Hourglassing, Element Formulation & Numerical Stability

Why Element Formulation Is a Structural Decision, Not a Solver Default

In an explicit dynamic analysis the element formulation determines how the continuum is discretised, how strain is sampled, and how the internal forces that resist deformation are computed. The formulation is not a housekeeping choice left to the solver default — it directly controls whether the model represents physical stiffness, whether it locks under bending or near-incompressibility, whether it can survive large distortion, and whether it introduces non-physical energy that contaminates the result. Every element type in an explicit model carries a formulation assumption, and that assumption must be justified against the deformation the element is expected to experience during the event.

Integration Points and the Cost–Accuracy Trade-off

An element evaluates its strain field at integration (Gauss) points. Full-integration elements sample the strain field at enough points to capture all physically admissible deformation modes of the element shape. Reduced-integration elements use fewer points — typically a single point at the element centre for low-order solids — which is dramatically cheaper per element and avoids certain locking pathologies, but at the cost of admitting non-physical deformation modes that cost no strain energy at the integration point. The choice between full and reduced integration is therefore a trade-off between computational cost, locking behaviour and the risk of zero-energy modes. For impact and crash models, where element counts are large and the timestep is small, reduced-integration elements are common — but they carry the obligation to control and verify the hourglass energy they can generate.

Zero-Energy (Hourglass) Modes

A zero-energy mode is a deformation pattern that produces no strain at the integration point and therefore generates no resisting internal force. The element can deform in this pattern without doing any physical work — it costs nothing numerically, so the solver has no natural resistance to it. The classic hourglass mode of an 8-node reduced-integration solid is a diagonal pattern of nodal displacements that alternates in sign around the element, producing the characteristic hourglass shape. Because the mode is not resisted, it can grow under load and pollute the deformation field with non-physical patterns that look wrong in a contour plot but, more seriously, exchange energy with the rest of the model in a way that has no physical basis.

ARTIFICIAL NUMERICAL CONTROL SHOULD STABILISE THE FORMULATION WITHOUT BECOMING A SIGNIFICANT SOURCE OF STRUCTURAL RESISTANCE.

Hourglass Control and Artificial Hourglass Energy

Hourglass control algorithms add an artificial resisting force or stiffness to the zero-energy modes so they cannot grow unbounded. These algorithms are numerical devices, not physical material behaviour. They dissipate or store energy — the hourglass energy — that has no counterpart in the real structure. A small hourglass energy indicates the control is doing its job quietly. A large hourglass energy indicates the control has become a load path: the artificial stiffness is carrying real load that should be carried by physical material deformation. The acceptable fraction of total energy depends on the solver, the control type and the problem, and no universal limit should be quoted without reference to a specific code and certification basis — but the principle is invariant: hourglass energy must be reported, it must be a small fraction of the total energy, and if it is not, the formulation or the mesh must be revised.

Shell and Solid Formulations in Impact Models

Shell formulations dominate thin-walled impact and crash models. The choice of shell formulation governs through-thickness integration, bending behaviour, membrane locking and large-rotation treatment. A formulation with too few through-thickness points will not capture the plastic bending moment distribution correctly; a formulation that locks will over-predict stiffness and under-predict deformation. Solid formulations are used where through-thickness gradients, contact penetration or three-dimensional stress states matter — at impactor noses, joints, thick laminates or regions of severe local distortion. Solid elements that lock under near-incompressible plastic flow will over-predict force and generate artificial hydrostatic stress. In both cases the formulation must be matched to the deformation it will see, not chosen for convenience.

Locking, Distortion and the Effect on the Structural Answer

Locking is the opposite failure mode from hourglassing. A locking element over-resists a legitimate deformation — typically bending or near-incompressible plastic flow — because the interpolation cannot represent the true strain field cheaply, so it generates excessive artificial stiffness. The consequence is an element that is too stiff, a structure that does not deform enough, a force history that is too high, and an energy absorption that is wrong. Distortion — element shape degradation during large deformation — compounds the problem: as an element squews, its integration sampling degrades and both locking and hourglass tendencies can change. An element that behaves well at the start of an event may behave badly once heavily distorted. Formulation choice must consider the end state of the element, not just the initial mesh.

Visible Deformation Is Not a Quantitative Check

A frequent and dangerous practice is to inspect the deformed shape animation, see no obvious hourglass pattern, and conclude the formulation is acceptable. This is a visual judgement on a quantity that must be measured. Hourglass energy can be significant even when the deformation looks smooth, because the control algorithm may suppress the visible pattern while still carrying load through artificial stiffness. Conversely, a small visible undulation may correspond to negligible energy. The only defensible check is quantitative: compare the hourglass energy history against the total energy and the internal energy, and judge whether the artificial energy is a negligible fraction. The threshold for "negligible" depends on the solver, the control type and the certification or acceptance basis — but the act of checking must always be quantitative.

ABSENCE OF VISIBLE HOURGLASS DEFORMATION DOES NOT MEAN ABSENCE OF HOURGLASS ENERGY. The hourglass energy must be checked quantitatively, not judged visually.

Valid Mode vs Non-Physical Zero-Energy Mode

The distinction between a physical deformation mode and a zero-energy hourglass mode is fundamental. A physical mode produces strain at the integration point and is resisted by real material stiffness — it costs energy to deform and that energy is stored or dissipated physically. A zero-energy mode produces no strain at the integration point and is resisted only by the artificial hourglass control — it costs no physical energy and any energy associated with it is artificial. Confusing the two leads to models where non-physical patterns are accepted as real response, or where artificial resistance is accepted as structural stiffness.

PHYSICAL DEFORMATION MODE                         NON-PHYSICAL ZERO-ENERGY (HOURGLASS) MODE
                                                   
  ●─────●          ●─────●                           ●\         /●
  │     │   →      │     │      (bending,           | \       / |
  │     │          │     │       strain at          |  \     /  |
  ●─────●          ●─────●       int. point)        ●\─/\─/\─/●  (alternating
                                                       \  ...  /    nodal pattern;
  Element resists via                               no strain at      no resisting
  real material stiffness                           int. point        internal force)
  → physical internal energy                        → artificial hourglass energy only

Verification Checks for Element Formulation

A formulation is verified not by visual inspection but by quantitative energy checks and, where feasible, by comparison against a reference. The checks below are the minimum expected for any explicit impact or crash model that uses reduced-integration elements.

• Report hourglass energy as a time history and as a fraction of total energy — The fraction must be small; the threshold depends on solver, control type and acceptance basis
• Confirm hourglass control type is appropriate for the element and deformation
• Check that artificial energy is not growing unbounded near failure or erosion
• Compare internal energy against hourglass energy — hourglass must not rival internal energy
• Verify no locking by comparing force/displacement against a benchmark or test where available
• Inspect heavily distorted elements late in the event for formulation degradation

Key Takeaways

• Element formulation determines stiffness representation, locking behaviour and zero-energy risk — it is a structural decision.
• Reduced-integration elements are efficient but require hourglass control and quantitative hourglass energy verification.
• Hourglass energy is artificial; it must be a small fraction of total energy or the formulation must be revised.
• A visually clean deformation does not prove the hourglass energy is acceptable — the check must be quantitative.
• Locking over-resists legitimate deformation and produces an overly stiff, overly strong answer — the opposite failure from hourglassing.