LRsplines

LRSplines, or locally refined splines, are a class of spline spaces that enable local refinement of tensor-product B-spline bases. They extend traditional B-splines by allowing higher resolution to be added only where needed, avoiding the global growth in degrees of freedom that comes with uniform refinement. LR-splines are built on LR-meshes, two- or three-dimensional subdivision grids obtained by inserting knot lines inside selected regions of the parameter domain. The resulting mesh contains irregular cells and sometimes T-junctions, but preserves a coherent global parametric structure.

Construction and basis. Starting from a tensor-product B-spline grid, refinement is performed by inserting knot lines in one or both parameter directions over a chosen region. The LR-mesh subdivides cells accordingly. The LR-spline basis consists of functions whose supports align with the refined mesh and can be expressed as linear combinations of the original tensor-product B-splines. The resulting spline space maintains a partition of unity and non-negativity, and its basis functions are designed to be locally supported.

Key properties. Local refinement is the principal advantage, with the degree of freedom growing only in refined regions. For analysis purposes, linear independence is guaranteed under the condition of analysis-suitable LR meshes. LR-splines are compatible with standard isogeometric analysis workflows and support adaptive refinement strategies. They provide a flexible alternative to global refinements and to hierarchical methods.

Variants and limitations. The most common variant is the LR B-spline basis. Ensuring independence and numerical stability relies on using analysis-suitable LR meshes. In three dimensions, mesh management and basis construction are more complex, and not all LR-meshes yield stable bases.

Applications. LR-splines are used in isogeometric analysis, computer-aided geometric design, and adaptive simulations where local features such as sharp gradients, cracks, or geometric details require targeted refinement.