دانلود رایگان مقاله کاهش مدل غیر خطی جهانی - محلی برای جریان در محیط متخلخل ناهمگن

Abstract

In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on a fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach significantly reduces the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

5. Conclusions

In this work, we present a global–local nonlinear model reduction approach to reduce the computational cost for solving high-contrast nonlinear parabolic PDEs. This is achieved through two main stages; offline and online. In the offline step, we use the generalized multiscale finite element method (GMsFEM) to represent the coarse-grid solutions through applying the local discrete empirical interpolation method (DEIM) to approximate the nonlinear functions that arise in the residual and Jacobian. Using the snapshots of the coarse-grid solutions, we compute the proper orthogonal decomposition (POD) modes. In the online step, we project the governing equation on the space spanned by the POD modes and use the global DEIM to approximate the nonlinear functions. Although one can perform global model reduction independently of GMsFEM, the computations of the global modes can be very expensive. Combining both local and global mode reduction methods along with applying DEIM to inexpensively compute the nonlinear function can allow a substantial speed-up. We demonstrate the effectiveness of the proposed global–local nonlinear model reduction method on several examples of nonlinear multiscale PDEs that are solved using a fully-implicit time marching schemes. The results show the great potential of the proposed approach to reproduce the flow field with good accuracy while reducing significantly the size of the original problem. Increasing the number of the local and global modes to improve the accuracy of the approximate solution is examined. Furthermore, the robustness of proposed model reduction approach with respect to variations in initial conditions, permeability fields, nonlinear-function’s parameters, and forcing terms is demonstrated.