optimum volumes presented herein were computed without employing a joint cost.
由于不确定区域的存在，我们发现所获得的数值布局往往形式复杂。这是因为用于解决基本 LP 问题（4）的内点法通常会找出一个结合了这些区域内所有可能设计的解决方案，例如见图 4b。为了解决这个问题，可以通过添加一个恒定的联合成本/长度（j = ± 10 ^− 6 单位长度）来修改长度向量 l，即 ${\tilde{l}}_i=l_i+j$ 。联接成本最早由 Parkes（1978 年）提出，是使最佳桁架合理化的一种简单方法。这里使用了很小的联合成本，以确保数值布局向基本 LP 解法推进，从而提高视觉清晰度。此外，数值测试表明，如果将 j 取为小正值用于阻挡梁，取为小负值用于下垂梁，则可获得最清晰的视觉结果。尽管如此，本文介绍的所有最佳体积都是在没有采用联合成本的情况下计算得出的。

Fig. 4 图 4

Square domain with clamped supports at corners: a analytical optimum layout according to Rozvany (1972b); new result obtained by numerical layout optimization for uniform pressure load; b without joint cost ($V_{\text{num}}=\hspace{0.17em}0.06250p{L}^4/m_{\mathrm{p}}$); c with joint cost
四角带夹紧支撑的方形域：a 根据 Rozvany ( 1972b) 的分析优化布局；通过对均匀压力载荷进行数值布局优化获得的新结果；b 不含连接成本 ( $V_{\text{num}}=\hspace{0.17em}0.06250p{L}^4/m_{\mathrm{p}}$ ) ；c 含连接成本

3.1 Benchmark examples 3.1 基准实例

A range of example problems are presented, starting with problems for which closed-form analytical solutions are available. It should however be noted that although countless analytical optimum layouts were presented in the papers by Rozvany et al., optimum volume values were rarely quoted. This is due to the fact that loads were generally not specified, since the layouts derived had universal applicability for arbitrary (downward) load. However, since the optimum displacement field can be recovered from an analytical layout, the optimum volume V can be computed from the virtual work done by given load W, although the process can be laborious and hence analytical volumes will only be provided for selected example problems.
文中介绍了一系列例题，首先介绍了可以用闭式分析法求解的问题。但应注意的是，尽管罗兹万尼等人的论文中提出了无数的最佳分析布局，但很少引用最佳体积值。这是因为一般都没有指定载荷，因为得出的布局普遍适用于任意（向下）载荷。不过，由于最佳位移场可以从分析布局中恢复，因此最佳体积 V 可以从给定载荷 W 所做的虚功中计算出来，不过计算过程可能比较费力，因此分析体积只适用于选定的示例问题。

3.1.1 Square domain with simple supports
3.1.1 带简单支撑的方形域

The first example considered herein involves a square design domain with simple supports, as shown on Fig. 3a. This problem is one of the oldest and simplest to derive analytically, e.g. see Morley (1966). The solution is optimum for arbitrary downward load; one square R⁺⁺ region is present along with four triangular R⁺⁻ regions. For comparison an optimum layout for a uniform pressure load was generated via the new layout optimization method, see Fig. 3b. Since the R⁺⁺ region is indeterminate the numerical solution presented is in fact one of an infinite number of possibilities, where here the pressure load is transferred to two beams of significantly larger cross section. The four triangular regions appear to be R⁺ type rather than R⁺⁻ as proposed analytically since only sagging beams are present, with the orthogonal hogging beams vanishing. This apparent discrepancy, along with other subtle issues associated with numerical layout optimization of grillages, will be considered in the next section.
本文考虑的第一个例子涉及一个具有简单支撑的正方形设计域，如图 3a 所示。这个问题是最古老也是最简单的分析推导问题之一，例如参见 Morley ( 1966)。对于任意向下载荷，该方案都是最佳方案；存在一个正方形 R ⁺⁺ 区域和四个三角形 R ⁺⁻ 区域。为了进行比较，通过新的布局优化方法生成了均匀压力载荷的最佳布局，见图 3b。由于 R ⁺⁺ 区域是不确定的，因此所提供的数值解决方案实际上是无数可能性中的一种，在这里，压力载荷被转移到两个横截面明显更大的横梁上。四个三角形区域似乎是 R ⁺ 类型，而不是分析中提出的 R ⁺⁻ 类型，因为只有下垂梁存在，而正交掘进梁消失了。下一节将讨论这一明显差异，以及与格栅数值布局优化相关的其他微妙问题。

3.1.2 Square domain with clamped supports at corners
3.1.2 四角有夹紧支撑的方形域

The second example involves a square design domain with clamped supports at the corners, as shown in Fig. 4. This serves to illustrate the effect of the joint cost used to rationalize the solution. The analytical layout shown in Fig. 4a is proposed based on the approach described by Rozvany (1972b) for arbitrary downward load. Aside from four R⁺⁻ regions the optimum grillage comprises five indeterminate regions, four R⁻⁻ regions and a single R⁺⁺ region. For comparison an optimum layout for a uniform pressure load was generated by layout optimization, initially without using a joint cost, as shown on Fig. 4b. The numerical representation of the R⁺⁻ regions coincides perfectly with the analytical design, whereas the indeterminate regions involve numerous overlapping beams in different orientations, thus rendering the numerical solution of little practical value. However, by re-running the problem with joint costs the solution is greatly simplified, as shown on Fig. 4c. The R⁺⁺ region is now transformed to a regular grid and the R⁻⁻ regions to cantilever fans radiating out from each of the four point supports; similar fans will be observed in the vicinities of clamped point supports (or concave corners of supported boundaries) in subsequent examples. However, it is evident that the introduction of a joint cost has appeared to transform the R⁺⁻ regions into R⁺ regions, as occurred in the previous example. This is because the optimum beam width distribution is not necessarily unique for a given applied load. Thus in the example shown in Fig. 4 the particular representation of the indeterminate square R⁺⁺ region influences whether or not hogging beams are present in the adjoining R⁺⁻ region. However, the lack of a unique beam width distribution can also be demonstrated in simpler problems, without R⁺⁺ or R⁻⁻ regions. For example, consider the case of two opposing cantilever beams of equal length subjected to a shared load at their tips; also Fig. 12 serves as a further example. Although a statically determinate optimum grillage is guaranteed to exist, the structure shown in Fig. 4b is clearly not statically determinate, due to the non-uniqueness of the solution. The use of a joint cost does not necessarily fully remedy this, e.g. see Fig. 4c.
第二个例子涉及一个四角有夹紧支撑的正方形设计域，如图 4 所示。这可以说明用于合理求解的连接成本的影响。图 4a 所示的分析布局是根据 Rozvany ( 1972b) 所描述的任意向下载荷的方法提出的。除了四个 R ⁺⁻ 区域外，最佳格栅还包括五个不确定区域、四个 R ⁻⁻ 区域和一个 R ⁺⁺ 区域。为了进行比较，我们通过布局优化生成了均匀压力载荷的最佳布局，最初没有使用联合成本，如图 4b 所示。R ⁺⁻ 区域的数值表示与分析设计完全吻合，而不确定区域则涉及大量不同方向的重叠梁，因此数值解决方案的实用价值不大。然而，如图 4c 所示，通过重新运行有联合成本的问题，解法得到了极大的简化。现在，R ⁺⁺ 区域转变为规则网格，R ⁻⁻ 区域转变为从四个点支撑中的每个点辐射出来的悬臂扇形；在后续示例中，夹紧点支撑（或支撑边界的凹角）附近也会出现类似的扇形。然而，显而易见的是，引入联合成本似乎将 R ⁺⁻ 区域转变为 R ⁺ 区域，如上一例中出现的情况。这是因为对于给定的外加载荷，最佳梁宽分布并不一定是唯一的。因此，在图 4 所示的示例中，不确定正方形 R ⁺⁺ 区域的特定表示方法会影响相邻的 R ⁺⁻ 区域是否存在曳引梁。 然而，在没有 R ⁺⁺ 或 R ⁻⁻ 区域的更简单问题中，也可以证明缺乏独特的梁宽分布。例如，考虑两个对置的等长悬臂梁，在其顶端承受共同荷载的情况；图 12 也是一个例子。虽然可以保证存在静力确定的最优格栅，但由于解的非唯一性，图 4b 所示结构显然不是静力确定的。使用联合成本并不一定能完全解决这个问题，例如见图 4c。

