In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we ...In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the "primal- dual" formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the "primal-dual" Reduced Basis approach with respect to an "only primal" one. Parametrized advection-reaction partial differential equations, Reduced Basis method, "primal-dual" reduced basis approach, Stabilized finite element method, a posteriori error estimation.展开更多
In this work,two approaches,based on the certified Reduced Basis method,have been developed for simulating the movement of nuclear reactor control rods,in time-dependent non-coercive settings featuring a 3D geometrica...In this work,two approaches,based on the certified Reduced Basis method,have been developed for simulating the movement of nuclear reactor control rods,in time-dependent non-coercive settings featuring a 3D geometrical framework.In particular,in a first approach,a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod.In the second approach,a“staircase”strategy has been adopted for simulating themovement of all the three rods featured by the nuclear reactor chosen as case study.The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion,which,in the present case,is a set of ten coupled parametrized parabolic equations(two energy groups for the neutron flux,and eight for the precursors).Both the reduced order models,developed according to the two approaches,provided a very good accuracy comparedwith high-fidelity results,assumed as“truth”solutions.At the same time,the computational speed-up in the Online phase,with respect to the fine“truth”finite element discretization,achievable by both the proposed approaches is at least of three orders of magnitude,allowing a real-time simulation of the rod movement and control.展开更多
In this article,an effective technique is developed to efficiently obtain the output responses of parameterized structural dynamic problems.This technique is based on the conception of reduced basis method and the usa...In this article,an effective technique is developed to efficiently obtain the output responses of parameterized structural dynamic problems.This technique is based on the conception of reduced basis method and the usage of linear interpolation principle.The original problem is projected onto the reduced basis space by linear interpolation projection,and subsequently an associated interpolation matrix is generated.To ensure the largest nonsingularity,the interpolation matrix needs to go through a timenode choosing process,which is developed by applying the angle of vector spaces.As a part of this technique,error estimation is recommended for achieving the computational error bound.To ensure the successful performance of this technique,the offline-online computational procedures are conducted in practical engineering.Two numerical examples demonstrate the accuracy and efficiency of the presented method.展开更多
In this paper,we propose a Static Condensation Reduced Basis Element(SCRBE)approach for the Reynolds Lubrication Equation(RLE).The SCRBEmethod is a computational tool that allows to efficiently analyze parametrized st...In this paper,we propose a Static Condensation Reduced Basis Element(SCRBE)approach for the Reynolds Lubrication Equation(RLE).The SCRBEmethod is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components.Here,we extend the methodology to allow for a more general domain decomposition,a typical example being a checkerboard-pattern assembled from similar components.To this end,we extend the formulation and associated a posteriori error bound procedure.Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE.However,the SCRBE approach presented is not limited to bearings and the RLE,but directly extends to other component-based systems.We show numerical results for plain bearings to demonstrate the validity of the proposed approach.展开更多
.A non-intrusive reduced order model(ROM)that combines a proper orthogonal decomposition(POD)and an artificial neural network(ANN)is primarily studied to investigate the applicability of the proposed ROM in recovering....A non-intrusive reduced order model(ROM)that combines a proper orthogonal decomposition(POD)and an artificial neural network(ANN)is primarily studied to investigate the applicability of the proposed ROM in recovering the solutions with shocks and strong gradients accurately and resolving fine-scale structures efficiently for hyperbolic conservation laws.Its accuracy is demonstrated by solving a high-dimensional parametrized ODE and the one-dimensional viscous Burgers’equation with a parameterized diffusion coefficient.The two-dimensional singlemode Rayleigh-Taylor instability(RTI),where the amplitude of the small perturbation and time are considered as free parameters,is also simulated.An adaptive sampling method in time during the linear regime of the RTI is designed to reduce the number of snapshots required for POD and the training of ANN.The extensive numerical results show that the ROM can achieve an acceptable accuracy with improved efficiency in comparison with the standard full order method.展开更多
基金support provided thorough the "Progetto Rocca", MIT-Politecnico di Milano collaboration
文摘In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the "primal- dual" formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the "primal-dual" Reduced Basis approach with respect to an "only primal" one. Parametrized advection-reaction partial differential equations, Reduced Basis method, "primal-dual" reduced basis approach, Stabilized finite element method, a posteriori error estimation.
