  The linear conjugate gradient method is widely used in physical simulation,
particularly for solving large-scale linear systems derived from Newton's
method. The nonlinear conjugate gradient method generalizes the conjugate
gradient method to nonlinear optimization, which is extensively utilized in
solving practical large-scale unconstrained optimization problems. However, it
is rarely discussed in physical simulation due to the requirement of multiple
vector-vector dot products. Fortunately, with the advancement of GPU-parallel
acceleration techniques, it is no longer a bottleneck. In this paper, we
propose a Jacobi preconditioned nonlinear conjugate gradient method for elastic
deformation using interior-point methods. Our method is straightforward,
GPU-parallelizable, and exhibits fast convergence and robustness against large
time steps. The employment of the barrier function in interior-point methods
necessitates continuous collision detection per iteration to obtain a
penetration-free step size, which is computationally expensive and challenging
to parallelize on GPUs. To address this issue, we introduce a line search
strategy that deduces an appropriate step size in a single pass, eliminating
the need for additional collision detection. Furthermore, we simplify and
accelerate the computations of Jacobi preconditioning and Hessian-vector
product for hyperelasticity and barrier function. Our method can accurately
simulate objects comprising over 100,000 tetrahedra in complex self-collision
scenarios at real-time speeds.