3.1.3 Square domain with external clamped and interior point supports
3.1.3 带外部夹紧和内部点支撑的方形域

The next example involves a square design domain with clamped external supports and four interior clamped point supports. In the paper by Rozvany et al. (1973) the problem was deemed load-independent and consequently an analytical layout was given for all downward loads; see Fig. 5a where sagging or hogging indeterminate regions are depicted in solid ink. Figure 5b shows the new numerical solution obtained, assuming a uniform pressure load is applied. Indeterminate R⁺⁺ and R⁻⁻ regions appear in the same form as those in the previous example; see Fig. 4c. Sagging R⁺-type fans are approximately represented by the given nodal discretization. For this example numerical results are presented in Table 1 of Appendix A, with the number of adaptive member adding iterations required to obtain a solution for a given nodal discretization shown together with associated CPU times.
下一个例子涉及一个方形设计域，该设计域具有夹紧外部支撑和四个内部夹紧点支撑。在 Rozvany 等人（1973 年）的论文中，该问题被认为与荷载无关，因此给出了所有向下荷载的分析布局；见图 5a，其中下垂或踌躇的不确定区域用实心墨水描绘。图 5b 显示了假设施加均匀压力载荷时获得的新数值解。不确定 R ⁺⁺ 和 R ⁻⁻ 区域的形式与上一示例中的相同；见图 4c。下垂的 R ⁺ 型扇形近似于给定的节点离散表示。本例的数值结果见附录 A 表 1，其中显示了获得给定节点离散化解决方案所需的自适应成员添加迭代次数以及相关的 CPU 时间。

Fig. 5 图 5

Square domain with external clamped and interior point supports: a optimum layout derived analytically by Rozvany et al. (1973); b new result obtained by numerical layout optimization, $V_{\text{num}}=\hspace{0.17em}0.004966p{L}^4/m_{\mathrm{p}}$
带外部夹紧和内部点支撑的方形域：a 由 Rozvany 等人（1973 年）分析得出的最佳布局；b 通过数值布局优化得到的新结果， $V_{\text{num}}=\hspace{0.17em}0.004966p{L}^4/m_{\mathrm{p}}$

3.1.4 Square domain with four column supports
3.1.4 四柱支撑方形域

The next example involves a square design domain with free external boundaries and four supporting columns in the interior, represented by square clamped supports. Assuming arbitrary downward load, Fig. 6a shows the optimum layout derived analytically by Rozvany (1972a) for this particular problem. An analytical solution u of the kinematic form can be uniquely derived based solely on the layout from Fig. 6a which is independent of the load, provided this is always downwards. For example, if a uniform pressure load of magnitude p is applied then, based on duality arguments, the exact volume V_exact of the optimum grillage can be computed from the virtual work done by the load as follows:
下一个例子涉及一个外部边界自由的正方形设计域，内部有四根支撑柱，由正方形夹紧支撑表示。假设荷载任意向下，图 6a 显示了罗兹万尼（1972a）针对这一特定问题分析得出的最佳布局。仅根据图 6a 中的布局，就可以唯一地推导出运动形式的分析解 u，该解与载荷无关，条件是载荷始终向下。例如，如果施加一个大小为 p 的均匀压力载荷，那么根据对偶论证，最佳格栅的精确体积 V _exact 可以通过载荷所做的虚功计算如下：

$$V_{\text{exact}}=W_{\text{exact}}={\int }_{\mathrm{\Omega }}pu\mathrm{d}x=\frac{10237}{768}\frac{p{b}^4}{m_p}\approx 13.3294\frac{p{b}^4}{m_p}$$

(7)

where Ω denotes the design domain. Both the function u and the integral are computed in Appendix B.
其中，Ω 表示设计域。函数 u 和积分的计算方法见附录 B。

Fig. 6 图 6

Square domain with four column supports: a optimum layout derived analytically by Rozvany (1972a), $V_{\text{exact}}=\hspace{0.17em}13.3294p{b}^4/m_p$; b new result obtained by numerical layout optimization, $V_{\text{num}}=\hspace{0.17em}13.33p{b}^4/m_p$
带有四个支柱支撑的方形域：a 由 Rozvany ( 1972a) 分析得出的最佳布局， $V_{\text{exact}}=\hspace{0.17em}13.3294p{b}^4/m_p$ ；b 通过数值布局优化得出的新结果， $V_{\text{num}}=\hspace{0.17em}13.33p{b}^4/m_p$

The numerical solution for this case is shown in Fig. 6b; the optimum volume V_num = 13.33pb⁴/m_p is derived from the values tabulated in Table 2 of Appendix A. The close correlation between the numerical and analytical solutions is clearly evident, both in terms of computed volume and grillage layout. Note that for this example only two adaptive member adding iterations are required to obtain a solution (see Table 2 in Appendix A) because the initial ground structure already contains most critical members; this also leads to relatively low associated CPU times.
这种情况下的数值解如图 6b 所示；最佳体积 V _num = 13.33pb ⁴ /m _p 是根据附录 A 表 2 中的数值得出的。无论是在计算体积还是格栅布局方面，数值解与分析解之间的紧密相关性都是显而易见的。需要注意的是，在本例中，由于初始地面结构已经包含了大部分关键构件，因此只需要进行两次自适应构件添加迭代就可以求得解（见附录 A 表 2）；这也导致相关的 CPU 时间相对较短。

