This paper is an introduction to mesh based generated reluctance network modeling using triangular elements.Many contributions on mesh based generated reluctance networks using rectangular shaped elements have been pu...This paper is an introduction to mesh based generated reluctance network modeling using triangular elements.Many contributions on mesh based generated reluctance networks using rectangular shaped elements have been published,but very few on those generated from a mesh using triangular elements.The use of triangular elements is aimed at extending the application of the approach to any shape of modeled devices.Basic concepts of the approach are presented in the case of electromagnetic devices.The procedure for coding the approach in the case of a flat linear permanent magnet machine is presented.Codes developed under MATLAB environment are also included.展开更多
The open-source finite element software,OpenSees,is widely used in the earthquake engineering community.However,the shell elements and explicit algorithm in OpenSees still require further improvements.Therefore,in thi...The open-source finite element software,OpenSees,is widely used in the earthquake engineering community.However,the shell elements and explicit algorithm in OpenSees still require further improvements.Therefore,in this work,a triangular shell element,NLDKGT,and an explicit algorithm are proposed and implemented in OpenSees.Specifically,based on the generalized conforming theory and the updated Lagrangian formulation,the proposed NLDKGT element is suitable for problems with complicated boundary conditions and strong nonlinearity.The accuracy and reliability of the NLDKGT element are validated through typical cases.Furthermore,by adopting the leapfrog integration method,an explicit algorithm in OpenSees and a modal damping model are developed.Finally,the stability and efficiency of the proposed shell element and explicit algorithm are validated through the nonlinear time-history analysis of a highrise building.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
Based on the first-order shear deformation theory,a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth,folded and multi-shell laminated composite struc...Based on the first-order shear deformation theory,a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth,folded and multi-shell laminated composite structures.The two smaller components of the mid-surface normal vector of shell at a node are defined as nodal rotational variables in the co-rotational local coordinate system.In the global coordinate system,two smaller components of one vector,together with the smallest or second smallest component of another vector,of an orthogonal triad at a node on a non-smooth intersection of plates and/or shells are defined as rotational variables,whereas the two smaller components of the mid-surface normal vector at a node on the smooth part of the plate or shell(away from non-smooth intersections)are defined as rotational variables.All these vectorial rotational variables can be updated in an additive manner during an incremental solution procedure,and thus improve the computational efficiency in the nonlinear solution of these composite shell structures.Due to the commutativity of all nodal variables in calculating of the second derivatives of the local nodal variables with respect to global nodal variables,and the second derivatives of the strain energy functional with respect to local nodal variables,symmetric tangent stiffness matrices in local and global coordinate systems are obtained.To overcome shear locking,the assumed transverse shear strains obtained from the line-integration approach are employed.The reliability and computational accuracy of the present 3-node triangular shell finite element are verified through modeling two patch tests,several smooth and non-smooth laminated composite shells undergoing large displacements and large rotations.展开更多
A numerical research on magnetohydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure at non-uniform heating of bottom wall has been studied numerically. The enclosure consists of insulated top wall ...A numerical research on magnetohydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure at non-uniform heating of bottom wall has been studied numerically. The enclosure consists of insulated top wall and cold side walls, too. It also contains a heated triangular block (<em>Rot</em> = 0<span style="font-family:Verdana, Helvetica, Arial;white-space:normal;background-color:#FFFFFF;">°</span> - 90<span style="font-family:Verdana, Helvetica, Arial;white-space:normal;background-color:#FFFFFF;">°</span>) located somewhere inside the enclosure. The boundary top wall of the enclosure is moving through uniform speed <em>U</em><sub>0</sub>. The geometry of the model has been represented mathematically by coupled governing equations in accordance with proper boundary conditions and then a two-dimensional Galerkin finite element based numerical approach has been adopted to solve this paper. The numerical computations have been carried out for the wide range of parameters Prandtl number (0.5 ≤ <em>Pr</em> ≤ 2), Reynolds number (60 ≤ <em>Re</em> ≤ 120), Rayleigh number (<em>Ra</em> = 10<sup>3</sup>) and Hartmann number (<em>Ha</em> = 20) taking with different rotations of heated triangular block. The results have been shown in the form of streamlines, temperature patterns or isotherms, average Nusselt number and average bulk temperature of the fluid in the enclosure at non-uniform heating of bottom wall. It is also indicated that both the streamlines, isotherm patterns strongly depend on the aforesaid governing parameters and location of the triangular block but the thermal conductivity of the triangular block has a noteworthy role on the isotherm pattern lines. Moreover, the variation of <em>Nu</em><sub>av</sub> of hot bottom wall and <em>θ</em><sub>av</sub> in the enclosure is demonstrated here to show the characteristics of heat transfer in the enclosure.展开更多
