TL框架中，一切积分、核函数梯度、邻域几何关系都在初始构形 （参考构形 Ω 0 \Omega_0 Ω 0
粒子间参考距离X i j = X i − X j \mathbf{X}_{ij} = \mathbf{X}_i - \mathbf{X}_j X ij = X i − X j 不随时间变
核函数W ( X i j , h 0 ) W(\mathbf{X}_{ij},h_0) W ( X ij , h 0 ) ∇ 0 W \nabla_0 W ∇ 0 W 不随时间变 （除非自适应h h h
与更新拉格朗日框架下的SPH相比，TLSPH可以避免邻域扭曲导致的核梯度噪声，并且更适合大变形固体。
每个粒子i i i
当前位置：x i \mathbf{x}_i x i v i \mathbf{v}_i v i
质量m i m_i m i ρ 0 i \rho_{0i} ρ 0 i V 0 i = m i / ρ 0 i V_{0i}=m_i/\rho_{0i} V 0 i = m i / ρ 0 i
变形梯度F i \mathbf{F}_i F i ∇ 0 u i \nabla_0 \mathbf{u}_i ∇ 0 u i
应力：P i \mathbf{P}_i P i S i \mathbf{S}_i S i
邻域由参考距离决定：
X i j = X i − X j , r i j = ∥ X i j ∥ . \mathbf{X}_{ij}=\mathbf{X}_i-\mathbf{X}_j,\quad r_{ij}=\|\mathbf{X}_{ij}\|. X ij = X i − X j , r ij = ∥ X ij ∥. 对任意矢量场v ( X ) \mathbf{v}(\mathbf{X}) v ( X ) SPH梯度近似的强形式和弱形式 - 知乎arrow-up-right 中强形式A2）：
⟨ ∇ 0 v ⟩ i = ∑ j V 0 j ( v j − v i ) ⊗ ∇ 0 W i j (1) \boxed{\langle\nabla_0 \mathbf{v}\rangle_i=\sum_{j} V_{0j}(\mathbf{v}_j - \mathbf{v}_i)\otimes\nabla_0 W_{ij}}\tag{1} ⟨ ∇ 0 v ⟩ i = j ∑ V 0 j ( v j − v i ) ⊗ ∇ 0 W ij ( 1 ) 重点：此处全部是 ∇ 0 \nabla_0 ∇ 0
由于模拟过程中粒子分布不均和边界截断的问题，SPH的粒子梯度近似可能与实际梯度相差较大。实践中，我们想着至少重构出线性（一阶）部分 ，也即保证梯度近似的一阶一致性。假设此标量场为
v ( X ) = a + M X + O ( X 2 ) . \mathbf{v}(\mathbf X)=a+\mathbf M\mathbf X+O(\mathbf X^2). v ( X ) = a + MX + O ( X 2 ) . 其梯度为
∇ 0 v = M . \nabla_0\mathbf{v}=\mathbf M. ∇ 0 v = M . 如果用一个修正矩阵B 0 \mathbf B_0 B 0 M \mathbf M M
[ ∑ j V 0 j ( v j − v i ) ⊗ ∇ 0 W i j ] B 0 i = M \left[\sum_{j} V_{0j}\,(\mathbf{v}_j - \mathbf{v}_i)\otimes\nabla_0 W_{ij}\right]\mathbf B_{0i}=\mathbf M [ j ∑ V 0 j ( v j − v i ) ⊗ ∇ 0 W ij ] B 0 i = M 将v ( X ) \mathbf{v}(\mathbf X) v ( X )