3.1.5 Domains with free and simply supported edges
3.1.5 具有自由边和简单支持边的域

Problems involving domains with both free and simply supported edges were investigated by Rozvany and Liebermann (1994). The associated optimization problems were challenging mathematically, with the formulas describing the optimum directions of the beams given in implicit integral form which had to be solved numerically.
Rozvany 和 Liebermann（1994 年）研究了涉及具有自由边缘和简单支撑边缘的域的问题。相关的优化问题在数学上具有挑战性，描述梁最佳方向的公式是以隐式积分形式给出的，必须通过数值求解。

Here a right-angled isosceles triangle domain with a simply supported base edge and a simple point support in the right-angle corner is considered, as shown in Fig. 7. The optimum grillage layout for this problem found by Rozvany and Liebermann (1994) for arbitrary downward loading is given in Fig. 7a. Note that the layout is not trivial as the beams do not radiate from the simple point support. A numerical solution was obtained for the uniform downward pressure load case and is presented in Fig. 7b. The close resemblance between the analytical and numerical solutions is clear.
如图 7 所示，这里考虑的是一个直角等腰三角形域，其底边为简单支撑，直角拐角处为简单点支撑。图 7a 给出了 Rozvany 和 Liebermann（1994 年）发现的任意向下荷载情况下的最佳格栅布局。需要注意的是，由于横梁并不从简单的点支撑上辐射开来，因此布局并不简单。图 7b 给出了均匀向下压力荷载情况下的数值解。分析解与数值解之间的相似性显而易见。

Fig. 7 图 7

Triangular domain with free and simply supported edges: a optimum layout derived analytically by Rozvany and Liebermann (1994); b new result obtained by numerical layout optimization for a uniform pressure load, $V_{\text{num}}=\hspace{0.17em}0.09505p{h}^4/m_{\mathrm{p}}$
具有自由边和简单支撑边的三角形域：a 由 Rozvany 和 Liebermann（1994 年）分析得出的最佳布局；b 在均匀压力负荷下，通过数值布局优化得出的新结果， $V_{\text{num}}=\hspace{0.17em}0.09505p{h}^4/m_{\mathrm{p}}$

3.2 Problems with clamped and free edges
3.2 夹边和自由边的问题

3.2.1 Square domain problem
3.2.1 方域问题

The next example involves a square domain comprising two clamped and two free edges, as shown in Fig. 8a. This pro- blem was previously considered by Rozvany (1972b), though an exact analytical solution was not derived (even then the class of optimum grillage problems for clamped / free boun- dary conditions was recognized as being difficult). The same problem was revisited by Sigmund et al. (1993), who presented numerical solutions obtained using a ground structure-based approach combined with FEM. Despite the insights these generated a general analytical method for grillages with clamped and free edges has still not been found, highlighting a clear gap in the grillage optimization theory developed by Rozvany et al. In Fig. 8 a range of problems are solved, for cases involving point and pressure loads.
下一个例子涉及由两条夹紧边和两条自由边组成的正方形域，如图 8a 所示。Rozvany ( 1972b) 曾考虑过这一问题，但没有得出精确的解析解（即使在当时，人们也认为夹紧/自由边条件下的最优格栅问题是一个难题）。Sigmund 等人（1993 年）对同一问题进行了重新研究，提出了基于地面结构的方法与有限元相结合的数值解决方案。尽管这些研究提出了一些见解，但仍未找到针对夹紧和自由边缘格栅的通用分析方法，这凸显了 Rozvany 等人开发的格栅优化理论中存在的明显缺陷。

Fig. 8 图 8

Square domain with two free and two clamped edges: a problem definition (domain has dimensions L × L, with point loads applied on the diagonal); bP₁ = P, P₂ = 0, $V_1=\hspace{0.17em}0.1408P{L}^2/m_p$ (which can be shown to coincide with the exact solution); cP₁ = 0, P₂ = P, $V_2=\hspace{0.17em}0.4093P{L}^2/m_p$; dP₁ = P, P₂ = P, $V_{1+2}=\hspace{0.17em}0.5315P{L}^2/m_p$; e uniform pressure load p = P/L², V = 0.07067PL²/m_p; f “beam-weave” phenomenon
具有两条自由边和两条夹紧边的正方形域：a 问题定义（域尺寸为 L × L，对角线上施加点载荷）； bP ₁ = P, P ₂ = 0, $V_1=\hspace{0.17em}0.1408P{L}^2/m_p$ （可证明与精确解相吻合）；cP ₁ = 0, P ₂ = P, $V_2=\hspace{0.17em}0.4093P{L}^2/m_p$ ; dP ₁ = P, P ₂ = P, $V_{1+2}=\hspace{0.17em}0.5315P{L}^2/m_p$ ; e 均压载荷 p = P/L ² , V = 0.07067PL ² /m _p ; f "编织梁 "现象

3.2.2 Domain with hole problem
3.2.2 带洞域问题

The next example involves a domain with a hole and free and clamped edges, as shown in Fig. 9a. Solutions were obtained for three different loading scenarios, involving either point or pressure loads. The optimum layouts for the problems involving point loads, presented in Fig. 9b,c, are perhaps of particular interest since they clearly indicate how the load finds its way through an optimum grillage around the hole back to the supports.
下一个例子涉及一个带孔、自由边缘和夹紧边缘的域，如图 9a 所示。针对点载荷或压力载荷这三种不同的载荷情况获得了解决方案。图 9b 和 c 所示的涉及点载荷问题的最佳布局可能尤其引人关注，因为它们清楚地表明了载荷是如何通过孔周围的最佳格栅返回支撑的。

Fig. 9 图 9

Domain with hole problem, including both free and clamped edges: a problem definition; b point load applied to the interior boundary; c point load applied to the exterior boundary; d uniform pressure load
带孔问题的域，包括自由边和夹紧边：a 问题定义；b 施加于内部边界的点载荷；c 施加于外部边界的点载荷；d 均布压力载荷

