In this paper we present the Projection Based Interpolation (PBI) technique for construction of continuous approximation of MRI scan data of the human head. We utilize the result of the PBI algorithm to perform three ...In this paper we present the Projection Based Interpolation (PBI) technique for construction of continuous approximation of MRI scan data of the human head. We utilize the result of the PBI algorithm to perform three dimensional (3D) Finite Element Method (FEM) simulations of the acoustics of the human head. The computational problem is a multi-physics problem modeled as acoustics coupled with linear elasticity. The computational grid contains tetrahedral finite elements with the number of equations and polynomial orders of approximation varying locally on finite element edges, faces, and interiors. We utilize our own out-of-core parallel direct solver for the solution of this multi-physics problem. The solver minimizes the memory usage by dumping out all local systems from all nodes of the entire elimination tree during the elimination phase.展开更多
We’ll consider the model of two-phase compressible miscible displacement in porous media which includes molecular diffusion and dispersion in one dimensional space. Time-discretization procedure is established and an...We’ll consider the model of two-phase compressible miscible displacement in porous media which includes molecular diffusion and dispersion in one dimensional space. Time-discretization procedure is established and analysed. The optimal error estimate in L2 norm is proved by introducing a new interpolation operator R.展开更多
The main purpose of this paper is to present numerical results of static bending and free vibration of functionally graded porous(FGP) variable-thickness plates by using an edge-based smoothed finite element method(ES...The main purpose of this paper is to present numerical results of static bending and free vibration of functionally graded porous(FGP) variable-thickness plates by using an edge-based smoothed finite element method(ES-FEM) associate with the mixed interpolation of tensorial components technique for the three-node triangular element(MITC3), so-called ES-MITC3. This ES-MITC3 element is performed to eliminate the shear locking problem and to enhance the accuracy of the existing MITC3 element. In the ES-MITC3 element, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains formed by two adjacent MITC3 triangular elements sharing an edge. Materials of the plate are FGP with a power-law index(k) and maximum porosity distributions(U) in the forms of cosine functions. The influences of some geometric parameters, material properties on static bending, and natural frequency of the FGP variable-thickness plates are examined in detail.展开更多
The present work proposed a new method for the modeling by the finite element method of the acoustic propagation problems in infinite axisymmetric cylindrical guides lined with locally reacting absorbent materials wit...The present work proposed a new method for the modeling by the finite element method of the acoustic propagation problems in infinite axisymmetric cylindrical guides lined with locally reacting absorbent materials without flow. The method deals with the development of an efficient transparent boundary condition based on DtN operators. The method developed in this study is successfully applied to a straight axisymmetric lined guide by imposing a mode on one of the artificial boundaries of the truncated guide. The results are in good agreement with analytical solutions. Applying the method for a non-uniform axisymmetric lined guide which is a complex case, proved its effectiveness and the results compared to those of PML layers are in very good agreement.展开更多
We develop the interpolated finite element method to solve second-order hy-perbolic equations. The standard linear finite element solution is used to generate a newsolution by quadratic interpolation over adjacent ele...We develop the interpolated finite element method to solve second-order hy-perbolic equations. The standard linear finite element solution is used to generate a newsolution by quadratic interpolation over adjacent elements. We prove that this interpo-lated finite element solution has superconvergence. This method can easily be applied togenerating more accurate gradient either locally or globally, depending on the applications.This method is also completely vectorizable and parallelizable to take the advantages ofmodern computer structures. Several numerical examples are presented to confirm ourtheoretical analysis.展开更多
Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation.In order to satisfy convergence criteria,the newly presented finite elements are modified using the Petrov-...Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation.In order to satisfy convergence criteria,the newly presented finite elements are modified using the Petrov-Galerkin method in which different interpolation is used for the test and trial functions.The elements are tested through four numerical examples consisting of a set of patch tests,a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation.In the higher-order patch test,the performance of the first-order element is significantly improved.However,since the problems analysed are already describable with quadratic polynomials,the enhancement due to linked interpolation for higher-order elements could not be highlighted.All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis,but the enhancement here is negligible with respect to standard Lagrangian elements,since the higher-order polynomials in this example are not needed.展开更多
