Analytically solving a three-dimensional （3-D） bioheat transfer problem with phase change during a freezing process is extremely difficult but theoretically important. The moving heat source model and the Green func...Analytically solving a three-dimensional （3-D） bioheat transfer problem with phase change during a freezing process is extremely difficult but theoretically important. The moving heat source model and the Green function method are introduced to deal with the cryopreservation process of in vitro biomaterials. Exact solutions for the 3-D temperature transients of tissues under various boundary conditions, such as totally convective cooling, totally fixed temperature cooling and a hybrid between them on tissue surfaces, are obtained. Furthermore, the cryosurgical process in living tissues subject to freezing by a single or multiple cryoprobes is also analytically solved. A closed-form analytical solution to the bioheat phase change process is derived by considering contributions from blood perfusion heat transfer, metabolic heat generation, and heat sink of a cryoprobe. The present method is expected to have significant value for analytically solving complex bioheat transfer problems with phase change.展开更多
We consider a parametric double phase problem with a reaction term which is only locally defined near zero and is not assumed to be odd.We show that for all big values of the parameter λ,the problem has infinitely ma...We consider a parametric double phase problem with a reaction term which is only locally defined near zero and is not assumed to be odd.We show that for all big values of the parameter λ,the problem has infinitely many nodal solutions.Our approach is based on variational methods combining upper-lower solutions and truncation techniques,and flow invariance arguments.展开更多
This paper is devoted to demonstrating several multiplicity results of nontrivial weak solutions to double phase problems of Kirchhoff type with Hardy potentials.The main features of the paper are the appearance of no...This paper is devoted to demonstrating several multiplicity results of nontrivial weak solutions to double phase problems of Kirchhoff type with Hardy potentials.The main features of the paper are the appearance of non-local Kirchhoff coefficients and the Hardy potential,the absence of the compactness condition of Palais-Smale,and the L^(∞)-bound for any possible weak solution.To establish multiplicity results,we utilize the fountain theorem and the dual fountain theorem as main tools.Also,we give the L^(∞)-bound for any possible weak solution by exploiting the De Giorgi iteration method and a truncated energy technique.As an application,we give the existence of a sequence of infinitely many weak solutions converging to zero in L^(∞)-norm.To derive this result,we employ the modified functional method and the dual fountain theorem.展开更多
Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a nov...Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.展开更多
This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start wi...This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.展开更多
The aim of this paper is the study of a double phase problems involving superlinear nonlinearities with a growth that need not satisfy the Ambrosetti-Rabinowitz condition.Using variational tools together with suitable...The aim of this paper is the study of a double phase problems involving superlinear nonlinearities with a growth that need not satisfy the Ambrosetti-Rabinowitz condition.Using variational tools together with suitable truncation and minimax techniques with Morse theory,the authors prove the existence of one and three nontrivial weak solutions,respectively.展开更多
A mathematical model for describing gas solid two phase steady mixed convection with phase change has been developed and numerical calculation methods presented.A melting liquid droplet failing a counter gas currenl e...A mathematical model for describing gas solid two phase steady mixed convection with phase change has been developed and numerical calculation methods presented.A melting liquid droplet failing a counter gas currenl expe- riences three processes,cooling of liquid droplet,solidification and cooling of the solid particle.The turbulent model used for Rayleigh number greater than 10~6 is a two equation(k—ε)model of turbulence.For phase change,an improved enthalpy method with varied time step is proposed.The gas particle two phase flow is described by using Eulerian-Lagrangian approach.Modified SIMPLE algorithm and Runge-Kutta method are used in interative calcu- lation.As an example of calculation,the flow in a special 2-dimensional axi-symmetrical prilling tower of diameter 20 m and height 50 m has been performed.Buoyancy effect is important for moving droplet with phase change. The model to be developed and analysis of results obtained in this paper are useful for engineering design in indus- try.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 50776097)
文摘Analytically solving a three-dimensional （3-D） bioheat transfer problem with phase change during a freezing process is extremely difficult but theoretically important. The moving heat source model and the Green function method are introduced to deal with the cryopreservation process of in vitro biomaterials. Exact solutions for the 3-D temperature transients of tissues under various boundary conditions, such as totally convective cooling, totally fixed temperature cooling and a hybrid between them on tissue surfaces, are obtained. Furthermore, the cryosurgical process in living tissues subject to freezing by a single or multiple cryoprobes is also analytically solved. A closed-form analytical solution to the bioheat phase change process is derived by considering contributions from blood perfusion heat transfer, metabolic heat generation, and heat sink of a cryoprobe. The present method is expected to have significant value for analytically solving complex bioheat transfer problems with phase change.
基金Guangdong Basic and Applied Basic Research Foundation(2023A1515010603)。
文摘We consider a parametric double phase problem with a reaction term which is only locally defined near zero and is not assumed to be odd.We show that for all big values of the parameter λ,the problem has infinitely many nodal solutions.Our approach is based on variational methods combining upper-lower solutions and truncation techniques,and flow invariance arguments.
文摘This paper is devoted to demonstrating several multiplicity results of nontrivial weak solutions to double phase problems of Kirchhoff type with Hardy potentials.The main features of the paper are the appearance of non-local Kirchhoff coefficients and the Hardy potential,the absence of the compactness condition of Palais-Smale,and the L^(∞)-bound for any possible weak solution.To establish multiplicity results,we utilize the fountain theorem and the dual fountain theorem as main tools.Also,we give the L^(∞)-bound for any possible weak solution by exploiting the De Giorgi iteration method and a truncated energy technique.As an application,we give the existence of a sequence of infinitely many weak solutions converging to zero in L^(∞)-norm.To derive this result,we employ the modified functional method and the dual fountain theorem.
文摘Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.
基金Research of R.Guo is supported by NSFC grant No.11601490Research of Y.Xu is supported by NSFC grant No.11722112,91630207.
文摘This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.
基金supported by the National Natural Science Foundation of China (No. 11201095)the Fundamental Research Funds for the Central Universities (No. 3072022TS2402)+1 种基金the Postdoctoral research startup foundation of Heilongjiang (No. LBH-Q14044)the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502)
文摘The aim of this paper is the study of a double phase problems involving superlinear nonlinearities with a growth that need not satisfy the Ambrosetti-Rabinowitz condition.Using variational tools together with suitable truncation and minimax techniques with Morse theory,the authors prove the existence of one and three nontrivial weak solutions,respectively.
文摘A mathematical model for describing gas solid two phase steady mixed convection with phase change has been developed and numerical calculation methods presented.A melting liquid droplet failing a counter gas currenl expe- riences three processes,cooling of liquid droplet,solidification and cooling of the solid particle.The turbulent model used for Rayleigh number greater than 10~6 is a two equation(k—ε)model of turbulence.For phase change,an improved enthalpy method with varied time step is proposed.The gas particle two phase flow is described by using Eulerian-Lagrangian approach.Modified SIMPLE algorithm and Runge-Kutta method are used in interative calcu- lation.As an example of calculation,the flow in a special 2-dimensional axi-symmetrical prilling tower of diameter 20 m and height 50 m has been performed.Buoyancy effect is important for moving droplet with phase change. The model to be developed and analysis of results obtained in this paper are useful for engineering design in indus- try.