A general method to construct locking free Reissner-Mindlin plate elements is presented. According to this method the shear strain is replaced by its proper interpolation polynomial, which corresponds to the Kirchoff ...A general method to construct locking free Reissner-Mindlin plate elements is presented. According to this method the shear strain is replaced by its proper interpolation polynomial, which corresponds to the Kirchoff conditions at the interpolation points as the thickness of plate tends to zero, so the element is locking free. We construct two triangular elements by this method - a 3-node element and a 6-node element. The numerical results are provided.展开更多
With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and re...With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and recombination of base functions. Hammer integral formulas are the special examples of those of the paper.展开更多
The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied ...The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.展开更多
The basic idea of quasi-conforming method is that the strain-dis-placement equations are weakened as well as the equilibrium equations.In this paper,an 18-DOF triangular element for couple stress theory is proposed wi...The basic idea of quasi-conforming method is that the strain-dis-placement equations are weakened as well as the equilibrium equations.In this paper,an 18-DOF triangular element for couple stress theory is proposed within the framework of quasi-conforming technique.The formulation starts from truncated Taylor expansion of strains and appropriate interpolation functions are chosen to calculate strain integration.This element satisfies C0 continuity with second order accuracy and weak C1 continuity simultaneously.Numerical examples demonstrate that the proposed model can pass the C0??1 patch test and has high accuracy.The element does not exhibit extra zero energy modes and can capture the scale effects of microstructure.展开更多
The application of the finite layer & triangular prism element method to the 3D ground subsidence and stress analysis caused by mining is presented. The layer elements and the triangular prism elements have been a...The application of the finite layer & triangular prism element method to the 3D ground subsidence and stress analysis caused by mining is presented. The layer elements and the triangular prism elements have been alternatively used in the numerical simulation system, the displacement pattern, strain matrix, elastic matrix, stiffness matrix, load matrix and the stress matrix of the layer element and triangular prism element have been presented. By means of the Fortran90 programming language, a numerical simulation system based on finite layer & triangular prism element have been built up, and this system is suitable for subsidence prediction and stress analysis of all mining condition and mining methods. Comparing with the infinite element method, this approach dramatically reduces the size of the set of equations that need to be solved, and greatly reduces the amount of data preparation required. It not only saves the internal storage, and the computation time, but also decreases the cost.展开更多
The phenomena of magneto-hydrodynamic natural convection in a two-dimensional semicircular top enclosure with triangular obstacle in the rectangular cavity were studied numerically. The governing differential equation...The phenomena of magneto-hydrodynamic natural convection in a two-dimensional semicircular top enclosure with triangular obstacle in the rectangular cavity were studied numerically. The governing differential equations are solved by using the most important method which is finite element method (weighted-residual method). The top wall is placed at cold T<sub>c</sub> and bottom wall is heated T<sub>h</sub>. Here the sidewalls of the cavity assumed adiabatic. Also all the wall are occupied to be no-slip condition. A heated triangular obstacle is located at the center of the cavity. The study accomplished for Prandtl number Pr = 0.71;the Rayleigh number Ra = 10<sup>3</sup>, 10<sup>5</sup>, 5 × 10<sup>5</sup>, 10<sup>6</sup> and for Hartmann number Ha = 0, 20, 50, 100. The results represent the streamlines, isotherms, velocity and temperature fields as well as local Nusselt number.展开更多
The family of Falk-Neilan P_(k)finite elements,combined with the Argyris P_(k+1)finite elements,solves the Reissner-Mindlin plate equation quasi-optimally and locking-free,on triangular meshes.The method is truly conf...The family of Falk-Neilan P_(k)finite elements,combined with the Argyris P_(k+1)finite elements,solves the Reissner-Mindlin plate equation quasi-optimally and locking-free,on triangular meshes.The method is truly conforming or consistent in the sense that no projection/reduction is introduced.Theoretical proof and numerical confirmation are presented.展开更多
A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulat...A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulation.This work is a continuation of our investigation of the SFWG method for the biharmonic equation.The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L^(2)norm on triangular grids.This new method also keeps the formulation that is symmetric,positive definite,and stabilizer-free.Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H^(2)norm.Superconvergence of four orders in the L^(2)norm is also derived for k≥3,where k is the degree of the approximation polynomial.The postprocessing is proved to lift a P_(k)SFWG solution to a P_(k+4)solution elementwise which converges at the optimal order.Numerical examples are tested to verify the theor ies.展开更多
文摘This paper is an introduction to mesh based generated reluctance network modeling using triangular elements.Many contributions on mesh based generated reluctance networks using rectangular shaped elements have been published,but very few on those generated from a mesh using triangular elements.The use of triangular elements is aimed at extending the application of the approach to any shape of modeled devices.Basic concepts of the approach are presented in the case of electromagnetic devices.The procedure for coding the approach in the case of a flat linear permanent magnet machine is presented.Codes developed under MATLAB environment are also included.
