The heterogeneous variational nodal method(HVNM)has emerged as a potential approach for solving high-fidelity neutron transport problems.However,achieving accurate results with HVNM in large-scale problems using high-...The heterogeneous variational nodal method(HVNM)has emerged as a potential approach for solving high-fidelity neutron transport problems.However,achieving accurate results with HVNM in large-scale problems using high-fidelity models has been challenging due to the prohibitive computational costs.This paper presents an efficient parallel algorithm tailored for HVNM based on the Message Passing Interface standard.The algorithm evenly distributes the response matrix sets among processors during the matrix formation process,thus enabling independent construction without communication.Once the formation tasks are completed,a collective operation merges and shares the matrix sets among the processors.For the solution process,the problem domain is decomposed into subdomains assigned to specific processors,and the red-black Gauss-Seidel iteration is employed within each subdomain to solve the response matrix equation.Point-to-point communication is conducted between adjacent subdomains to exchange data along the boundaries.The accuracy and efficiency of the parallel algorithm are verified using the KAIST and JRR-3 test cases.Numerical results obtained with multiple processors agree well with those obtained from Monte Carlo calculations.The parallelization of HVNM results in eigenvalue errors of 31 pcm/-90 pcm and fission rate RMS errors of 1.22%/0.66%,respectively,for the 3D KAIST problem and the 3D JRR-3 problem.In addition,the parallel algorithm significantly reduces computation time,with an efficiency of 68.51% using 36 processors in the KAIST problem and 77.14% using 144 processors in the JRR-3 problem.展开更多
SP3 （simplified P3） theory is widely used in LWR （light water reactor） analyses to partly capture the transport effect, especially for pin-by-pin core analysis with pin size homogenization. In this paper, a SP3 co...SP3 （simplified P3） theory is widely used in LWR （light water reactor） analyses to partly capture the transport effect, especially for pin-by-pin core analysis with pin size homogenization. In this paper, a SP3 code named STELLA is developed and verified at SNERDI （Shanghai Nuclear Engineering Research and Design Institute）. For SP3 method, neutron transport equation can be transformed into two coupled equations in the same mathematical form as diffusion equation. In this work, SANM （semi-analytic nodal method） is used to solve diffusion-like equation, due to its easy to handle multi-group problem. Whole core nodal boundary net current coupling is used to improve convergence stability in SANM, instead of solving two-node problem. CMFD （coarse-mesh finite difference） acceleration method is employed for 0-th SP3 equation, which represents the neutron balance relationship. Three benchmarks are used to verify the SP3 code, STELLA. The first one is a self-defined one dimensional problem, which demonstrates SP3 method is extremely accurate, due to no academic approximation in one dimensional for SP3. The second one is a two dimensional one-group problem cited from Larsen＇s paper, which is usually used to verify and prove the SP3 code correct and accurate. And the third one is modified from 2D C5G7-MOX benchmark, whose numerical results indicate that STELLA is accurate and efficient in pin size level, compared to diffusion model.展开更多
This paper describes a numerical solution for two dimensional partial differential equations modeling (or arising from) a fluid flow and transport phenomena. The diffusion equation is discretized by the Nodal methods....This paper describes a numerical solution for two dimensional partial differential equations modeling (or arising from) a fluid flow and transport phenomena. The diffusion equation is discretized by the Nodal methods. The saturation equation is solved by a finite volume method. We start with incompressible single-phase flow and move step-by-step to the black-oil model and compressible two phase flow. Numerical results are presented to see the performance of the method, and seem to be interesting by comparing them with other recent results.展开更多
In this paper, we prove the convergence of the nodal expansion method, a new numerical method for partial differential equations and provide the error estimates of approximation solution.
