2. Hourglass modes in reduced-integration solid and shell elements

Reduced integration is one of the most consequential modeling decisions in explicit structural analysis. By evaluating element stiffness at fewer integration points than the polynomial order strictly requires, LS-DYNA cuts the cost of element internal-force evaluation and, more importantly, avoids volumetric and shear locking in nearly incompressible or thin-walled problems. The price is that certain deformation patterns produce no strain at any integration point, and therefore no resisting internal force. These are the hourglass modes: zero-energy, non-physical displacement fields that the element cannot detect. Understanding their geometry, their origin, and their physical meaning is the prerequisite for choosing a control strategy rather than blindly accepting a default.

Where the zero-energy modes come from

Consider the eight-node solid element (ELFORM=1, the classic constant-stress hexahedron). Its displacement field is trilinear, so the strain field is linear in each natural coordinate. A single integration point at the element center samples only the constant part of that strain field. Any displacement pattern whose strain vanishes at the center point is invisible to the element: it generates nodal forces of zero, contributes nothing to internal energy, and is restrained only by neighboring elements or by hourglass control. The same logic applies to the four-node shell with one in-plane integration point (ELFORM=2, Belytschko-Tsay), where the through-thickness integration is separate and the in-plane reduced rule leaves bending and membrane hourglass patterns unconstrained.

The number of such modes follows from a simple count. For the hex, 24 nodal degrees of freedom minus 6 rigid-body modes leaves 18 deformation modes; a single integration point provides 6 independent strain constraints, so 12 zero-energy modes remain. For the shell, the count depends on the formulation and on whether drilling and transverse-shear terms are included, but the practical outcome is the same: several independent hourglass patterns exist per element, and they can be excited by any load that is not perfectly smooth across the mesh.

Deformation patterns and their physical interpretation

The hourglass patterns are best understood as alternating, checkerboard-like distortions. In the hex, the canonical mode is a "bowtie" or "sandwich" pattern: the top face translates in +x while the bottom face translates in −x, with the mid-plane nodes stationary, or the four side faces rotate in alternating senses. In the shell, the best-known pattern is the "butterfly" or "diamond" mode, in which the four corner nodes move out of plane in an alternating up-down-up-down sequence while the element center stays put. These are not rigid-body motions and not genuine bending; they are purely kinematic artifacts of under-sampling.

Physically, an hourglass mode represents a deformation that a real continuum would resist through higher-order strain gradients, but which the reduced-integration element cannot see because those gradients vanish at the sampled point. In a real structure, such a pattern would require local warping and would be energetically expensive. In the discretized model, it is free. The practical consequence is that hourglassing is a numerical instability, not a material instability, and it appears most readily where the mesh is coarse, the loading is localized (point loads, contact impacts, blast pressures), or the material is nearly incompressible and the element is already close to locking.

Recognizing hourglassing in an LS-DYNA run

The primary quantitative indicator is the hourglass energy reported in the ASCII file glstat and in the matsum file. LS-DYNA tracks the energy absorbed by hourglass control separately; the ratio of hourglass energy to total internal energy is the standard acceptance metric. A commonly used rule of thumb is that hourglass energy should stay below roughly 5% of internal energy for the result to be considered trustworthy, and below 10% at the absolute limit; values above that indicate the control is doing too much work and the solution is being stiffened artificially. The glstat line for hourglass energy appears alongside kinetic, internal, and total energy, so the check is a single column comparison rather than a separate post-processing exercise.

Visually, the pattern is unmistakable once you know what to look for. Plot the displacement or stress contour on the deformed mesh and look for element-to-element alternation. In a shell, the butterfly mode shows up as adjacent elements bending in opposite directions; in a solid, the bowtie mode shows as alternating expansion and contraction of element faces. Animating the deformation at a few frames per millisecond makes the oscillation obvious, because hourglass modes typically grow monotonically once excited and do not decay the way physical vibrations do.

Why the modes matter for the rest of the course

The existence of zero-energy modes is the reason hourglass control exists at all. Every control scheme — viscous, stiffness, or the combined formulations available through CONTROL_HOURGLASS and the IHQ flag on SECTION_SOLID and SECTION_SHELL — works by adding a small artificial stiffness or damping that penalizes the hourglass deformation without affecting the physical response. The choice of coefficient, the type of control, and the element formulation are therefore coupled decisions: a fully integrated element (ELFORM=2 for solids, ELFORM=16 for shells) eliminates the modes but reintroduces locking and cost; a reduced-integration element with well-tuned control is usually the better engineering compromise. The next sections examine how each control type behaves and how to select its parameters.

Practical example: detecting hourglassing in a drop test

As a concrete illustration, consider a 2 mm thick steel sheet-metal bracket modeled with ELFORM=2 shells at a 3 mm target element size, impacted by a rigid drop mass at 5 m/s. The model uses *CONTROL_HOURGLASS with IHQ=4 (stiffness-based, Flanagan-Belytschko) and a default coefficient of 0.1. After the run, open glstat and locate the hourglass energy column. Suppose the peak internal energy is 42 J and the peak hourglass energy is 1.1 J — a ratio of about 2.6%, comfortably inside the 5% guideline. Now refine the mesh to 1.5 mm without changing the control coefficient: the hourglass energy typically drops further because the finer mesh resolves the deformation gradient that the coarse mesh could not. Conversely, if you coarsen to 6 mm and the ratio climbs to 12%, the correct response is not to raise the coefficient blindly but to first check whether the loading is being applied as a point load on a single node; distributing the impact over a small patch often removes the excitation entirely.

The failure symptom to watch for is a solution that looks plausible in a contour plot but has hourglass energy growing without bound while kinetic energy decays faster than the physical damping would explain. In that case the artificial stiffness is absorbing energy that should have gone into plastic work, and the reported forces and displacements are unreliable. The remedy sequence is: verify the energy ratio, inspect the deformation pattern for mesh-following alternation, then adjust element formulation, mesh density, or control type in that order of preference.

The key takeaway is that hourglass modes are a property of the discretization, not of the material or the loading. They arise whenever the integration rule samples too few points to see a deformation pattern, and they manifest as checkerboard or butterfly distortions that carry no physical meaning. Recognizing them requires both a visual habit — looking for mesh-following alternation — and a quantitative habit — reading the hourglass energy column in glstat against the internal energy. With those two habits in place, the choice of control scheme becomes an informed engineering decision rather than a default setting.