SciencePediaIn the world of computational engineering, the Finite Element Method (FEM) stands as a titan, allowing us to simulate everything from skyscraper stability to the mechanics of human tissue. However, this powerful tool has a critical vulnerability: numerical locking. Under certain common conditions, such as modeling thin, bending structures or nearly incompressible materials, standard finite elements can become pathologically stiff, yielding results that are wildly inaccurate and physically meaningless. This article addresses this fundamental challenge by exploring the elegant and powerful family of assumed strain methods, uncovering how these techniques provide a 'smarter' approach to element formulation that bypasses the root causes of numerical locking.
The journey begins in the "Principles and Mechanisms" chapter, where we will demystify the core idea of replacing problematic strains with carefully constructed 'assumed' ones, exploring foundational techniques like the B-bar and Enhanced Assumed Strain (EAS) methods. Following this, the "Applications and Interdisciplinary Connections" chapter will showcase these methods in action, revealing how they solve real-world problems across engineering and science, from preventing shear locking in beams to enabling accurate modeling of plastic deformation in metals.
Imagine you are building a model of the Eiffel Tower using Lego bricks. You build the base, the pillars, and the arches. But when you get to the delicate, curved upper sections, you run into a problem. Your standard, rectangular Lego bricks are too rigid. You can't capture the graceful bend of the real structure; your model becomes a clunky, stiff approximation. You can try using smaller and smaller bricks, but the fundamental problem remains: the bricks themselves are not designed for that kind of deformation. They "lock up."
This is precisely the challenge engineers face when simulating complex structures on a computer. In the Finite Element Method (FEM), we break down a complex object—be it a bridge, an airplane wing, or a biological tissue—into a collection of simple shapes, or finite elements. For simple problems, this works beautifully. But when we try to model thin structures that bend, or nearly incompressible materials (like rubber) that deform at constant volume, these simple digital "bricks" can become pathologically stiff. This phenomenon is called numerical locking, and it's not a small error; it can render a simulation completely useless, predicting a structure to be thousands of times stiffer than it actually is.
Why does this happen? The computer element calculates its internal stress based on the positions of its corners. For a thin element asked to bend, a simple interpolation of its corner movements can create a large amount of spurious shear strain—a kind of internal grinding that real, thin plates avoid by bending gracefully. For a nearly incompressible element asked to deform, the same simple interpolation might force a change in volume, which the material resists with immense force. In both cases, the element locks because its limited "vocabulary" of deformation is inconsistent with the physics it's trying to represent. Making the elements smaller (mesh refinement) doesn't always solve the problem, because the flaw is in the very nature of the element itself. To build a better model, we need smarter bricks. This is where the beautiful idea of assumed strain methods comes in.
The core idea behind assumed strain methods is both simple and profound: if the strain calculated directly from the element's distorted shape is causing problems, then let's not use it. Instead, let's assume a simpler, better-behaved strain field inside the element. We are, in a sense, telling the element a "lie" about its internal state of deformation—but it's a carefully constructed lie, a consistent lie that ultimately tells a deeper truth.
Let's see how this works in the simplest possible case: a straight, uniform bar being pulled at its ends. Using a standard finite element, we can model this with a single two-node element. The displacement varies linearly from one end to the other, which results in a perfectly constant strain throughout the element. The standard method gets this exactly right. Now, let's try an "assumed stress" approach, which is a close cousin to assumed strain. We start with a mixed principle (the Hellinger-Reissner principle) and assume the stress inside the element is just a single, constant value. We then use the principles of mechanics to find what this constant stress must be to be consistent with the pulling at the ends. When we solve the equations, we find that this procedure gives us the exact same, correct stiffness for the bar. This is a crucial first check. It tells us that this philosophy of assuming internal fields isn't just a random hack; it's rooted in the same fundamental variational principles of mechanics as the standard method, and it can reproduce correct results.
Now, let's tackle a real locking problem: modeling a block of rubber. Rubber is nearly incompressible; you can change its shape easily, but it's incredibly hard to change its volume. Any deformation can be mathematically split into two parts: a deviatoric part (change in shape) and a volumetric part (change in volume). Volumetric locking happens because a simple element's geometry might force it to calculate non-zero volume changes, even when it's just trying to shear or bend. The simulated material, being nearly incompressible, responds with enormous resistance, and the element locks.
