体积约束的非局部扩散问题的加罚有限元方法
|
摘要
针对非齐次和齐次体积约束的非局部扩散问题设计了新的有限元方法——加罚有限元方法,并给出该方法的误差分析.数值算例验证了加罚有限元方法的稳定性和有效性.
We propose a finite element method,penalized finite element method,to solve a homogenous and inhomogenous nonlocal diffusion problem with volume constraints,and give error estimate of the method. Finally,we show numerical experiments to verify the theoretical results.
We propose a finite element method,penalized finite element method,to solve a homogenous and inhomogenous nonlocal diffusion problem with volume constraints,and give error estimate of the method. Finally,we show numerical experiments to verify the theoretical results.
引文
[1]KRONER E.Elasticity theory of materials with long range forces[J].International Journal of Solids and Structures,1967,3(5):731-742.
[2]ROGULA D.Nonlocal theory of material media[M].Berlin:Springer,1982.
[3]ALTAN B.Uniqueness of initial-boundary value problems in nonlocal elasticity[J].International Journal of Solids and Structures,1989,25(11):1271-1278.
[4]ALTAN B.Uniqueness in nonlocal thermoelasticity[J].Journal of Thermal Stresses,1991,14(2):121-128.
[5]WANG J,DHALIWAL R.Uniqueness in generalized nonlocal thermoelasticity[J].Journal of Thermal Stresses,1993,16(1):71-77.
[6]WANG J,DHALIWAL R.On some theorems in the nonlocal theory of micropolar elasticity[J].International Journal of Solids and Structures,1993,30(10):1331-1338.
[7]SILLING S.Reformulation of elasticity theory for discontinuities and long-range forces[J].Journal of the Mechanics and Physics of Solids,2000,48(1):175-209.
[8]BOBARU F,SILLING S.Peridynamic 3D problems of nanofiber networks and carbon nanotube-reinforced composites[J]∥Proceedings of the International Conference on Numerical Methods in Industrial Forming Processes,American Institute of Physics,2004:1565-1570.
[9]BOBARU F,SILLING S,JIANG H.Peridynamic frature and damage modeling of membranes and nanofiber networks[J].Proceedings of the XI International Conference on Fracture,Turin,2005,5748:1-6.
[10]SILLING S.Dynamic frature modeling with a meshfree peridynamic code[J].Computational Fluid and Solid Mechanics,2003,641-644.
[11]GILBOA G,OSHER S.Nonlocal operators with applications to image processing[J].Siam Journal on Multiscale Modeling and Simulation,2008,7(3):1005-1028.
[12]LOU Y,ZHANG X,OSHER S,BERTOZZI A.Image recovery via nonlocal operators[J].Journal of Scientific Computing,2010,42(2):185-197.
[13]SILLING S A,LEHOUCQ R B.Convergence of peridynamics to classical elasticity theory[J].Journal of Elasticity,2008,93(1):13-37.
[14]DU Q,ZHUO K.Mathematical analysis for the peridynamics nonlocal continuum theory[J].Esaim Mathematical Modelling and Numerical Analysis,2011,45(2):217-234.
[15]EMMRICH E,WECKNER O.On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity[J].Communications in Mathematical Sciences,2007 5(4):851-864.
[16]EMMRICH E,WECKNER O.Analysis and numerical approximation of an integro-differential equation modeling nonlocal effects in linear elasticity[J].Mathematics and Mechanics of Solids,2007,4(4):363-384.
[17]LEHOUCQ R B,SILLING S.Force flux and the peridynamic stress tensor[J].Journal of the Mechanics and Physics of Solids,2008,56(4):1566-1577.
[18]GUNZBURGER M,LEHOUCQ R B.A nonlocal vector calculus with application to nonlocal boundary value problems[J].SIAM Journal on Multiscale Modeling and Simulation,2010,8(5):1581-1598.
[19]DU Q,GUNZBUEGER M,LEHOUCQ R B,et al.A nonlocal vector caculus,nonlocal volume-constrained problems,and the nonlocal balance laws[J].Mathematical Models and Methods in Applied Sciences,2013,23(3):493-540.
[20]AKSOYLU B,PARKS M.Variational theory and domain decomposition for nonlocal problems[J].Applied Mathematics and Computation,2011,217(14):6498-6515.
[21]ZHU X G,NIE Y F,WANG J G,et al.A characteristic finite element method for fractional convection-diffusion equations[J].Chinese Journal of Computational Physics,2017,34(4):417-424.
[22]CHEN X,GUNZBURGER M.Continuous and discontinuous finite element methods for a peridynamics model of mechanics[J].Computer Methods in Applied Mechanics and Engineering,2011,200(9-12):1237-1250.
[23]MACEK R W,SILLING S.Peridynamics via finite element analysis[J].Finite Elements in Analysis and Design,2007,43(15):1169-1178.