基金The authors would like to acknowledge the financial supports of Beijing Natural Science Foundation(No.8182025).
文摘The open-source finite element software,OpenSees,is widely used in the earthquake engineering community.However,the shell elements and explicit algorithm in OpenSees still require further improvements.Therefore,in this work,a triangular shell element,NLDKGT,and an explicit algorithm are proposed and implemented in OpenSees.Specifically,based on the generalized conforming theory and the updated Lagrangian formulation,the proposed NLDKGT element is suitable for problems with complicated boundary conditions and strong nonlinearity.The accuracy and reliability of the NLDKGT element are validated through typical cases.Furthermore,by adopting the leapfrog integration method,an explicit algorithm in OpenSees and a modal damping model are developed.Finally,the stability and efficiency of the proposed shell element and explicit algorithm are validated through the nonlinear time-history analysis of a highrise building.
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This work was supported by National Natural Science Foundation of China under Grant 11672266.
文摘Based on the first-order shear deformation theory,a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth,folded and multi-shell laminated composite structures.The two smaller components of the mid-surface normal vector of shell at a node are defined as nodal rotational variables in the co-rotational local coordinate system.In the global coordinate system,two smaller components of one vector,together with the smallest or second smallest component of another vector,of an orthogonal triad at a node on a non-smooth intersection of plates and/or shells are defined as rotational variables,whereas the two smaller components of the mid-surface normal vector at a node on the smooth part of the plate or shell(away from non-smooth intersections)are defined as rotational variables.All these vectorial rotational variables can be updated in an additive manner during an incremental solution procedure,and thus improve the computational efficiency in the nonlinear solution of these composite shell structures.Due to the commutativity of all nodal variables in calculating of the second derivatives of the local nodal variables with respect to global nodal variables,and the second derivatives of the strain energy functional with respect to local nodal variables,symmetric tangent stiffness matrices in local and global coordinate systems are obtained.To overcome shear locking,the assumed transverse shear strains obtained from the line-integration approach are employed.The reliability and computational accuracy of the present 3-node triangular shell finite element are verified through modeling two patch tests,several smooth and non-smooth laminated composite shells undergoing large displacements and large rotations.
文摘A numerical research on magnetohydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure at non-uniform heating of bottom wall has been studied numerically. The enclosure consists of insulated top wall and cold side walls, too. It also contains a heated triangular block (<em>Rot</em> = 0<span style="font-family:Verdana, Helvetica, Arial;white-space:normal;background-color:#FFFFFF;">°</span> - 90<span style="font-family:Verdana, Helvetica, Arial;white-space:normal;background-color:#FFFFFF;">°</span>) located somewhere inside the enclosure. The boundary top wall of the enclosure is moving through uniform speed <em>U</em><sub>0</sub>. The geometry of the model has been represented mathematically by coupled governing equations in accordance with proper boundary conditions and then a two-dimensional Galerkin finite element based numerical approach has been adopted to solve this paper. The numerical computations have been carried out for the wide range of parameters Prandtl number (0.5 ≤ <em>Pr</em> ≤ 2), Reynolds number (60 ≤ <em>Re</em> ≤ 120), Rayleigh number (<em>Ra</em> = 10<sup>3</sup>) and Hartmann number (<em>Ha</em> = 20) taking with different rotations of heated triangular block. The results have been shown in the form of streamlines, temperature patterns or isotherms, average Nusselt number and average bulk temperature of the fluid in the enclosure at non-uniform heating of bottom wall. It is also indicated that both the streamlines, isotherm patterns strongly depend on the aforesaid governing parameters and location of the triangular block but the thermal conductivity of the triangular block has a noteworthy role on the isotherm pattern lines. Moreover, the variation of <em>Nu</em><sub>av</sub> of hot bottom wall and <em>θ</em><sub>av</sub> in the enclosure is demonstrated here to show the characteristics of heat transfer in the enclosure.
