This paper focuses on the study of the stability of explicit time integration algorithm for dynamic problem by the Extended Finite Element Method(XFEM).A new enrichment scheme of crack tip is proposed within the frame...This paper focuses on the study of the stability of explicit time integration algorithm for dynamic problem by the Extended Finite Element Method(XFEM).A new enrichment scheme of crack tip is proposed within the framework of XFEM.Then the governing equations are derived and evolved into the discretized form.For dynamic problem,the lumped mass and the explicit time algorithm are applied.With different grid densities and different forms of Newmark scheme,the Dynamic Stress Intensity Factor(DSIF)is computed by using interaction integral approach to reflect the dynamic response.The effectiveness of the proposed scheme is demonstrated through the numerical examples,and the critical time stepping in different situations are listed and analyzed to illustrate the factors that affect the numerical stability.展开更多
We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, ...We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.展开更多
We devise hybrid high-order(HHO)methods for the acoustic wave equation in the time domain.We frst consider the second-order formulation in time.Using the Newmark scheme for the temporal discretization,we show that the...We devise hybrid high-order(HHO)methods for the acoustic wave equation in the time domain.We frst consider the second-order formulation in time.Using the Newmark scheme for the temporal discretization,we show that the resulting HHO-Newmark scheme is energy-conservative,and this scheme is also amenable to static condensation at each time step.We then consider the formulation of the acoustic wave equation as a frst-order system together with singly-diagonally implicit and explicit Runge-Kutta(SDIRK and ERK)schemes.HHO-SDIRK schemes are amenable to static condensation at each time step.For HHO-ERK schemes,the use of the mixed-order formulation,where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns,is key to beneft from the explicit structure of the scheme.Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order O(hk+1)in the H1-norm and of order O(h^(k+2))in the L^(2)-norm.Moreover,test cases on wave propagation in heterogeneous media indicate the benefts of using high-order methods.展开更多
基金the National Natural Science Foundation of China(No.11672101,No.11372099)the 12th Five-Year Supporting Plan Issue(No.2015 BAB07B10)+1 种基金Jiangsu Province Natural Science Fund Project(No.BK 20151493)the Postgraduate Research and Innovation Projects in Jiangsu Province(No.2014B 31614)for the financial support.
文摘This paper focuses on the study of the stability of explicit time integration algorithm for dynamic problem by the Extended Finite Element Method(XFEM).A new enrichment scheme of crack tip is proposed within the framework of XFEM.Then the governing equations are derived and evolved into the discretized form.For dynamic problem,the lumped mass and the explicit time algorithm are applied.With different grid densities and different forms of Newmark scheme,the Dynamic Stress Intensity Factor(DSIF)is computed by using interaction integral approach to reflect the dynamic response.The effectiveness of the proposed scheme is demonstrated through the numerical examples,and the critical time stepping in different situations are listed and analyzed to illustrate the factors that affect the numerical stability.
基金Project supported by the National Natural Science Foundation of China(Grant No.11471262)the National Basic Research Program of China(Grant No.2012CB025904)the State Key Laboratory of Science and Engineering Computing and the Center for High Performance Computing of Northwestern Polytechnical University,China
文摘We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.
基金The authors would like to thank L.Guillot(CEA/DAM)for insightful discussions and CEA/DAM for partial fnancial support.EB was partially supported by the EPSRC grants EP/P01576X/1 and EP/P012434/1.
文摘We devise hybrid high-order(HHO)methods for the acoustic wave equation in the time domain.We frst consider the second-order formulation in time.Using the Newmark scheme for the temporal discretization,we show that the resulting HHO-Newmark scheme is energy-conservative,and this scheme is also amenable to static condensation at each time step.We then consider the formulation of the acoustic wave equation as a frst-order system together with singly-diagonally implicit and explicit Runge-Kutta(SDIRK and ERK)schemes.HHO-SDIRK schemes are amenable to static condensation at each time step.For HHO-ERK schemes,the use of the mixed-order formulation,where the polynomial degree of the cell unknowns is one order higher than that of the face unknowns,is key to beneft from the explicit structure of the scheme.Numerical results on test cases with analytical solutions show that the methods can deliver optimal convergence rates for smooth solutions of order O(hk+1)in the H1-norm and of order O(h^(k+2))in the L^(2)-norm.Moreover,test cases on wave propagation in heterogeneous media indicate the benefts of using high-order methods.
