实际工作：优化桥梁结构

本教程旨在使用 truss 包裹。 * In order to use it, download the file ``truss.py`` and put it into your current work directory.

%load_ext autoreload
%autoreload 2

The autoreload extension is already loaded. To reload it, use:
  %reload_ext autoreload

import numpy as np
import matplotlib.pyplot as plt
import copy, os
from scipy import optimize
import truss
%matplotlib nbagg

建造桥梁结构

在本课程中，我们将使用桁架对桥梁结构进行建模，并使用各种标准对其进行优化。基本结构介绍如下。它是由钢筋制成的，并承受一个垂直力 \(G\) .这座桥是对称的，所以只有左半边是模型。

E       = 210.e9   # Young Modulus [Pa]
rho     = 7800.    # Density       [kg/m**3]
A       = 5.e-2    # Cross section [m**2]
sigmay  = 400.e6   # Yield Stress  [Pa]

# Model definition
model = truss.Model() # Model definition

# NODES
nA = model.add_node((0.,0.), label = "A")
nC = model.add_node((3.,0.), label = "C")
nD = model.add_node((3.,3.), label = "D")
nE = model.add_node((6.,0.), label = "E")
nF = model.add_node((6.,3.), label = "F")
nG = model.add_node((9.,0.), label = "G")
nH = model.add_node((9.,3.), label = "H")

# BOUNDARY CONDITIONS
nA.block[1] = True
nG.block[0] = True
nH.block[0] = True

# BARS
AC = model.add_bar(nA, nC, modulus = E, density = rho, section = A, yield_stress = sigmay)
CD = model.add_bar(nC, nD, modulus = E, density = rho, section = A, yield_stress = sigmay)
AD = model.add_bar(nA, nD, modulus = E, density = rho, section = A, yield_stress = sigmay)
CE = model.add_bar(nC, nE, modulus = E, density = rho, section = A, yield_stress = sigmay)
DF = model.add_bar(nD, nF, modulus = E, density = rho, section = A, yield_stress = sigmay)
DE = model.add_bar(nD, nE, modulus = E, density = rho, section = A, yield_stress = sigmay)
EF = model.add_bar(nE, nF, modulus = E, density = rho, section = A, yield_stress = sigmay)
EG = model.add_bar(nE, nG, modulus = E, density = rho, section = A, yield_stress = sigmay)
FH = model.add_bar(nF, nH, modulus = E, density = rho, section = A, yield_stress = sigmay)
FG = model.add_bar(nF, nG, modulus = E, density = rho, section = A, yield_stress = sigmay)
GH = model.add_bar(nG, nH, modulus = E, density = rho, section = A, yield_stress = sigmay)

# STRUCTURAL LOADING
nG.force = np.array([0., -1.e6])


%timeit model.solve()
model

3.17 ms ± 395 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

<Model: 7 nodes, 11 bars>

xlim, ylim = model.bbox(deformed = False)
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
ax.set_aspect("equal")
#ax.axis("off")
model.draw(ax, deformed = False, field = "stress", label = True, force_scale = 1.e-6, forces = True)
plt.xlim(xlim)
plt.ylim(ylim)
plt.grid()
plt.xlabel("Axe $x$")
plt.ylabel("Axe $y$")

<IPython.core.display.Javascript object>

<matplotlib.text.Text at 0x7f79009258d0>

节点的详细结果

model.data(at = "nodes")

	block		coords		disp		force		label
	bx	by	x	y	ux	uy	Fx	Fy	o
0	False	True	0	0	-0.00114286	0	0	1e+06	A
1	False	False	3	0	-0.000857143	-0.00337955	0	0	C
2	False	False	3	3	0.00142857	-0.00337955	0	0	D
3	False	False	6	0	-0.000571429	-0.00618767	0	0	E
4	False	False	6	3	0.000857143	-0.00647339	0	0	F
5	True	False	9	0	0	-0.00813865	3e+06	-1e+06	G
6	True	False	9	3	0	-0.00813865	-3e+06	0	H

钢筋的详细结果

model.data(at = "bars")

	conn		direction		geometry		props			state
	c1	c2	dx	dy	length	volume	density	mass	section	elongation	failure	strain	stress	tension
0	A	C	1	0	3	0.15	7800	1170	0.05	0.000285714	False	9.52381e-05	2e+07	1e+06
1	C	D	0	1	3	0.15	7800	1170	0.05	0	False	0	0	0
2	A	D	0.707107	0.707107	4.24264	0.212132	7800	1654.63	0.05	-0.000571429	False	-0.000134687	-2.82843e+07	-1.41421e+06
3	C	E	1	0	3	0.15	7800	1170	0.05	0.000285714	False	9.52381e-05	2e+07	1e+06
4	D	F	1	0	3	0.15	7800	1170	0.05	-0.000571429	False	-0.000190476	-4e+07	-2e+06
5	D	E	0.707107	-0.707107	4.24264	0.212132	7800	1654.63	0.05	0.000571429	False	0.000134687	2.82843e+07	1.41421e+06
6	E	F	0	1	3	0.15	7800	1170	0.05	-0.000285714	False	-9.52381e-05	-2e+07	-1e+06
7	E	G	1	0	3	0.15	7800	1170	0.05	0.000571429	False	0.000190476	4e+07	2e+06
8	F	H	1	0	3	0.15	7800	1170	0.05	-0.000857143	False	-0.000285714	-6e+07	-3e+06
9	F	G	0.707107	-0.707107	4.24264	0.212132	7800	1654.63	0.05	0.000571429	False	0.000134687	2.82843e+07	1.41421e+06
10	G	H	0	1	3	0.15	7800	1170	0.05	0	False	0	0	0

死亡（也就是说。结构）质量

m0 = model.mass()
m0 * 1.e-3 # Mass in tons !

14.323889603929565

问题

问题1 ：验证是否没有超过屈服应力，您认为此结构的重量是否为最佳？您可以使用 state/failure 整个模型的可用数据。

# Example:
model.data(at = "bars").state.failure.values

#...

array([False, False, False, False, False, False, False, False, False,
       False, False], dtype=object)

问题2 ：同时修改所有横截面，以使重量最小化，同时保持可接受的应力水平。

问题3 ：我们想沿着 \(\vec y\) 点的轴 \(D\) ， \(F\) 和 \(H\) 为了使节点的垂直位移最小化 \(G\) 乘以结构的质量 \(\alpha\) ：

\[\α=u_y（g）m\]

哪里 \(u_y(G)\) 是节点的位移。 \(G\) 沿着 \(\vec y\) 轴和 \(m\) 整个结构的质量。

不要进一步修改问题4中确定的部分。评论解决方案。

问题4 ：同样的问题，也有位移 \(\vec x\) 属于 \(C\) ， \(D\) ， \(E\) 和 \(F\) .更好吗？

问题5 ：您现在可以尝试通过删除/合并精心选择的梁和节点来执行拓扑优化。为了使结构更加高效。

问题6 ：现在要求您优化横截面以及 \(C\) ， \(D\) ， \(E\) 和 \(F\) 以达到每个梁的屈服应力。