文摘A general method to construct locking free Reissner-Mindlin plate elements is presented. According to this method the shear strain is replaced by its proper interpolation polynomial, which corresponds to the Kirchoff conditions at the interpolation points as the thickness of plate tends to zero, so the element is locking free. We construct two triangular elements by this method - a 3-node element and a 6-node element. The numerical results are provided.
文摘With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and recombination of base functions. Hammer integral formulas are the special examples of those of the paper.
文摘The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.
基金the Fundamental Research Funds for the Central Universities(DUT14RC(3)092)the National Natural Science Foundation of China(No.11272075,11472071).
文摘The basic idea of quasi-conforming method is that the strain-dis-placement equations are weakened as well as the equilibrium equations.In this paper,an 18-DOF triangular element for couple stress theory is proposed within the framework of quasi-conforming technique.The formulation starts from truncated Taylor expansion of strains and appropriate interpolation functions are chosen to calculate strain integration.This element satisfies C0 continuity with second order accuracy and weak C1 continuity simultaneously.Numerical examples demonstrate that the proposed model can pass the C0??1 patch test and has high accuracy.The element does not exhibit extra zero energy modes and can capture the scale effects of microstructure.
文摘The application of the finite layer & triangular prism element method to the 3D ground subsidence and stress analysis caused by mining is presented. The layer elements and the triangular prism elements have been alternatively used in the numerical simulation system, the displacement pattern, strain matrix, elastic matrix, stiffness matrix, load matrix and the stress matrix of the layer element and triangular prism element have been presented. By means of the Fortran90 programming language, a numerical simulation system based on finite layer & triangular prism element have been built up, and this system is suitable for subsidence prediction and stress analysis of all mining condition and mining methods. Comparing with the infinite element method, this approach dramatically reduces the size of the set of equations that need to be solved, and greatly reduces the amount of data preparation required. It not only saves the internal storage, and the computation time, but also decreases the cost.
文摘The phenomena of magneto-hydrodynamic natural convection in a two-dimensional semicircular top enclosure with triangular obstacle in the rectangular cavity were studied numerically. The governing differential equations are solved by using the most important method which is finite element method (weighted-residual method). The top wall is placed at cold T<sub>c</sub> and bottom wall is heated T<sub>h</sub>. Here the sidewalls of the cavity assumed adiabatic. Also all the wall are occupied to be no-slip condition. A heated triangular obstacle is located at the center of the cavity. The study accomplished for Prandtl number Pr = 0.71;the Rayleigh number Ra = 10<sup>3</sup>, 10<sup>5</sup>, 5 × 10<sup>5</sup>, 10<sup>6</sup> and for Hartmann number Ha = 0, 20, 50, 100. The results represent the streamlines, isotherms, velocity and temperature fields as well as local Nusselt number.
文摘The family of Falk-Neilan P_(k)finite elements,combined with the Argyris P_(k+1)finite elements,solves the Reissner-Mindlin plate equation quasi-optimally and locking-free,on triangular meshes.The method is truly conforming or consistent in the sense that no projection/reduction is introduced.Theoretical proof and numerical confirmation are presented.
文摘A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulation.This work is a continuation of our investigation of the SFWG method for the biharmonic equation.The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L^(2)norm on triangular grids.This new method also keeps the formulation that is symmetric,positive definite,and stabilizer-free.Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H^(2)norm.Superconvergence of four orders in the L^(2)norm is also derived for k≥3,where k is the degree of the approximation polynomial.The postprocessing is proved to lift a P_(k)SFWG solution to a P_(k+4)solution elementwise which converges at the optimal order.Numerical examples are tested to verify the theor ies.