In this paper,we consider the energy conserving numerical scheme for coupled nonlinear Klein-Gordon equations.We propose energy conserving finite element method and get the unconditional superconvergence resultO(h^(2)...In this paper,we consider the energy conserving numerical scheme for coupled nonlinear Klein-Gordon equations.We propose energy conserving finite element method and get the unconditional superconvergence resultO(h^(2)+Dt^(2))by using the error splitting technique and postprocessing interpolation.Numerical experiments are carried out to support our theoretical results.展开更多
A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces f...A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators.The new algorithm is simple in formulation and the computational complexity is also reduced.The corresponding approximating spaces consist of piecewise polynomials of degree k≥1 for the velocity and k-1 for the pressure,respectively.Optimal order error estimates have been derived for the velocity in both H^(1) and L^(2) norms and for the pressure in L^(2) norm.Numerical examples are presented to illustrate the accuracy and convergency of the method.展开更多
We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization...We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.展开更多
The operator splitting method is used to deal with the Navier-Stokes equation, in which the physical process described by the equation is decomposed into two processes： a diffusion process and a convection process; a...The operator splitting method is used to deal with the Navier-Stokes equation, in which the physical process described by the equation is decomposed into two processes： a diffusion process and a convection process; and the finite element equation is established. The velocity field in the element is described by the shape function of the isoparametric element with nine nodes and the pressure field is described by the interpolation function of the four nodes at the vertex of the isoparametric element with nine nodes. The subroutine of the element and the integrated finite element code are generated by the Finite Element Program Generator （FEPG） successfully. The numerical simulation about the incompressible viscous liquid flowing over a cylinder is carded out. The solution agrees with the experimental results very well.展开更多
For a class of two-point boundary value problems, by virtue of onedimensional projection interpolation, it is proved that the nodal recovery derivative obtained by Yuan＇s element energy projection （EEP） method has ...For a class of two-point boundary value problems, by virtue of onedimensional projection interpolation, it is proved that the nodal recovery derivative obtained by Yuan＇s element energy projection （EEP） method has the accuracy O（h^min{2k,k＋4}） The theoretical analysis coincides the reported numerical results.展开更多
Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential ...Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.展开更多
This article is concerned with finite element implementations of the three- dimensional geometrically exact rod. The special attention is paid to identifying the con- dition that ensures the frame invariance of the re...This article is concerned with finite element implementations of the three- dimensional geometrically exact rod. The special attention is paid to identifying the con- dition that ensures the frame invariance of the resulting discrete approximations. From the perspective of symmetry, this requirement is equivalent to the commutativity of the employed interpolation operator I with the action of the special Euclidean group SE（3）, or I is SE（3）-equivariant. This geometric criterion helps to clarify several subtle issues about the interpolation of finite rotation. It leads us to reexamine the finite element for- mulation first proposed by Simo in his work on energy-momentum conserving algorithms. That formulation is often mistakenly regarded as non-objective. However, we show that the obtained approximation is invariant under the superposed rigid body motions, and as a corollary, the objectivity of the continuum model is preserved. The key of this proof comes from the observation that since the numerical quadrature is used to compute the integrals, by storing the rotation field and its derivative at the Gauss points, the equiv- ariant conditions can be relaxed only at these points. Several numerical examples are presented to confirm the theoretical results and demonstrate the performance of this al- gorithm.展开更多
The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differe...The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.展开更多
Rotor blade is one of the most significant components of helicopters. But due to its highspeed rotation characteristics, it is difficult to collect the vibration signals during the flight stage.Moreover, sensors are h...Rotor blade is one of the most significant components of helicopters. But due to its highspeed rotation characteristics, it is difficult to collect the vibration signals during the flight stage.Moreover, sensors are highly susceptible to damage resulting in the failure of the measurement.In order to make signal predictions for the damaged sensors, an operational modal analysis(OMA) together with the virtual sensing(VS) technology is proposed in this paper. This paper discusses two situations, i.e., mode shapes measured by all sensors(both normal and damaged) can be obtained using OMA, and mode shapes measured by some sensors(only including normal) can be obtained using OMA. For the second situation, it is necessary to use finite element(FE) analysis to supplement the missing mode shapes of damaged sensor. In order to