Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine fini...Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.展开更多
In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of fini...In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.展开更多
For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlinear iterations is only to execute on the coarse grid an...For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlinear iterations is only to execute on the coarse grid and the fine-grid solution can be obtained in a single linear step. For the nonlinear convection-dominated diffusion equation, this method can not only stabilize the numerical oscillation but also accelerate the convergence and improve the computational efficiency. The error analysis demonstrates if the mesh sizes between coarse-grid and fine-grid satisfy the certain relationship, the two-grid solution and the characteristics finite-element solution have the same order of accuracy. The numerical is more efficient than that of characteristics example confirms that the two-grid method finite-element method.展开更多
In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and ...In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.展开更多
In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the m...In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.展开更多
Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow pheno...Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.展开更多
A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leave...A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leaves only small region adjacent to the aerodynamic bodies to be filled with the cloud of points used in the gridless methods,which results in a better combination of the computational efficiency of the Cartesian grid and the flexibility of the gridless method in handling complex geometries.The clouds of points in the local gridless region are implemented in an anisotropic way according to the features of the thin boundary layer of the viscous flows over the airfoils,and the clouds of points at the vicinity of the interface between the grid and the gridless regions are also controlled by using an adaptive refinement technique during the generation of the unstructured Cartesian grid.An implementation of the resulting hybrid method is presented for solving two-dimensional compressible Navier-Stokes(NS)equations.The simulations of the viscous flows over a RAE2822airfoil or a two-element airfoil are successfully carried out,and the obtained results agree well with the available experimental data.展开更多
In this paper we describe a multi-grid algorithm for mixed problems with penalty by the linear finite element approximation. It is proved that the convergence rate of the algorithm is bound ed away from 1 independentl...In this paper we describe a multi-grid algorithm for mixed problems with penalty by the linear finite element approximation. It is proved that the convergence rate of the algorithm is bound ed away from 1 independently of the meshsize. For convenience, we only discuss Jacobi relaxation as smoothing operator in detail.展开更多
The emerging of false data injection attacks(FDIAs)can fool the traditional detection methods by injecting false data,which has brought huge risks to the security of smart grids.For this reason,a resilient active defe...The emerging of false data injection attacks(FDIAs)can fool the traditional detection methods by injecting false data,which has brought huge risks to the security of smart grids.For this reason,a resilient active defense control scheme based on interval observer detection is proposed in this paper to protect smart grids.The proposed active defense highlights the integration of detection and defense against FDIAs in smart girds.First,a dynamic physical grid model under FDIAs is modeled,in which model uncertainty and parameter uncertainty are taken into account.Then,an interval observer-based detection method against FDIAs is proposed,where a detection criteria using interval residual is put forward.Corresponding to the detection results,the resilient defense controller is triggered to defense the FDIAs if the system states are affected by FDIAs.Linear matrix inequality(LMI)approach is applied to design the resilient controller with H_(∞)performance.The system with the resilient defense controller can be robust to FDIAs and the gain of the resilient controller has a certain gain margin.Our active resilient defense approach can be built in real time and show accurate and quick respond to the injected FDIAs.The effectiveness of the proposed defense scheme is verified by the simulation results on an IEEE 30-bus grid system.展开更多
This paper proposes a hybrid vertex-centered fi- nite volume/finite element method for solution of the two di- mensional （2D） incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fra...This paper proposes a hybrid vertex-centered fi- nite volume/finite element method for solution of the two di- mensional （2D） incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by join- ing the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an im- plicit second-order scheme is utilized to enhance the com- putational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method （FVM） and the pressure Poisson equation is solved by the Galerkin finite el- ement method （FEM）. The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both veloc- ity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.展开更多
