基于自适应松弛Picard法的高效非饱和渗流有限元分析
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摘要
有效地模拟非饱和渗流过程对土质边坡稳定性分析、土石坝渗流、污染物迁移等众多领域有着重要的意义。描述非饱和渗流的Richards方程是具有强烈非线性的偏微分方程,通常需要采用有限元等数值方法并结合有效的迭代方法进行求解。Picard迭代法是实用的非线性计算方法,在非饱和渗流领域应用广泛,但经常会出现收敛震荡、速度缓慢和精度降低的问题。为提高计算性能,结合有限元法提出了一种高效的自适应松弛Picard法。通过模拟一维和二维渗流算例,并与传统方法的结果进行对比,对算法和程序的准确性、高效性和鲁棒性进行了验证。测试结果表明,该方法可以在保证计算精度的同时有效地减少数值震荡,提高收敛速度。研究成果对非饱和渗流有限元程序的开发和应用有一定的参考价值。
Effectively simulating unsaturated flow is of great significance in many areas such as the soil slope stability analysis, seepage of earth-rockfill dam, contaminant transport. Due to the strongly nonlinear characteristics of Richards equation for the unsaturated flow, numerical solution schemes such as the finite element method with an efficient iterative algorithm usually have to be employed. Picard method as a practical nonlinear iterative method is widely applied to the unsaturated flow field; but usually suffers from convergence oscillation, slow convergence rate and inaccurate solution. In order to improve the computing performance, an efficient finite element procedure based on the adaptive relaxed Picard method is developed. Through the simulations of 1D and 2D unsaturated seepage problems, accuracy, efficiency and robustness of the proposed procedure are validated by comparing with the traditional methods. It is shown that the adaptive relaxed Picard method can effectively reduce the convergence oscillation and significantly improve the convergence rate with the accuracy guaranteed. The proposed procedure provides a helpful reference for the program development and application to unsaturated flow.
Effectively simulating unsaturated flow is of great significance in many areas such as the soil slope stability analysis, seepage of earth-rockfill dam, contaminant transport. Due to the strongly nonlinear characteristics of Richards equation for the unsaturated flow, numerical solution schemes such as the finite element method with an efficient iterative algorithm usually have to be employed. Picard method as a practical nonlinear iterative method is widely applied to the unsaturated flow field; but usually suffers from convergence oscillation, slow convergence rate and inaccurate solution. In order to improve the computing performance, an efficient finite element procedure based on the adaptive relaxed Picard method is developed. Through the simulations of 1D and 2D unsaturated seepage problems, accuracy, efficiency and robustness of the proposed procedure are validated by comparing with the traditional methods. It is shown that the adaptive relaxed Picard method can effectively reduce the convergence oscillation and significantly improve the convergence rate with the accuracy guaranteed. The proposed procedure provides a helpful reference for the program development and application to unsaturated flow.
引文
[1]吴梦喜,高莲士.饱和-非饱和土体非稳定渗流数值分析[J].水利学报,1999,30(12):38-42.WU Meng-xi,GAO Lian-shi.Saturated-unsaturated unsteady seepage numerical analysis[J].Journal of Hydraulic Engineering,1999,30(12):38-42.
[2]周桂云.饱和-非饱和非稳定渗流有限元分析方法的改进[J].水利水电科技进展,2009,29(1):5-11.ZHOU Gui-yun.Improvement of finite element analysis of unstable saturated-unsaturated seepage[J].Advances in Science and Technology of Water Resources,2009,29(1):5-11.
[3]PAN L,WIERENGA P J.A transformed pressure head-based approach to solve Richards’equation for variably saturated soils[J].Water Resources Research,1995,31(4):925-931.
[4]WILLIAMS G A,MILLER C T,KELLEY C.Transformation approaches for simulating flow in variably saturated porous media[J].Water Resources Research,2000,36(4):923-934.