3.3 Uplift effect 3.3 上升效应

One of limitations of the computer software tool produced by Hill and Rozvany (1985) was that it could not model internal simple supports because of the potential for uplift, rendering the optimum grillage layout dependent on the position of the load(s) involved. An example is shown in Fig. 10. In this case the internal support divides the design domain into two parts: one subjected to pressure load p₁, and the other p₂. When p₁ and p₂ are both applied, the optimum grillage layout is shown in Fig. 10b. If only one of them is applied, different results are obtained, as shown in Fig. 10c and d, indicating the load dependant nature of the problem. Uplift effects are present in the problems shown in Fig. 10b and d, where in (b) the load p₁ effectively cancels out some of the bending effects caused by p₂, leading to a lower volume than in (d).
Hill 和 Rozvany（1985 年）制作的计算机软件工具有一个局限性，那就是它无法对内部简支梁进行建模，因为内部简支梁可能会发生上翘，从而使格栅的最佳布局取决于相关荷载的位置。图 10 显示了一个例子。在这种情况下，内部支撑将设计域分为两部分：一部分承受压力荷载 p ₁ ，另一部分承受压力荷载 p ₂ 。当同时施加 p ₁ 和 p ₂ 时，最佳格栅布局如图 10b 所示。如图 10c 和 d 所示，如果只应用其中一个，则会得到不同的结果，这表明了问题的负载依赖性。在图 10b 和 d 所示的问题中存在上浮效应，在（b）中，载荷 p ₁ 有效地抵消了 p ₂ 造成的一些弯曲效应，导致体积比（d）中的小。

Fig. 10 图 10

Square domain supported internally along its diagonal: a problem definition; bp₁ = p, p₂ = p, V_{1 + 2} = 0.03754pL⁴/m_p; cp₁ = p, p₂ = 0, V₁ = 0.007889pL⁴/m_p; dp₁ = 0, p₂ = p, V₂ = 0.04217pL⁴/m_p
沿对角线内部支撑的正方形域：问题定义； bp ₁ = p, p ₂ = p, V _{1 + 2} = 0.03754pL ⁴ /m _p ; cp ₁ = p, p ₂ = 0, V ₁ = 0.007889pL ⁴ /m _p ； dp ₁ = 0, p ₂ = p, V ₂ = 0.04217pL ⁴ /m _p

3.4 Partially downward and partially upward load
3.4 部分向下和部分向上的负载

Applying mixed downward and upward loads yields yet another class of problem for which load-independent optimal layouts cannot be found. Analytical results for a modest range of such problems were published in a short paper by Rozvany (1997); however a general method of treating such problems analytically has not yet been developed. The example presented in Fig. 11 gives a good insight into the nature of such problems; here two point loads are to be transferred to four simple point supports. Figure 11a shows the solution for all-downward load; in this case two separate simply supported beams of total volume V_dd = 0.25PL²/m_p prove to be optimal. However the optimum layout shown in Fig. 11b for the case when one of the loads is upward clearly involves interaction between the two forces, thus considerably reducing the optimum volume, to V_du = 0.1875PL²/m_p.
应用混合向下和向上载荷会产生另一类问题，即无法找到与载荷无关的最佳布局。Rozvany （1997 年）在一篇短文中发表了对少量此类问题的分析结果；然而，目前尚未开发出分析处理此类问题的通用方法。图 11 中的示例很好地揭示了此类问题的本质；在这里，两个点载荷将被传递到四个简单的点支撑上。图 11a 显示了全向下荷载的解决方案；在这种情况下，总体积为 V _dd = 0.25PL ² /m _p 的两个独立简支梁被证明是最佳的。然而，图 11b 中所示的其中一个荷载向上时的最佳布局显然涉及两个力之间的相互作用，从而大大减少了最佳体积，为 V _du = 0.1875PL ² /m _p 。

Fig. 11 图 11

Partially downward and upward load problem: a both loads P applied downwards, V_dd = 0.25PL²/m_p (exact solution); b one load P applied downwards and the other upwards, V_du = 0.1875PL²/m_p
部分向下和向上的载荷问题：a 两个载荷 P 都向下施加，V _dd = 0.25PL ² /m _p （精确解）；b 一个载荷 P 向下施加，另一个向上施加，V _du = 0.1875PL ² /m _p

3.5 Point moment load 3.5 点力矩荷载

Formulation (4) permits point moments to be applied directly, thus yielding another class of load-dependent problem. An illustrative example involving a rectangular domain with a point moment load remote from a support is shown in Fig. 12; domains of constant width and varying height are considered. The key observation is that, despite filling the entire height of the domain with an optimum layout, the optimum volume remains constant, at V = M L/m_p. This indicates the indeterminacy of the layout in each case (since e.g. design (c) is also a viable solution to problems (a) and (b)).
公式 ( 4) 允许直接施加点力矩，从而产生另一类与载荷有关的问题。图 12 显示了一个矩形域的示例，该域的点力矩载荷远离支撑物；域的宽度不变，高度变化。主要观察结果是，尽管最佳布局填充了整个域的高度，但最佳体积保持不变，即 V = M L/m _p 。这表明在每种情况下布局都是不确定的（因为例如设计 (c) 也是问题 (a) 和 (b) 的可行解决方案）。

Fig. 12 图 12

Rectangular domain with a point moment load M and simple line support: aH/L = 1/2; bH/L = 1/4; cH ≪ L; the optimum volume V = M L/m_p (exact value) is independent of the height of the domain
矩形域，带点弯矩载荷 M 和简单线支撑：aH/L = 1/2; bH/L = 1/4; cH ≪ L; 最佳体积 V = M L/m _p （精确值）与域的高度无关

4 Discussion 4 讨论

4.1 Non-optimal design of beams with end moments of different signs
4.1 具有不同端弯矩的梁的非优化设计

As mentioned in section 2.3 the solutions obtained via the proposed numerical method will overestimate the true solution in cases where the bending moment function changes in sign along the length of one or more beams. However, none of the optimum layouts presented in section 3 contained any beams where this was the case. Numerical experiments involving other problems showed that when such beams were present, use of a higher nodal refinement remedied this. This is to be expected, since two shorter beams can always be chosen to meet at the point of contraflexure in a long beam, at least approximately.
如第 2.3 节所述，在弯矩函数沿一个或多个梁的长度方向发生符号变化的情况下，通过建议的数值方法获得的解会高估真实解。然而，第 3 节中介绍的最佳布局中没有任何梁出现这种情况。涉及其他问题的数值实验表明，当出现此类梁时，使用更高的节点细化可以解决这一问题。这是意料之中的，因为总是可以选择两个较短的梁在长梁的反折点相交，至少是近似相交。

4.2 Load dependent layouts in grillage optimization
4.2 格栅优化中与载荷有关的布局

The class of grillage optimization problems solved fully and analytically by Rozvany et al. share an essential property: independence of the layout from load, providing the latter is always applied in a downward direction. This means that there is a displacement vector u that solves dual problem (5) for every downward load, or alternatively, that there exists a displacement vector u that maximizes the out-of-plane dis- placement of every point (node) simultaneously. In contrast the optimal layouts for the problems considered in Sections 3.2 to 3.5 are load dependent, and there are currently no analytical methods that can be applied to such problems.
罗兹万尼等人通过完全分析解决的格栅优化问题有一个共同的基本特性：布局与载荷无关，条件是载荷总是向下施加。这意味着存在一个位移矢量 u，可以解决每个向下负载的对偶问题 ( 5)，或者说，存在一个位移矢量 u，可以同时使每个点（节点）的平面外位移最大化。相比之下，第 3.2 至 3.5 节中考虑的问题的最佳布局取决于荷载，目前还没有分析方法可用于此类问题。

