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L12 — Nonlinearity（非线性）🔴 样卷仅略微涉及

Q89 🔴 三类非线性来源

EN: The three major sources of nonlinearity in FEM:

  1. Material nonlinearity（材料非线性）: The stress-strain relationship is not linear — plasticity, hyperelasticity (rubber), creep, damage. The tangent modulus changes with strain and loading history. σ=f(ε,history)\boldsymbol{\sigma} = f(\boldsymbol{\varepsilon}, \text{history}).

  2. Geometric nonlinearity（几何非线性）: Large displacements, large rotations, or large strains. The strain-displacement relationship includes quadratic terms (Green-Lagrange strain). Equilibrium must be established on the deformed configuration. Stiffness depends on displacement.

  3. Boundary nonlinearity（边界非线性）: Contact and impact problems. The contact status (open/closed/sticking/sliding) changes with loading and is unknown a priori. The response is non-smooth.

中文： 三大非线性来源——(1)材料非线性：应力-应变关系不是直线（塑性、橡胶超弹性、蠕变、损伤）；(2)几何非线性：大位移/大转动/大应变，应变-位移关系含二次项，平衡方程需写在变形后构形上；(3)边界非线性：接触/碰撞/摩擦，接触状态随加载变化且未知。

Q90 🔴 线性 vs 非线性分析的关键区别

EN:

Feature Linear Nonlinear
Governing equation Ku=P\mathbf{Ku} = \mathbf{P} K(u,t)u=P(u,t)\mathbf{K}(\mathbf{u},t)\mathbf{u} = \mathbf{P}(\mathbf{u},t)
Stiffness matrix Constant Depends on displacement, time, state
Scaling of results Yes (proportional) No (cannot simply scale)
Superposition Valid Not valid
Solution method Solve once Incremental + Iterative
Cost 1× 10× to 1000× linear analysis

中文： 线性：K为常数，结果可比例缩放，叠加原理成立，直接求解；非线性：K随位移/时间/状态变化，不可比例缩放，叠加原理不成立，需增量+迭代求解。非线性分析成本 = 几十到上千次线性分析。

Q91 🔴 几何非线性的Green-Lagrange应变

EN: For large deformation problems, the linear strain εij=12(ui,j+uj,i)\varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i}) is insufficient. The Green-Lagrange strain tensor includes quadratic terms:

εij=12(∂ui∂xj+∂uj∂xi+∂um∂xi∂um∂xj)\boxed{\varepsilon_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} + \frac{\partial u_m}{\partial x_i}\frac{\partial u_m}{\partial x_j}\right)}

The underlined term ∂um∂xi∂um∂xj\frac{\partial u_m}{\partial x_i}\frac{\partial u_m}{\partial x_j} is the nonlinear (quadratic) term that accounts for large deformation effects. It couples different displacement gradient components.

中文： 大变形下，线性应变不够，需用Green-Lagrange应变（含位移梯度的二次项 ∂um∂xi∂um∂xj\frac{\partial u_m}{\partial x_i}\frac{\partial u_m}{\partial x_j}）。这个二次项是大变形分析的核心特征。小变形时二次项可忽略，退化为线性应变。

Q92 🔴 接触问题为什么是最困难的非线性问题？

EN: Contact problems are among the most difficult nonlinear problems because:

  1. Unknown contact region — the contacting area is not known a priori and changes during loading
  2. Discontinuous normal velocity — when contact is established or lost, the normal velocity jumps
  3. Non-smooth friction — stick-slip transitions cause tangential velocity discontinuities; friction force changes direction abruptly
  4. No-penetration constraint — the inequality constraint gN≤0g_N \leq 0 (gap cannot be negative) must be enforced
  5. Multiple status changes — each contact point can switch between open, sticking, and sliding states

中文： 接触问题五大难点——(1)接触区域未知且随加载变化；(2)法向速度不连续（接触建立/丧失时跳变）；(3)摩擦非光滑（粘着-滑动转换时切向力突变）；(4)不可穿透约束 gN≤0g_N\leq0 需强制满足；(5)每个接触点可在开/粘/滑三态间切换。

Q93 🔴 主面(Master) vs 从面(Slave)

EN: In contact modeling:

  • Master surface（主面）: Typically the stiffer, rigid, or coarser-meshed surface
  • Slave surface（从面）: Typically the softer, deformable, or finer-meshed surface

Rule: Slave nodes cannot penetrate master segments. The slave surface should have a finer mesh to accurately resolve the contact pressure distribution. The master should be the surface with the stiffer material behavior.

