Strong & Weak Forms

The transformation from strong form to weak form is the mathematical key that makes finite element analysis possible. This lesson explains why we can't directly solve the strong form and how the weak form opens the door to numerical solutions.

The Strong Form: What Physics Demands

Consider a simple 1D problem: a bar fixed at one end, with a distributed load along its length.

The equilibrium equation (balance of forces) in differential form is:

$$-\frac{d}{dx}\left(EA\frac{du}{dx}\right) = f(x)$$

Where:

• $u(x)$ = displacement at position $x$
• $E$ = Young's modulus (material stiffness)
• $A$ = cross-sectional area
• $f(x)$ = distributed load (force per unit length)

This is the strong form — a differential equation that must be satisfied at every point in the domain.

Why "Strong"?

The strong form demands:

• The solution $u(x)$ must be twice differentiable (smooth enough for second derivatives)
• The equation must hold exactly at every point
• Boundary conditions must be satisfied exactly

For simple geometries (uniform bar, constant load), we can solve this analytically. But for complex geometries, variable cross-sections, or multiple materials — analytical solutions don't exist.

The Problem with Direct Numerical Solution

Why not just discretize the strong form directly using finite differences?

$$\frac{d^2u}{dx^2} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{h^2}$$

This works for simple domains but fails for:

• Complex geometries: Irregular boundaries don't fit a regular grid
• Material discontinuities: Derivatives don't exist at interfaces
• Stress concentrations: Local refinement is awkward

We need a different approach — one that relaxes the requirement for pointwise satisfaction.

The Weak Form: Trading Precision for Solvability

The weak form is obtained by:

• Multiplying the strong form by a test function $v(x)$
• Integrating over the domain
• Using integration by parts to move derivatives from $u$ to $v$

Step 1: Multiply by Test Function and Integrate

Start with the strong form:

$$-\frac{d}{dx}\left(EA\frac{du}{dx}\right) = f(x)$$

Multiply both sides by a test function $v(x)$ and integrate:

$$-\int_0^L v \frac{d}{dx}\left(EA\frac{du}{dx}\right) dx = \int_0^L v \cdot f \, dx$$

Step 2: Integration by Parts

Apply integration by parts to the left side. Recall:

$$\int u \, dv = uv - \int v \, du$$

Here, let $u = v$ and $dv = \frac{d}{dx}(EA\frac{du}{dx})dx$:

$$-\left[v \cdot EA\frac{du}{dx}\right]_0^L + \int_0^L \frac{dv}{dx} \cdot EA\frac{du}{dx} \, dx = \int_0^L v \cdot f \, dx$$

Step 3: Apply Boundary Conditions

If $v(0) = 0$ (test function vanishes where displacement is prescribed), the boundary term simplifies:

$$\int_0^L EA\frac{du}{dx}\frac{dv}{dx} \, dx = \int_0^L v \cdot f \, dx + \left[v \cdot EA\frac{du}{dx}\right]_L$$

The last term represents the natural boundary condition (applied force at the free end).

The Final Weak Form

$$\int_0^L EA\frac{du}{dx}\frac{dv}{dx} \, dx = \int_0^L v \cdot f \, dx + v(L) \cdot F_L$$

Where $F_L$ is the applied force at $x = L$.

Why "Weak"?

The weak form is "weaker" because:

• The equation is satisfied on average (integral), not pointwise
• Only first derivatives are required (not second)
• Discontinuities in material properties are naturally handled

This relaxation is precisely what we need for numerical methods!

The Weighted Residual Method

The weak form comes from a broader framework called the method of weighted residuals.

The residual is the error when we substitute our approximation into the strong form:

$$R(x) = -\frac{d}{dx}\left(EA\frac{d\tilde{u}}{dx}\right) - f(x)$$

For an exact solution, $R(x) = 0$ everywhere. For an approximation, $R(x) \neq 0$.

Making the Residual Small

Instead of making $R(x) = 0$ everywhere (impossible for most approximations), we make the weighted integral of the residual zero:

$$\int_0^L w(x) \cdot R(x) \, dx = 0$$

Different choices of weight function $w(x)$ give different methods.

Types of Weighted Residual Methods

The Galerkin Method

In the Galerkin method, we choose the weight functions to be the same as the shape functions used to approximate $u$:

$$\tilde{u}(x) = \sum_j N_j(x) \cdot u_j$$

$$w_i(x) = N_i(x)$$

This is the foundation of the finite element method. The Galerkin method produces symmetric stiffness matrices (for self-adjoint problems), which are computationally efficient to solve.

Physical Interpretation

The weak form has a beautiful physical meaning. Consider:

$$\int_0^L EA\frac{du}{dx}\frac{dv}{dx} \, dx = \int_0^L v \cdot f \, dx + v(L) \cdot F_L$$

$$\text{Internal Work} = \int \sigma \cdot \delta\varepsilon \, dV = \int EA\varepsilon \cdot \delta\varepsilon \, dx$$

$$\text{External Work} = \int f \cdot \delta u \, dx + F \cdot \delta u|_{\text{boundary}}$$

The weak form is equivalent to the Principle of Virtual Work:

For a body in equilibrium, the virtual work done by internal stresses equals the virtual work done by external forces for any admissible virtual displacement.

This is energy balance, and it's why FEA works!

From Weak Form to Matrix Equation

Here's the crucial step that leads to FEA:

• Approximate $u(x)$ using shape functions: $u(x) = \sum_j N_j(x) u_j$
• Choose $v(x) = N_i(x)$ (Galerkin method)
• Substitute into the weak form
• Rearrange to get: $\sum_j \left[\int EA \frac{dN_i}{dx}\frac{dN_j}{dx} dx\right] u_j = \int N_i f \, dx + N_i(L) F_L$

This is the matrix equation:

$$[K]\{u\} = \{F\}$$

Where:

• $K_{ij} = \int EA \frac{dN_i}{dx}\frac{dN_j}{dx} dx$ — Stiffness matrix
• $F_i = \int N_i f \, dx + N_i(L) F_L$ — Force vector

We'll derive this explicitly in Lesson 4.

Essential vs. Natural Boundary Conditions

The weak form reveals two types of boundary conditions:

Essential (Dirichlet) Boundary Conditions

• Prescribed values of the primary variable (displacement)
• Example: $u(0) = 0$ (fixed end)
• Must be explicitly enforced on the solution space
• Test functions must satisfy $v = 0$ where $u$ is prescribed

Natural (Neumann) Boundary Conditions

• Prescribed derivatives (flux, traction, force)
• Example: $EA\frac{du}{dx}|_{x=L} = F_L$ (applied force)
• Automatically satisfied through the boundary integral term
• "Natural" because they appear naturally in the weak form

This distinction is fundamental in FEA:

• Essential BCs: Modify the system by removing/constraining DOFs
• Natural BCs: Add terms to the force vector

Key Takeaways

• Strong form requires pointwise satisfaction of the PDE — hard for complex problems
• Weak form requires integral (average) satisfaction — enables numerical methods
• The transformation uses integration by parts to move derivatives to test functions
• Galerkin method: Choose test functions = shape functions
• Weak form = Virtual Work Principle — energy balance
• Essential BCs (prescribed values) are enforced explicitly
• Natural BCs (prescribed forces) appear automatically in the weak form

What's Next

In the next lesson, we'll introduce shape functions — the polynomial functions used to approximate the field variable within each element. Shape functions are the practical tool that turns the weak form into a computable system.