M [ ∑ j V 0 j ( X j − X i ) ⊗ ∇ 0 W i j ] B 0 i = M \mathbf M\left[\sum_{j} V_{0j}\,(\mathbf X_j - \mathbf X_i)\otimes\nabla_0 W_{ij}\right]\mathbf B_{0i}=\mathbf M M [ j ∑ V 0 j ( X j − X i ) ⊗ ∇ 0 W ij ] B 0 i = M 记A 0 i = ∑ j V 0 j ( X j − X i ) ⊗ ∇ 0 W i j \mathbf A_{0i}=\sum_{j} V_{0j}\,(\mathbf X_j - \mathbf X_i)\otimes\nabla_0 W_{ij} A 0 i = ∑ j V 0 j ( X j − X i ) ⊗ ∇ 0 W ij B 0 i \mathbf B_{0i} B 0 i
B 0 i = A 0 − 1 = [ ∑ j V 0 j ( X j − X i ) ⊗ ∇ 0 W i j ] − 1 (2) \boxed{\mathbf B_{0i}=\mathbf A_0^{-1}=\left[\sum_{j} V_{0j}(\mathbf X_j - \mathbf X_i)\otimes\nabla_0 W_{ij}\right]^{-1}}\tag{2} B 0 i = A 0 − 1 = [ j ∑ V 0 j ( X j − X i ) ⊗ ∇ 0 W ij ] − 1 ( 2 ) 因此，我们对于任何一个矢量场v ( X ) \mathbf{v}(\mathbf{X}) v ( X ) B 0 \mathbf B_0 B 0
⟨ ∇ 0 v ⟩ i = [ ∑ j V 0 j ( v j − v i ) ⊗ ∇ 0 W i j ] B 0 i (3) \boxed{\langle\nabla_0 \mathbf{v}\rangle_i=
\left[\sum_{j} V_{0j}(\mathbf{v}_j - \mathbf{v}_i)\otimes\nabla_0 W_{ij}\right]\mathbf B_{0i}
}\tag{3} ⟨ ∇ 0 v ⟩ i = [ j ∑ V 0 j ( v j − v i ) ⊗ ∇ 0 W ij ] B 0 i ( 3 ) 前提是B 0 i \mathbf B_{0i} B 0 i B 0 i \mathbf B_{0i} B 0 i A 0 i \mathbf A_{0i} A 0 i
B 0 i † = ( A 0 i T A 0 i + ε I ) − 1 A 0 i T . \mathbf B_{0i}^{\dagger}=(\mathbf A_{0i}^T\mathbf A_{0i}+\varepsilon\mathbf I)^{-1}\mathbf A_{0i}^T. B 0 i † = ( A 0 i T A 0 i + ε I ) − 1 A 0 i T . 然后通过A 0 i \mathbf A_{0i} A 0 i ∣ A 0 i ∣ < α |\mathbf A_{0i}| < \alpha ∣ A 0 i ∣ < α I \mathbf I I B 0 i † \mathbf B_{0i}^{\dagger} B 0 i †
B 0 i = { ∣ A 0 i ∣ α B 0 i † + α − ∣ A 0 i ∣ α I , ∣ A 0 i ∣ < α ; B 0 i † , ∣ A 0 i ∣ ≥ α . \mathbf B_{0i}=\begin{cases}
\dfrac{|\mathbf A_{0i}|}{\alpha}\mathbf B_{0i}^{\dagger}+\dfrac{\alpha-|\mathbf A_{0i}|}{\alpha}\mathbf I,&|\mathbf A_{0i}| < \alpha;\\
\mathbf B_{0i}^{\dagger},& |\mathbf A_{0i}| \geq\alpha.
