Applying the standard Galerkin finite element method for solving flow problems in porous media encounters some difficulties such as numerical oscillation at the shock front and discontinuity of the velocity field on e...Applying the standard Galerkin finite element method for solving flow problems in porous media encounters some difficulties such as numerical oscillation at the shock front and discontinuity of the velocity field on element faces.Discontinuity of velocity field leads this method not to conserve mass locally.Moreover,the accuracy and stability of a solution is highly affected by a non-conservative method.In this paper,a three dimensional control volume finite element method is developed for twophase fluid flow simulation which overcomes the deficiency of the standard finite element method,and attains high-orders of accuracy at a reasonable computational cost.Moreover,this method is capable of handling heterogeneity in a very rational way.A fully implicit scheme is applied to temporal discretization of the governing equations to achieve an unconditionally stable solution.The accuracy and efficiency of the method are verified by simulating some waterflooding experiments.Some representative examples are presented to illustrate the capability of the method to simulate two-phase fluid flow in heterogeneous porous media.展开更多
Deep target hydrocarbon detection is still challenging and expensive. Direct hydrocarbon indicators (DHIs) in seismic data do not correspond to economical hydrocarbon exploration. Due to unreliability in seismic data ...Deep target hydrocarbon detection is still challenging and expensive. Direct hydrocarbon indicators (DHIs) in seismic data do not correspond to economical hydrocarbon exploration. Due to unreliability in seismic data for the detection of DHIs, new methods have been investigated. Marine controlled source electromagnet (MCSEM) or Sea bed logging (SBL) is new method for the detection of deep target hydrocarbon reservoir. Sea bed logging has also the potential to reduce the risks of DHIs in deep sea environment. Modelling of real sea environment helps to reduce the further risks before drilling the oil wells. 3D electromagnetic (EM) modelling of seabed logging requires more accurate methods for the detection of hydrocarbon reservoir. Finite element method (FEM) is chosen for the modelling of seabed logging to get more precise EM response from hydrocarbon reservoir below 4000 m from seabed. FEM allows to investigate the total electric and magnetic fields instead of scattered electric and magnetic fields, which shows accurate and precise resistivity contrast below the seabed. From the modelling results, It was investigated that Hz field shows higher magni- tude with 342% than the Ex field. It was observed that 0.125 Hz frequency can be able to show better resistivity contrast of Hz field (31.30%) and Ex field (16.49%) at target depth of 1000 m below seafloor for our proposed model. Hz and Ex field delineation was found to decrease as target depth increased from 1000 m to 4000 m. At the target depth of 4000 m, no field delineation response was seen from the current electromagnetic (EM) antenna used by the industry. New EM antenna has been used to see the EM response for deep target hydrocarbon detection. It was investigated that novel EM antenna shows better delineation at 4000 m target depth for Ex and Hz field up to 10.3% and 15.1% respectively. Novel EM antenna also shows better Hz phase response (128.4%) than the Ex phase response (38.3%) at the target depth of 4000 m below the seafloor.展开更多
In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existenc...In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.展开更多
In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of thi...In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.展开更多
This paper presents the optimal control variational principle for Perzyna modelwhich is one of the main constitutive relation of viscoplasticity in dynamics. And itcould also be transformed to solve the parametric qua...This paper presents the optimal control variational principle for Perzyna modelwhich is one of the main constitutive relation of viscoplasticity in dynamics. And itcould also be transformed to solve the parametric quadratic programming problem.The FEM form of this problem and its implementation have also been discussed in thepaper.展开更多
A physically accurate and computationally effective pure finite element method (FEM) was developed to simulate the isothermal resin infusing process. The FEM was based on conservation of resin muss at and instant of...A physically accurate and computationally effective pure finite element method (FEM) was developed to simulate the isothermal resin infusing process. The FEM was based on conservation of resin muss at and instant of time and was objective of resin film infusion (RFI) fiber impregnation and mold filling . The developed computer code was able to simulate the resin infusing visually. A numerical example presented here demonstrated that compared with traditional finite element/ control-volume (FE/CV), and FEM was physically accurate and computationally efficient.展开更多
In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and ...In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.展开更多
