Optimization Based On Energy Splitting Methods For Two Dimensional Discrete Dislocation Dynamics

Crystals are applied so widely that their plastic deformation attracts much interest from academia and industry.The motion of line defects,or dislocations,is believed to be the main reason for plastic deformation in crystals.Dislocation dynamics is established to describe the plasticity behaviors of crystals,and it has been used in many fields,including phase evolution and fracture problems.However,the model is computationally expensive,which restricts its applicability.The Runge-Kutta method is a classical algorithm for dislocation dynamics,which does not take the energy between dislocations into account.In this paper,we will reformulate the dislocation dynamics from a thermodynamic point of view and introduce two optimal algorithms,including an energy splitting method and the nonlinear conjugate gradient method,in order to speed up the simulations.At first,the total energy of dislocation systems is proposed derived from the interacting energy between dislocations.Regrettably,the interacting energy is derived from classical stress field theory for dislocations,which cannot be applied at the core of dislocations,and so the energy total becomes singular in this region as well.In order to remove the singularity of dislocation dynamics,we introduce a parameter to define the width of the dislocation core region and give the non-singular stress field theory of dislocations.Based on this theory,the new energy minimization problem is proposed,which is equivalent to the dynamic problem,making it solvable.To solve the energy minimization problems in dislocation dynamics,two optimal algorithms are introduced — the energy splitting method based on the proximal point algorithm and the nonlinear conjugate gradient method.The former has a proven convergence rate,suggesting its capability to deal with dislocation dynamics,while the latter is widely used in solving nonlinear minimization problems and excels in handling complex onesIn order to analyze the new algorithms for dislocation dynamics,the Runge-KuttaFehlberg method is presented.In a given scenario,three approaches are used to simulate the motion of dislocations.The simulations show that the nonlinear conjugate gradient method spends less time than other algorithms to find the stationary state of dislocation systems.When it comes to some complex numerical experiments,the energy splitting method is less time-consuming compared to the Runge-Kutta-Fehlberg method.The stress-strain curves obtained by three methods are a little bit different from each other,and it is difficult to analyze it from a mathematical point of view.