\end{cases} B 0 i = ⎩ ⎨ ⎧ α ∣ A 0 i ∣ B 0 i † + α α − ∣ A 0 i ∣ I , B 0 i † , ∣ A 0 i ∣ < α ; ∣ A 0 i ∣ ≥ α . 变形梯度的连续定义是
F = ∂ x ∂ X = ∇ 0 x \mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}}=\nabla_0\mathbf x F = ∂ X ∂ x = ∇ 0 x 也就是将v ( X ) = x \mathbf v(\mathbf X)=\mathbf x v ( X ) = x
⟨ F ⟩ i = ⟨ ∇ 0 x ⟩ i = [ ∑ j V 0 j ( x j − x i ) ⊗ ∇ 0 W i j ] B 0 i \langle\mathbf{F}\rangle_i=\langle\nabla_0\mathbf x\rangle_i=\left[\sum_{j} V_{0j}(\mathbf{x}_j - \mathbf{x}_i)\otimes\nabla_0 W_{ij}\right]\mathbf B_{0i} ⟨ F ⟩ i = ⟨ ∇ 0 x ⟩ i = [ j ∑ V 0 j ( x j − x i ) ⊗ ∇ 0 W ij ] B 0 i 因为F = ∇ 0 u + I \mathbf F=\nabla_0\mathbf u+\mathbf I F = ∇ 0 u + I u \mathbf u u
⟨ F ⟩ i = ⟨ ∇ 0 u ⟩ i + I = [ ∑ j V 0 j ( u j − u i ) ⊗ ∇ 0 W i j ] B 0 i + I \langle\mathbf{F}\rangle_i=\langle\nabla_0\mathbf u\rangle_i+\mathbf I
=\left[\sum_{j} V_{0j}(\mathbf{u}_j - \mathbf{u}_i)\otimes\nabla_0 W_{ij}\right]\mathbf B_{0i}+\mathbf I ⟨ F ⟩ i = ⟨ ∇ 0 u ⟩ i + I = [ j ∑ V 0 j ( u j − u i ) ⊗ ∇ 0 W ij ] B 0 i + I 动量方程的连续形式为：
ρ 0 x ¨ = ∇ 0 ⋅ P + ρ 0 b \rho_0 \ddot{\mathbf{x}} = \nabla_0\cdot \mathbf{P} + \rho_0\mathbf{b} ρ 0 x ¨ = ∇ 0 ⋅ P + ρ 0 b 参考SPH梯度近似的强形式和弱形式 - 知乎arrow-up-right 中的弱形式B2，我们可以将∇ 0 ⋅ P \nabla_0\cdot \mathbf{P} ∇ 0 ⋅ P
⟨ ∇ 0 ⋅ P ⟩ i = ∑ j V j ( P i + P j ) ⋅ ∇ 0 W i j \langle\nabla_0\cdot\mathbf P\rangle_i=\sum_j V_j(\mathbf P_i+\mathbf P_j)\cdot\nabla_0 W_{ij} ⟨ ∇ 0 ⋅ P ⟩ i = j ∑ V j ( P i + P j ) ⋅ ∇ 0 W ij 有核梯度修正的版本为：
⟨ ∇ 0 ⋅ P ⟩ i = ∑ j V j ( P i B 0 i + P j B 0 j ) ⋅ ∇ 0 W i j \langle\nabla_0\cdot\mathbf P\rangle_i=\sum_j V_j(\mathbf P_i\mathbf B_{0i}+\mathbf P_j\mathbf B_{0j})\cdot\nabla_0 W_{ij} ⟨ ∇ 0 ⋅ P ⟩ i = j ∑ V j ( P i B 0 i + P j B 0 j ) ⋅ ∇ 0 W ij 因此完整的动量方程离散为：
ρ 0 i x ¨ i = ∑ j V j ( P i B 0 i + P j B 0 j ) ⋅ ∇ 0 W i j + ρ 0 i b i . \boxed{\rho_{0i} \ddot{\mathbf{x}}_i = \sum_j V_j(\mathbf P_i\mathbf B_{0i}+\mathbf P_j\mathbf B_{0j})\cdot\nabla_0 W_{ij} + \rho_{0i}\mathbf{b}_i.} ρ 0 i x ¨ i = j ∑ V j ( P i B 0 i + P j B 0 j ) ⋅ ∇ 0 W ij + ρ 0 i b i . 见Ye等（2025）：
Ye, M. et al. DRLinSPH: an open-source platform using deep reinforcement learning and SPHinXsys for fluid-structure-interaction problems. Engineering Applications of Computational Fluid Mechanics 19 , 2460677 (2025).