improve the correlation between the FE model and the real structure, the FE mode shapes are corrected using the local correspondence(LC) principle and mode shapes measured by some sensors(only including normal).Then, based on the VS technology, the vibration signals of the damaged sensors during the flight stage can be accurately predicted using the identified mode shapes(obtained based on OMA and FE analysis) and the normal sensors signals. Given the high degrees of freedom(DOFs) in the FE mode shapes, this approach can also be used to predict vibration data at locations without sensors. The effectiveness and robustness of the proposed method is verified through finite element simulation, experiment as well as the actual flight test. The present work can be further used in the fault diagnosis and damage identification for rotor blade of helicopters.展开更多
Dynamic stability equations of bearingless rotor blades were investigated using a simplified model.The aerodynamic loads of blades were evaluated using two-dimensional airfoil theory.Perturbation equations were obtain...Dynamic stability equations of bearingless rotor blades were investigated using a simplified model.The aerodynamic loads of blades were evaluated using two-dimensional airfoil theory.Perturbation equations were obtained by linearization of the perturbation.A normal-mode approach was used to transform the equations expressed by nodal degrees of freedom into equations expressed by modal degrees of freedom,which can reduce the dimension of the equations.The stability results of rotor blades were presented using eigenvalue analysis.The shape function matrix was obtained using spline interpolation,which simplified the analysis and made assembly of the inertial matrix,damping matrix,and stiffness matrix a simple mathematical summation.The results indicate that the method is efficient and greatly simplifies the analysis.展开更多
This paper is concerned with the superconvergence error estimates of a classical mixed finite element method for a nonlinear parabolic/elliptic coupled thermistor equations.The method is based on a popular combination...This paper is concerned with the superconvergence error estimates of a classical mixed finite element method for a nonlinear parabolic/elliptic coupled thermistor equations.The method is based on a popular combination of the lowest-order rectangular Raviart-Thomas mixed approximation for the electric potential/field(φ,θ)and the bilinear Lagrange approximation for temperature u.In terms of the special properties of these elements above,the superclose error estimates with order O(h^(2))are obtained firstly for all three components in such a strongly coupled system.Subsequently,the global superconvergence error estimates with order O(h^(2))are derived through a simple and effective interpolation post-processing technique.As by a product,optimal error estimates are acquired for potential/field and temperature in the order of O(h)and O(h^(2)),respectively.Finally,some numerical results are provided to confirm the theoretical analysis.展开更多
In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant...In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.展开更多
In many deformation analyses,the partial derivatives at the interpolated scattered data points are required.In this paper,the Gaussian Radial Basis Functions(GRBF)is proposed for the interpolation and differentiation ...In many deformation analyses,the partial derivatives at the interpolated scattered data points are required.In this paper,the Gaussian Radial Basis Functions(GRBF)is proposed for the interpolation and differentiation of the scattered data in the vertical deformation analysis.For the optimal selection of the shape parameter,which is crucial in the GRBF interpolation,two methods are used:the Power Gaussian Radial Basis Functions(PGRBF)and Leave One Out Cross Validation(LOOCV)(LGRBF).We compared the PGRBF and LGRBF to the traditional interpolation methods such as the Finite Element Method(FEM),polynomials,Moving Least Squares(MLS),and the usual GRBF in both the simulated and actual Interferometric Synthetic Aperture Radar(InSAR)data.The estimated results showed that the surface interpolation accuracy was greatly improved by LGRBF and PGRBF methods in comparison withFEM,polynomial,and MLS methods.Finally,LGRBF and PGRBF interpolation methods are used to compute invariant vertical deformation parameters,i.e.,changes in Gaussian and mean Curvatures in the Groningen area in the North of Netherlands.展开更多
Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions...Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the trian- gular area coordinates and the B-net method, which can ex- actly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The nu- merical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.展开更多
文摘In this paper we present the Projection Based Interpolation (PBI) technique for construction of continuous approximation of MRI scan data of the human head. We utilize the result of the PBI algorithm to perform three dimensional (3D) Finite Element Method (FEM) simulations of the acoustics of the human head. The computational problem is a multi-physics problem modeled as acoustics coupled with linear elasticity. The computational grid contains tetrahedral finite elements with the number of equations and polynomial orders of approximation varying locally on finite element edges, faces, and interiors. We utilize our own out-of-core parallel direct solver for the solution of this multi-physics problem. The solver minimizes the memory usage by dumping out all local systems from all nodes of the entire elimination tree during the elimination phase.