基金We acknowledge CINECA and Regione Lombardia LISA computational initiative,for the availability of high performance computing resources and support.G.Rozza acknowledges INDAM-GNCS national activity group and NOFYSAS program of SISSA.
文摘In this work,two approaches,based on the certified Reduced Basis method,have been developed for simulating the movement of nuclear reactor control rods,in time-dependent non-coercive settings featuring a 3D geometrical framework.In particular,in a first approach,a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod.In the second approach,a“staircase”strategy has been adopted for simulating themovement of all the three rods featured by the nuclear reactor chosen as case study.The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion,which,in the present case,is a set of ten coupled parametrized parabolic equations(two energy groups for the neutron flux,and eight for the precursors).Both the reduced order models,developed according to the two approaches,provided a very good accuracy comparedwith high-fidelity results,assumed as“truth”solutions.At the same time,the computational speed-up in the Online phase,with respect to the fine“truth”finite element discretization,achievable by both the proposed approaches is at least of three orders of magnitude,allowing a real-time simulation of the rod movement and control.
基金supported by the National Natural Science Foundation of China (10802028)the Major State Basic Research Development Program of China (2010CB832705)the National Science Fund for Distinguished Young Scholars (10725208)
文摘In this article,an effective technique is developed to efficiently obtain the output responses of parameterized structural dynamic problems.This technique is based on the conception of reduced basis method and the usage of linear interpolation principle.The original problem is projected onto the reduced basis space by linear interpolation projection,and subsequently an associated interpolation matrix is generated.To ensure the largest nonsingularity,the interpolation matrix needs to go through a timenode choosing process,which is developed by applying the angle of vector spaces.As a part of this technique,error estimation is recommended for achieving the computational error bound.To ensure the successful performance of this technique,the offline-online computational procedures are conducted in practical engineering.Two numerical examples demonstrate the accuracy and efficiency of the presented method.
基金We would like to thank Prof.A.T.Patera and Dr.J.Eftang for helpful discussions on the SCRBE method as well as Prof.G.Knoll and Dr.R.Schönen from ISTmbH for providing the specific application.This work was supported by the Excellence Initiative of the German federal and state governments and the German Research Foundation through Grant GSC 111.
文摘In this paper,we propose a Static Condensation Reduced Basis Element(SCRBE)approach for the Reynolds Lubrication Equation(RLE).The SCRBEmethod is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components.Here,we extend the methodology to allow for a more general domain decomposition,a typical example being a checkerboard-pattern assembled from similar components.To this end,we extend the formulation and associated a posteriori error bound procedure.Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE.However,the SCRBE approach presented is not limited to bearings and the RLE,but directly extends to other component-based systems.We show numerical results for plain bearings to demonstrate the validity of the proposed approach.
基金funding support of this research by the National Natural Science Foundation of China(11871443)Shandong Provincial Qingchuang Science and Technology Project(2019KJI002)the Ocean University of China for providing the startup funding(201712011)that is used in supporting this work.
文摘.A non-intrusive reduced order model(ROM)that combines a proper orthogonal decomposition(POD)and an artificial neural network(ANN)is primarily studied to investigate the applicability of the proposed ROM in recovering the solutions with shocks and strong gradients accurately and resolving fine-scale structures efficiently for hyperbolic conservation laws.Its accuracy is demonstrated by solving a high-dimensional parametrized ODE and the one-dimensional viscous Burgers’equation with a parameterized diffusion coefficient.The two-dimensional singlemode Rayleigh-Taylor instability(RTI),where the amplitude of the small perturbation and time are considered as free parameters,is also simulated.An adaptive sampling method in time during the linear regime of the RTI is designed to reduce the number of snapshots required for POD and the training of ANN.The extensive numerical results show that the ROM can achieve an acceptable accuracy with improved efficiency in comparison with the standard full order method.