XI International Conference on Adaptive Modeling and Simulation
We consider a hybrid approach for the approximation of the solution to parametric partial differe... more We consider a hybrid approach for the approximation of the solution to parametric partial differential equations based on finite elements and deep neural networks. Finite element simulations with adaptive mesh refinement are used to generate input data for the training of a neural network. A deep feedforward neural network is then used to approximate the solution of the partial differential equation. We aim at balancing the numerical errors introduced by the finite element method and the neural network approximation respectively. Numerical results are presented for the transport equation.
process. We present a new model which allows to investigate the influence of convection, both for... more process. We present a new model which allows to investigate the influence of convection, both forced and natural in the evolution of physical system. We use a diffusive interface model, where the sharp interface is replaced by a smooth phase-field function. We perform 2D simulations using a finite element method with mesh adaptation with high aspect ratio based on error estimator for the concentration equation (validity of this approach has been studied and proven for the simpler cases). We focus on the liquid flow induced by solidification shrinkage ( l < s) and show that this phenomena gives quantitatively different results and cannot be neglected. Numerical results predict a pressure drop which can give rise to the microporosity.
Order one finite elements together with GLS (Galerkin Least Squares) methods are used for solving... more Order one finite elements together with GLS (Galerkin Least Squares) methods are used for solving a three fields Stokes problem arising from viscoelastic flows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity-pressure and extra-stress calculations is proposed. The link with the modified EVSS (Elastic-Viscous-SplitExtra -Stress) method studied in [6, 9] is presented. Numerical results are in agreement with theoretical predictions, and with those presented in [6]. Extensions to viscoelastic flows are discussed. 1 Introduction Numerical simulation of viscoelastic flows is of interest for many industrial applications. Modelling of viscoelastic flows usually consists in supplementing the mass and momentum equations with a di#erential constitutive equation involving the extra-stress and the velocity field. When considering this set of partial di#erent...
Numerical Mathematics and Advanced Applications 2009, 2010
A posteriori error estimates are presented for the Laplace equation and meshes with large aspect ... more A posteriori error estimates are presented for the Laplace equation and meshes with large aspect ratio. Error estimates are presented in the natural H 1 seminorm or in the framework of goal oriented error control. The proposed estimator relies on anisotropic interpolation estimates derived by Formaggia and Perotto [19, and on Zienckiewicz-Zhu post-processing techniques, thus avoiding the use of numerical techniques in order to compute approximations of the Hessian of the solution. All the constant involved in the error estimates are independent of the mesh size and aspect ratio, which enables the use of anisotropic, adaptive finite elements.
A stabilized semidiscrete finite element discretization of the transient transport equation is st... more A stabilized semidiscrete finite element discretization of the transient transport equation is studied in the framework of anisotropic meshes. A priori and a posteriori error estimates are derived, the involved constants being independent of the mesh aspect ratio, only space discretization being considered. Numerical results on nonadapted, anisotropic meshes and small time steps confirm the sharpness of the theoretical predictions. An anisotropic, adaptive finite element algorithm is then proposed with the goal to control the L 2 error in space at final time, the time step being kept constant. Numerical results are then presented on anisotropic, adapted meshes and small time steps. Three different methods are proposed to interpolate the solution between two adapted meshes. Numerical results indicate that both the conservative interpolation method of [F. Alauzet and M. Mehrenberger, Internat.
Modelling and numerical simulation of the dynamics of glaciers process of a 2D glacier tongue sub... more Modelling and numerical simulation of the dynamics of glaciers process of a 2D glacier tongue submerged by water.
A numerical model is presented for the simulation of complex fluid flows with free surfaces. The ... more A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain.
International Journal for Numerical Methods in Fluids, 2009
This paper describes the use of an a posteriori error estimator to control anisotropic mesh adapt... more This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single-parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size h min is enforced across solution discontinuities. This h min condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and h min . We verify numerically that the effect of varying h min is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of h min with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods.
