In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy o...In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.展开更多
The complex structure and strong heterogeneity of advanced nuclear reactor systems pose challenges for high-fidelity neutron-shielding calculations. Unstructured meshes exhibit strong geometric adaptability and can ov...The complex structure and strong heterogeneity of advanced nuclear reactor systems pose challenges for high-fidelity neutron-shielding calculations. Unstructured meshes exhibit strong geometric adaptability and can overcome the deficiencies of conventionally structured meshes in complex geometry modeling. A multithreaded parallel upwind sweep algorithm for S_(N) transport was proposed to achieve a more accurate geometric description and improve the computational efficiency. The spatial variables were discretized using the standard discontinuous Galerkin finite-element method. The angular flux transmission between neighboring meshes was handled using an upwind scheme. In addition, a combination of a mesh transport sweep and angular iterations was realized using a multithreaded parallel technique. The algorithm was implemented in the 2D/3D S_(N) transport code ThorSNIPE, and numerical evaluations were conducted using three typical benchmark problems:IAEA, Kobayashi-3i, and VENUS-3. These numerical results indicate that the multithreaded parallel upwind sweep algorithm can achieve high computational efficiency. ThorSNIPE, with a multithreaded parallel upwind sweep algorithm, has good reliability, stability, and high efficiency, making it suitable for complex shielding calculations.展开更多
We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems.For the dynamic responses of continuous medium structures,the traditional numerical algorithm is ...We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems.For the dynamic responses of continuous medium structures,the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation.Thus,a symplectic finite element method with energy conservation is constructed in this paper.A linear elastic system can be discretized into multiple elements,and a Hamiltonian system of each element can be constructed.The single element is discretized by the Galerkin method,and then the Hamiltonian system is constructed into the Birkhoffian system.Finally,all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme.Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate,it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm.The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems.展开更多
The discrete ordinates(S N)method requires numerous angular unknowns to achieve the desired accu-racy for shielding calculations involving strong anisotropy.Our objective is to develop an angular adaptive algorithm in...The discrete ordinates(S N)method requires numerous angular unknowns to achieve the desired accu-racy for shielding calculations involving strong anisotropy.Our objective is to develop an angular adaptive algorithm in the S N method to automatically optimize the angular distribution and minimize angular discretization errors with lower expenses.The proposed method enables linear dis-continuous finite element quadrature sets over an icosahe-dron to vary their quadrature orders in a one-twentieth sphere so that fine resolutions can be applied to the angular domains that are important.An error estimation that operates in conjunction with the spherical harmonics method is developed to determine the locations where more refinement is required.The adaptive quadrature sets are applied to three duct problems,including the Kobayashi benchmarks and the IRI-TUB research reactor,which emphasize the ability of this method to resolve neutron streaming through ducts with voids.The results indicate that the performance of the adaptive method is more effi-cient than that of uniform quadrature sets for duct transport problems.Our adaptive method offers an appropriate placement of angular unknowns to accurately integrate angular fluxes while reducing the computational costs in terms of unknowns and run times.展开更多
A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order...A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.展开更多
In the last decade, three dimensional discontin- uous deformation analyses (3D DDA) has attracted more and more attention of researchers and geotechnical engineers worldwide. The original DDA formulation utilizes a ...In the last decade, three dimensional discontin- uous deformation analyses (3D DDA) has attracted more and more attention of researchers and geotechnical engineers worldwide. The original DDA formulation utilizes a linear displacement function to describe the block movement and deformation, which would cause block expansion under rigid body rotation and thus limit its capability to model block de- formation. In this paper, 3D DDA is coupled with tetrahe- dron finite elements to tackle these two problems. Tetrahe- dron is the simplest in the 3D domain and makes it easy to implement automatic discretization, even for complex topol- ogy shape. Furthermore, element faces will remain planar and element edges will remain straight after deformation for tetrahedron finite elements and polyhedral contact detection schemes can be used directly. The matrices of equilibrium equations for this coupled method are given in detail and an effective contact searching algorithm is suggested. Valida- tion is conducted by comparing the results of the proposed coupled method with that of physical model tests using one of the most common failure modes, i.e., wedge failure. Most of the failure modes predicted by the coupled method agree with the physical model results except for 4 cases out of the total 65 cases. Finally, a complex rockslide example demon- strates the robustness and versatility of the coupled method.展开更多
An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same t...An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.展开更多
The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the correspond...The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.展开更多