Finite element method is based on element matrix, so regardless of whetherthe mesh is structured or unstructured, it Possesses an unified fashion of treatment. Finiteelement method in conjunction with unstructured gri...Finite element method is based on element matrix, so regardless of whetherthe mesh is structured or unstructured, it Possesses an unified fashion of treatment. Finiteelement method in conjunction with unstructured grid will improve the ability of numericalsimulation for complicated now field. In this paper, a 3D unstructured grid generationtechno1ogy is developed and the Euler equation on the unstructured mesh for real compli-cated aircraft configurations is solved by the finite e1ement method. Numerical results in-dicate that the method presented is reliable end efficient.展开更多
How the outer substance could penetrate through the skin lies in the stratum corneum, because it is the main barrier in the multi-layers of the skin. Supposing the keratin cell with a special geometry as tetrakaidecah...How the outer substance could penetrate through the skin lies in the stratum corneum, because it is the main barrier in the multi-layers of the skin. Supposing the keratin cell with a special geometry as tetrakaidecahedron, the penetration property of stratum corneum was the key problem which was numerically simulated with finite element method. At first the discretization of the stratum corneum region was given in two steps： first, the discretization of the keratin cell; second, the discretization of fattiness that surrounds the keratin. Then there was the work of numerical simulation. In this procedure, the finite element method and the multi-grid method were used. The former was to obtain the discretization of basic elements; the latter was to decrease the high frequency error. At last the visualization of the numerical simulation was shown.展开更多
This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on t...This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.展开更多
Four process parameters, pad diameter, stencil thickness, ball diameter and stand-off were chosen as four control factors. By using an L25 （5^6 ） orthogonal array the ceramic ball grid array （ CBGA ） solder joints...Four process parameters, pad diameter, stencil thickness, ball diameter and stand-off were chosen as four control factors. By using an L25 （5^6 ） orthogonal array the ceramic ball grid array （ CBGA ） solder joints which have 25 different combinations of process parameters were designed. The numerical models of all the 25 CBGA solder joints were developed using the Sugrace Evolver. Utilizing the sugrace coordinate exported from the 25 CBGA solder joints numerical models, the finite element analysis models were set up and the nonlinear finite element analysis of the CBGA solder joints under thermal cycles were pegrormed by ANSYS. The thermal fatigue life of CBGA solder joint was calculated using Coffin-Manson equation. Based on the calculated thermal fatigue life results, the range analysis and the variance analysis were pegrormed. The results show that the fatigue life of CBGA solder joint is affected by the pad diameter, the stencil thickness, the ball diameter and the stand-off in a descending order, the best combination of process parameters results in the longest fatigue life is 0.07 mm stand-off, 0.125 mm stencil thickness of, 0.85 mm ball diameter and 0. 89 mm pad diameter. With 95% confidence the pad diameter has a significant effect on the reliability of CBGA solder joints whereas the stand-off, the stencil thickness and the ball diameter have little effect on the reliability of CBGA solder joints.展开更多
This is the second paper of a series where we introduce a control volume based finite element method （CVFEM） to simulate multiphase flow in porous media. This is a fully conservative method able to deal with unstruc...This is the second paper of a series where we introduce a control volume based finite element method （CVFEM） to simulate multiphase flow in porous media. This is a fully conservative method able to deal with unstructured grids which can be used for representing any complexity of reservoir geometry and its geological objects in an accurate and efficient manner. In order to deal with the inherent heterogeneity of the reservoirs, all operations related to discretization are performed at the element level in a manner similar to classical finite element method （FEM）. Moreover, the proposed method can effectively reduce the so-called grid orientation effects. In the first paper of this series, we presented this method and its application for incompressible and immiscible two-phase flow simulation in homogeneous and heterogeneous porous media. In this paper, we evaluate the capability of the method in the solution of highly nonlinear and coupled partial differential equations by simulating hydrocarbon reservoirs using the black-oil model. Furthermore, the effect of grid orientation is investigated by simulating a benchmark waterflooding problem. The numerical results show that the formulation presented here is efficient and accurate for solving the bubble point and three-phase coning problems.展开更多
Wave motion in finite element models presents some characteristics different from those of wave motion in continuum, which leads to the errors and other special phenomena in finite element simulation of wave motion. T...Wave motion in finite element models presents some characteristics different from those of wave motion in continuum, which leads to the errors and other special phenomena in finite element simulation of wave motion. The wave propagation in a 3-D finite element model is studied by utilizing the formal solution in the paper, and the corresponding dispersion relations are derived. Then the main properties of wave motion in 3-D grids such as dispersion, cut-off frequency and polarization drift are discussed. Characteristics different from those of wave motion in 2-D grids are revealed.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10371096)
文摘Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.