Computer Methods in Applied Mechanics and Engineering, 1995
An efficient adaptive algorithm is presented for a stationary regularized Stefan problem in 2D. T... more An efficient adaptive algorithm is presented for a stationary regularized Stefan problem in 2D. The adaptive criteria relies upon a posteriori estimates based on the residual equation. Since the problem we are studying is a non-linear diffusion-convection problem, these error estimates are relatively crude in the sense that the constant relating the true error to the error indicator is unknown,
We consider Chorin-Temam scheme (the simplest pressure-correction projection method) for the time... more We consider Chorin-Temam scheme (the simplest pressure-correction projection method) for the time-discretization of an unstationary Stokes problem in D ⊂ R d (d = 2, 3) given µ, f : (P) find (u, p) solution to u| t=0 = u 0 , u| ∂D = 0 and Inspired by the analyses of the Backward Euler scheme performed by C.Bernardi and R.Verfürth, we derive a posteriori estimators for the error on ∇u in L 2 (0, T ; L 2 (D))-norm. Our invesigation is supported by numerical experiments. French version: On discrétise en temps par le schéma Chorin-Temam un problème de Stokes non-stationnaire posé dans D ⊂ R d (d = 2, 3) étant donnés µ, f : (P) trouver (u, p) solution de u| t=0 = u 0 , u| ∂D = 0 et (1). En s'insipirant des analyses de C.Bernardi and R.Verfürth pour le schéma Euler rétrograde, nous construisons des estimateurs a posteriori pour l'erreur commise sur ∇u en norme L 2 (0, T ; L 2 (D)). Notre étude est étayée par des expériences numériques.
In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in ... more In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in an aluminum production cell as well as its numerical approximation by a finite element method. We put the emphasis on the magnetic effects which live in the whole three dimensional space and which are solved numerically with a domain decomposition method.
... Schotzau, James Warsa 687 Modeling of Multi-Phase Flows with a Level-Set Method Sander P. van... more ... Schotzau, James Warsa 687 Modeling of Multi-Phase Flows with a Level-Set Method Sander P. van der Pijl, A. Segal, C. Vuik 698 Numerical Modeling of Bypass Flow Vladimir Prokop, Karel Kozel 708 A Posteriori Estimation of Dimension Reduction Errors Sergey Repin, Stefan ...
XI International Conference on Adaptive Modeling and Simulation
We consider a hybrid approach for the approximation of the solution to parametric partial differe... more We consider a hybrid approach for the approximation of the solution to parametric partial differential equations based on finite elements and deep neural networks. Finite element simulations with adaptive mesh refinement are used to generate input data for the training of a neural network. A deep feedforward neural network is then used to approximate the solution of the partial differential equation. We aim at balancing the numerical errors introduced by the finite element method and the neural network approximation respectively. Numerical results are presented for the transport equation.
process. We present a new model which allows to investigate the influence of convection, both for... more process. We present a new model which allows to investigate the influence of convection, both forced and natural in the evolution of physical system. We use a diffusive interface model, where the sharp interface is replaced by a smooth phase-field function. We perform 2D simulations using a finite element method with mesh adaptation with high aspect ratio based on error estimator for the concentration equation (validity of this approach has been studied and proven for the simpler cases). We focus on the liquid flow induced by solidification shrinkage ( l < s) and show that this phenomena gives quantitatively different results and cannot be neglected. Numerical results predict a pressure drop which can give rise to the microporosity.
Order one finite elements together with GLS (Galerkin Least Squares) methods are used for solving... more Order one finite elements together with GLS (Galerkin Least Squares) methods are used for solving a three fields Stokes problem arising from viscoelastic flows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity-pressure and extra-stress calculations is proposed. The link with the modified EVSS (Elastic-Viscous-SplitExtra -Stress) method studied in [6, 9] is presented. Numerical results are in agreement with theoretical predictions, and with those presented in [6]. Extensions to viscoelastic flows are discussed. 1 Introduction Numerical simulation of viscoelastic flows is of interest for many industrial applications. Modelling of viscoelastic flows usually consists in supplementing the mass and momentum equations with a di#erential constitutive equation involving the extra-stress and the velocity field. When considering this set of partial di#erent...