In this context it is worth revisiting the triangular domain problem initially investigated in Section 3.1.5. According to Rozvany and Liebermann (1994) the analytical layout shown in Fig. 7a should be universal for all downward loads. However, this can be checked by using the numerical method developed herein to explore a range of different loading scenarios. Thus consider for example the case of a single point load applied midway along one of the free edges, leading to the numerical optimum layout shown in Fig. 13a. Comparing Fig. 13a with Fig. 7a it is evident that the beam directions differ, suggesting that this problem is not load independent after all. To verify this finding the theory of grillage-like slabs can be invoked; see e.g. Rozvany (1972a). With the given point load P duality theorems can be used to show that the analytically derived layout of Fig. 7a yields a lower bound volume $\underset{\_}{V}$ ≈ 0.18Ph²/m_p. Conversely the numerical solution of Fig. 13a is associated with a one-line bending moment field $\overline{M}$ that furnishes an upper bound volume $\overline{V}=0.25P{h}^2/m_p$. In order to prove that the exact volume $V_{\text{exact}}=\overline{V}$, and the associated exact moment field $M_{\text{exact}}=\overline{M}$, it is sufficient to guess a displacement function $\underset{\_}{u}$ such that curvature constraints are met and the optimality relation between $\overline{M}$ and $\underset{\_}{u}$ holds, i.e.
在这种情况下，值得重新审视第 3.1.5 节中最初研究的三角形域问题。根据 Rozvany 和 Liebermann（1994 年）的研究，图 7a 所示的分析布局应适用于所有向下荷载。不过，可以使用本文开发的数值方法来检验这一点，以探索一系列不同的荷载情况。例如，考虑在一条自由边的中间位置施加单点荷载，从而得出图 13a 所示的最佳数值布局。对比图 13a 和图 7a，可以明显看出横梁的方向不同，这说明这个问题与载荷无关。为了验证这一结论，可以引用格栅状板的理论，例如参见 Rozvany ( 1972a)。在给定点荷载 P 的情况下，可以使用对偶定理来证明图 7a 中通过分析推导的布局可以得到体积 $\underset{\_}{V}$ ≈ 0.18Ph ² /m _p 的下限。相反，图 13a 的数值解与单线弯矩场 $\overline{M}$ 相关联，它提供了上界体积 $\overline{V}=0.25P{h}^2/m_p$ 。为了证明精确体积 $V_{\text{exact}}=\overline{V}$ 以及相关的精确力矩场 $M_{\text{exact}}=\overline{M}$ ，只需猜测位移函数 $\underset{\_}{u}$ ，从而满足曲率约束条件，并且 $\overline{M}$ 和 $\underset{\_}{u}$ 之间的最优关系成立，即

• the principal curvatures κ_I, κ_II produced by $\underset{\_}{u}$ satisfy the point-wise inequalities: − 1/m_p ≤ κ_I, κ_II ≤ + 1/m_p and
  由 $\underset{\_}{u}$ 产生的主曲率 κ _I , κ _II 满足点向不等式：- 1/m _p ≤ κ _I , κ _II ≤ + 1/m _p 和

• the left free edge is one of the principal trajectories of curvature field κ and the principal curvature κ_I is equal to + 1/m_p along this edge
  左侧自由边是曲率场 κ 的主轨迹之一，且这条边上的主曲率 κ _I 等于 + 1/m _p

respectively. Naturally the function $\underset{\_}{u}$ must also satisfy the support conditions. It can easily be verified that the function in question can be given by an extremely simple closed-form expression (where x,y are Cartesian coordinates, as indicated in Fig. 13a):
分别为当然，函数 $\underset{\_}{u}$ 也必须满足支持条件。我们可以很容易地验证，有关函数可以用一个极其简单的闭式表达式给出（其中 x、y 为直角坐标，如图 13a 所示）：

$$\underset{\_}{u}(x,y)=\frac{1}{m_p}x(h-y),$$

(8)

which implies $M_{\text{exact}}=\overline{M}$, $u_{\text{exact}}=\underset{\_}{u}$ and $V_{\text{exact}}=\overline{V}=0.25P{h}^2/m_p$, and further that the numerical solution given in Fig. 13a is in fact the exact solution for the grillage-like slab problem with a single point load. The $\underset{\_}{u}$ field resembles a slab being twisted around the y axis, as shown in Fig. 13b. In fact, the same field $\underset{\_}{u}$ is also found when a point moment is applied at the point support; Fig. 13c shows the corresponding layout. (It now becomes clear that a function of the form of (8) also furnishes a solution to the problem described in Section 3.5.)
这意味着 $M_{\text{exact}}=\overline{M}$ 、 $u_{\text{exact}}=\underset{\_}{u}$ 和 $V_{\text{exact}}=\overline{V}=0.25P{h}^2/m_p$ ，而且图 13a 中给出的数值解实际上是单点载荷下格栅状板块问题的精确解。如图 13b 所示， $\underset{\_}{u}$ 场类似于绕 y 轴扭转的板。事实上，当在点支座上施加点力矩时，也会出现相同的 $\underset{\_}{u}$ 场；图 13c 显示了相应的布局。(现在可以清楚地看到， ( 8) 形式的函数也提供了第 3.5 节所述问题的解决方案）。

Fig. 13 图 13

a Numerical optimum layout when a mid-edge point load applied, V = 0.250Ph²/m_p (exact solution); b associated exact displacement field; c numerical optimum layout for a point moment problem, which shares the displacement field shown in b; d numerical optimum layout for two symmetrically positioned point loads of equal magnitude, V = 0.3549Ph²/m_p; e numerical optimum layout for two symmetrically positioned point loads of unequal magnitude, V = 0.7633Ph²/m_p; f numerical optimum layout for the mid-edge point load problem with the simple point support replaced by a short simply supported edge, V = 0.1812Ph²/m_p
a 施加中边点荷载时的最佳数值布置，V = 0.250Ph ² /m _p （精确解）；b 相关的精确位移场；c 点力矩问题的最佳数值布置，它与 b 中所示的位移场相同；d 两个对称定位的等量点荷载的最佳数值布置，V = 0.3549Ph ² /m _p ； e 两个对称布置的不等大点荷载的最佳数值布置，V = 0.7633Ph ² /m _p ； f 中边点荷载问题的最佳数值布置，简单点支承由简单支承短边代替，V = 0.1812Ph ² /m _p