Through the construction of a new ramp function, the element-flee Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the inte...Through the construction of a new ramp function, the element-flee Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the interface regions, both satisfying the essential boundary conditions and deploying meshless nodes and finite elements in a convenient and flexible way, which can meet the requirements of computation for complicated field. The comparison between the results of the present study and the corresponding analytical solutions shows this method is feasible and effective.展开更多
An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any co...An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfiirth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.展开更多
In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume (RKCV) discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian co...In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume (RKCV) discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin (RKDG) method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.展开更多
A new membrane finite element method for modeling fluid flow in a porous medium is presented in order to quickly and accurately simulate the geo-membrane fabric used in civil engineering. It is based on discontinuous ...A new membrane finite element method for modeling fluid flow in a porous medium is presented in order to quickly and accurately simulate the geo-membrane fabric used in civil engineering. It is based on discontinuous finite element theory, and can be easily coupled with the normal Galerkin finite element method. Based on the saturated seepage equation, the element coefficient matrix of the membrane element method is derived, and a geometric transform relation for the membrane element between a global coordinate system and a local coordinate system is obtained. A method for the determination of the fluid flux conductivity of the membrane element is presented. This method provides a basis for determining discontinuous parameters in discontinuous finite element theory. An anti-seepage problem regarding the foundation of a building is analyzed by coupling the membrane finite element method with the normal Galerkin finite element method. The analysis results demonstrate the utility and superiority of the membrane finite element method in fluid flow analysis of a porous medium.展开更多
In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by ener...In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.展开更多
A streamline upwind/Petrov-Galerkin (SUPG) finite element method based on a penalty function is pro- posed for steady incompressible Navier-Stokes equations. The SUPG stabilization technique is employed for the for-...A streamline upwind/Petrov-Galerkin (SUPG) finite element method based on a penalty function is pro- posed for steady incompressible Navier-Stokes equations. The SUPG stabilization technique is employed for the for- mulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pres- sure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.展开更多
A discontinuity-capturing scheme of finite element method(FEM)is proposed.The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unstea...A discontinuity-capturing scheme of finite element method(FEM)is proposed.The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unsteady flows,which exhibits the capability of capturing the shock waves and/or thin shear layers accurately in an unsteady viscous flow at high Reynolds number. In particular,a new testing variable,i.e.,the disturbed kinetic energy E,is suggested and used in the adaptive mesh computation,which is universally applicable to the capturing of both shock waves and shear layers in the inviscid flow and viscous flow at high Reynolds number.Based on several calculated examples,this approach has been proved to be effective and efficient for the calculations of compressible and incompressible flows.展开更多
Discontinuous deformation problems are common in rock engineering. Numerical analysis methods based on system models of the discrete body can better solve these problems. One of the most effective solutions is discont...Discontinuous deformation problems are common in rock engineering. Numerical analysis methods based on system models of the discrete body can better solve these problems. One of the most effective solutions is discontinuous deformation analysis (DDA) method, but the DDA method brings about rock embedding problems when it uses the strain assumption in elastic deformation and adopts virtual springs to simulate the contact problems. The multi-body finite element method (FEM) proposed in this paper can solve the problems of contact and deformation of blocks very well because it integrates the FEM and multi-body system dynamics theory. It is therefore a complete method for solving discontinuous deformation problems through balance equations of the contact surface and for simulating the displacement of whole blocks. In this study, this method was successfully used for deformation analysis of underground caverns in stratified rock. The simulation results indicate that the multi-body FEM can show contact forces and the stress states on contact surfaces better than DDA, and that the results calculated with the multi-body FEM are more consistent with engineering practice than those calculated with DDA method.展开更多
The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite ...The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.展开更多
A meshless local discontinuous Petrov-Galerkin (MLDPG) method based on the local symmetric weak form (LSWF) is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge...A meshless local discontinuous Petrov-Galerkin (MLDPG) method based on the local symmetric weak form (LSWF) is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin (RKDG) method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional non-overlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method (MM), and it can handle the problems with extremely complicated physics and geometries easily. Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.展开更多
文摘In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.
