Handling locking in isogeometric analysis

I frequently get asked how structural locking phenomena is handled in isogeometric analysis (IGA). By locking I mean parasitic stiffness resulting from certain incompatibilities or deficiencies in a finite element’s kinematic description. The practical consequence of this is that the structure will become overly stiff, especially for coarse meshes, and will require many more degrees of freedom than practical to converge to a correct solution.

The most common types of locking are shear and membrane locking that occur in reduced kinematic descriptions like beam and shell elements and volumetric or pressure locking that occurs in incompressible elastic materials. However, it’s important to realize that locking can occur for any type of constraint that is applied incorrectly in a finite element formulation. A good example of this occurs in finite element contact formulations. When a contact constraint is applied incorrectly locking will occur causing the classical contact chattering phenomena that many FEA practitioners have experienced.

Mathematically, locking can be described as an instability resulting from a certain kinematic constraint (e.g., the incompressibility constraint) being enforced too strongly. In other words, too many constraints are being enforced resulting in a stiffening of the element’s response. You will sometimes hear more mathematically inclined FEA people describe locking as an inf-sup instability or an element not being inf-sup stable, referring to a classical mathematical result that indicates when the appropriate number of constraints are being enforced for a given element formulation. The application of constraints in a finite element formulation result in what are often called mixed finite element methods. A very nice overview of inf-sup stability and mixed finite element methods can be found here.

To “fix” locking in structural finite elements there are two primary lines of attack: (1) construct a mixed finite element method where the correct number of constraints is enforced exactly by constructing the correct function space for the Lagrange multipliers (that are used to enforce the constraint) or (2) mimic the behavior of a mixed method by using a “reduced” quadrature scheme to integrate the terms of the formulation that correspond to the constraint enforcement. Since quadrature can be viewed as a type of projection into a polynomial space the number of quadrature points can be tailored to replace the need to construct a Lagrange multiplier space and solve a mixed linear system, that can have many more degrees of freedom due to the presence of additional Lagrange multiplier degrees of freedom.

In industrial finite element codes, reduced quadrature schemes dominate due to their simplicity to implement, efficiency due to fewer degrees of freedom, robustness, and straight-forward application to nonlinear materials. However, it should be noted that for C^0 finite elements, mixed (or closely related projection) methods sometimes masquerade as standard nodal finite elements since the degrees of freedom associated with the Lagrange multipliers are “condensed” out analytically at the element level without disrupting the sparsity and accuracy of the resulting linear system.

For smooth finite elements, like those enountered in IGA, this condensation of degrees of freedom at the element level is much more difficult to achieve due to the increased inter-element smoothness (a dense stiffness matrix results), further placing emphasis on reduced quadrature as the appropriate mechanism for alleviating locking in isogeometric finite elements.

It should be noted, however, that sometimes smoothness can be leveraged to change the finite element formulation such that locking is precluded a priori. The classic example of this in IGA is using Kirchhoff-Love shells, that require C^1 smoothness, instead of Reissner-Mindlin shells, that only require C^0 smoothness, to eliminate shear locking at the level of the shell theory (Kirchhoff-Love shells do not have shear strains).

Many academic studies have been performed on locking phenomena in IGA. I will only mention a few of the most prominent, pioneering or those works currently influencing Coreform IGA. Note that the interested reader can follow the citation chains for these papers to find many related works and to develop a more well-rounded feel for the subject.