论文标题
在金属材料中位错密度演变的数学方面
On mathematical aspects of evolution of dislocation density in metallic materials
论文作者
论文摘要
本文介绍了描述金属材料中位错密度演变的延迟微分方程的解决方案。表征位错种群演变的硬化，恢复和重结晶提供了模型的必要方程式。最后一项将普通微分方程（ODE）转换为具有强（通常是Hölder）非线性的延迟微分方程（DDE）。在假设右侧函数是hölder连续和单调的假设下，我们证明了euler方法的上误差界限，这使我们能够比较模型中其他数值方法的准确性（例如runge-kutta）（例如，尤其是何时何时何时何时进行解决方案的明确公式进行解决方案公式）。最后，我们通过模拟实际工业过程来测试上述结果。
This paper deals with the solution of delay differential equations describing evolution of dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing the evolution of dislocation populations provide the essential equation of the model. The last term transforms ordinary differential equation (ODE) into delay differential equation (DDE) with strong (in general, Hölder) nonlinearity. We prove upper error bounds for the explicit Euler method, under the assumption that the right-hand side function is Hölder continuous and monotone which allows us to compare accuracy of other numerical methods in our model (e.g. Runge-Kutta), in particular when explicit formulas for solutions are not known. Finally, we test the above results in simulations of real industrial process.
