We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard(CH)equation in this paper.It is a well-known fact that the maximum principle does not hold for the CH equati...We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard(CH)equation in this paper.It is a well-known fact that the maximum principle does not hold for the CH equation.Therefore,a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time.To overcome this drawback,we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow.We consider the temporal profiles of numerical results in the one-,two-,and three-dimensional spaces to examine the effect of extra mass source.Numerical solutions are obtained using a finite difference multigrid solver.Moreover,we perform numerical tests for tumor growth simulation,which is a typical application of generalized CH equations in biology.We apply the proposed cut-off logistic source term and have good results.展开更多
In this study,we propose a mathematical model and perform numerical simulations for the antibubble dynamics.An antibubble is a droplet of liquid sur-rounded by a thin film of a lighter liquid,which is also in a heavie...In this study,we propose a mathematical model and perform numerical simulations for the antibubble dynamics.An antibubble is a droplet of liquid sur-rounded by a thin film of a lighter liquid,which is also in a heavier surrounding fluid.The model is based on a phase-field method using a conservative Allen–Cahn equa-tion with a space-time dependent Lagrange multiplier and a modified Navier–Stokes equation.In this model,the inner fluid,middle fluid and outer fluid locate in specific diffusive layer regions according to specific phase filed(order parameter)values.If we represent the antibubble with conventional binary or ternary phase-field models,then it is difficult to have stable thin film.However,the proposed approach can prevent nonphysical breakup of fluid film during the simulation.Various numerical tests are performed to verify the efficiency of the proposed model.展开更多
基金The first author(C.Lee)was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(NRF-2019R1A6A3A13094308)The corresponding author(J.S.Kim)was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(NRF-2019R1A2C1003053).
文摘We investigate the effect of cut-off logistic source on evolutionary dynamics of a generalized Cahn-Hilliard(CH)equation in this paper.It is a well-known fact that the maximum principle does not hold for the CH equation.Therefore,a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time.To overcome this drawback,we propose the cut-off logistic source such that only the positive value greater than a given critical concentration can grow.We consider the temporal profiles of numerical results in the one-,two-,and three-dimensional spaces to examine the effect of extra mass source.Numerical solutions are obtained using a finite difference multigrid solver.Moreover,we perform numerical tests for tumor growth simulation,which is a typical application of generalized CH equations in biology.We apply the proposed cut-off logistic source term and have good results.
基金The author(D.Jeong)was supported by the National Research Foundation of Korea(NRF)grant funded by the Korea government(MSIP)(NRF-2017R1E1A1A03070953)The author(Y.B.Li)is supported by National Natural Sci-ence Foundation of China(Nos.11601416,11631012)+1 种基金the China Postdoctoral Science Foundation(No.2018M640968)The corresponding author(J.S.Kim)was supported by Basic Science Research Program through the National Research Founda-tion of Korea(NRF)funded by the Ministry of Education(NRF-2019R1A2C1003053).
文摘In this study,we propose a mathematical model and perform numerical simulations for the antibubble dynamics.An antibubble is a droplet of liquid sur-rounded by a thin film of a lighter liquid,which is also in a heavier surrounding fluid.The model is based on a phase-field method using a conservative Allen–Cahn equa-tion with a space-time dependent Lagrange multiplier and a modified Navier–Stokes equation.In this model,the inner fluid,middle fluid and outer fluid locate in specific diffusive layer regions according to specific phase filed(order parameter)values.If we represent the antibubble with conventional binary or ternary phase-field models,then it is difficult to have stable thin film.However,the proposed approach can prevent nonphysical breakup of fluid film during the simulation.Various numerical tests are performed to verify the efficiency of the proposed model.
