A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element...A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.展开更多
A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element...A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.展开更多
We enhance a robust parallel finite element model for coasts and estuaries cases with the use of N-Best refinement algorithms,in multilevel partitioning scheme.Graph partitioning is an important step to construct the ...We enhance a robust parallel finite element model for coasts and estuaries cases with the use of N-Best refinement algorithms,in multilevel partitioning scheme.Graph partitioning is an important step to construct the parallel model,in which computation speed is a big concern.The partitioning strategy includes the division of the research domain into several semi-equal-sized sub-domains,minimizing the sum weight of edges between different sub-domains.Multilevel schemes for graph partitioning are divided into three phases:coarsening,partitioning,and uncoarsening.In the uncoarsening phase,many refinement algorithms have been proposed previously,such as KL,Greedy,and Boundary refinements.In this study,we propose an N-Best refinement algorithm and show its advantages in our case study of Xiamen Bay.Compared with original partitioning algorithm in previous models,the N-Best algorithm can speed up the computation by 1.9 times,and the simulation results are in a good match with the in-situ data.展开更多
In this article,two kinds of expandable parallel finite element methods,based on two-grid discretizations,are given to solve the linear elliptic problems.Compared with the classical local and parallel finite element m...In this article,two kinds of expandable parallel finite element methods,based on two-grid discretizations,are given to solve the linear elliptic problems.Compared with the classical local and parallel finite element methods,there are two attractive features of the methods shown in this article:1)a partition of unity is used to generate a series of local and independent subproblems to guarantee the final approximation globally continuous;2)the computational domain of each local subproblem is contained in a ball with radius of O(H)(H is the coarse mesh parameter),which means methods in this article are more suitable for parallel computing in a large parallel computer system.Some a priori error estimation are obtained and optimal error bounds in both H^1-normal and L^2-normal are derived.Finally,numerical results are reported to test and verify the feasibility and validity of our methods.展开更多
We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems.The solution of this class of problems often exhibits non-smooth features such as st...We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems.The solution of this class of problems often exhibits non-smooth features such as steep gradients and boundary layers which can be very challenging to recover using the conventional low-order finite element methods.A class of steady-state exponential functions has been widely used for enrichment and its performance to numerically solve these challenges has been demonstrated.However,these enrichment functions have been used only in context of the standard h-version refinement or the so-called q-version refinement.In this paper we demonstrate that the p-version refinement can also be a very attractive option in terms of the efficiency and the accuracy in the enriched partition of unity finite element method.First,the transient diffusion problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the p-version enriched finite elements.Numerical results are presented for three test examples of timedependent diffusion problems in both homogeneous and heterogeneous media.The computed results show the significant improvement when using the p-version refined enriched approximations in the finite element analysis.In addition,these results support our expectations for a robust and high-order accurate enriched partition of unity finite element method.展开更多
This study proposed a partitioned method to analyze maneuvering of insects during flapping flight.This method decomposed the insect flapping flight into wing and body subsystems and then coupled them via boundary cond...This study proposed a partitioned method to analyze maneuvering of insects during flapping flight.This method decomposed the insect flapping flight into wing and body subsystems and then coupled them via boundary conditions imposed on the wing’s base using one-way coupling.In the wing subsystem,the strong coupling of the flexible wings and surrounding fluid was accurately analyzed using the finite element method to obtain the thrust forces acting on the insect’s body.The resulting thrust forces were passed from the wing subsystem to the body subsystem,and then rigid body motion was analyzed in the body subsystem.The rolling,yawing,and pitching motions were simulated using the proposed method as follows:In the rolling simulation,the difference of the stroke angle between the right and left wings caused a roll torque.In the yawing simulation,the initial feathering angle in the right wing only caused a yaw torque.In the pitching simulation,the difference between the front-and back-stroke angles in both the right and left wings caused a pitch torque.All three torques generated maneuvering motion comparable with that obtained in actual observations of insect flight.These results demonstrate that the proposed method can adequately simulate the fundamental maneuvers of insect flapping flight.In the present simulations,the maneuvering mechanisms were investigated at the governing equation level,which might be difficult using other approaches.Therefore,the proposed method will contribute to revealing the underlying insect flight mechanisms.展开更多
A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid m...A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid mechanics problems. With a polynomial form, the VNM achieves better results than those of traditional PFEMs, including the Wachspress method and the mean value method in standard patch tests. Compared with the standard triangular FEM, the VNM can achieve better accuracy. With the ability to construct shape functions on polygonal elements, the VNM provides greater flexibility in mesh generation. Therefore, several fracture problems are studied to demonstrate the potential implementation. With the advantage of the VNM, the convenient refinement and remeshing strategy are applied.展开更多