基金This work was supported by National Science Foundation and China State Major Key Project for Basic Research
文摘We’ll consider the model of two-phase compressible miscible displacement in porous media which includes molecular diffusion and dispersion in one dimensional space. Time-discretization procedure is established and analysed. The optimal error estimate in L2 norm is proved by introducing a new interpolation operator R.
基金funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02-2019.330。
文摘The main purpose of this paper is to present numerical results of static bending and free vibration of functionally graded porous(FGP) variable-thickness plates by using an edge-based smoothed finite element method(ES-FEM) associate with the mixed interpolation of tensorial components technique for the three-node triangular element(MITC3), so-called ES-MITC3. This ES-MITC3 element is performed to eliminate the shear locking problem and to enhance the accuracy of the existing MITC3 element. In the ES-MITC3 element, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains formed by two adjacent MITC3 triangular elements sharing an edge. Materials of the plate are FGP with a power-law index(k) and maximum porosity distributions(U) in the forms of cosine functions. The influences of some geometric parameters, material properties on static bending, and natural frequency of the FGP variable-thickness plates are examined in detail.
文摘The present work proposed a new method for the modeling by the finite element method of the acoustic propagation problems in infinite axisymmetric cylindrical guides lined with locally reacting absorbent materials without flow. The method deals with the development of an efficient transparent boundary condition based on DtN operators. The method developed in this study is successfully applied to a straight axisymmetric lined guide by imposing a mode on one of the artificial boundaries of the truncated guide. The results are in good agreement with analytical solutions. Applying the method for a non-uniform axisymmetric lined guide which is a complex case, proved its effectiveness and the results compared to those of PML layers are in very good agreement.
基金This research is supported in part by NSF Grant No.DMS-8922865,and by Funding from the Institute of Scientific Computations at the University of Wyoming through NSF Grant.
文摘We develop the interpolated finite element method to solve second-order hy-perbolic equations. The standard linear finite element solution is used to generate a newsolution by quadratic interpolation over adjacent elements. We prove that this interpo-lated finite element solution has superconvergence. This method can easily be applied togenerating more accurate gradient either locally or globally, depending on the applications.This method is also completely vectorizable and parallelizable to take the advantages ofmodern computer structures. Several numerical examples are presented to confirm ourtheoretical analysis.
基金The research presented in this paper has been financially supported by the Croatian Science Foundation(Grants HRZZ-IP-11-2013-1631 and HRZZ-IP-2018-01-1732)Young Researchers'Career Development—Training of Doctoral Students,as well as a French Government Scholarship.
文摘Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation.In order to satisfy convergence criteria,the newly presented finite elements are modified using the Petrov-Galerkin method in which different interpolation is used for the test and trial functions.The elements are tested through four numerical examples consisting of a set of patch tests,a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation.In the higher-order patch test,the performance of the first-order element is significantly improved.However,since the problems analysed are already describable with quadratic polynomials,the enhancement due to linked interpolation for higher-order elements could not be highlighted.All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis,but the enhancement here is negligible with respect to standard Lagrangian elements,since the higher-order polynomials in this example are not needed.
基金The work is supported by the National Natural Science Foundation of China(No.11871441)Beijing Natural Science Foundation(No.1192003).
