Element Formulation & Degrees of Freedom

Technical provenance

Applicable standards / specifications

• ASME V&V 10 (2019) — Standard for Verification and Validation in Computational Solid Mechanics
• ASME VVUQ 10.2 (2021) — The Role of Uncertainty Quantification in Verification and Validation of Computational Solid Mechanics Models

References

• ASME Verification, Validation and Uncertainty Quantification (VVUQ) — Authoritative standards family for computational-model verification and validation.
• Bathe, K.-J. — Finite Element Procedures — Background reference for finite-element formulation, discretisation, solution and verification.

What Is It?

Element formulation is the mathematical definition of how an element deforms — the shape functions that interpolate the displacement field, the integration scheme that computes the element stiffness, and the degrees of freedom that the element uses. The formulation determines what deformation modes the element can represent and how accurately. Two elements with the same shape (e.g. a hexahedron) can have different formulations (reduced integration, full integration, enhanced strain) that produce different behaviour. Understanding element formulation is essential for choosing the right element for the problem and for diagnosing issues like locking, hourglassing and poor bending behaviour. An element type is not just a mesh shape — it is a mathematical model of deformation.

Why It Matters

The element formulation determines what the element can do. A beam element with only translational DOFs cannot carry bending. A shell element with a membrane-only formulation cannot carry bending. A solid element with full integration may lock in bending or in near-incompressible behaviour. A solid element with reduced integration may develop hourglass modes (zero-energy deformation patterns). The analyst who does not understand the element formulation may choose an element that cannot represent the governing behaviour — and the result is wrong without any visible warning. The solver solves the model; the element formulation determines what the model can represent. Understanding the formulation is essential for credible finite element analysis.

AN ELEMENT TYPE IS A MATHEMATICAL MODEL OF DEFORMATION, NOT JUST A MESH SHAPE. A hexahedron is a shape; a hexahedron with reduced integration, enhanced strain or full integration is a formulation. The formulation determines the deformation modes, the accuracy and the potential issues. Choose the formulation, not just the shape.

Translational and Rotational Degrees of Freedom

The degrees of freedom (DOFs) are the variables that the solver computes at each node. A 3D translational DOF is the displacement in one direction (ux, uy, uz). A 3D rotational DOF is the rotation about one axis (θx, θy, θz). The element type determines which DOFs are active. Solid elements typically have only translational DOFs (3 per node) — they represent the displacement field, and the rotation is derived from the displacement gradients. Beam and shell elements have both translational and rotational DOFs (6 per node) — the rotational DOFs allow the element to carry bending and torsion directly. The DOF set determines what loads and constraints can be applied — a moment load can only be applied to a node with rotational DOFs (beam, shell); a solid node cannot carry a moment directly.

Shape Functions and Interpolation

The shape functions (also called interpolation functions) define how the displacement field varies within the element. They are polynomial functions of the element coordinates, with coefficients determined by the nodal displacements. For a first-order element, the shape functions are linear — the displacement varies linearly between nodes. For a second-order element, the shape functions are quadratic — the displacement varies quadratically, with midside nodes providing the curvature. The shape functions determine the strain field within the element: linear shape functions produce constant strain (first-order); quadratic shape functions produce linear strain (second-order). The strain field, in turn, determines the stress field through the material law. The choice of shape function (element order) affects the accuracy of the displacement, strain and stress — and the computational cost.

Membrane, Bending, Shear, Axial and Torsional Behaviour

Different element types are formulated to represent different deformation modes. Beam elements represent axial (tension/compression along the beam), bending (curvature about two axes), shear (transverse to the beam) and torsion (twist about the beam axis). Shell elements represent membrane (in-plane stretching and shear), bending (out-of-plane curvature) and transverse shear. Solid elements represent the full 3D strain field — all normal and shear components. The element formulation must be able to represent the deformation mode that governs the structural response. A beam element without a torsion DOF cannot represent a shaft in torsion. A shell element without bending stiffness cannot represent a plate in bending. The analyst must know what deformation modes the element supports and must choose an element that covers the governing mode.

Reduced vs Full Integration

The integration scheme determines how the element stiffness is computed. Full integration evaluates the stiffness at enough integration points to integrate the shape functions exactly. Reduced integration uses fewer integration points — typically one point less in each direction. Reduced integration is cheaper (fewer integration points) and avoids certain locking issues (shear locking in bending, volumetric locking in near-incompressible materials). However, reduced integration can produce hourglass modes — zero-energy deformation patterns that deform without producing strain at the integration points. The hourglass modes are non-physical — they represent deformation that the element cannot resist. Hourglass control (artificial stiffness or viscous damping) is used to suppress the hourglass modes. The choice between full and reduced integration depends on the element type, the material and the problem — each has advantages and disadvantages.

Locking — Shear and Volumetric

Locking is a phenomenon where the element is too stiff because the formulation cannot represent the true deformation mode. Shear locking occurs in bending-dominated problems with first-order elements — the linear shape functions cannot represent the quadratic bending displacement, and the element develops artificial shear strain that makes it too stiff in bending. Volumetric locking occurs in near-incompressible materials (Poisson's ratio near 0.5, such as rubber) — the element cannot accommodate the nearly incompressible deformation and becomes too stiff. Both types of locking produce elements that are excessively stiff, leading to under-prediction of displacements and over-prediction of natural frequencies. The remedies include reduced integration, enhanced strain formulations, higher-order elements or specialised element formulations (hybrid elements for incompressibility).

Hourglass Modes

Hourglass modes are zero-energy deformation patterns that occur in under-integrated elements (reduced integration with too few integration points). The element deforms in a pattern that produces no strain at the integration points — the element has no stiffness against this deformation. The hourglass mode is non-physical — it does not represent a real deformation, but the solver cannot distinguish it from a real deformation. The result is a model that deforms in a zig-zag or hourglass pattern that is not physical. Hourglass control — artificial stiffness or viscous damping that resists the hourglass mode — is used to suppress the non-physical deformation. The analyst should check for hourglass modes by examining the deformation shape and the hourglass energy (if reported by the solver). Significant hourglass energy relative to the internal energy indicates that the hourglass control is not adequate and the mesh or the formulation should be changed.

NUMERICAL CONSIDERATION: Reduced integration elements are efficient and avoid locking but require hourglass control. Check the hourglass energy — if it is a significant fraction of the internal energy, the hourglass control is not adequate and the results are unreliable. The remedy may be a finer mesh, a different hourglass control parameter or a different element formulation.

Key Takeaways

• Element formulation determines what deformation modes the element can represent
• DOFs: solid elements have 3 translations; beam and shell have 6 (3 translations + 3 rotations)
• Shape functions interpolate the displacement; first-order = linear, second-order = quadratic
• Reduced integration avoids locking but may produce hourglass modes requiring control
• Locking (shear, volumetric) makes elements artificially stiff — choose the right formulation