Metal forming plays an important role in manufacturing industry and is widely applied in industries.The tradi- tional finite element method(FEM)numerical simulation is commonly used to predict metal forming process.Co...Metal forming plays an important role in manufacturing industry and is widely applied in industries.The tradi- tional finite element method(FEM)numerical simulation is commonly used to predict metal forming process.Conventional finite element analysis of metal forming processes often breaks down due to severe mesh distortion,therefore time-consuming remeshing is necessary.Meshfree methods have been developed since 1977 and can avoid this problem.This new generation of computational methods reduces time-consuming model generation and refinement effort,and its shape function has higher order connectivity than FEM’s.In this paper the velocity shape functions are developed from a reproducing kernel approximation that satisfies consistency conditions and is used to analyze metal tension rigid viscoplastic deforming and Magnesium Alloy(MB 15)sheet superplastic ten- sion forming.A meshfree method metal forming modeling program is set up,the partition of unity method is used to compute the integrations in weak form equations and penalty method is used to impose the essential boundary condition exactly.Metal forming examples,such as sheet metal superplastic tension forming and metal rigid viscoplastic tension forming,are analyzed to demon- strate the performance of mesh free method.展开更多
A class of nonlinear parabolic equation on a polygonal domain Ω R2 is inves- tigated in this paper. We introduce a finite element method on overlapping non-matching grids for the nonlinear parabolic equation based o...A class of nonlinear parabolic equation on a polygonal domain Ω R2 is inves- tigated in this paper. We introduce a finite element method on overlapping non-matching grids for the nonlinear parabolic equation based on the partition of unity method. We give the construction and convergence analysis for the semi-discrete and the fully discrete finite element methods. Moreover, we prove that the error of the discrete variational problem has good approximation properties. Our results are valid for any spatial dimensions. A numerical example to illustrate the theoretical results is also given.展开更多
This paper establishes a new finite volume element scheme for Poisson equation on trian- gular meshes. The trial function space is taken as Lagrangian cubic finite element space on triangular partition, and the test f...This paper establishes a new finite volume element scheme for Poisson equation on trian- gular meshes. The trial function space is taken as Lagrangian cubic finite element space on triangular partition, and the test function space is defined as piecewise constant space on dual partition. Under some weak condition about the triangular meshes, the authors prove that the stiffness matrix is uni- formly positive definite and convergence rate to be O（h3） in Hi-norm. Some numerical experiments confirm the theoretical considerations.展开更多
Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadrati...Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.展开更多
We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element methods.We use the lowest order Raviart-Thomas mix...We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element methods.We use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control variable.We obtain the superconvergence of O(h^(1+s))(0<s≤1)for the control variable.Finally,we present two numerical examples to confirm our superconvergence results.展开更多
The multi-physics simulation of coupled fluid-structure interaction problems, with disjoint fluid and solid domains, requires one to choose a method for enforcing the fluid-structure coupling at the interface between ...The multi-physics simulation of coupled fluid-structure interaction problems, with disjoint fluid and solid domains, requires one to choose a method for enforcing the fluid-structure coupling at the interface between solid and fluid. While it is common knowledge that the choice of coupling technique can be very problem dependent, there exists no satisfactory coupling comparison methodology that allows for conclusions to be drawn with respect to the comparison of computational cost and solution accuracy for a given scenario. In this work, we develop a computational framework where all aspects of the computation can be held constant, save for the method in which the coupled nature of the fluid-structure equations is enforced. To enable a fair comparison of coupling methods, all simulations presented in this work are implemented within a single numerical framework within the deal.ii [1] finite element library. We have chosen the two-dimensional benchmark test problem of Turek and Hron [2] as an example to examine the relative accuracy of the coupling methods studied;however, the comparison technique is equally applicable to more complex problems. We show that for the specific case considered herein the monolithic approach outperforms partitioned and quasi-direct methods;however, this result is problem dependent and we discuss computational and modeling aspects which may affect other comparison studies.展开更多
Examines a nonlinear partial differential equation of elliptic type with the homogeneous Dirichlet boundary conditions; Proof of the comparison and maximum principles; Approximation of the finite element; Introduction...Examines a nonlinear partial differential equation of elliptic type with the homogeneous Dirichlet boundary conditions; Proof of the comparison and maximum principles; Approximation of the finite element; Introduction of a discrete analogue of the maximum principle for linear elements.展开更多
By combination of iteration methods with the partition of unity method(PUM),some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics(MHD)with different physical parameters are pre...By combination of iteration methods with the partition of unity method(PUM),some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics(MHD)with different physical parameters are presented and analyzed.These algorithms are highly efficient.At first,a global solution is obtained on a coarse grid for all approaches by one of the iteration methods.By parallelized residual schemes,local corrected solutions are calculated on finer meshes with overlapping sub-domains.The subdomains can be achieved flexibly by a class of PUM.The proposed algorithm is proved to be uniformly stable and convergent.Finally,one numerical example is presented to confirm the theoretical findings.展开更多
基金Project supported by the National Basic Research Program of China (973Project) (No.2002CB412709) and the National Natural Science Foundation of China (Nos.50278012,10272027,19832010)
文摘A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.