[24]SILLING S,ASKARI E.A meshfree method based on the peridynamic model of solid mechanics[J].Computers and Structures,2005,83(17-18):1526-1535.
[25]ZHOU K,DU Q.Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions[J].SIAM Journal on Numerical Analysis,2010,48(5):1759-1780.
[26]RIVIERE B.Discontinuous Galerkin methods for solving elliptic and parabolic equations:Theory and implementation[M].SIAM,Philadelphia,2008.
[2]ROGULA D.Nonlocal theory of material media[M].Berlin:Springer,1982.
[3]ALTAN B.Uniqueness of initial-boundary value problems in nonlocal elasticity[J].International Journal of Solids and Structures,1989,25(11):1271-1278.
[4]ALTAN B.Uniqueness in nonlocal thermoelasticity[J].Journal of Thermal Stresses,1991,14(2):121-128.
[5]WANG J,DHALIWAL R.Uniqueness in generalized nonlocal thermoelasticity[J].Journal of Thermal Stresses,1993,16(1):71-77.
[6]WANG J,DHALIWAL R.On some theorems in the nonlocal theory of micropolar elasticity[J].International Journal of Solids and Structures,1993,30(10):1331-1338.
[7]SILLING S.Reformulation of elasticity theory for discontinuities and long-range forces[J].Journal of the Mechanics and Physics of Solids,2000,48(1):175-209.
[8]BOBARU F,SILLING S.Peridynamic 3D problems of nanofiber networks and carbon nanotube-reinforced composites[J]∥Proceedings of the International Conference on Numerical Methods in Industrial Forming Processes,American Institute of Physics,2004:1565-1570.
[9]BOBARU F,SILLING S,JIANG H.Peridynamic frature and damage modeling of membranes and nanofiber networks[J].Proceedings of the XI International Conference on Fracture,Turin,2005,5748:1-6.
[10]SILLING S.Dynamic frature modeling with a meshfree peridynamic code[J].Computational Fluid and Solid Mechanics,2003,641-644.
[11]GILBOA G,OSHER S.Nonlocal operators with applications to image processing[J].Siam Journal on Multiscale Modeling and Simulation,2008,7(3):1005-1028.
[12]LOU Y,ZHANG X,OSHER S,BERTOZZI A.Image recovery via nonlocal operators[J].Journal of Scientific Computing,2010,42(2):185-197.
[13]SILLING S A,LEHOUCQ R B.Convergence of peridynamics to classical elasticity theory[J].Journal of Elasticity,2008,93(1):13-37.
[14]DU Q,ZHUO K.Mathematical analysis for the peridynamics nonlocal continuum theory[J].Esaim Mathematical Modelling and Numerical Analysis,2011,45(2):217-234.
[15]EMMRICH E,WECKNER O.On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity[J].Communications in Mathematical Sciences,2007 5(4):851-864.
[16]EMMRICH E,WECKNER O.Analysis and numerical approximation of an integro-differential equation modeling nonlocal effects in linear elasticity[J].Mathematics and Mechanics of Solids,2007,4(4):363-384.
[17]LEHOUCQ R B,SILLING S.Force flux and the peridynamic stress tensor[J].Journal of the Mechanics and Physics of Solids,2008,56(4):1566-1577.
[18]GUNZBURGER M,LEHOUCQ R B.A nonlocal vector calculus with application to nonlocal boundary value problems[J].SIAM Journal on Multiscale Modeling and Simulation,2010,8(5):1581-1598.
[19]DU Q,GUNZBUEGER M,LEHOUCQ R B,et al.A nonlocal vector caculus,nonlocal volume-constrained problems,and the nonlocal balance laws[J].Mathematical Models and Methods in Applied Sciences,2013,23(3):493-540.
[20]AKSOYLU B,PARKS M.Variational theory and domain decomposition for nonlocal problems[J].Applied Mathematics and Computation,2011,217(14):6498-6515.
[21]ZHU X G,NIE Y F,WANG J G,et al.A characteristic finite element method for fractional convection-diffusion equations[J].Chinese Journal of Computational Physics,2017,34(4):417-424.
[22]CHEN X,GUNZBURGER M.Continuous and discontinuous finite element methods for a peridynamics model of mechanics[J].Computer Methods in Applied Mechanics and Engineering,2011,200(9-12):1237-1250.
[23]MACEK R W,SILLING S.Peridynamics via finite element analysis[J].Finite Elements in Analysis and Design,2007,43(15):1169-1178.
[24]SILLING S,ASKARI E.A meshfree method based on the peridynamic model of solid mechanics[J].Computers and Structures,2005,83(17-18):1526-1535.
[25]ZHOU K,DU Q.Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions[J].SIAM Journal on Numerical Analysis,2010,48(5):1759-1780.
[26]RIVIERE B.Discontinuous Galerkin methods for solving elliptic and parabolic equations:Theory and implementation[M].SIAM,Philadelphia,2008.