中文： 主面 = 刚性/较硬/网格较粗的面；从面 = 较软/网格较细的面。从面节点不能穿透主面。从面网格应更细以准确解析接触压力。主面通常是较硬一侧。

Q94 🔴 Coulomb摩擦模型

EN: The Coulomb friction model states that the tangential contact traction tTt_T is bounded by the friction coefficient μ\mu times the normal contact pressure:

∣tT∣≤μ∣tN∣\boxed{|t_T| \leq \mu |t_N|}

Two states:

  • Sticking（粘着）: ∣tT∣<μ∣tN∣|t_T| < \mu|t_N| — no relative tangential motion
  • Sliding（滑移）: ∣tT∣=μ∣tN∣|t_T| = \mu|t_N| — relative tangential motion occurs; friction force opposes the motion direction

The stick-slip transition makes the tangential response non-smooth.

中文： Coulomb摩擦：切向力不超过 μ\mu × 法向力。粘着时 ∣tT∣<μ∣tN∣|t_T|<\mu|t_N|（无相对滑动）；滑移时 ∣tT∣=μ∣tN∣|t_T|=\mu|t_N|（摩擦力达极限）。粘着-滑移转换导致切向不光滑。

Q95 🔴 Newton-Raphson方法

EN: The Newton-Raphson (N-R) method solves nonlinear equations R(u)=P−Fint(u)=0\mathbf{R}(\mathbf{u}) = \mathbf{P} - \mathbf{F}_{int}(\mathbf{u}) = \mathbf{0} iteratively:

KT(i)Δu(i)=R(i)\boxed{\mathbf{K}_T^{(i)} \Delta\mathbf{u}^{(i)} = \mathbf{R}^{(i)}}

u(i+1)=u(i)+Δu(i)\boxed{\mathbf{u}^{(i+1)} = \mathbf{u}^{(i)} + \Delta\mathbf{u}^{(i)}}

where KT=∂Fint/∂u\mathbf{K}_T = \partial\mathbf{F}_{int}/\partial\mathbf{u} is the tangent stiffness matrix and R\mathbf{R} is the residual (imbalance between external and internal forces). Iteration continues until ∥R∥<tolerance\|\mathbf{R}\| < \text{tolerance}.

Advantages: Quadratic convergence near the solution (very fast).
Disadvantages: Must recalculate and factorize KT\mathbf{K}_T each iteration — computationally expensive. May diverge if initial guess is poor.

中文： N-R迭代：用切线刚度矩阵 KT\mathbf{K}_T 求解位移增量，更新位移，直至残差收敛。优点：近解时二次收敛（极快）；缺点：每次迭代需更新切线刚度（计算量大），初值不好可能发散。

Q96 🔴 Modified Newton-Raphson vs N-R

EN:

Feature N-R Modified N-R
Tangent stiffness update Every iteration Less frequently (e.g., once per increment)
Cost per iteration High Lower
Convergence rate Fast (quadratic) Slower (linear)
Total iterations Fewer More
Best for Highly nonlinear problems Mildly nonlinear problems

The trade-off is between the cost of updating KT\mathbf{K}_T and the number of iterations needed.

中文： N-R每次迭代都更新切线刚度（贵但收敛快）；mN-R少更新（便宜但需更多迭代）。本质是计算成本与收敛速度的权衡。强烈非线性用N-R，弱非线性用mN-R可能更高效。

Q97 🔴 增量-迭代求解策略

EN: The standard approach for nonlinear FEM analysis is the incremental-iterative strategy:

  1. Divide the total load into increments: P1,P2,…,Pn\mathbf{P}_1, \mathbf{P}_2, \ldots, \mathbf{P}_n
  2. For each increment, apply the load step ΔP\Delta\mathbf{P}
  3. Iterate within the increment using N-R or modified N-R until convergence
  4. Update state variables (stress, plastic strain, contact status, etc.)
  5. Proceed to the next load increment

This avoids the divergence that would occur if the full load were applied in one step. Each increment can be thought of as a “small” nonlinear problem.

中文： 增量+迭代 = FEM非线性求解的标准策略。将总载荷分段施加，每段内用N-R/mN-R迭代至收敛。这种策略可避免一次性施加全载荷导致的发散。这是所有商业FEM软件处理非线性的基本方法。

Q98 🟡 N-R可能发散的情况

EN: Newton-Raphson may fail to converge (diverge) when:

  1. The tangent stiffness KT\mathbf{K}_T becomes zero or negative (e.g., snap-through buckling, strain softening)
  2. The initial guess is too far from the solution
  3. The load increment is too large for the degree of nonlinearity

Special techniques: arc-length methods (Riks method) for snap-through; automatic time stepping; line search methods.

中文： N-R在以下情况可能发散：切线刚度变零或变负（跳跃屈曲snap-through、应变软化）、初值太差、载荷增量太大。对策：弧长法(Riks)、自适应步长、线搜索。

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