避免跨界单元建模形式
Quad
1
Quad
2 3
17
USE OF SYMMETRY
不同类型的对称:
Mirror symmetry
Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error.
10
Element distortion
对于面积和体积的要求
不能存在负面积， 物理坐标和自然坐 标之间的转换
11
Element distortion
对于面积和体积的要求
12
Element distortion
中部节点位置
可能导致应力场的奇异
a b >b/4 >a/4
13
MESH COMPATIBILITY
•No rotational components w.r.t. axis parallel to symmetry plane（平行于
对称面）
Plane of
u
v
w
x
y
z
symmetry
xy
Free Free Fix Fix Fix Free
yz
Fix Free Free Free Fix Fix
3
Element distortion
单元会存在不规则的情况，但是不能逾越有限 元法的基本原理.
The distortions are measured against the basic shape of the element
Square Quadrilateral elements Isosceles triangle Triangle elements Cube Hexahedron elements Isosceles tetrahedron Tetrahedron elements
Axial symmetry
Cyclic symmetry
Repetitive symmetry
18
Mirror symmetry
特殊面的对称形式
Planes of symmetry
Modelling of quarter model is sufficient
19
Mirror symmetry
zx
Free Fix Free Fix Free Fix
23
Mirror symmetry
Anti-symmetric 反对称
•No translational displacement parallel to symmetry plane •No rotational components w.r.t. axis normal to symmetry plane
任何加载可以分解为对称和反对称的组合
y
P/2
P/2
x
y
Symmetric loading
P
Asymmetric loading
a
b
b
a
x=
a
b
b
a
+
y
P/2
P/2
Anti-Symmetric loading
a
b
b
a
x
25
Mirror symmetry
Y
P/2
P
= P /2
+
X Full frame structure
P /2
Sym.
P /2
Anti-sym.
26
Mirror symmetry
Y
Y
P
P
2
2
Properties are halved for this member
X
X
All nodes on this line fixed against the horizontal displacement and rotation.
Mirror symmetry
Anti-symmetric loading 反对称加载
y
P
P
x
a
b
b
a
Deflection = 0
P
偏移为0
Rotation = Free
转角自由
22
Mirror symmetry
Symmetric 对称
•No translational displacement normal to symmetry plane（垂直于对称 面）
最小势能原理的要求 单元边界的协调性
14
不同阶数的单元组合
1
Quad 2 Linear
3
Quad
1
Linear
2
3
单元间隙，造成应力场的奇异
15
不同阶数的单元组合
解决方式:
Use same type of elements throughout
1
1
Linear Linear
2
Use transition elements
Plane of symmetry
xy
yz
zx
u
v
w
x
y
z
Fix Fix Free Free Free Fix
Free Fix Fix Fix Free Free
Fix Free Fix Free Fix Free
24
Mirror symmetry
Any load can be decomposed to a symmetric and an anti-symmetric load
Quad 2 Quad
3
Quad Transition Linear
Vary quadratically along this edge
Vary linearly along this edge
Transition Element
Use MPC equations 多点约束方程
16
Straddling elements 跨界单元模式
7
Element distortion
单元的横纵比
a
Rule of thumb:
b
b a
3 10
Stress analysis Displacement analysis
8
Element distortion
角度的要求
skew
b
Taper
b<5a
a
9
Element distortion
曲率的要求
第九章-有限元法中相 关问题的处理
INTRODUCTION
保证有限元计算的结果可靠，稳定 提高求解的精度和效率需要考虑的主要因素：
计算量和计算规模的大小； 明确需求和问题的特点； 根据物理性质和几何特征选择合理的单元配置； 边界条件的施加； 初始条件的加载。
考虑二维问题，如何施加约束:
y
u1x = 0
1
u2x = 0
u3x = 0
2
Single point
constraints (SPC)
3
单点约束
1
2
x
3
20
Mirror symmetry
对称加载
y
P
P
x
a
b
Deflection = Free 法向偏移无约束
Rotation = 0 转角为0
b
a
P
21