摘要
多领域统一建模中,为求解DAEs (Differential Algebraic Equations)需要解决初始相容性问题。结构化分析可以验证初始相容性,求解部分初值并确定方程自由度,但无法确定需要初始化的变量。错误地选择初始化变量可能极大地降低求解与仿真的效率,甚至导致失效。分析了DAEs初始相容性问题,提出了基于DM分解(Dulmage and Mendelsohn Decomposition)的初始化变量选择策略,通过DM分解将结构化分析得到的系统分解为两部分,对欠约束部分进行可达性分析,确定需要初始化的变量。该方法可以有效地确定初始化变量,避免错误的选择。
In multi-domain modeling, the problem of initial consistency must be settled to solve differential algebraic equations(DAEs). Structural analysis is capable of verifying the initial consistency,solving some initial values and determining the degree of freedom, but not capable of affirming the variables that need to be initialized. If improper selection is made for the initiation variables, it will become low efficient and even fail to solve and simulate the model. After analyzing the initial consistency of DAEs, a strategy based on DM decomposition is proposed for selecting the initialization variables. The method decomposes the system into two parts with the use of DM decomposition after the system is structurally analyzed. By analyzing the reachability of under constraint part, the initialization variables can be affirmed. This method is efficient to affirm the initialization variables and avoids wrong selections.
引文
[1]Otter M.Multi-domain Modeling and Simulation[M].Encyclopedia of Systems and Control.London:Springer London,2015:805-816.
[2]Burge M,Gerdts M.A Survey on Numerical Methods for the Simulation of Initial Value Problems with sDAEs[M].Surveys in Differential-Algebraic Equations IV.Springer International Publishing,2017.
[3]Pryce J D.Solving high-index DAEs by Taylor series[J].Numerical Algorithms(S1017-1398),1998,19(1/4):195-211.
[4]唐卷,杨文强,吴文渊,等.微分代数系统中分块快速指标约简的启发式算法[J].四川大学学报(工程科学版),2014,46(4):67-74.Tang Juan,Yang Wenqiang,Wu Wenyuan,et al.AHeuristic Algorithm for Block Fast Index Reduction in Differential Algebraic Systems[J].Journal of Sichuan University(Engineering Science Edition),2014,46(4):67-74.
[5]Qin Xiaolin,Tang Juan,Feng Yong,et al.Efficient index reduction algorithm for large scale systems of differential algebraic equations[J].Applied Mathematics and Computation(S0096-3003),2016,277(3):10-22.
[6]H?ger C.Faster Structural Analysis of DifferentialAlgebraic Equations by Graph Compression.IFAC-Papers On Line(S2405-8963),2015,48(1):135-140.
[7]Scholz L,Steinbrecher A.Regularization of DAEs based on the Signature method[J].Bit Numerical Mathematics(S0006-3835),2016,56(1):319-340.
[8]Scholz L,Steinbrecher A.Structural-algebraic regularization for coupled systems of DAEs[J].BITNumerical Mathematics(S0006-3835),2016,56(2):777-804.
[9]Pantelides C.The consistent initialization of differentialalgebraic systems[J].Siam Journal on Scientific Computing(S1064-8275),1988,9(2):213-231.
[10]Mattsson S E,Derlind G.Index reduction in differential-algebraic equations using dummy derivatives[J].Siam Journal on Scientific Computing(S1064-8275),1993,14(3):677-692.
[11]Safdarnejad S M,Hedengren J D,Lewis N R,et al.Initialization strategies for optimization of dynamic systems[J].Computers&Chemical Engineering(S0098-1354),2015,78(4):39-50.
[12]李光远,冯勇.微分代数系统结构化分析[J].控制理论与应用,2017,34(8):1019-1027.Li Guangyuan,Feng Yong.Structural analysis for differential algebraic systems[J].Control Theory&Applications,2017,34(8):1019-1027.
[13]Thota S.Initial value problems for system of differential-algebraic equations in Maple[J].BMCResearch Notes(S1756-0500),2018,11(1).
[14]Estévez Schwarz,Diana,Lamour,René.A new approach for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization[J].Numerical Algorithms(S1017-1398),2017,78(2):355-377.
[15]Pryce J D.A simple structural analysis method for DAEs[J].BIT Numerical Mathematics(S0006-3835),2001,41(2):364-394.
[16]Dulmage A L,Mendelsohn N S.Coverings of bipartite graphs[J].Canadian Journal of Mathematics(S1496-4279),1958,10(4):516-534.
