Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. For many problems this can result in severe...
Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. For many problems this can result in severe time-step constraints. The techniques used to develop explicit positivity preserving DG discretizations can,however, not easily be combined with implicit time integration methods. In this paper we therefore present a different approach. Using Lagrange multipliers the conditions imposed by the positivity preserving limiters are directly coupled to the DG discretization combined with a Diagonally Implicit Runge-Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush-Kuhn-Tucker (KKT) problem, which is frequently encountered in constrained optimization. The resulting nonsmooth nonlinear algebraic equations have,however, a different structure compared to most constrained optimization *** therefore developed an efficient active set semi-smooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG *** of this semi-smooth Newton method can be proven using a specially designed quasi-directional derivative of the timeimplicit positivity preserving DG discretization. The time-implicit positivity preserving DG discretization is demonstrated for nonlinear scalar conservation laws. Since the limiter is only active in areas where positivity must be enforced it does not affect the higher order DG discretization elsewhere. Numerical experiments for the advection, Burgers,Allen-Cahn, Barenblatt, and Buckley-Leverett equations are provided.
离散单元法(Discrete element method,DEM)是颗粒流模拟的主流方法,并且能够适应不同颗粒物性以及设备操作条件。但DEM方法本身存在计算量巨大、模拟时间过长等问题,因而难以针对工业规模的设备进行颗粒流动混合过程的模拟。为了克服此...
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离散单元法(Discrete element method,DEM)是颗粒流模拟的主流方法,并且能够适应不同颗粒物性以及设备操作条件。但DEM方法本身存在计算量巨大、模拟时间过长等问题,因而难以针对工业规模的设备进行颗粒流动混合过程的模拟。为了克服此困难,本论文在前人工作基础上拓展并优化了基于多尺度异构超级计算系统的高效大规模离散单元模拟方法和程序,实现了胶囊和圆柱等非球形颗粒的模拟,并以此考察了滚筒混合器以及料斗内颗粒物性对其流动混合性能的影响,加深了对相应设备中颗粒流动混合机理的认识。论文研究表明在水平滚筒内采用交错抄板能够促进颗粒的径向抛洒和沿抄板的轴向导流。对所研究的滚筒,在最优抄板结构下,其轴向混合效率分别为无抄板滚筒和全抄板滚筒的16倍和5倍。模拟还表明该效应对不同颗粒物性(粒径、形状、密度和摩擦系数)和滚筒尺寸均存在,预示了其广泛的应用前景。论文研究还表明,滚筒内胶囊颗粒的形状能够影响其堆积密度和空间取向,从而影响其混合性能。而在所考察的物性范围内,颗粒密度对偏析结构的影响较形状更为显著。利用颗粒形状与密度的共同作用可进一步提高混合性能。论文通过与实验数据的对比表明,三维非规则颗粒碰撞力模型较传统的线性模型对料斗中落料过程模拟的准确性更高。以此进行的模拟表明:Beverloo方程一般能够较好地预测多粒径颗粒体系的落料行为,但在颗粒尺寸差异较大时,会高估整体落料速度。分析表明,颗粒体系的力链强度明显增加导致的内部流动阻力增大和动态拱桥的形成是其重要成因。论文最后总结了对DEM模型与模拟方法的改进,及其在颗粒流动与混合模拟研究中的应用,并展望了其后续工作与工业应用前景。
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