文摘A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.
基金Supported by the National Natural Science Foundation of China (Nos. 40406005,41076001,40440420596)
文摘We enhance a robust parallel finite element model for coasts and estuaries cases with the use of N-Best refinement algorithms,in multilevel partitioning scheme.Graph partitioning is an important step to construct the parallel model,in which computation speed is a big concern.The partitioning strategy includes the division of the research domain into several semi-equal-sized sub-domains,minimizing the sum weight of edges between different sub-domains.Multilevel schemes for graph partitioning are divided into three phases:coarsening,partitioning,and uncoarsening.In the uncoarsening phase,many refinement algorithms have been proposed previously,such as KL,Greedy,and Boundary refinements.In this study,we propose an N-Best refinement algorithm and show its advantages in our case study of Xiamen Bay.Compared with original partitioning algorithm in previous models,the N-Best algorithm can speed up the computation by 1.9 times,and the simulation results are in a good match with the in-situ data.
基金Subsidized by NSFC (11701343)partially supported by NSFC (11571274,11401466)
文摘In this article,two kinds of expandable parallel finite element methods,based on two-grid discretizations,are given to solve the linear elliptic problems.Compared with the classical local and parallel finite element methods,there are two attractive features of the methods shown in this article:1)a partition of unity is used to generate a series of local and independent subproblems to guarantee the final approximation globally continuous;2)the computational domain of each local subproblem is contained in a ball with radius of O(H)(H is the coarse mesh parameter),which means methods in this article are more suitable for parallel computing in a large parallel computer system.Some a priori error estimation are obtained and optimal error bounds in both H^1-normal and L^2-normal are derived.Finally,numerical results are reported to test and verify the feasibility and validity of our methods.
文摘We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems.The solution of this class of problems often exhibits non-smooth features such as steep gradients and boundary layers which can be very challenging to recover using the conventional low-order finite element methods.A class of steady-state exponential functions has been widely used for enrichment and its performance to numerically solve these challenges has been demonstrated.However,these enrichment functions have been used only in context of the standard h-version refinement or the so-called q-version refinement.In this paper we demonstrate that the p-version refinement can also be a very attractive option in terms of the efficiency and the accuracy in the enriched partition of unity finite element method.First,the transient diffusion problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the p-version enriched finite elements.Numerical results are presented for three test examples of timedependent diffusion problems in both homogeneous and heterogeneous media.The computed results show the significant improvement when using the p-version refined enriched approximations in the finite element analysis.In addition,these results support our expectations for a robust and high-order accurate enriched partition of unity finite element method.
文摘This study proposed a partitioned method to analyze maneuvering of insects during flapping flight.This method decomposed the insect flapping flight into wing and body subsystems and then coupled them via boundary conditions imposed on the wing’s base using one-way coupling.In the wing subsystem,the strong coupling of the flexible wings and surrounding fluid was accurately analyzed using the finite element method to obtain the thrust forces acting on the insect’s body.The resulting thrust forces were passed from the wing subsystem to the body subsystem,and then rigid body motion was analyzed in the body subsystem.The rolling,yawing,and pitching motions were simulated using the proposed method as follows:In the rolling simulation,the difference of the stroke angle between the right and left wings caused a roll torque.In the yawing simulation,the initial feathering angle in the right wing only caused a yaw torque.In the pitching simulation,the difference between the front-and back-stroke angles in both the right and left wings caused a pitch torque.All three torques generated maneuvering motion comparable with that obtained in actual observations of insect flight.These results demonstrate that the proposed method can adequately simulate the fundamental maneuvers of insect flapping flight.In the present simulations,the maneuvering mechanisms were investigated at the governing equation level,which might be difficult using other approaches.Therefore,the proposed method will contribute to revealing the underlying insect flight mechanisms.
文摘A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid mechanics problems. With a polynomial form, the VNM achieves better results than those of traditional PFEMs, including the Wachspress method and the mean value method in standard patch tests. Compared with the standard triangular FEM, the VNM can achieve better accuracy. With the ability to construct shape functions on polygonal elements, the VNM provides greater flexibility in mesh generation. Therefore, several fracture problems are studied to demonstrate the potential implementation. With the advantage of the VNM, the convenient refinement and remeshing strategy are applied.