The famous B-bar () method offers an elegant solution. It says: let's keep the complex, shape-changing (deviatoric) strain that our element calculates from its corners. But for the problematic, volume-changing (volumetric) strain, let's throw out the complicated local variations. Instead, we'll calculate the average volume change over the entire element and assume that this single, constant value applies everywhere inside.
It's like telling the digital brick: "I see you think you're being squeezed in this corner and stretched in that one. Forget it. On average, your volume hasn't changed, so just relax." This simple act of averaging, or projecting the volumetric strain onto a constant, provides just enough flexibility for the element to deform freely without locking. It's a targeted strike, fixing only the part of the strain that causes the problem. And once again, this practical "trick" isn't an arbitrary fix. It can be rigorously shown to be equivalent to a more complex mixed variational formulation (a specific form of the Hu-Washizu principle) where an independent pressure field is introduced to manage the incompressibility constraint. This beautiful connection between a simple, implementable idea and a deep theoretical principle is a hallmark of great engineering science.
The method is a specialized tool for volumetric locking. Can we generalize this idea to tackle other forms of locking, like shear locking in plates and shells? The answer is yes, and the framework is known as the Enhanced Assumed Strain (EAS) method.
The philosophy is to augment, or "enhance," the strain field. The total strain inside the element is no longer just the compatible strain () derived from the movement of its corners. Instead, we add an extra piece, an enhanced strain (), that lives entirely inside the element.
This enhanced strain acts as a kind of internal "slack," giving the element extra deformation modes that aren't tied to the global movement of its nodes. This extra flexibility allows the element to satisfy the physical constraints (like zero shear in pure bending) locally, without transmitting spurious stiffness to the overall structure. For instance, in a quadrilateral element trying to model a nearly incompressible material, we can add an extra volumetric strain mode that allows the incompressibility constraint to be met without locking the element's primary deformation.
But how do we add this slack without tearing the model apart? The key is that the enhanced strain field is designed to be zero on the boundaries of the element. It's often defined using a so-called "bubble function," which has a value of zero on the element's edges and "bubbles up" in the interior. This ensures that while the element has extra flexibility inside, it still connects perfectly and continuously with its neighbors. The enhancement is a purely internal affair.
Of course, this extra freedom can't be a free-for-all. It must be governed by a physical principle. The consistency condition that ties the whole theory together is a statement of orthogonality. Derived from the variational principles of mechanics, the condition states that the work done by the final stress field on the enhanced strain modes must be zero.
This beautiful condition ensures that the enhanced modes are only there to alleviate constraints and cannot spuriously generate or absorb energy. They are "ghost" deformations that serve their purpose and then disappear from the energy balance. Furthermore, to ensure the method is consistent, the formulation must still be able to correctly represent the simplest cases, like a state of constant strain. This is guaranteed by passing a condition known as the patch test, which imposes further mathematical constraints on the choice of the enhanced strain fields.
The assumed strain method is not the only way to build smarter bricks. It belongs to a family of advanced formulations, each with a different philosophy but a shared goal.
Reduced Integration: This method is like squinting. Instead of checking for strain everywhere in the element, you only check it at a few select points. This makes the element less sensitive to the spurious strains that cause locking. It's computationally cheap and effective but can sometimes be too flexible, leading to non-physical "hourglass" motions that need to be controlled.
Incompatible Modes: This approach adds extra, "incompatible" displacement modes inside the element, rather than adding strain directly. These modes are also designed to be zero on the boundary, providing internal flexibility much like the EAS method.
Hybrid Stress Methods: This is perhaps the most different philosophy. Instead of starting with displacements and deriving strains, this method starts by assuming an independent stress field inside the element. It then uses a different variational principle (the Hellinger-Reissner principle) to enforce equilibrium and compatibility in a weak, integral sense. This approach also very effectively decouples the variables that cause locking but comes with its own set of stability requirements, known as the Ladyzhenskaya–Babuška–Brezzi (LBB) conditions, that must be carefully satisfied.