A nodal discontinuous Galerkin formulation based on Lagrange polynomials basis is used to simulate the acoustic wave propagation. Its dispersion and dissipation properties for the advection equation are investigated b...A nodal discontinuous Galerkin formulation based on Lagrange polynomials basis is used to simulate the acoustic wave propagation. Its dispersion and dissipation properties for the advection equation are investigated by utilizing an eigenvalue analysis. Two test problems of wave propagation with initial disturbance consisting of a Gaussian profile or rectangular pulse are performed. And the performance of the schemes in short,intermediate,and long waves is evaluated. Moreover,numerical results between the nodal discontinuous Galerkin method and finite difference type schemes are compared,which indicate that the numerical solution obtained using nodal discontinuous Galerkin method with a pure central flux has obviously high frequency oscillations for initial disturbance consisting of a rectangular pulse,which is the same as those obtained using finite difference type schemes without artificial selective damping. When an upwind flux is adopted,spurious waves are eliminated effectively except for the location of discontinuities. When a limiter is used,the spurious short waves are almost completely removed. Therefore,the quality of the computed solution has improved.展开更多
There are vast constraint equations in conventional dynamics analysis of deployable structures,which lead to differential-algebraic equations(DAEs)solved hard.To reduce the difficulty of solving and the amount of equa...There are vast constraint equations in conventional dynamics analysis of deployable structures,which lead to differential-algebraic equations(DAEs)solved hard.To reduce the difficulty of solving and the amount of equations,a new flexible multibody dynamics analysis methodology of deployable structures with scissor-like elements(SLEs)is presented.Firstly,a precise model of a flexible bar of SLE is established by the higher order shear deformable beam element based on the absolute nodal coordinate formulation(ANCF),and the master/slave freedom method is used to obtain the dynamics equations of SLEs without constraint equations.Secondly,according to features of deployable structures,the specification matrix method(SMM)is proposed to eliminate the constraint equations among SLEs in the frame of ANCF.With this method,the inner and the boundary nodal coordinates of element characteristic matrices can be separated simply and efficiently,especially on condition that there are vast nodal coordinates.So the element characteristic matrices can be added end to end circularly.Thus,the dynamic model of deployable structure reduces dimension and can be assembled without any constraint equation.Next,a new iteration procedure for the generalized-a algorithm is presented to solve the ordinary differential equations(ODEs)of deployable structure.Finally,the proposed methodology is used to analyze the flexible multi-body dynamics of a planar linear array deployable structure based on three scissor-like elements.The simulation results show that flexibility has a significant influence on the deployment motion of the deployable structure.The proposed methodology indeed reduce the difficulty of solving and the amount of equations by eliminating redundant degrees of freedom and the constraint equations in scissor-like elements and among scissor-like elements.展开更多
For the purpose of achieving high-resolution optimal solutions this paper proposes a nodal design variablebased adaptive method for topology optimization of continuum structures. The analysis mesh-independent density ...For the purpose of achieving high-resolution optimal solutions this paper proposes a nodal design variablebased adaptive method for topology optimization of continuum structures. The analysis mesh-independent density field, interpolated by the nodal design variables at a given set of density points, is adaptively refined/coarsened accord- ing to a criterion regarding the gray-scale measure of local regions. New density points are added into the gray regions and redundant ones are removed from the regions occupied by purely solid/void phases for decreasing the number of de- sign variables. A penalization factor adaptivity technique is employed-to prevent premature convergence of the optimiza- tion iterations. Such an adaptive scheme not only improves the structural boundary description quality, but also allows for sufficient further topological evolution of the structural layout in higher adaptivity levels and thus essentially enables high-resolution solutions. Moreover, compared with the case with uniformly and finely distributed density points, the proposed adaptive method can achieve a higher numerical efficiency of the optimization process.展开更多
The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts high...The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.展开更多
基金supported by the National Key Research and Development Program of China(No.2020YFB1901900)the National Natural Science Foundation of China(Nos.U20B2011,12175138)the Shanghai Rising-Star Program。
文摘The heterogeneous variational nodal method(HVNM)has emerged as a potential approach for solving high-fidelity neutron transport problems.However,achieving accurate results with HVNM in large-scale problems using high-fidelity models has been challenging due to the prohibitive computational costs.This paper presents an efficient parallel algorithm tailored for HVNM based on the Message Passing Interface standard.The algorithm evenly distributes the response matrix sets among processors during the matrix formation process,thus enabling independent construction without communication.Once the formation tasks are completed,a collective operation merges and shares the matrix sets among the processors.For the solution process,the problem domain is decomposed into subdomains assigned to specific processors,and the red-black Gauss-Seidel iteration is employed within each subdomain to solve the response matrix equation.Point-to-point communication is conducted between adjacent subdomains to exchange data along the boundaries.The accuracy and efficiency of the parallel algorithm are verified using the KAIST and JRR-3 test cases.Numerical results obtained with multiple processors agree well with those obtained from Monte Carlo calculations.The parallelization of HVNM results in eigenvalue errors of 31 pcm/-90 pcm and fission rate RMS errors of 1.22%/0.66%,respectively,for the 3D KAIST problem and the 3D JRR-3 problem.In addition,the parallel algorithm significantly reduces computation time,with an efficiency of 68.51% using 36 processors in the KAIST problem and 77.14% using 144 processors in the JRR-3 problem.
文摘SP3 （simplified P3） theory is widely used in LWR （light water reactor） analyses to partly capture the transport effect, especially for pin-by-pin core analysis with pin size homogenization. In this paper, a SP3 code named STELLA is developed and verified at SNERDI （Shanghai Nuclear Engineering Research and Design Institute）. For SP3 method, neutron transport equation can be transformed into two coupled equations in the same mathematical form as diffusion equation. In this work, SANM （semi-analytic nodal method） is used to solve diffusion-like equation, due to its easy to handle multi-group problem. Whole core nodal boundary net current coupling is used to improve convergence stability in SANM, instead of solving two-node problem. CMFD （coarse-mesh finite difference） acceleration method is employed for 0-th SP3 equation, which represents the neutron balance relationship. Three benchmarks are used to verify the SP3 code, STELLA. The first one is a self-defined one dimensional problem, which demonstrates SP3 method is extremely accurate, due to no academic approximation in one dimensional for SP3. The second one is a two dimensional one-group problem cited from Larsen＇s paper, which is usually used to verify and prove the SP3 code correct and accurate. And the third one is modified from 2D C5G7-MOX benchmark, whose numerical results indicate that STELLA is accurate and efficient in pin size level, compared to diffusion model.
