符号表 Notation
以下是本论文中使用的符号的非详尽列表。我们将采用粗体字符表示矢量的惯例。
欧拉描述
| Explanation | 中文解释 |
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Ω | An open subset of Rn | 背景区域 |
∂Ω | boundary of Ω | 区域边界 |
u | Velocity field | 速度场 |
p | Pressure field | 压强场 |
n | Unit vector normal to ∂Ω | 单位外法向 |
∇⋅u | Divergence of u | 速度散度 |
∇u | Gradient of u | 速度梯度 |
Δu | Laplacian of u, defined by Δ=∇⋅∇ | 拉普拉斯算子 |
⟨⋅,⋅⟩ | L2-inner product on Ω | L2内积 |
∥⋅∥k | Norm defined by the Hk-inner product on Ω if k=0,1 | Hk范数 |
∥⋅∥2 | Euclidian norm | 欧几里得范数 |
x | x=(x1,x2,x3) Eulerian coordinates | 欧拉坐标 |
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拉格朗日描述
代数
| Explanation | 中文解释 |
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J | | 雅可比矩阵 |
F | | 形变梯度 |
参数
| Explanation | 中文解释 |
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μ | (Dynamic) Viscosity | 动力粘性 |
ρ | Density | 密度 |
ν | Kinematic viscosity, defined as nu:=mu//rho | 运动粘性 |
Δt | Timestep | 时间步长 |
Re | Reynolds number | 雷诺数 |
h | Discretization parameter1 | 空间离散参数 |
矩阵的性质
| Explanation | 中文解释 |
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L2(Ω) | Space of square-integrable functions on Ω | Ω上的平方可积函数 |
Hk(Ω) | Space of L2-functions on Ω with derivatives up to order k which are also in L2(Ω) | Hk空间 |
H0k(Ω) | Space of functions in Hk(Ω) which equal zero on ∂Ω | H1空间 |
H−1 | Dual space of H1 | H1的对偶空间 |
固体力学
| Explanation | 中文解释 |
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σ | The Cauchy stress tensor | 柯西应力张量 |
P | the first Piola–Kirchhoff stress tensor | 第一PK应力张量 |
W | Strain energy density function | 应变能密度函数 |
E | Elastic potential energy function | 弹性势能函数 |
Ek | Kinetic energy function | 动能函数 |
B | A region in three-dimensional Euclidean point space(Rn), called configuration of B | 固体的构型 |
Br | A specific configuration of B | 参考构型 |
Bt | Current configutation of B | 现时构型,当前构型 |
B | A body, considered as a set of elements, or particles, or material points, whatever | 固体 |
X(X,t) | bijection mapping between Br and Bt | |
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X | X=(X1,X2,X3)Coordinates of material points at reference configuration | 固体粒子在参考构型的坐标 |
t | current time | 当前时间 |
T | total time interval | 总时间 |
多孔弹性体
浸没边界法
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S(X,t) | prolongation operation, distribution operator | 延拓算子 |
J(X,t) | restriction operator, interpolation operator | 插值算子 |
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