What's truly wonderful is that all these sophisticated techniques—assumed strains, assumed stresses, and others—can be seen through a single, unifying lens. They are all ingenious ways to reparameterize the constitutive constraint. The standard method locks because it insists on a rigid, sequential chain: nodal displacements dictate the compatible strain, which in turn dictates the stress. Assumed strain and stress methods break this rigid chain. They introduce new, independent variables—the assumed strains or stresses—and rewrite the laws of physics in a more flexible, but equally valid, variational form. They create new pathways for the simulation to express the underlying physics, bypassing the roadblocks that cause numerical locking and allowing our digital models to behave with the same grace and subtlety as the real world.
We have spent some time learning the formal machinery of assumed strain methods—the variational principles, the interpolations, and the projections. It can feel like a rather abstract exercise in numerical calculus. But to leave it there would be like learning the rules of chess without ever seeing the beauty of a grandmaster's game. The real magic of these methods lies not in their mathematical formalism, but in the vast and varied landscape of physical problems they allow us to explore with fidelity and confidence.
In this chapter, we embark on a journey to see these methods in action. We will discover that the strange numerical pathologies they are designed to cure are not just isolated glitches, but manifestations of a deep conflict between simple numerical approximations and the rich kinematics of the physical world. And in seeing how assumed strain methods resolve these conflicts, we will appreciate them as more than mere "fixes"; they are profound statements of physical intuition encoded into our computational tools.
At the heart of our story is a trio of numerical illusions known collectively as "locking." These are situations where a perfectly reasonable physical structure, when modeled with simple finite elements, appears to be absurdly, non-physically stiff. It "locks up" and refuses to deform. It turns out there are three main ways an element can be fooled, and for each, assumed strain methods provide the key to disillusionment.
Imagine a thin, flexible ruler. You can easily bend it. Now, suppose you build a computer model of this ruler using a line of simple, low-order "Timoshenko" beam elements. You apply a bending load, and... nothing happens. The computed ruler remains stubbornly straight, as if it were made of diamond instead of plastic. This is shear locking.
The problem arises from a simple confusion. The Timoshenko beam theory, unlike simpler theories, accounts for the shear deformation of the beam's cross-section. In the limit of a very thin beam, this shear deformation must vanish. The finite element, however, with its simple linear interpolation for displacement and rotation, gets confused. It finds that the only way it can represent a bent shape is by also introducing a large, spurious shear strain. To minimize the total energy, the element sees this huge (and artificial) shear energy and decides the best course of action is to prevent bending altogether.
How do we teach the element the truth? The assumed strain method, in its Mixed Interpolation of Tensorial Components (MITC) guise, provides an elegant answer. Instead of letting the element compute a complicated, spatially varying, and incorrect shear strain, we tell it to "assume" that the shear strain is just a single constant value over its entire length. This single value is chosen to be the average of what the displacements would have implied. By replacing a complex, pointwise constraint with a single, weaker, averaged constraint, the element is freed from its self-imposed straitjacket. It can now bend freely without creating parasitic shear, and it even begins to pass fundamental consistency checks, like the "shear patch test," which verifies that it can represent a state of pure, constant shear exactly. This same principle extends beautifully from one-dimensional beams to two-dimensional plates and shells, where low-order elements trying to model the behavior of thin structures would otherwise lock up in the exact same way.
Our next illusion is more subtle and appears when we move from flat to curved objects. Consider a thin, curved shell, like a piece of an eggshell. You can gently bend it, changing its curvature, but it is very difficult to stretch its surface—this is called an "inextensional" deformation. The energy required to bend the shell scales with the cube of its thickness, , while the energy to stretch it scales linearly with thickness, . For a very thin shell, the penalty for stretching is vastly higher than the penalty for bending.
A simple finite element, placed on this curved surface, often cannot distinguish between the two. Its limited vocabulary of shapes means that when it tries to represent a pure bending mode, it inadvertently introduces spurious stretching, or "membrane" strains. Just as with shear locking, the element sees the enormous energy penalty associated with this artificial stretching and locks up, refusing to bend.
This is not just a numerical curiosity; it has dangerous practical consequences. For instance, predicting the buckling load of a curved dome or fuselage requires an accurate calculation of its bending stiffness. A membrane-locked model will be artificially stiff, dangerously over-predicting the load at which the structure will catastrophically fail.