It is also of interest to now consider the case of two point loads applied symmetrically midway along each the free edges; this yields a volume V= 0.354Ph²/m_p and the layout shown in Fig. 13d. Here the optimum layout appears to be inscribed within the analytical layout proposed by Rozvany and Liebermann (1994); see Fig. 7a. Note that the optimum volume is considerably smaller than double the volume of the one-beam solution. However, if the magnitudes of the applied loads are changed, a non-symmetrical numerical layout is obtained; see Fig. 13e. Here the orientation of the sagging beams forming the fans noticeably diverge from the analytical layout shown on Fig. 7a.
现在，我们还可以考虑在每个自由边缘的中间对称施加两个点载荷的情况；这将产生体积 V= 0.354Ph ² /m _p 和图 13d 所示的布局。这里的最佳布局似乎与 Rozvany 和 Liebermann（1994 年）提出的分析布局相吻合；见图 7a。请注意，最佳布局的体积比单梁方案的两倍要小得多。然而，如果改变外加载荷的大小，就会得到非对称的数值布局；见图 13e。在这里，形成风扇的下垂横梁的方向与图 7a 所示的分析布局明显不同。

These numerical experiments, together with the analytically proposed function $\underset{\_}{u}$ ultimately show the load-dependence of the triangular domain problem initially investigated in Section 3.1.5. The load-dependence appears to be due to the same uplift effect that occurs in the problems considered in Section 3.3, where in that case the axis of uplift was an internal line of simple support. In the triangular domain problem every straight line passing through the point support and the interior of the domain is a potential uplift axis. From this argument it can be concluded that the presence of a simple point support, either placed on the boundary or in the interior of the design domain, is likely to lead to load-dependence in the grillage optimization problem. Note that the triangular domain problem was the only example given in Rozvany and Liebermann (1994) that considered a simple point support; the authors’ focus was originally a class of problems with free and simply supported edges only, so all other optimum layouts derived therein can be assumed to be truly load-independent and hence correct.
这些数值实验以及分析提出的函数 $\underset{\_}{u}$ 最终显示了第 3.1.5 节中最初研究的三角域问题与荷载的关系。这种荷载依赖性似乎是由于与第 3.3 节中考虑的问题相同的上浮效应造成的，在第 3.3 节中，上浮轴是简单支撑的内部线。在三角形域问题中，通过点支撑和域内部的每一条直线都是潜在的上浮轴。由此可以得出结论，无论是在设计域的边界上还是在设计域的内部，简支点的存在都有可能导致格栅优化问题中的荷载依赖性。请注意，三角形域问题是 Rozvany 和 Liebermann（1994 年）中唯一考虑简单点支撑的示例；作者最初关注的是一类仅有自由边和简单支撑边的问题，因此可以认为其中得出的所有其他最优布局都真正与荷载无关，因而是正确的。

Finally, suppose that the triangular domain of this problem is transformed into a trapezium to allow the point support to be replaced with a very short simply supported edge. The solution when a single point load applied midway along the free edge is shown in Fig. 13f. It is evident that the new optimum layout now appears to be in agreement with the analytical layout derived by Rozvany and Liebermann (1994). This suggests that the anomaly in this case stemmed from an assumption that an infinitely short line of simple support could be taken to be equivalent to a point support. However, the former prevents rotation about the y axis, and allows a reaction moment about the same axis to be generated. This appears to be crucial in order for the optimum grillage to comprise beams which coincide with the analytical solution shown in Fig. 7a.
最后，假设将此问题的三角形域转换为梯形，以便用很短的简单支撑边代替点支撑。图 13f 显示了在自由边中段施加单点载荷时的解。很明显，新的最佳布局似乎与 Rozvany 和 Liebermann（1994 年）得出的分析布局一致。这表明，这种情况下的异常源于一个假设，即无限短的简支撑线可以等同于点支撑。然而，前者可以防止绕 y 轴旋转，并允许产生绕同一轴的反作用力矩。这一点对于最佳格栅由梁组成似乎至关重要，而梁与图 7a 所示的分析解相吻合。

4.3 Beam-weave phenomenon and including torsion
4.3 梁波现象和包括扭转

In the solutions shown in e.g. Fig. 8e,f a thin region of orthogonally intersecting sagging and hogging beams occurs along each free edge. This phenomenon has previously been identified in the optimum grillage layouts found analytically for problems involving mixed free and simply supported edges (Rozvany and Liebermann 1994); the result from Fig. 7 with R⁺⁻-type free edges serves as an example. This “beam-weave”, as it was called therein, is particularly difficult to approximate using the ground structure approach since, theoretically, it is supposed to be infinitely thin. Similarly, a beam-weave turns out to be an optimum means of transferring load along free edges; e.g., see Fig. 9. Here the role of the beam-weave becomes more apparent; essentially it attempts to mimic a single beam capable of transferring torsion. Consequently, one can observe that limiting the height of the domain in the problem shown in Fig. 12 would provide an infinitely thin, beam-weave-like design which is essentially equivalent to a single member in pure torsion.
在图 8e,f 等所示的解法中，每个自由边缘都有一个正交下垂和踌躇梁的薄区域。这种现象以前在分析涉及混合自由边缘和简单支撑边缘的问题时发现过（Rozvany 和 Liebermann，1994 年）；图 7 中 R ⁺⁻ 型自由边缘的结果就是一个例子。这种被称为 "梁式编织 "的结构尤其难以用地面结构方法来近似，因为从理论上讲，它应该是无限薄的。同样，梁式编织结构也是沿自由边缘传递荷载的最佳方式，例如，见图 9。在这里，编织梁的作用变得更加明显；从本质上讲，它试图模仿能够传递扭力的单梁。因此，我们可以看到，在图 12 所示的问题中，限制域的高度将提供一种无限薄的、类似于编织梁的设计，这种设计基本上等同于纯扭转的单个构件。

The above suggests that it may be worthwhile to include torsion in the problem formulation after all, since the beam-weave regions degrade the quality of the numerical layouts. However, the underlying problem formulation then becomes nonlinear, and means of obtaining a suitable linearized approxi- mation of the problem will be the subject of future research.
上述结果表明，由于编织梁区域会降低数值布局的质量，因此在问题表述中加入扭转可能是值得的。然而，基本问题的表述就变成了非线性问题，如何获得问题的适当线性化近似值将是未来研究的主题。