[5]TAN T S,PHOON K K,CHONG P C.Numerical study of finite element method based solutions for propagation of wetting fronts in unsaturated soil[J].Journal of Geotechnical and Geoenvironmental Engineering,2004,130(3):254-263.
[6]PHOON K K,TAN T S,CHONG P C.Numerical simulation of Richards equation in partially saturated porous media:Under-relaxation and mass balance[J].Geotechnical and Geological Engineering,2007,25(5):525-541.
[7]陈曦,于玉贞,程勇刚.非饱和渗流Richards方程数值求解的欠松弛方法[J].岩土力学,2012,33(1):238-243.CHEN Xi,YU Yu-zhen,CHENG Yong-gang.Underrelaxation methods for numerical solution of Richards’equation of variably saturated flow[J].Rock and Soil Mechanics,2012,33(1):238-243.
[8]吴梦喜.饱和-非饱和土中渗流Richards方程有限元算法[J].水利学报,2009,40(10):1274-1279.WU Meng-xi.Finite element algorithm for Richards’equation for saturated-unsaturated seepage flow[J].Journal of Hydraulic Engineering,2009,40(10):1274-1279.
[9]MILLER C T,ABHISHEK C,FARTHING M W.A spatially and temporally adaptive solution of Richards’equation[J].Advances in Water Resources,2006,29(4):525-545.
[10]DURBIN T,DELEMOS D.Adaptive under-relaxation of Picard iterations in ground water models[J].Ground Water,2007,45(5):648-651.
[11]PANICONI C,PUTTI M.A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[J].Water Resources Research,1994,30(12):3357-3374.
[12]VAN GENUCHTEN M T.A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[J].Soil Science Society of America Journal,1980,44(5):892-898.
[13]BROOKS R H,COREY A T.Properties of porous media affecting fluid flow[J].Journal of the Irrigation and Drainage Division,1966,92(2):61-90.
[14]ZIENKIEWICZ O,TAYLOR R.The finite element method,volume 1:The basis[M].Oxford:ButterworthHeinemann,2000.
[15]LI C.A simplified Newton iteration method with linear finite elements for transient unsaturated flow[J].Water Resources Research,1993,29(4):965-971.
[16]HUYAKORN P S,SPRINGER E P,GUVANASEN V,et al.A three-dimensional finite-element model for simulating water flow in variably saturated porous media[J].Water Resources Research,1986,22(13):1790-1808.
[17]DE SMEDT B,PATTYN F,DE GROEN P.Using the unstable manifold correction in a Picard iteration to solve the velocity field in higher-order ice-flow models[J].Journal of Glaciology,2010,56(196):257-261.
[18]RIB R,DE RIERA PASENAU M,ESCOLANO E.Gid reference manual,pre and post processing system for FEM calculations[M].Barcelona:International Center For Numerical Methods in Engineering(CIMNE),2011.
[19]SRIVASTAVA R,YEH T C J.Analytical solutions for one-dimensional,transient infiltration toward the water table in homogeneous and layered soils[J].Water Resources Research,1991,27(5):753-762.
[20]VAUCLIN M,KHANJI D,VACHAUD G.Experimental and numerical study of a transient,two-dimensional unsaturated-saturated water table recharge problem[J].Water Resources Research,1979,15(5):1089-1101.
[21]CLEMENT T,WISE W R,MOLZ F J.A physically based,two-dimensional,finite-difference algorithm for modeling variably saturated flow[J].Journal of Hydrology,1994,161(1):71-90.
[22]MUALEM Y.A new model for predicting the hydraulic conductivity of unsaturated porous media[J].Water Resources Research,1976,12(3):513-522.
[2]周桂云.饱和-非饱和非稳定渗流有限元分析方法的改进[J].水利水电科技进展,2009,29(1):5-11.ZHOU Gui-yun.Improvement of finite element analysis of unstable saturated-unsaturated seepage[J].Advances in Science and Technology of Water Resources,2009,29(1):5-11.