Once again, assumed strain and mixed methods come to the rescue. Formulations like the Enhanced Assumed Strain (EAS) method enrich the element's kinematic description. They add extra, "enhanced" strain modes—mathematical degrees of freedom that exist only inside the element—specifically to absorb the spurious membrane strains. This allows the element to bend without paying the artificial stretching penalty, restoring the correct, flexible response and enabling accurate predictions of complex behaviors like nonlinear buckling.
The final and perhaps most profound form of locking occurs when a material refuses to be squeezed. This volumetric locking arises whenever a material is, or becomes, nearly incompressible (its Poisson's ratio approaches ). A standard displacement-based element tries to enforce this incompressibility constraint at every single integration point. For a low-order element, this is an impossible task; the only way it can satisfy so many constraints is by not deforming at all.
This illusion of incompressibility can arise from several fascinating sources:
Inherent Material Properties: Some materials, like rubber, are naturally nearly incompressible. Modeling a rubber seal or engine mount with standard elements would lead to a completely useless, overly stiff result. Advanced formulations, including mixed displacement-pressure (u-p) methods and assumed strain variants, are essential. The mixed method introduces pressure as a new, independent variable to handle the constraint, but this requires satisfying the tricky "inf-sup" stability condition. Assumed strain methods like the ("B-bar") or ("F-bar") methods offer a clever alternative. They modify the kinematics directly, typically by assuming the volumetric part of the strain is constant throughout the element, effectively relaxing the pointwise constraint to a single average constraint.
Emergent Physical Behavior: This is where things get truly interesting. Consider a standard piece of metal, like steel or aluminum. It is normally quite compressible. However, when it begins to yield and deform plastically, the fundamental physics of dislocation motion dictates that this plastic flow occurs at constant volume. The material, through its own behavior, becomes nearly incompressible. A simulation of metal forming or a crash scenario will therefore inevitably encounter volumetric locking unless an appropriate formulation, such as the method, is used to account for this emergent incompressibility.
System-Level Constraints: The constraint doesn't even have to come from the material itself. Imagine a flexible structure interacting with an incompressible fluid, like a heart valve leaflet opening and closing in blood, or a ship's hull vibrating in water. Because the fluid's total volume cannot change, the motion of the solid structure at the interface is globally constrained. This externally imposed constraint can induce a severe form of locking in the solid model. The solution is a holistic one: one must use stable, locking-free elements for both the solid (perhaps using a mixed or assumed strain method) and the fluid, and ensure their coupling at the interface is also stable and consistent. This demonstrates the power of the assumed strain concept to solve problems that span multiple physical domains.
The journey does not end with simply "unlocking" our elements so they give the right displacement. The deeper reward of using methods like EAS is that they produce more accurate, physically meaningful stress and strain fields inside the elements. This is crucial for many areas of engineering and science where we need to know not just how much something deforms, but why it might fail.
A beautiful example comes from fracture mechanics. A key parameter for predicting crack growth is the -integral, a quantity calculated from the stress and strain fields in a region surrounding a crack tip. For this quantity to be physically meaningful, it must be "path-independent"—its value should not depend on the precise contour over which it is calculated. If one uses a simple element with a cheap fix for stability, like reduced integration without proper hourglass control, the resulting stress fields can be noisy and polluted. This pollution violates the local equilibrium conditions upon which path-independence is founded, and the computed -integral becomes meaningless garbage.
However, a well-formulated element, whether it's a higher-order element or a lower-order one stabilized with an assumed strain approach, produces a clean stress field that better approximates local equilibrium. This "truthful" field yields a nearly path-independent -integral, providing a reliable prediction of the energy flowing into the crack tip and, ultimately, a reliable assessment of the structure's safety.
From the simple bending of a beam to the buckling of a spacecraft, from the plastic flow of metal to the flutter of a heart valve, we have seen the same story play out. A simple computational model, when faced with a subtle physical constraint, creates a numerical illusion. And in each case, the key to seeing the truth is to imbue the model with a little more physical intuition—to "assume" a simpler, more physically appropriate form for the strains that are causing the trouble. This is the unifying power of assumed strain methods. They are a testament to the idea that the most robust and elegant computational tools are those that have the laws and symmetries of physics woven directly into their very fabric.