5 Conclusions 5 结论

A new numerical layout optimization method capable of identifying the minimum volume and associated optimal layout of a grillage has been proposed. Beam members which are tapered along their lengths between nodes have been employed to maintain the linear character of the problem. This means that highly efficient linear programming algorithms can be used to obtain solutions, with the adaptive “member adding” technique previously applied to truss layout problems enabling solution of large-scale problems, containing large numbers of nodes and interconnecting members. A key feature of the new method is its generality; it can be applied to problems involving arbitrary domain geometries and loading and support configurations, and can faithfully capture important phenomena such as “beam-weaves”, which provide resistance to torsion when individual beams have negligible torsional resistance.
我们提出了一种新的数值布局优化方法，该方法能够确定格栅的最小体积和相关优化布局。为了保持问题的线性特征，我们采用了节点之间沿长度方向呈锥形的梁构件。这意味着可以使用高效的线性规划算法来获得解决方案，而之前应用于桁架布局问题的自适应 "添加构件 "技术则可以解决包含大量节点和互连构件的大型问题。新方法的一个主要特点是其通用性；它可以应用于涉及任意域几何形状、载荷和支撑配置的问题，并能忠实捕捉重要现象，如 "梁波"，当单个梁的抗扭能力可忽略不计时，"梁波 "可提供抗扭能力。

When applied to problems for which exact analytical solutions exist it has been found that close approxima- tions of these solutions can be found. However, analytical methods developed to date by workers such as Rozvany et al. can only be applied to problems for which the optimal layout is independent of loading. Thus the proposed method has also been applied to a range of load dependent problems, for which analytical solutions are currently not available. Interestingly the new method revealed that one problem in the literature which had been thought to be load independent (providing the load was always applied in a downward direc- tion), is in fact load dependent, rendering the proposed analy- tical solution less generally applicable than previously thought.
当应用于存在精确分析解的问题时，我们发现可以找到这些解的近似值。然而，Rozvany 等人迄今为止开发的分析方法只能应用于最佳布局与载荷无关的问题。因此，所提出的方法也被应用于一系列与载荷相关的问题，而这些问题目前还没有分析解决方案。有趣的是，新方法揭示了文献中的一个问题，该问题曾被认为与载荷无关（条件是载荷总是向下施加），但实际上与载荷有关，这使得所提出的分析解决方案不如以前想象的那么普遍适用。

Appendices 附录

Appendix A: Computing extrapolated volumes
附录 A：计算外推体积

As described in Darwich et al. (2010), numerical solutions obtained from numerical layout optimization runs appear to follow a relation of the form:
正如 Darwich 等人（2010 年）所描述的那样，通过数值布局优化运行获得的数值解似乎遵循着这样一种关系：

$$V_n=V_{\mathrm{\infty }}+k{n}^{-\alpha },$$

(9)

where V_n is the numerically computed volume for n equally spaced nodal divisions, V_∞ is the volume when n →∞, and k and α are constants. Using (9), a weighted least-square approach can be used to find the best-fit values for V_∞, k and α, with the weighting coefficient taken as n. Numerical solutions are given in Tables 1–3.
其中，V _n 是 n 个等间距节点划分时的数值计算体积，V _∞ 是 n →∞ 时的体积，k 和 α 是常数。利用 ( 9) 可以用加权最小二乘法找到 V _∞ 、k 和 α 的最佳拟合值，加权系数取 n。表 1- 3 给出了数值解。

Table 1 Numerical solutions and extrapolated volume for example given in Fig. 5 (A quarter domain was used due to symmetry)
表 1 图 5 所示例子的数值解和推断体积（由于对称性，使用的是四分之一域）

Table 2 Numerical solutions and extrapolated volume for the four-column example given in Fig. 6 (A quarter domain was used due to symmetry)
表 2 图 6 所示四柱示例的数值解和推断体积（由于对称性，使用了四分之一域）

Table 3 Numerical solutions and extrapolated volumes for other examples
表 3 其他例子的数值解和推断体积

Appendix B: Computing the exact optimum volume for the square domain with four column supports problem
附录 B：计算四柱支撑方形域问题的精确最佳体积

By duality arguments one can compute the volume of the optimum grillage (grillage-like slab) from the following equality
通过对偶论证，我们可以根据下面的等式计算出最佳格栅（格栅状板块）的体积

$$V_{\text{exact}}=W_{\text{exact}}={\int }_{\mathrm{\Omega }}pu\mathrm{d}x$$

(10)

where u is an out of plane displacement function that solves the dual displacement form and p is a load function. As implied in Rozvany (1972a) the four column slab problem considered in section 3.1.4 enjoys a solution u that is independent of the load p provided the latter is always downwards. The analytical layout given in this paper and presented in Fig. 6a furnishes principal trajectories of curvature field κ associated with u and of principal curvatures being equal to ± 1/m_p. To facilitate comparison of the volume V_exact with numerical results, the load p is assumed to be uniformly distributed. Now, by making use of information on the curvature function, the solution u can be recovered region by region; in addition the function u must be continuous and continuously differentiable in each point of the domain. As the layout is bisymmetrical, 1/4 of the domain is considered; for region partition and coordinate systems see Fig. 14. For sake of simplicity m_p is taken as unity.
其中 u 是平面外位移函数，用于求解二元位移形式，p 是荷载函数。正如 Rozvany ( 1972a) 所暗示的，第 3.1.4 节中考虑的四柱板问题有一个与荷载 p 无关的解 u，条件是荷载 p 始终向下。本文给出的分析布局如图 6a 所示，提供了与 u 相关的曲率场 κ 的主轨迹，主曲率等于 ± 1/m _p 。为了便于将体积 V _exact 与数值结果进行比较，假定载荷 p 是均匀分布的。现在，通过利用曲率函数的信息，可以逐个区域恢复解 u；此外，函数 u 必须是连续的，并且在域的每个点上都是连续可微的。由于布局是双对称的，因此考虑了 1/4 的区域；区域划分和坐标系见图 14。为简单起见，m _p 取为一。

Fig. 14 图 14

Region partition of the design domain and coordinate systems in each region
设计域的区域划分和每个区域的坐标系

The function u is of identical form for R⁻ regions no. 1,2,3 and is
对于 R ⁻ 区域 1、2、3，函数 u 的形式完全相同。1、2、3 区域的函数 u 的形式相同，并且

$$\begin{array}{rcl}& & u_{1,2,3}(x,y)=\frac{1}{2}{y}^2\end{array}$$

(11)

which gives κ_xx = 0,κ_yy = − 1,κ_xy = 0 as desired. The volumes below follow:
由此得出 κ _xx = 0，κ _yy = - 1，κ _xy = 0。下面的体积如下：

$$\begin{array}{rcl}& & V_1={\int }_0^{\frac{b}{2}}{\int }_0^{\frac{b}{2}}p{u}_1(x,y)\mathrm{d}y\mathrm{d}x=\frac{1}{96}p{b}^4,\end{array}$$

(12)

$$\begin{array}{rcl}& & V_2={\int }_0^{\frac{b}{2}}{\int }_0^{\frac{b}{2}+\frac{x}{2}}p{u}_2(x,y)\mathrm{d}y\mathrm{d}x=\frac{65}{3072}p{b}^4,\end{array}$$

