Dynamic simulation of elastic bodies is a longstanding task in engineering and computer graphics. In graphics, numerical integrators like implicit Euler and BDF2 are preferred due to their stability at large time steps, but they tend to dissipate energy uncontrollably. In contrast, symplectic methods like implicit midpoint can conserve energy but are not unconditionally stable and fail on moderately stiff problems. To address these limitations, we propose a general class of numerical integrators for Hamiltonian problems which are symplectic on linear problems, yet have superior stability on nonlinear problems. With this, we derive a novel energy-controllable time integrator, A-search, a simple modification of implicit Euler that can follow user-specified energy targets, enabling flexible control over energy dissipation or conservation while maintaining stability and physical fidelity. Our method integrates seamlessly with barrier-type energies and allows for inversion-free and penetration-free guarantees, making it well-suited for handling large deformations and complex collisions. Extensive evaluations over a wide range of material parameters and scenes demonstrate that A-search has biases to keep energy in low frequency motion rather than dissipation, and A-search outperforms traditional methods such as BDF2 at similar total running times by maintaining energy and leading to more visually desirable simulations.

翻译：弹性体的动态模拟是工程学和计算机图形学中长期存在的任务。在图形学中，隐式欧拉法和BDF2等数值积分器因其在大时间步长下的稳定性而受到青睐，但它们往往会导致能量不可控地耗散。相比之下，隐式中点法等辛方法能够保持能量守恒，但并非无条件稳定，且在中等刚度问题上会失效。为解决这些局限性，我们提出了一类适用于哈密顿问题的通用数值积分器，该类方法在线性问题上具有辛特性，同时在非线性问题上具有优越的稳定性。基于此，我们推导出一种新颖的能量可控时间积分器——A-search，它是对隐式欧拉法的简单修改，能够遵循用户指定的能量目标，在保持稳定性和物理保真度的同时，实现对能量耗散或守恒的灵活控制。我们的方法能与势垒型能量无缝集成，并提供无求逆和无穿透保证，使其特别适用于处理大变形和复杂碰撞。通过对广泛材料参数和场景的大量评估表明，A-search倾向于将能量保持在低频运动而非耗散，并且在相似总运行时间下，A-search通过保持能量并产生视觉上更理想的模拟效果，优于BDF2等传统方法。