Efficient B-Spline Finite Elements for Cloth Simulation

TL;DR

This work presents an efficient B-spline finite element framework for cloth simulation that achieves a globally $C^1$-continuous displacement field by using quadratic B-spline surfaces. By separating membrane and bending energies and applying a tailored reduced integration scheme, the method reduces Hessian density and quadrature cost while maintaining or improving accuracy. Coupled with fast, parallel Hessian assembly and a partial-factorization-based linear solve, the approach delivers substantial speedups—up to about $2\times$ faster than optimized linear FEM and significantly faster than prior high-order methods—across a wide range of tests including contact-rich scenarios. The combination of smooth geometry representation, robust contact handling via IPC, and solver optimizations yields a practical, scalable, high-fidelity cloth simulator with strong wrinkling detail and robustness for garment animation and non-rectangular patches.

Abstract

We present an efficient B-spline finite element method (FEM) for cloth simulation. While higher-order FEM has long promised higher accuracy, its adoption in cloth simulators has been limited by its larger computational costs while generating results with similar visual quality. Our contribution is a full algorithmic pipeline that makes cloth simulation using quadratic B-spline surfaces faster than standard linear FEM in practice while consistently improving accuracy and visual fidelity. Using quadratic B-spline basis functions, we obtain a globally $C^1$-continuous displacement field that supports consistent discretization of both membrane and bending energies, effectively reducing locking artifacts and mesh dependence common to linear elements. To close the performance gap, we introduce a reduced integration scheme that separately optimizes quadrature rules for membrane and bending energies, an accelerated Hessian assembly procedure tailored to the spline structure, and an optimized linear solver based on partial factorization. Together, these optimizations make high-order, smooth cloth simulation competitive at scale, yielding an average $2\times$ speedup over highly-optimized linear FEM in our tests. Extensive experiments demonstrate improved accuracy, wrinkle detail, and robustness, including contact-rich scenarios, relative to linear FEM and recent higher-order approaches. Our method enables realistic wrinkling dynamics across a wide range of material parameters and supports practical garment animation, providing a new promising spatial discretization for high-quality cloth simulation.

Figures (30)

• Figure 1: Open vs. non-open B-spline curves. Uniform quadratic B-spline curves and basis functions with non-open (top) and open (bottom) knot vectors. The associated knot vector is specified besides each curve. Control points and their corresponding basis functions (plotted in the parametric space) are color-matched. An open knot vector ensures the curve passes through the control points at the endpoints, while a non-open one does not.
• Figure 2: B-spline surfaces. Left: Equidistant knots placed at integer lattice points in parametric space. Top right: 2D B-spline surface in material space $\Omega^0$, allowing for curvy rest shapes. Bottom right: 3D B-spline surface undergoing deformation in world space $\Omega^t$. Here, knot spans are drawn as grid cells with alternating color, and control points are shown as purple dots.
• Figure 3: Reduced integration. Illustration of reduced quadrature schemes on a B-spline surface over the parametric space $[0, 5] \times [0, 5]$. Quadrature points are marked with "$\bm{\times}$". Left: Quadrature points for membrane energy. In interior knot spans, we apply alternating $2 \times 1$ and $1 \times 2$ Gaussian quadrature on the dual grid. For boundary knot spans (shaded in gray), the quadrature pattern follows that of the corresponding reference cell, shown on the right with matching colored borders. Right: Quadrature points for bending energy. Points are placed at the center of each knot span along the boundary, and at the center of each dual grid cell in the interior.
• Figure 4: Mapping control points to quadrature points. The example control point (purple dot) and its associated quadrature points ($\times$) in an interior $3\times3$ knot span are highlighted.
• Figure 5: Control points influenced by contact mesh DoFs. A $5 \times 5$ B-spline surface with its contact mesh. A representative contact vertex (black square) and the 9 control points (purple circles) within its support are highlighted.