Work, Energy, and the Virtual Work Principle in FEM

1. Work, Energy, and Why FEM Uses the Weak Form

If Newton's Second Law is the physical foundation of FEM, then the principle of virtual work is the mathematical bridge that makes it numerically solvable. Understanding this connection transforms FEM from a black box into a tool you can reason about.

The core idea: instead of requiring the differential equation of equilibrium to hold at every single point in a body (which is very restrictive when dealing with element boundaries and discontinuities), we instead require that the work done by all forces through any admissible virtual displacement is zero. This "integral form" is weaker than the point-wise form — hence "weak form" — but mathematically equivalent for the solutions we care about.

2. Work: W = F·d and the Dot Product

Work is done when a force moves its point of application through a displacement. For a constant force $\mathbf{F}$ over displacement $\mathbf{d}$:

The dot product means only the component of force in the direction of displacement does work. This has profound implications for FEM boundary conditions:

• A fixed boundary (zero displacement) does zero work — reactions at fixed nodes contribute nothing to the energy balance
• A symmetry boundary (zero normal displacement, free tangential) does zero normal work — symmetry conditions are energy-conservative by construction

For a varying force over a curved path, work is the line integral:

In FEM, this integral is evaluated numerically using Gauss quadrature within each element.

Units and Scaling

3. Kinetic Energy and the Work-Energy Theorem

The work-energy theorem: the net work done on an object equals its change in kinetic energy.

This is derived directly from Newton's Second Law by integrating $F = ma$ over displacement. In FEM terms, the kinetic energy is:

Where $\dot{\mathbf{u}}$ is the global velocity vector and $[\mathbf{M}]$ is the global mass matrix. In Abaqus, this is the "ALLKE" energy output variable.

4. Elastic Strain Energy in Structures

When a structure deforms elastically, work is stored as elastic strain energy. For a simple spring (Hooke's Law: $F = ku$):

For a 3D solid body with stress $\boldsymbol{\sigma}$ and strain $\boldsymbol{\varepsilon}$:

This equivalence between the continuum integral and the matrix product is why the stiffness matrix $[\mathbf{K}]$ can be built by integrating material properties over element volumes — the two formulations are mathematically identical.

Strain Energy Density

The strain energy density $u_0 = \frac{1}{2}\boldsymbol{\sigma}:\boldsymbol{\varepsilon}$ (J/m³) is a local quantity available at each integration point in FEM. It's useful for identifying which parts of a structure carry the most elastic energy — a guide to weight optimization. Material can be efficiently removed from regions with low strain energy density (topology optimization).

5. The Principle of Virtual Work

A virtual displacement $\delta\mathbf{u}$ is an infinitesimally small, kinematically admissible (compatible with boundary conditions) imaginary displacement. It's not a real motion — it's a mathematical device.

The Principle of Virtual Work states: for a body in equilibrium, the total virtual work done by all internal and external forces through any virtual displacement is zero:

Expanding:

Where $\mathbf{b}$ is the body force (gravity, etc.), $\mathbf{t}$ is the surface traction, $\boldsymbol{\sigma}$ is the Cauchy stress tensor, and $\delta\boldsymbol{\varepsilon}$ is the virtual strain field corresponding to $\delta\mathbf{u}$.

This equation is the weak form of the equilibrium equation. It integrates over the volume instead of requiring the equilibrium condition to hold pointwise — making it compatible with piecewise polynomial approximations over finite elements.

6. From Virtual Work to FEM Weak Form

FEM approximates the displacement field using shape functions $\mathbf{N}$ and nodal displacements $\mathbf{d}$:

Where $\mathbf{B}$ is the strain-displacement matrix (derivatives of $\mathbf{N}$). Substituting into the virtual work equation with $\delta\mathbf{u} = \mathbf{N}\delta\mathbf{d}$:

Since this must hold for any $\delta\mathbf{d}$:

This is the fundamental FEM system of equations — derived rigorously from the principle of virtual work. The stiffness matrix $[\mathbf{K}] = \int_V \mathbf{B}^T \mathbf{D} \mathbf{B} \, dV$ is computed by Gauss quadrature over each element's volume.

7. Energy Quantities in FEM Output

Most FEM solvers output a comprehensive energy balance. Understanding these is essential for validating simulation quality:

8. Energy Conservation Checks in Simulations

The energy balance is one of the most powerful quality checks for any FEM simulation:

If this doesn't balance, something is wrong — hourglass energy (in reduced-integration elements), artificial damping, or contact penetration.

Hourglass Energy: A Common Explicit FEM Problem

Reduced-integration elements (C3D8R, S4R) use one Gauss point instead of eight. This is much faster but introduces "hourglass modes" — zero-energy deformation modes that don't register strain. The solver adds artificial hourglass stiffness to suppress them. Monitor the hourglass energy (ALLAE in Abaqus): if it exceeds 5% of ALLIE, your mesh is too coarse or element aspect ratios are poor.

9. Cross-Topics