[3]PAN L,WIERENGA P J.A transformed pressure head-based approach to solve Richards’equation for variably saturated soils[J].Water Resources Research,1995,31(4):925-931.
[4]WILLIAMS G A,MILLER C T,KELLEY C.Transformation approaches for simulating flow in variably saturated porous media[J].Water Resources Research,2000,36(4):923-934.
[5]TAN T S,PHOON K K,CHONG P C.Numerical study of finite element method based solutions for propagation of wetting fronts in unsaturated soil[J].Journal of Geotechnical and Geoenvironmental Engineering,2004,130(3):254-263.
[6]PHOON K K,TAN T S,CHONG P C.Numerical simulation of Richards equation in partially saturated porous media:Under-relaxation and mass balance[J].Geotechnical and Geological Engineering,2007,25(5):525-541.
[7]陈曦,于玉贞,程勇刚.非饱和渗流Richards方程数值求解的欠松弛方法[J].岩土力学,2012,33(1):238-243.CHEN Xi,YU Yu-zhen,CHENG Yong-gang.Underrelaxation methods for numerical solution of Richards’equation of variably saturated flow[J].Rock and Soil Mechanics,2012,33(1):238-243.
[8]吴梦喜.饱和-非饱和土中渗流Richards方程有限元算法[J].水利学报,2009,40(10):1274-1279.WU Meng-xi.Finite element algorithm for Richards’equation for saturated-unsaturated seepage flow[J].Journal of Hydraulic Engineering,2009,40(10):1274-1279.
[9]MILLER C T,ABHISHEK C,FARTHING M W.A spatially and temporally adaptive solution of Richards’equation[J].Advances in Water Resources,2006,29(4):525-545.
[10]DURBIN T,DELEMOS D.Adaptive under-relaxation of Picard iterations in ground water models[J].Ground Water,2007,45(5):648-651.
[11]PANICONI C,PUTTI M.A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems[J].Water Resources Research,1994,30(12):3357-3374.
[12]VAN GENUCHTEN M T.A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[J].Soil Science Society of America Journal,1980,44(5):892-898.
[13]BROOKS R H,COREY A T.Properties of porous media affecting fluid flow[J].Journal of the Irrigation and Drainage Division,1966,92(2):61-90.
[14]ZIENKIEWICZ O,TAYLOR R.The finite element method,volume 1:The basis[M].Oxford:ButterworthHeinemann,2000.
[15]LI C.A simplified Newton iteration method with linear finite elements for transient unsaturated flow[J].Water Resources Research,1993,29(4):965-971.
[16]HUYAKORN P S,SPRINGER E P,GUVANASEN V,et al.A three-dimensional finite-element model for simulating water flow in variably saturated porous media[J].Water Resources Research,1986,22(13):1790-1808.
[17]DE SMEDT B,PATTYN F,DE GROEN P.Using the unstable manifold correction in a Picard iteration to solve the velocity field in higher-order ice-flow models[J].Journal of Glaciology,2010,56(196):257-261.
[18]RIB R,DE RIERA PASENAU M,ESCOLANO E.Gid reference manual,pre and post processing system for FEM calculations[M].Barcelona:International Center For Numerical Methods in Engineering(CIMNE),2011.
[19]SRIVASTAVA R,YEH T C J.Analytical solutions for one-dimensional,transient infiltration toward the water table in homogeneous and layered soils[J].Water Resources Research,1991,27(5):753-762.
[20]VAUCLIN M,KHANJI D,VACHAUD G.Experimental and numerical study of a transient,two-dimensional unsaturated-saturated water table recharge problem[J].Water Resources Research,1979,15(5):1089-1101.
[21]CLEMENT T,WISE W R,MOLZ F J.A physically based,two-dimensional,finite-difference algorithm for modeling variably saturated flow[J].Journal of Hydrology,1994,161(1):71-90.
[22]MUALEM Y.A new model for predicting the hydraulic conductivity of unsaturated porous media[J].Water Resources Research,1976,12(3):513-522.