Discrete bending forces and their Jacobians

Computation of bending forces on triangle meshes is required for numerous simulation and geometry processing applications. In particular it is a key component in cloth simulation. A common quantity in many bending models is the hinge angle between two adjacent triangles. This angle is straightforward to compute, and its gradient with respect to vertex positions (required for the forces) is easily found in the literature. However, the Hessian of the bend angle, which is required to compute the associated force Jacobians is not documented in the literature. Force Jacobians are required for efficient numerics (e.g., implicit time stepping, Newton-based energy minimization) and are thus highly desirable. Readily available computations of the force Jacobian, such as those produced by symbolic algebra systems, or by autodifferentiation codes, are expensive to compute and therefore less useful. We present compact, easily reproducible, closed form expressions for the Hessian of the bend angle. Compared to automatic differentiation, we measure up to 7× speedup for the evaluation of the bending forces and their Jacobians.

Figure optionsDownload as PowerPoint slide