文摘The complex structure and strong heterogeneity of advanced nuclear reactor systems pose challenges for high-fidelity neutron-shielding calculations. Unstructured meshes exhibit strong geometric adaptability and can overcome the deficiencies of conventionally structured meshes in complex geometry modeling. A multithreaded parallel upwind sweep algorithm for S_(N) transport was proposed to achieve a more accurate geometric description and improve the computational efficiency. The spatial variables were discretized using the standard discontinuous Galerkin finite-element method. The angular flux transmission between neighboring meshes was handled using an upwind scheme. In addition, a combination of a mesh transport sweep and angular iterations was realized using a multithreaded parallel technique. The algorithm was implemented in the 2D/3D S_(N) transport code ThorSNIPE, and numerical evaluations were conducted using three typical benchmark problems:IAEA, Kobayashi-3i, and VENUS-3. These numerical results indicate that the multithreaded parallel upwind sweep algorithm can achieve high computational efficiency. ThorSNIPE, with a multithreaded parallel upwind sweep algorithm, has good reliability, stability, and high efficiency, making it suitable for complex shielding calculations.
基金supported by the National Natural Science Foundation of China(Nos.12132001 and 52192632)。
文摘We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems.For the dynamic responses of continuous medium structures,the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation.Thus,a symplectic finite element method with energy conservation is constructed in this paper.A linear elastic system can be discretized into multiple elements,and a Hamiltonian system of each element can be constructed.The single element is discretized by the Galerkin method,and then the Hamiltonian system is constructed into the Birkhoffian system.Finally,all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme.Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate,it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm.The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems.
基金supported by the National Natural Science Foundation of China(No.11975097)the Fundamental Research Funds for the Central Universities(No.2019MS038).
文摘The discrete ordinates(S N)method requires numerous angular unknowns to achieve the desired accu-racy for shielding calculations involving strong anisotropy.Our objective is to develop an angular adaptive algorithm in the S N method to automatically optimize the angular distribution and minimize angular discretization errors with lower expenses.The proposed method enables linear dis-continuous finite element quadrature sets over an icosahe-dron to vary their quadrature orders in a one-twentieth sphere so that fine resolutions can be applied to the angular domains that are important.An error estimation that operates in conjunction with the spherical harmonics method is developed to determine the locations where more refinement is required.The adaptive quadrature sets are applied to three duct problems,including the Kobayashi benchmarks and the IRI-TUB research reactor,which emphasize the ability of this method to resolve neutron streaming through ducts with voids.The results indicate that the performance of the adaptive method is more effi-cient than that of uniform quadrature sets for duct transport problems.Our adaptive method offers an appropriate placement of angular unknowns to accurately integrate angular fluxes while reducing the computational costs in terms of unknowns and run times.
基金supported by the National Natural Science Foundation of China (No. 10601022)NSF ofInner Mongolia Autonomous Region of China (No. 200607010106)513 and Science Fund of InnerMongolia University for Distinguished Young Scholars (No. ND0702)
文摘A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.
基金supported by the Key Project of Chinese National Programs for Fundamental Research and Development(2010CB731502)the National Natural Science Foundation of China(50978745)
文摘In the last decade, three dimensional discontin- uous deformation analyses (3D DDA) has attracted more and more attention of researchers and geotechnical engineers worldwide. The original DDA formulation utilizes a linear displacement function to describe the block movement and deformation, which would cause block expansion under rigid body rotation and thus limit its capability to model block de- formation. In this paper, 3D DDA is coupled with tetrahe- dron finite elements to tackle these two problems. Tetrahe- dron is the simplest in the 3D domain and makes it easy to implement automatic discretization, even for complex topol- ogy shape. Furthermore, element faces will remain planar and element edges will remain straight after deformation for tetrahedron finite elements and polyhedral contact detection schemes can be used directly. The matrices of equilibrium equations for this coupled method are given in detail and an effective contact searching algorithm is suggested. Valida- tion is conducted by comparing the results of the proposed coupled method with that of physical model tests using one of the most common failure modes, i.e., wedge failure. Most of the failure modes predicted by the coupled method agree with the physical model results except for 4 cases out of the total 65 cases. Finally, a complex rockslide example demon- strates the robustness and versatility of the coupled method.
基金Supported by the National Natural Science Foundation of China (10601022)Natural Science Foundation of Inner Mongolia Autonomous Region (200607010106)Youth Science Foundation of Inner Mongolia University(ND0702)
文摘An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.
文摘The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.