(13)

$$\begin{array}{rcl}& & V_3={\int }_0^b{\int }_0^{\frac{b}{2}+\frac{x}{2}}p{u}_3(x,y)\mathrm{d}y\mathrm{d}x=\frac{5}{64}p{b}^4.\end{array}$$

(14)

Moving on to two R⁻⁻ regions no. 4, 5 adjacent to the corners of the columns, the following functions
接下来是两个 R ⁻⁻ 区域编号。4, 5 相邻的柱角，函数如下

$$\begin{array}{rcl}& & u_{4,5}(x,y)=\frac{1}{2}{x}^2+\frac{1}{2}{y}^2\end{array}$$

(15)

result in κ_xx = − 1,κ_yy = − 1,κ_xy = 0. Hence
结果是 κ _xx = - 1,κ _yy = - 1,κ _xy = 0。

$$\begin{array}{rcl}& & V_4={\int }_0^{\frac{b}{2}}{\int }_0^{\frac{b}{2}-x}p{u}_4(x,y)\mathrm{d}y\mathrm{d}x=\frac{1}{192}p{b}^4,\end{array}$$

(16)

$$\begin{array}{rcl}& & V_5={\int }_0^{\frac{b}{2}}{\int }_0^{b+x}p{u}_5(x,y)\mathrm{d}y\mathrm{d}x=\frac{19}{96}p{b}^4.\end{array}$$

(17)

The bisymmetry of the layout imposes the form of displacement function in R⁺⁺ region no. 6 as follows
由于布局的两对称性，R ⁺⁺ 6 号区域的位移函数形式如下

$$\begin{array}{rcl}& & u_6(x,y)=-\frac{1}{2}{x}^2-\frac{1}{2}{y}^2+C_6\end{array}$$

(18)

where constant C₆ is chosen such that u is continuous at points where regions no. 2 and 6 touch, i.e.
其中常数 C ₆ 的选择是为了使 u 在第 2 和第 6 区域相交的点上是连续的，即在第 2 和第 6 区域相交的点上是连续的。即

$$\begin{array}{rcl}& & u_2(b/2,a/4)=u_6(a/4,0)\end{array}$$

(19)

which gives C₆ = 9/16 b². It can be verified that slope continuity also holds. Eventually
由此得出 C ₆ = 9/16 b ² 。可以验证斜率连续性也是成立的。最终

$$\begin{array}{rcl}& & V_6={\int }_0^{\frac{a}{4}}{\int }_0^{\frac{a}{4}-x}p{u}_6(x,y)\mathrm{d}y\mathrm{d}x=\frac{135}{1024}p{b}^4.\end{array}$$

(20)

Regions no. 7, 8, 9, 10 lie on the diagonal axis of symmetry, since κ_xx = 1 for all these regions a universal formulae follows
区域编号第 7、8、9、10 号区域位于对角对称轴上，由于所有这些区域的 κ _xx = 1，因此可以得出一个通用公式

$$\begin{array}{rcl}& & u_i(x,y)=-\frac{1}{2}{x}^2+f_i(y)i=7,8,9,10\end{array}$$

(21)

where functions f₇, f₈, f₉, f₁₀ are such that continuity of u holds on the interfaces of regions no. 2-7, 4-8, 3-9, 5-10 respectively, thus
其中函数 f ₇ , f ₈ , f ₉ , f ₁₀ 使 u 的连续性在 2-7 号区域的界面上成立。分别为 2-7、4-8、3-9、5-10，因此

$$\begin{array}{rcl}& & f_7(y)=\frac{1}{6}{y}^2+\frac{b}{2\sqrt{2}}y+\frac{3b^2}{16},\end{array}$$

(22)

$$\begin{array}{rcl}& & f_8(y)=\frac{1}{2}{y}^2-\frac{b}{2\sqrt{2}}y+\frac{3b^2}{16},\end{array}$$

(23)

$$\begin{array}{rcl}& & f_9(y)=\frac{1}{6}{y}^2+\frac{b}{2\sqrt{2}}y+\frac{3b^2}{16},\end{array}$$

(24)

$$\begin{array}{rcl}& & f_{10}(y)=\frac{1}{2}{y}^2+\frac{b}{\sqrt{2}}y+\frac{3b^2}{4}.\end{array}$$

(25)

Note that κ_yy;7,9 = − 1/3 > − 1 and κ_yy;8,10 = − 1 which agrees with the optimum layout. The volume of the last four regions can now be readily computed:
请注意，κ _yy;7,9 = - 1/3 > - 1，κ _yy;8,10 = - 1，这与最佳布局一致。现在可以轻松计算出最后四个区域的体积：

$$\begin{array}{rcl}& & V_7={\int }_0^{\frac{3b}{4\sqrt{2}}}{\int }_{-(\frac{b}{2\sqrt{2}}+\frac{y}{3})}^{\frac{b}{2\sqrt{2}}+\frac{y}{3}}p{u}_7(x,y)\mathrm{d}x\mathrm{d}y=\frac{65}{512}p{b}^4,\end{array}$$

(26)

$$\begin{array}{rcl}& & V_8={\int }_0^{\frac{b}{\sqrt{2}}}{\int }_{-\frac{b}{2\sqrt{2}}}^{\frac{b}{2\sqrt{2}}}p{u}_8(x,y)\mathrm{d}x\mathrm{d}y=\frac{1}{16}p{b}^4,\end{array}$$

(27)

$$\begin{array}{rcl}& & V_9={\int }_0^{\frac{3b}{2\sqrt{2}}}{\int }_{-(\frac{b}{2\sqrt{2}}+\frac{y}{3})}^{\frac{b}{2\sqrt{2}}+\frac{y}{3}}p{u}_9(x,y)\mathrm{d}x\mathrm{d}y=\frac{15}{32}p{b}^4\end{array}$$

(28)

and, splitting region no. 10 into two symmetrical parts,
并将 10 号区域分成两个对称部分、

$$\begin{array}{rcl}& & V_{10}=\hspace{0.17em}2{\int }_0^{\frac{b}{\sqrt{2}}}{\int }_0^{\sqrt{2}b-x}p{u}_{10}(x,y)\mathrm{d}y\mathrm{d}x=\frac{23}{12}p{b}^4.\end{array}$$

(29)

The volume of 1/4 of optimum grillage reads
最佳格栅 1/4的体积为

$$\begin{array}{rcl}& & V/4=\sum _{i=1}^{5}2V_i+\sum _{i=6}^{10}{V}_i=\frac{10237}{3072}p{b}^4\end{array}$$

(30)

and finally, again allowing arbitrary m_p, one arrives at
最后，同样允许任意 m _p ，得出

$$\begin{array}{rcl}& & V=\frac{10237}{768}\frac{p{b}^4}{m_p}.\end{array}$$

(31)