文摘In this paper,we consider the energy conserving numerical scheme for coupled nonlinear Klein-Gordon equations.We propose energy conserving finite element method and get the unconditional superconvergence resultO(h^(2)+Dt^(2))by using the error splitting technique and postprocessing interpolation.Numerical experiments are carried out to support our theoretical results.
基金supported in part by China Natural National Science Foundation(Nos.91630201,U1530116,11726102,11771179,93K172018Z01,11701210,JJKH20180113KJ,20190103029JH)by the Program for Cheung Kong Scholars of Ministry of Education of China,Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education.The research of Liu was partially supported by China Natural National Science Foundation(No.12001306)Guangdong Provincial Natural Science Foundation(No.2017A030310285).
文摘A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators.The new algorithm is simple in formulation and the computational complexity is also reduced.The corresponding approximating spaces consist of piecewise polynomials of degree k≥1 for the velocity and k-1 for the pressure,respectively.Optimal order error estimates have been derived for the velocity in both H^(1) and L^(2) norms and for the pressure in L^(2) norm.Numerical examples are presented to illustrate the accuracy and convergency of the method.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11271145）, the Foundation for Talent Introduction of Guangdong Provincial University, the Specialized Research Fund for the Doctoral Program of Higher Education （20114407110009）, and the Project of Department of Education of Guangdong Province （2012KJCX0036）.
文摘We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.
文摘The operator splitting method is used to deal with the Navier-Stokes equation, in which the physical process described by the equation is decomposed into two processes： a diffusion process and a convection process; and the finite element equation is established. The velocity field in the element is described by the shape function of the isoparametric element with nine nodes and the pressure field is described by the interpolation function of the four nodes at the vertex of the isoparametric element with nine nodes. The subroutine of the element and the integrated finite element code are generated by the Finite Element Program Generator （FEPG） successfully. The numerical simulation about the incompressible viscous liquid flowing over a cylinder is carded out. The solution agrees with the experimental results very well.
基金Project supported by the National Natural Science Foundation of China (Nos. 10571046, 10371038)
文摘For a class of two-point boundary value problems, by virtue of onedimensional projection interpolation, it is proved that the nodal recovery derivative obtained by Yuan＇s element energy projection （EEP） method has the accuracy O（h^min{2k,k＋4}） The theoretical analysis coincides the reported numerical results.
文摘Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.
文摘This article is concerned with finite element implementations of the three- dimensional geometrically exact rod. The special attention is paid to identifying the con- dition that ensures the frame invariance of the resulting discrete approximations. From the perspective of symmetry, this requirement is equivalent to the commutativity of the employed interpolation operator I with the action of the special Euclidean group SE（3）, or I is SE（3）-equivariant. This geometric criterion helps to clarify several subtle issues about the interpolation of finite rotation. It leads us to reexamine the finite element for- mulation first proposed by Simo in his work on energy-momentum conserving algorithms. That formulation is often mistakenly regarded as non-objective. However, we show that the obtained approximation is invariant under the superposed rigid body motions, and as a corollary, the objectivity of the continuum model is preserved. The key of this proof comes from the observation that since the numerical quadrature is used to compute the integrals, by storing the rotation field and its derivative at the Gauss points, the equiv- ariant conditions can be relaxed only at these points. Several numerical examples are presented to confirm the theoretical results and demonstrate the performance of this al- gorithm.
文摘The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.
基金supported by grants from the High-Level Oversea Talent Introduction Plan,Chinathe Special Fund for Basic Scientific Research in Central Universities of China-Doctoral Research and Innovation Fund Project,China(No.3072023CFJ0206).