In this paper, we present the numerical solution for the optimal control problem of monodomain modelwith Rogers-modified FitzHugh-Nagumo ion kinetic. The monodomain model, which is a well-known mathematical model for ...In this paper, we present the numerical solution for the optimal control problem of monodomain modelwith Rogers-modified FitzHugh-Nagumo ion kinetic. The monodomain model, which is a well-known mathematical model for simulation of cardiac electrical activity, appears as the constraint in our problem. Our control objective is to dampen the excitation wavefront of the transmembrane potential in the observation domain using optimal applied current. Various conjugate gradient methods have been applied by researchers for solving this type of optimal control problem. For the present paper, we adopt the modified Fletcher-Reeves method and modified Dai-Yuan methodfor computing the optimal applied current. Numerical results show that the excitation wavefront is successfully dampened out by the optimal applied current when the modified Fletcher-Reeves method is used. However, this is not the case when the modified Dai-Yuan method is employed. Numerical results indicate that the modified Dai-Yuan method failed to converge to the optimal solution when the Armijo line search is used.展开更多
Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal co...Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal control problems which contain a linear elliptic boundary value problem as a state equation, control in the righthand side of the equation or in the boundary conditions, and point-wise constraints for both state and control functions. The convergence of the constructed iterative methods is proved, the implementation problems are discussed, and the numerical comparison of the methods is executed.展开更多
In this paper, we provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for a quasi-variational inequalities related to ergodic control problems studied by M. Boulbrachene [1], where t...In this paper, we provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for a quasi-variational inequalities related to ergodic control problems studied by M. Boulbrachene [1], where the “discount factor” (i.e., the zero order term) is set to 0, we use an overlapping Schwarz method on nonmatching grid which consists in decomposing the domain in two sub domains, where the discrete alternating Schwarz sequences in sub domains converge to the solution of the ergodic control IQV for the zero order term. For and under a discrete maximum principle we show that the discretization on each sub domain converges quasi-optimally in the norm to 0.展开更多
We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,ex...We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.展开更多
This paper aims to present a high accuracy approximation and superconvergence for the distributed convex optimal control problem. In the basis of integral identity technique, we discuss the superconvergence of the rec...This paper aims to present a high accuracy approximation and superconvergence for the distributed convex optimal control problem. In the basis of integral identity technique, we discuss the superconvergence of the rectangular finite element and the uniform triangular finite element for the optimal control problem. Using interpolation postprocessing technique, we construct a high accuracy finite element approximation scheme. The numerical examples demonstrating these results airs also presented.展开更多
In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element method...In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods.The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k≥0).A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained.Finally,we present some numerical examples which confirm our theoretical results.展开更多
Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolat...Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.展开更多
The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approx...The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.展开更多
This paper analyzes two extended finite element methods(XFEMs)for linear quadratic optimal control problems governed by Poisson equation in non-convex domains.We follow the variational discretization concept to discre...This paper analyzes two extended finite element methods(XFEMs)for linear quadratic optimal control problems governed by Poisson equation in non-convex domains.We follow the variational discretization concept to discretize the continuous problems,and apply an XFEM with a cut-off function and a classic XFEM with a fixed enrichment area to discretize the state and co-state equations.Optimal error estimates are derived for the state,co-state and control.Numerical results confirm our theoretical results.展开更多
In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables a...In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant functions.The time discretization of the state and co-state are based on finite difference methods.First,we derive a priori error estimates for the control variable,the state variables and the adjoint state variables.Second,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.展开更多
基金Iranian Offshore Oil Company (IOOC) for financial support of this work
文摘Applying the standard Galerkin finite element method for solving flow problems in porous media encounters some difficulties such as numerical oscillation at the shock front and discontinuity of the velocity field on element faces.Discontinuity of velocity field leads this method not to conserve mass locally.Moreover,the accuracy and stability of a solution is highly affected by a non-conservative method.In this paper,a three dimensional control volume finite element method is developed for twophase fluid flow simulation which overcomes the deficiency of the standard finite element method,and attains high-orders of accuracy at a reasonable computational cost.Moreover,this method is capable of handling heterogeneity in a very rational way.A fully implicit scheme is applied to temporal discretization of the governing equations to achieve an unconditionally stable solution.The accuracy and efficiency of the method are verified by simulating some waterflooding experiments.Some representative examples are presented to illustrate the capability of the method to simulate two-phase fluid flow in heterogeneous porous media.
