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L15b — Weighted Residual Method（加权残值法）

Q123 🔴 FEM方程推导方法有哪几种？

EN: Methods to derive FEM element equations:

  1. Direct equilibrium method — suitable for simple 1D elements (springs, bars)
  2. Energy methods — Principle of minimum potential energy (elastic materials), Castigliano’s theorem, Principle of virtual work (any material)
  3. Variational method — find a functional equivalent to the PDE
  4. Methods of weighted residuals (MWR) — most general, applicable to any PDE-BC system

The energy method and MWR are the most widely used. Galerkin’s method (a type of MWR) produces identical results to the variational method for self-adjoint problems.

中文： 四种方法——(1)直接平衡法（简单1D单元）；(2)能量法（最小势能原理、虚功原理）；(3)变分法（需泛函）；(4)加权残值法（最通用，适用于任何PDE）。Galerkin法（加权残值法的一种）对自伴随问题等价于变分法。

Q124 🔴 强形式 vs 弱形式

EN: Strong form — the governing PDE must be satisfied at EACH material point. This is a very strict requirement. Exact analytical solutions are often impossible.

Weak form — the PDE is satisfied in an integral (weighted average) sense: ∫VW⋅R dV=0\int_V \mathbf{W} \cdot \mathbf{R} \, dV = \mathbf{0}, where R\mathbf{R} is the residual (error) and W\mathbf{W} is a weight function. This is a “big release” — it allows approximate solutions where the residual at individual points may be non-zero, as long as the weighted integral over the domain vanishes.

FEM is based on the weak form. The weak form also naturally introduces boundary conditions through integration by parts.

中文： 强形式 = 每个物质点精确满足PDE（太严格，解析解常不可得）；弱形式 = 在加权积分意义上满足PDE（∫W⋅R dV=0\int\mathbf{W}\cdot\mathbf{R}\,dV=0），允许各点有残差只要加权积分为零。FEM基于弱形式。分部积分将边界条件自然引入弱形式。

Q125 🔴 四种加权残值法

EN: Four variants of MWR, differing by the choice of weight function WW:

Method Weight Function WW Description
Collocation Dirac delta Set R=0R = 0 at specific points (collocation points)
Subdomain W=1W = 1 on subdomain Set integral of RR over subdomains = 0
Least Squares W=∂R/∂aiW = \partial R/\partial a_i Minimize ∫R2dV\int R^2 dV w.r.t. unknown coefficients
Galerkin Wi=NiW_i = N_i (shape functions) Use same functions for trial and weight

Galerkin is the most commonly used in FEM — it produces symmetric matrices and, for self-adjoint problems, yields the same result as the variational method.

中文： 四种方法——配点法（设R在特定点为0）、子域法（子域内R积分为0）、最小二乘法（极小化R²积分）、Galerkin法（权函数=形函数）。Galerkin最常用，产生对称矩阵，等价于变分法。

Q126 ⭐ Galerkin方法的定义

EN: In Galerkin’s method, the weight functions are the same shape functions used to approximate the solution:

∫VNi⋅R(u~) dV=0,i=1,2,…,n\boxed{\int_V N_i \cdot R(\tilde{\mathbf{u}}) \, dV = 0, \quad i = 1, 2, \ldots, n}

where R(u~)R(\tilde{\mathbf{u}}) is the residual obtained by substituting the approximate solution u~=∑Njdj\tilde{\mathbf{u}} = \sum N_j \mathbf{d}_j into the governing PDE.

This yields nn algebraic equations for the nn unknown coefficients. For the bar element and beam element, Galerkin’s method produces the same stiffness matrix as the energy method.

中文： Galerkin法 = 用形函数做权函数，要求残差的加权积分为零：∫NiR dV=0\int N_i R\,dV=0。得到关于未知系数的n个代数方程。对杆和梁，Galerkin法产生的刚度矩阵与能量法完全一致。

Q127 🔴 加权残值法的通用性

EN: The weighted residual method is the most general approach to deriving FEM equations. Unlike the energy/variational method (which requires a functional — only exists for certain PDE types like elasticity and heat transfer), MWR can be applied to ANY partial differential equation with ANY boundary conditions, even those without a known variational principle.

This generality is a major advantage of MWR/Galerkin’s method and why it is the preferred approach in modern FEM.

中文： 加权残值法是最通用的FEM推导方法。能量法/变分法需要有泛函（仅存在于弹性问题、热传导等特定类型），而MWR适用于任何PDE和边界条件组合。这种通用性是MWR的最大优势。

Q128 🟡 分部积分在Galerkin法中的作用

EN: Integration by parts in Galerkin’s method serves two critical purposes:

  1. Reduces differentiation order — transfers derivatives from the trial function to the weight function, allowing the use of lower-continuity shape functions (e.g., C⁰ elements for a 2nd-order PDE)

  2. Introduces boundary conditions naturally — the boundary terms from integration by parts correspond to natural (Neumann/force) boundary conditions. Essential (Dirichlet/displacement) BCs must still be enforced separately.

Example for bar element: ∫0LNiddx(AEdudx)dx=NiAEdudx∣0L−∫0LAEdudxdNidxdx\int_0^L N_i \frac{d}{dx}(AE\frac{du}{dx})dx = N_i AE\frac{du}{dx}\big|_0^L - \int_0^L AE\frac{du}{dx}\frac{dN_i}{dx}dx. The boundary term NiAEdudxN_i AE\frac{du}{dx} corresponds to the nodal force.

中文： 分部积分的两大作用——(1)降阶：将对试函数的导数转移到权函数上，降低对形函数连续性的要求；(2)引入边界条件：边界项对应自然（力）边界条件。杆单元例子：边界项 NiAE du/dxN_i AE\,du/dx 对应节点力。

Q129 🟡 Galerkin vs 配点法精度对比

EN: Galerkin’s method generally provides better accuracy than the collocation method for the same number of unknown coefficients. In a standard 1D example (u′′+u+x=0u''+u+x=0, u(0)=u(1)=0u(0)=u(1)=0):

  • Collocation (1 point): ~21.7% error
  • Galerkin (1 unknown): ~0.2% error

This is because Galerkin uses information from the entire domain (through integration), while collocation only enforces the equation at isolated points. Galerkin’s method “averages” the error in a way that minimizes it globally.

中文： Galerkin精度远高于配点法。原因：Galerkin利用整个域的信息（积分），配点法只在孤立点满足方程。Galerkin通过加权积分"平均"误差，实现全局误差最小化。

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