基金Project supported by the National Natural Science Foundation of China(Nos.11671157 and11826212)
文摘In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.
文摘For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlinear iterations is only to execute on the coarse grid and the fine-grid solution can be obtained in a single linear step. For the nonlinear convection-dominated diffusion equation, this method can not only stabilize the numerical oscillation but also accelerate the convergence and improve the computational efficiency. The error analysis demonstrates if the mesh sizes between coarse-grid and fine-grid satisfy the certain relationship, the two-grid solution and the characteristics finite-element solution have the same order of accuracy. The numerical is more efficient than that of characteristics example confirms that the two-grid method finite-element method.
文摘In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.
文摘In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.
基金King Mongkut’s University of Technology North Bangkok (KMUTNB)the Office of the Higher Education Commission (OHEC)the National Metal and Materials Technology Center (MTEC) for supporting this research work
文摘Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.
基金Supported by the National Natural Science Foundation of China(11172134)the Funding of Jiangsu Innovation Program for Graduate Education(CXZZ110192)the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leaves only small region adjacent to the aerodynamic bodies to be filled with the cloud of points used in the gridless methods,which results in a better combination of the computational efficiency of the Cartesian grid and the flexibility of the gridless method in handling complex geometries.The clouds of points in the local gridless region are implemented in an anisotropic way according to the features of the thin boundary layer of the viscous flows over the airfoils,and the clouds of points at the vicinity of the interface between the grid and the gridless regions are also controlled by using an adaptive refinement technique during the generation of the unstructured Cartesian grid.An implementation of the resulting hybrid method is presented for solving two-dimensional compressible Navier-Stokes(NS)equations.The simulations of the viscous flows over a RAE2822airfoil or a two-element airfoil are successfully carried out,and the obtained results agree well with the available experimental data.
基金This work was supported by China State Major Key Project for Basic Researches
文摘In this paper we describe a multi-grid algorithm for mixed problems with penalty by the linear finite element approximation. It is proved that the convergence rate of the algorithm is bound ed away from 1 independently of the meshsize. For convenience, we only discuss Jacobi relaxation as smoothing operator in detail.
基金supported by the National Nature Science Foundation of China(Nos.62103357,62203376)the Science and Technology Plan of Hebei Education Department(No.QN2021139)+1 种基金the Nature Science Foundation of Hebei Province(Nos.F2021203043,F2022203074)the Open Research Fund of Jiangsu Collaborative Innovation Center for Smart Distribution Network,Nanjing Institute of Technology(No.XTCX202203).
文摘The emerging of false data injection attacks(FDIAs)can fool the traditional detection methods by injecting false data,which has brought huge risks to the security of smart grids.For this reason,a resilient active defense control scheme based on interval observer detection is proposed in this paper to protect smart grids.The proposed active defense highlights the integration of detection and defense against FDIAs in smart girds.First,a dynamic physical grid model under FDIAs is modeled,in which model uncertainty and parameter uncertainty are taken into account.Then,an interval observer-based detection method against FDIAs is proposed,where a detection criteria using interval residual is put forward.Corresponding to the detection results,the resilient defense controller is triggered to defense the FDIAs if the system states are affected by FDIAs.Linear matrix inequality(LMI)approach is applied to design the resilient controller with H_(∞)performance.The system with the resilient defense controller can be robust to FDIAs and the gain of the resilient controller has a certain gain margin.Our active resilient defense approach can be built in real time and show accurate and quick respond to the injected FDIAs.The effectiveness of the proposed defense scheme is verified by the simulation results on an IEEE 30-bus grid system.
基金supported by the Natural Science Foundation of China (11061021)the Program of Higher-level talents of Inner Mongolia University (SPH-IMU,Z200901004)the Scientific Research Projection of Higher Schools of Inner Mongolia(NJ10016,NJ10006)
文摘This paper proposes a hybrid vertex-centered fi- nite volume/finite element method for solution of the two di- mensional （2D） incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by join- ing the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an im- plicit second-order scheme is utilized to enhance the com- putational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method （FVM） and the pressure Poisson equation is solved by the Galerkin finite el- ement method （FEM）. The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both veloc- ity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.