文摘Deep target hydrocarbon detection is still challenging and expensive. Direct hydrocarbon indicators (DHIs) in seismic data do not correspond to economical hydrocarbon exploration. Due to unreliability in seismic data for the detection of DHIs, new methods have been investigated. Marine controlled source electromagnet (MCSEM) or Sea bed logging (SBL) is new method for the detection of deep target hydrocarbon reservoir. Sea bed logging has also the potential to reduce the risks of DHIs in deep sea environment. Modelling of real sea environment helps to reduce the further risks before drilling the oil wells. 3D electromagnetic (EM) modelling of seabed logging requires more accurate methods for the detection of hydrocarbon reservoir. Finite element method (FEM) is chosen for the modelling of seabed logging to get more precise EM response from hydrocarbon reservoir below 4000 m from seabed. FEM allows to investigate the total electric and magnetic fields instead of scattered electric and magnetic fields, which shows accurate and precise resistivity contrast below the seabed. From the modelling results, It was investigated that Hz field shows higher magni- tude with 342% than the Ex field. It was observed that 0.125 Hz frequency can be able to show better resistivity contrast of Hz field (31.30%) and Ex field (16.49%) at target depth of 1000 m below seafloor for our proposed model. Hz and Ex field delineation was found to decrease as target depth increased from 1000 m to 4000 m. At the target depth of 4000 m, no field delineation response was seen from the current electromagnetic (EM) antenna used by the industry. New EM antenna has been used to see the EM response for deep target hydrocarbon detection. It was investigated that novel EM antenna shows better delineation at 4000 m target depth for Ex and Hz field up to 10.3% and 15.1% respectively. Novel EM antenna also shows better Hz phase response (128.4%) than the Ex phase response (38.3%) at the target depth of 4000 m below the seafloor.
基金supported by the National Basic Research Program under the Grant 2005CB321701the National Natural Science Foundation of China under the Grants 60474027 and 10771211.
文摘In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.
基金supported by the National Natural Science Foundation of China(Nos.11701253,11971259,11801216)Natural Science Foundation of Shandong Province(No.ZR2017BA010)。
文摘In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.
文摘This paper presents the optimal control variational principle for Perzyna modelwhich is one of the main constitutive relation of viscoplasticity in dynamics. And itcould also be transformed to solve the parametric quadratic programming problem.The FEM form of this problem and its implementation have also been discussed in thepaper.
基金Funded bythe National Natural Science Foundation of China(No50573060)
文摘A physically accurate and computationally effective pure finite element method (FEM) was developed to simulate the isothermal resin infusing process. The FEM was based on conservation of resin muss at and instant of time and was objective of resin film infusion (RFI) fiber impregnation and mold filling . The developed computer code was able to simulate the resin infusing visually. A numerical example presented here demonstrated that compared with traditional finite element/ control-volume (FE/CV), and FEM was physically accurate and computationally efficient.
文摘In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.