文摘This paper describes a numerical solution for two dimensional partial differential equations modeling (or arising from) a fluid flow and transport phenomena. The diffusion equation is discretized by the Nodal methods. The saturation equation is solved by a finite volume method. We start with incompressible single-phase flow and move step-by-step to the black-oil model and compressible two phase flow. Numerical results are presented to see the performance of the method, and seem to be interesting by comparing them with other recent results.
基金This project is supported by the National Science Foundation of China
文摘In this paper, we prove the convergence of the nodal expansion method, a new numerical method for partial differential equations and provide the error estimates of approximation solution.
基金Supported by the National Natural Science Foundation of China(51106099,50976072)the Leading Academic Discipline Project of Shanghai Municipal Education Commission(J50501)
文摘A nodal discontinuous Galerkin formulation based on Lagrange polynomials basis is used to simulate the acoustic wave propagation. Its dispersion and dissipation properties for the advection equation are investigated by utilizing an eigenvalue analysis. Two test problems of wave propagation with initial disturbance consisting of a Gaussian profile or rectangular pulse are performed. And the performance of the schemes in short,intermediate,and long waves is evaluated. Moreover,numerical results between the nodal discontinuous Galerkin method and finite difference type schemes are compared,which indicate that the numerical solution obtained using nodal discontinuous Galerkin method with a pure central flux has obviously high frequency oscillations for initial disturbance consisting of a rectangular pulse,which is the same as those obtained using finite difference type schemes without artificial selective damping. When an upwind flux is adopted,spurious waves are eliminated effectively except for the location of discontinuities. When a limiter is used,the spurious short waves are almost completely removed. Therefore,the quality of the computed solution has improved.
基金Supported by National Natural Science Foundation of China(Grant No.51175422)
文摘There are vast constraint equations in conventional dynamics analysis of deployable structures,which lead to differential-algebraic equations(DAEs)solved hard.To reduce the difficulty of solving and the amount of equations,a new flexible multibody dynamics analysis methodology of deployable structures with scissor-like elements(SLEs)is presented.Firstly,a precise model of a flexible bar of SLE is established by the higher order shear deformable beam element based on the absolute nodal coordinate formulation(ANCF),and the master/slave freedom method is used to obtain the dynamics equations of SLEs without constraint equations.Secondly,according to features of deployable structures,the specification matrix method(SMM)is proposed to eliminate the constraint equations among SLEs in the frame of ANCF.With this method,the inner and the boundary nodal coordinates of element characteristic matrices can be separated simply and efficiently,especially on condition that there are vast nodal coordinates.So the element characteristic matrices can be added end to end circularly.Thus,the dynamic model of deployable structure reduces dimension and can be assembled without any constraint equation.Next,a new iteration procedure for the generalized-a algorithm is presented to solve the ordinary differential equations(ODEs)of deployable structure.Finally,the proposed methodology is used to analyze the flexible multi-body dynamics of a planar linear array deployable structure based on three scissor-like elements.The simulation results show that flexibility has a significant influence on the deployment motion of the deployable structure.The proposed methodology indeed reduce the difficulty of solving and the amount of equations by eliminating redundant degrees of freedom and the constraint equations in scissor-like elements and among scissor-like elements.
基金supported by the Key Project of Chinese National Programs for Fundamental Research and Development(2010CB832703)the National Natural Science Foundation of China(11072047 and 91130025)
文摘For the purpose of achieving high-resolution optimal solutions this paper proposes a nodal design variablebased adaptive method for topology optimization of continuum structures. The analysis mesh-independent density field, interpolated by the nodal design variables at a given set of density points, is adaptively refined/coarsened accord- ing to a criterion regarding the gray-scale measure of local regions. New density points are added into the gray regions and redundant ones are removed from the regions occupied by purely solid/void phases for decreasing the number of de- sign variables. A penalization factor adaptivity technique is employed-to prevent premature convergence of the optimiza- tion iterations. Such an adaptive scheme not only improves the structural boundary description quality, but also allows for sufficient further topological evolution of the structural layout in higher adaptivity levels and thus essentially enables high-resolution solutions. Moreover, compared with the case with uniformly and finely distributed density points, the proposed adaptive method can achieve a higher numerical efficiency of the optimization process.
基金supported by the National Key R&D Program of China (Grant No.2018YFC0407002)the National Natural Science Foundation of China(Grant Nos.11502033 and 51879014)
文摘The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.