Numerical Mathematics and Advanced Applications 2009, 2010
A posteriori error estimates are presented for the Laplace equation and meshes with large aspect ... more A posteriori error estimates are presented for the Laplace equation and meshes with large aspect ratio. Error estimates are presented in the natural H 1 seminorm or in the framework of goal oriented error control. The proposed estimator relies on anisotropic interpolation estimates derived by Formaggia and Perotto [19, and on Zienckiewicz-Zhu post-processing techniques, thus avoiding the use of numerical techniques in order to compute approximations of the Hessian of the solution. All the constant involved in the error estimates are independent of the mesh size and aspect ratio, which enables the use of anisotropic, adaptive finite elements.
A stabilized semidiscrete finite element discretization of the transient transport equation is st... more A stabilized semidiscrete finite element discretization of the transient transport equation is studied in the framework of anisotropic meshes. A priori and a posteriori error estimates are derived, the involved constants being independent of the mesh aspect ratio, only space discretization being considered. Numerical results on nonadapted, anisotropic meshes and small time steps confirm the sharpness of the theoretical predictions. An anisotropic, adaptive finite element algorithm is then proposed with the goal to control the L 2 error in space at final time, the time step being kept constant. Numerical results are then presented on anisotropic, adapted meshes and small time steps. Three different methods are proposed to interpolate the solution between two adapted meshes. Numerical results indicate that both the conservative interpolation method of [F. Alauzet and M. Mehrenberger, Internat.
Modelling and numerical simulation of the dynamics of glaciers process of a 2D glacier tongue sub... more Modelling and numerical simulation of the dynamics of glaciers process of a 2D glacier tongue submerged by water.
A numerical model is presented for the simulation of complex fluid flows with free surfaces. The ... more A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain.
International Journal for Numerical Methods in Fluids, 2009
This paper describes the use of an a posteriori error estimator to control anisotropic mesh adapt... more This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single-parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size h min is enforced across solution discontinuities. This h min condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and h min . We verify numerically that the effect of varying h min is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of h min with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods.
Computer Methods in Applied Mechanics and Engineering, 1995
An efficient adaptive algorithm is presented for a stationary regularized Stefan problem in 2D. T... more An efficient adaptive algorithm is presented for a stationary regularized Stefan problem in 2D. The adaptive criteria relies upon a posteriori estimates based on the residual equation. Since the problem we are studying is a non-linear diffusion-convection problem, these error estimates are relatively crude in the sense that the constant relating the true error to the error indicator is unknown,
We consider Chorin-Temam scheme (the simplest pressure-correction projection method) for the time... more We consider Chorin-Temam scheme (the simplest pressure-correction projection method) for the time-discretization of an unstationary Stokes problem in D ⊂ R d (d = 2, 3) given µ, f : (P) find (u, p) solution to u| t=0 = u 0 , u| ∂D = 0 and Inspired by the analyses of the Backward Euler scheme performed by C.Bernardi and R.Verfürth, we derive a posteriori estimators for the error on ∇u in L 2 (0, T ; L 2 (D))-norm. Our invesigation is supported by numerical experiments. French version: On discrétise en temps par le schéma Chorin-Temam un problème de Stokes non-stationnaire posé dans D ⊂ R d (d = 2, 3) étant donnés µ, f : (P) trouver (u, p) solution de u| t=0 = u 0 , u| ∂D = 0 et (1). En s'insipirant des analyses de C.Bernardi and R.Verfürth pour le schéma Euler rétrograde, nous construisons des estimateurs a posteriori pour l'erreur commise sur ∇u en norme L 2 (0, T ; L 2 (D)). Notre étude est étayée par des expériences numériques.
In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in ... more In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in an aluminum production cell as well as its numerical approximation by a finite element method. We put the emphasis on the magnetic effects which live in the whole three dimensional space and which are solved numerically with a domain decomposition method.
... Schotzau, James Warsa 687 Modeling of Multi-Phase Flows with a Level-Set Method Sander P. van... more ... Schotzau, James Warsa 687 Modeling of Multi-Phase Flows with a Level-Set Method Sander P. van der Pijl, A. Segal, C. Vuik 698 Numerical Modeling of Bypass Flow Vladimir Prokop, Karel Kozel 708 A Posteriori Estimation of Dimension Reduction Errors Sergey Repin, Stefan ...
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