The letter presents a penalized version of the time-discretization local minimization scheme first proposed by Efendiev and Mielke in 2006 to resolve time discontinuities in rate-independent systems with nonconvex energies. In order to penalize inequality constrains enforcing the local minimality, the Moreau–Yosida approximation is employed. We prove the convergence of time-discrete solutions to functions that are parametrized BV solutions of the time-continuous problem (in an abstract infinite-dimensional setting), provided that the discretization and approximation parameters are chosen appropriately. We test our scheme on a one-dimensional example and find a notable improvement compared with the original version.

这封信介绍了Efendiev和Mielke于2006年首次提出的时间离散局部最小化方案的一种惩罚形式，用于解决具有非凸能量的速率无关系统中的时间不连续性。为了惩罚强制局部极小值的不平等约束，采用了Moreau-Yosida近似。我们证明了离散函数和逼近参数的时间离散解的收敛性，这些函数是时间连续问题的参数化BV解（在抽象的无穷维设置中），前提是适当地选择了离散化和近似参数。我们在一个一维示例上测试了我们的方案，并发现与原始版本相比有显着改进。