文摘Through the construction of a new ramp function, the element-flee Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the interface regions, both satisfying the essential boundary conditions and deploying meshless nodes and finite elements in a convenient and flexible way, which can meet the requirements of computation for complicated field. The comparison between the results of the present study and the corresponding analytical solutions shows this method is feasible and effective.
基金supported by the National Natural Science Foundation of China(Nos.10871156 and 11171269)the Fund of Xi'an Jiaotong University(No.2009xjtujc30)
文摘An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfiirth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11261035 and 11171038)the Science Research Foundation of the Institute of Higher Education of Inner Mongolia Autonomous Region, China (Grant No. NJZZ12198)the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2012MS0102)
文摘In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume (RKCV) discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin (RKDG) method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.
基金supported by the National Natural Science Foundation of China (Grant No. 50779012)
文摘A new membrane finite element method for modeling fluid flow in a porous medium is presented in order to quickly and accurately simulate the geo-membrane fabric used in civil engineering. It is based on discontinuous finite element theory, and can be easily coupled with the normal Galerkin finite element method. Based on the saturated seepage equation, the element coefficient matrix of the membrane element method is derived, and a geometric transform relation for the membrane element between a global coordinate system and a local coordinate system is obtained. A method for the determination of the fluid flux conductivity of the membrane element is presented. This method provides a basis for determining discontinuous parameters in discontinuous finite element theory. An anti-seepage problem regarding the foundation of a building is analyzed by coupling the membrane finite element method with the normal Galerkin finite element method. The analysis results demonstrate the utility and superiority of the membrane finite element method in fluid flow analysis of a porous medium.
文摘In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.
基金the National Natural Science Foundation of China (Grants 41372301 and 51349011)the Preeminent Youth Talent Project of Southwest University of Science and Technology (Grant 13zx9109)
文摘A streamline upwind/Petrov-Galerkin (SUPG) finite element method based on a penalty function is pro- posed for steady incompressible Navier-Stokes equations. The SUPG stabilization technique is employed for the for- mulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pres- sure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.
基金The project supported by the National Natural Science Foundation of China (10125210),the Hundred-Talent Programme of the Chinese Academy of Sciences and the Innovation Project of the Chinese Academy of Sciences (KJCX-SW-L04,KJCX2-SW-L2)
文摘A discontinuity-capturing scheme of finite element method(FEM)is proposed.The unstructured-grid technique combined with a new type of adaptive mesh approach is developed for both compressible and incompressible unsteady flows,which exhibits the capability of capturing the shock waves and/or thin shear layers accurately in an unsteady viscous flow at high Reynolds number. In particular,a new testing variable,i.e.,the disturbed kinetic energy E,is suggested and used in the adaptive mesh computation,which is universally applicable to the capturing of both shock waves and shear layers in the inviscid flow and viscous flow at high Reynolds number.Based on several calculated examples,this approach has been proved to be effective and efficient for the calculations of compressible and incompressible flows.
文摘Discontinuous deformation problems are common in rock engineering. Numerical analysis methods based on system models of the discrete body can better solve these problems. One of the most effective solutions is discontinuous deformation analysis (DDA) method, but the DDA method brings about rock embedding problems when it uses the strain assumption in elastic deformation and adopts virtual springs to simulate the contact problems. The multi-body finite element method (FEM) proposed in this paper can solve the problems of contact and deformation of blocks very well because it integrates the FEM and multi-body system dynamics theory. It is therefore a complete method for solving discontinuous deformation problems through balance equations of the contact surface and for simulating the displacement of whole blocks. In this study, this method was successfully used for deformation analysis of underground caverns in stratified rock. The simulation results indicate that the multi-body FEM can show contact forces and the stress states on contact surfaces better than DDA, and that the results calculated with the multi-body FEM are more consistent with engineering practice than those calculated with DDA method.
基金This work was supported in part by the National Science Foundation under grant DMS-1620288。
文摘The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.
基金Supported by New Century Excellent Talents in University in China(NCET),National"973" Program(No.61338)Innovative Research Project of Xi'an Hi-Tech Institute(EPXY0806)
文摘A meshless local discontinuous Petrov-Galerkin (MLDPG) method based on the local symmetric weak form (LSWF) is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin (RKDG) method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional non-overlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method (MM), and it can handle the problems with extremely complicated physics and geometries easily. Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.