文摘Finite element method is based on element matrix, so regardless of whetherthe mesh is structured or unstructured, it Possesses an unified fashion of treatment. Finiteelement method in conjunction with unstructured grid will improve the ability of numericalsimulation for complicated now field. In this paper, a 3D unstructured grid generationtechno1ogy is developed and the Euler equation on the unstructured mesh for real compli-cated aircraft configurations is solved by the finite e1ement method. Numerical results in-dicate that the method presented is reliable end efficient.
文摘How the outer substance could penetrate through the skin lies in the stratum corneum, because it is the main barrier in the multi-layers of the skin. Supposing the keratin cell with a special geometry as tetrakaidecahedron, the penetration property of stratum corneum was the key problem which was numerically simulated with finite element method. At first the discretization of the stratum corneum region was given in two steps： first, the discretization of the keratin cell; second, the discretization of fattiness that surrounds the keratin. Then there was the work of numerical simulation. In this procedure, the finite element method and the multi-grid method were used. The former was to obtain the discretization of basic elements; the latter was to decrease the high frequency error. At last the visualization of the numerical simulation was shown.
基金supported by the National Natural Science Foundation of China (10771134).
文摘This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.
基金This work was supported by Science Foundation of Guangxi Zhuang Autonomous Region (Contract No. 02336060).
文摘Four process parameters, pad diameter, stencil thickness, ball diameter and stand-off were chosen as four control factors. By using an L25 （5^6 ） orthogonal array the ceramic ball grid array （ CBGA ） solder joints which have 25 different combinations of process parameters were designed. The numerical models of all the 25 CBGA solder joints were developed using the Sugrace Evolver. Utilizing the sugrace coordinate exported from the 25 CBGA solder joints numerical models, the finite element analysis models were set up and the nonlinear finite element analysis of the CBGA solder joints under thermal cycles were pegrormed by ANSYS. The thermal fatigue life of CBGA solder joint was calculated using Coffin-Manson equation. Based on the calculated thermal fatigue life results, the range analysis and the variance analysis were pegrormed. The results show that the fatigue life of CBGA solder joint is affected by the pad diameter, the stencil thickness, the ball diameter and the stand-off in a descending order, the best combination of process parameters results in the longest fatigue life is 0.07 mm stand-off, 0.125 mm stencil thickness of, 0.85 mm ball diameter and 0. 89 mm pad diameter. With 95% confidence the pad diameter has a significant effect on the reliability of CBGA solder joints whereas the stand-off, the stencil thickness and the ball diameter have little effect on the reliability of CBGA solder joints.
基金Iranian Offshore OilCompany (IOOC) for financial support of this work
文摘This is the second paper of a series where we introduce a control volume based finite element method （CVFEM） to simulate multiphase flow in porous media. This is a fully conservative method able to deal with unstructured grids which can be used for representing any complexity of reservoir geometry and its geological objects in an accurate and efficient manner. In order to deal with the inherent heterogeneity of the reservoirs, all operations related to discretization are performed at the element level in a manner similar to classical finite element method （FEM）. Moreover, the proposed method can effectively reduce the so-called grid orientation effects. In the first paper of this series, we presented this method and its application for incompressible and immiscible two-phase flow simulation in homogeneous and heterogeneous porous media. In this paper, we evaluate the capability of the method in the solution of highly nonlinear and coupled partial differential equations by simulating hydrocarbon reservoirs using the black-oil model. Furthermore, the effect of grid orientation is investigated by simulating a benchmark waterflooding problem. The numerical results show that the formulation presented here is efficient and accurate for solving the bubble point and three-phase coning problems.
文摘Wave motion in finite element models presents some characteristics different from those of wave motion in continuum, which leads to the errors and other special phenomena in finite element simulation of wave motion. The wave propagation in a 3-D finite element model is studied by utilizing the formal solution in the paper, and the corresponding dispersion relations are derived. Then the main properties of wave motion in 3-D grids such as dispersion, cut-off frequency and polarization drift are discussed. Characteristics different from those of wave motion in 2-D grids are revealed.