文摘Rotor blade is one of the most significant components of helicopters. But due to its highspeed rotation characteristics, it is difficult to collect the vibration signals during the flight stage.Moreover, sensors are highly susceptible to damage resulting in the failure of the measurement.In order to make signal predictions for the damaged sensors, an operational modal analysis(OMA) together with the virtual sensing(VS) technology is proposed in this paper. This paper discusses two situations, i.e., mode shapes measured by all sensors(both normal and damaged) can be obtained using OMA, and mode shapes measured by some sensors(only including normal) can be obtained using OMA. For the second situation, it is necessary to use finite element(FE) analysis to supplement the missing mode shapes of damaged sensor. In order to improve the correlation between the FE model and the real structure, the FE mode shapes are corrected using the local correspondence(LC) principle and mode shapes measured by some sensors(only including normal).Then, based on the VS technology, the vibration signals of the damaged sensors during the flight stage can be accurately predicted using the identified mode shapes(obtained based on OMA and FE analysis) and the normal sensors signals. Given the high degrees of freedom(DOFs) in the FE mode shapes, this approach can also be used to predict vibration data at locations without sensors. The effectiveness and robustness of the proposed method is verified through finite element simulation, experiment as well as the actual flight test. The present work can be further used in the fault diagnosis and damage identification for rotor blade of helicopters.
基金National Hi-tech Research and Development Program of China(2012AA112201)National Natural Science Foundation of China(10772013)+1 种基金The Fundamental Research Funds for the Central UniversitiesAeronautical Science Foundation of China(20100251007)
文摘Dynamic stability equations of bearingless rotor blades were investigated using a simplified model.The aerodynamic loads of blades were evaluated using two-dimensional airfoil theory.Perturbation equations were obtained by linearization of the perturbation.A normal-mode approach was used to transform the equations expressed by nodal degrees of freedom into equations expressed by modal degrees of freedom,which can reduce the dimension of the equations.The stability results of rotor blades were presented using eigenvalue analysis.The shape function matrix was obtained using spline interpolation,which simplified the analysis and made assembly of the inertial matrix,damping matrix,and stiffness matrix a simple mathematical summation.The results indicate that the method is efficient and greatly simplifies the analysis.
基金supported by the National Natural Science Foundation of China(Grant Nos.12101568,12071443).
文摘This paper is concerned with the superconvergence error estimates of a classical mixed finite element method for a nonlinear parabolic/elliptic coupled thermistor equations.The method is based on a popular combination of the lowest-order rectangular Raviart-Thomas mixed approximation for the electric potential/field(φ,θ)and the bilinear Lagrange approximation for temperature u.In terms of the special properties of these elements above,the superclose error estimates with order O(h^(2))are obtained firstly for all three components in such a strongly coupled system.Subsequently,the global superconvergence error estimates with order O(h^(2))are derived through a simple and effective interpolation post-processing technique.As by a product,optimal error estimates are acquired for potential/field and temperature in the order of O(h)and O(h^(2)),respectively.Finally,some numerical results are provided to confirm the theoretical analysis.
基金supported by the National Natural Science Foundation of China (60533060, 10672032, 10726067)Science Foundation of Dalian University of Technology (SFDUT07001)
文摘In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.
文摘In many deformation analyses,the partial derivatives at the interpolated scattered data points are required.In this paper,the Gaussian Radial Basis Functions(GRBF)is proposed for the interpolation and differentiation of the scattered data in the vertical deformation analysis.For the optimal selection of the shape parameter,which is crucial in the GRBF interpolation,two methods are used:the Power Gaussian Radial Basis Functions(PGRBF)and Leave One Out Cross Validation(LOOCV)(LGRBF).We compared the PGRBF and LGRBF to the traditional interpolation methods such as the Finite Element Method(FEM),polynomials,Moving Least Squares(MLS),and the usual GRBF in both the simulated and actual Interferometric Synthetic Aperture Radar(InSAR)data.The estimated results showed that the surface interpolation accuracy was greatly improved by LGRBF and PGRBF methods in comparison withFEM,polynomial,and MLS methods.Finally,LGRBF and PGRBF interpolation methods are used to compute invariant vertical deformation parameters,i.e.,changes in Gaussian and mean Curvatures in the Groningen area in the North of Netherlands.
基金supported by the National Natural Science Foundation of China(11001037,11102037 and 11290143)the Fundamental Research Funds for the Central Universities
文摘Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the trian- gular area coordinates and the B-net method, which can ex- actly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The nu- merical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.