文摘In this paper, we present the numerical solution for the optimal control problem of monodomain modelwith Rogers-modified FitzHugh-Nagumo ion kinetic. The monodomain model, which is a well-known mathematical model for simulation of cardiac electrical activity, appears as the constraint in our problem. Our control objective is to dampen the excitation wavefront of the transmembrane potential in the observation domain using optimal applied current. Various conjugate gradient methods have been applied by researchers for solving this type of optimal control problem. For the present paper, we adopt the modified Fletcher-Reeves method and modified Dai-Yuan methodfor computing the optimal applied current. Numerical results show that the excitation wavefront is successfully dampened out by the optimal applied current when the modified Fletcher-Reeves method is used. However, this is not the case when the modified Dai-Yuan method is employed. Numerical results indicate that the modified Dai-Yuan method failed to converge to the optimal solution when the Armijo line search is used.
文摘Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal control problems which contain a linear elliptic boundary value problem as a state equation, control in the righthand side of the equation or in the boundary conditions, and point-wise constraints for both state and control functions. The convergence of the constructed iterative methods is proved, the implementation problems are discussed, and the numerical comparison of the methods is executed.
文摘In this paper, we provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for a quasi-variational inequalities related to ergodic control problems studied by M. Boulbrachene [1], where the “discount factor” (i.e., the zero order term) is set to 0, we use an overlapping Schwarz method on nonmatching grid which consists in decomposing the domain in two sub domains, where the discrete alternating Schwarz sequences in sub domains converge to the solution of the ergodic control IQV for the zero order term. For and under a discrete maximum principle we show that the discretization on each sub domain converges quasi-optimally in the norm to 0.
基金supported by the National Natural Science Foundation of China(11971276,12171287)Natural Science Foundation of Shandong Province(ZR2016JL004)+1 种基金supported by the China Postdoctoral Science Foundation(2021TQ0017,2021M700244)International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(YJ20210019)。
文摘We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.
基金the National Natural Science Foundation of China.
文摘This paper aims to present a high accuracy approximation and superconvergence for the distributed convex optimal control problem. In the basis of integral identity technique, we discuss the superconvergence of the rectangular finite element and the uniform triangular finite element for the optimal control problem. Using interpolation postprocessing technique, we construct a high accuracy finite element approximation scheme. The numerical examples demonstrating these results airs also presented.
基金supported by the Foundation for Talent Introduction of Guangdong Provincial Universities and CollegesPearl River Scholar Funded Scheme(2008)National Science Foundation of China(10971074).
文摘In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods.The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k≥0).A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained.Finally,we present some numerical examples which confirm our theoretical results.
基金supported in part by the National Basic Research Program (2007CB814906)the National Natural Science Foundation of China (10471103 and 10771158)+4 种基金Social Science Foundation of the Ministry of Education of China (06JA630047)Tianjin Natural Science Foundation (07JCYBJC14300)Tianjin University of Finance and Economicssupported by the National Basic Research Program under the Grant 2005CB321701the National Natural Science Foundation of China under the Grant 10771211
文摘Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.
文摘The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.
基金supported by National Natural Science Foundation of China(Grant No.11771312)。
文摘This paper analyzes two extended finite element methods(XFEMs)for linear quadratic optimal control problems governed by Poisson equation in non-convex domains.We follow the variational discretization concept to discretize the continuous problems,and apply an XFEM with a cut-off function and a classic XFEM with a fixed enrichment area to discretize the state and co-state equations.Optimal error estimates are derived for the state,co-state and control.Numerical results confirm our theoretical results.
基金This work was supported by National Natural Science Foundation of China(11601014,11626037,11526036)China Postdoctoral Science Foundation(2016M 601359)+4 种基金Scientific and Technological Developing Scheme of Jilin Province(20160520108 JH,20170101037JC)Science and Technology Research Project of Jilin Provincial Depart-ment of Education(201646)Special Funding for Promotion of Young Teachers of Beihua University,Natural Science Foundation of Hunan Province(14JJ3135)the Youth Project of Hunan Provincial Education Department(15B096)the construct program of the key discipline in Hunan University of Science and Engineering.
文摘In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant functions.The time discretization of the state and co-state are based on finite difference methods.First,we derive a priori error estimates for the control variable,the state variables and the adjoint state variables.Second,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.