文摘Metal forming plays an important role in manufacturing industry and is widely applied in industries.The tradi- tional finite element method(FEM)numerical simulation is commonly used to predict metal forming process.Conventional finite element analysis of metal forming processes often breaks down due to severe mesh distortion,therefore time-consuming remeshing is necessary.Meshfree methods have been developed since 1977 and can avoid this problem.This new generation of computational methods reduces time-consuming model generation and refinement effort,and its shape function has higher order connectivity than FEM’s.In this paper the velocity shape functions are developed from a reproducing kernel approximation that satisfies consistency conditions and is used to analyze metal tension rigid viscoplastic deforming and Magnesium Alloy(MB 15)sheet superplastic ten- sion forming.A meshfree method metal forming modeling program is set up,the partition of unity method is used to compute the integrations in weak form equations and penalty method is used to impose the essential boundary condition exactly.Metal forming examples,such as sheet metal superplastic tension forming and metal rigid viscoplastic tension forming,are analyzed to demon- strate the performance of mesh free method.
基金Supported by the Natural Science Foundation of Hunan under Grant No. 06C713.
文摘A class of nonlinear parabolic equation on a polygonal domain Ω R2 is inves- tigated in this paper. We introduce a finite element method on overlapping non-matching grids for the nonlinear parabolic equation based on the partition of unity method. We give the construction and convergence analysis for the semi-discrete and the fully discrete finite element methods. Moreover, we prove that the error of the discrete variational problem has good approximation properties. Our results are valid for any spatial dimensions. A numerical example to illustrate the theoretical results is also given.
基金This research is supported by the ＇985＇ programme of Jilin University, the National Natural Science Foundation of China under Grant Nos. 10971082 and 11076014.
文摘This paper establishes a new finite volume element scheme for Poisson equation on trian- gular meshes. The trial function space is taken as Lagrangian cubic finite element space on triangular partition, and the test function space is defined as piecewise constant space on dual partition. Under some weak condition about the triangular meshes, the authors prove that the stiffness matrix is uni- formly positive definite and convergence rate to be O（h3） in Hi-norm. Some numerical experiments confirm the theoretical considerations.
基金supported by the National Natural Science Foundation of China(Nos.11871009,12271055)the Foundation of LCP and the Foundation of CAEP(CX20210044).
文摘Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.
基金supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China 10971074+1 种基金the National Basic Research Program under the Grant 2005CB321703Hunan Provincial Innovation Foundation For Postgraduate CX2009B119.
文摘We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element methods.We use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control variable.We obtain the superconvergence of O(h^(1+s))(0<s≤1)for the control variable.Finally,we present two numerical examples to confirm our superconvergence results.
文摘The multi-physics simulation of coupled fluid-structure interaction problems, with disjoint fluid and solid domains, requires one to choose a method for enforcing the fluid-structure coupling at the interface between solid and fluid. While it is common knowledge that the choice of coupling technique can be very problem dependent, there exists no satisfactory coupling comparison methodology that allows for conclusions to be drawn with respect to the comparison of computational cost and solution accuracy for a given scenario. In this work, we develop a computational framework where all aspects of the computation can be held constant, save for the method in which the coupled nature of the fluid-structure equations is enforced. To enable a fair comparison of coupling methods, all simulations presented in this work are implemented within a single numerical framework within the deal.ii [1] finite element library. We have chosen the two-dimensional benchmark test problem of Turek and Hron [2] as an example to examine the relative accuracy of the coupling methods studied;however, the comparison technique is equally applicable to more complex problems. We show that for the specific case considered herein the monolithic approach outperforms partitioned and quasi-direct methods;however, this result is problem dependent and we discuss computational and modeling aspects which may affect other comparison studies.
基金This paper was supported by the common Czech-US cooperative research project of the programmeKONTACT No. ME 148 (1998).
文摘Examines a nonlinear partial differential equation of elliptic type with the homogeneous Dirichlet boundary conditions; Proof of the comparison and maximum principles; Approximation of the finite element; Introduction of a discrete analogue of the maximum principle for linear elements.
基金supported by the National Natural Science Foundation of China(Grant Nos.12071404,12271465,12026254)by the Young Elite Scientist Sponsorship Program by CAST(Grant No.2020QNRC001)+3 种基金by the China Postdoctoral Science Foundation(Grant No.2018T110073)by the Natural Science Foundation of Hunan Province(Grant No.2019JJ40279)by the Excellent Youth Program of Scientific Research Project of Hunan Provincial Department of Education(Grant No.20B564)by the International Scientific and Technological Innovation Cooperation Base of Hunan Province for Computational Science(Grant No.2018WK4006).
文摘By combination of iteration methods with the partition of unity method(PUM),some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics(MHD)with different physical parameters are presented and analyzed.These algorithms are highly efficient.At first,a global solution is obtained on a coarse grid for all approaches by one of the iteration methods.By parallelized residual schemes,local corrected solutions are calculated on finer meshes with overlapping sub-domains.The subdomains can be achieved flexibly by a class of PUM.The proposed algorithm is proved to be uniformly stable and convergent.Finally,one numerical example is presented to confirm the theoretical findings.
