An adaptive finite element heterogeneous multiscale method for Stokes flow in porous media

A finite element heterogeneous multiscale method is proposed for solving the Stokes problem in porous media. The method is based on the coupling of an effective Darcy equation on a macroscopic mesh with unknown permeabilities recovered from micro finite element calculations for Stokes problems on sampling domains centered at quadrature points in each macro element. The numerical method accounts for nonperiodic microscopic geometry that can be obtained from a smooth deformation of a reference pore sampling domain. The computational work is nevertheless independent of the small size of the pore structure. A priori error estimates reveal that the overall accuracy of the numerical scheme is limited by the regularity of the solutions of the Stokes microproblems. This regularity is low for a typical situation of nonconvex microscopic pore geometries. We therefore propose an adaptive scheme with micro- macro mesh refinement driven by residual-based indicators that quantify both the macro- and microerrors. A posteriori error analysis is derived for the new method. Two- and three-dimensional numerical experiments confirm the robustness and the accuracy of the adaptive method

Reduced basis heterogeneous multiscale methods

Numerical methods for partial differential equations with multiple scales that combine numerical homogenization methods with reduced order modeling techniques are discussed. These numerical methods can be applied to a variety of problems including multiscale nonlinear elliptic and parabolic problems or Stokes flow in heterogenenous media

Free form deformation techniques applied to 3D shape optimization problems

The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation

MATHICSE Technical Report : Analytic regularity and collocation approximation for PDEs with random domain deformations

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped on to a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in CN with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and standard deviation of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments

A discontinuous Galerkin reduced basis numerical homogenization method for fluid flow in porous media

We present a new conservative multiscale method for Stokes flow in heterogeneous porous media. The method couples a discontinuous Galerkin finite element method (DG-FEM) at the macroscopic scale for the solution of an effective Darcy equation with a Stokes solver at the pore scale to recover effective permeabilities at macroscopic quadrature points. To avoid costly computation of numerous Stokes problems throughout the macroscopic computational domain, the pore geometry is parametrized and a model order reduction algorithm is used to select representative microscopic geometries. Accurate Stokes solutions and related permeabilities are obtained for these representative geometries in an offline stage. In an online stage, the DG-FEM is computed with permeabilities recovered at the desired macroscopic quadrature points from the precomputed Stokes solutions. The multiscale method is shown to be mass conservative at the macro scale and the computational cost for the online stage is similar to the cost of solving a single scale Darcy problem. Numerical experiments for two and three dimensional problems illustrate the efficiency and the performance of the proposed method

A reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media

A reduced basis Darcy-Stokes finite element heterogeneous multiscale method (RB-DS-FE-HMM) is proposed for the Stokes problem in porous media. The multiscale method is based on the Darcy-Stokes finite element heterogeneous multiscale method (DS-FE-HMM) introduced in Abdulle and Budac (2015) that couples a Darcy equation solved on a macroscopic mesh, with missing permeability data extracted from the solutions of Stokes micro problems at each macroscopic quadrature point. To overcome the increasingly growing cost of repeatedly solving the Stokes micro problems as the macroscopic mesh is refined, we parametrize the microscopic solid geometry and approximate the infinite-dimensional manifold of parameter dependent solutions of Stokes problems by a low-dimensional space. This low-dimensional (reduced basis) space is obtained in an offline stage by a greedy algorithm and used in an online stage to compute the effective Darcy permeability at a cost independent of the microscopic mesh. The discretization of the parametrized Stokes problems relies on a Petrov-Galerkin formulation that allows for a stable and fast online evaluation of the required permeabilities. A priori and a posteriori estimates of the RB-DS-FE-HMM are derived and a residual-based adaptive algorithm is proposed. Two- and three-dimensional numerical experiments confirm the accuracy of the RB-DS-FE-HMM and illustrate the speedup compared to the DS-FE-HMM. (C) 2016 Elsevier B.V. All rights reserved

Mathematical Modeling of Lithium-ion Batteries and Improving Mathematics Learning Experience for Engineering Students

Increase in the world’s energy consumption along with the environmental impacts of conventional sources of energy (gas, petroleum, and coal) makes the shift to clean energy sources unavoidable. To address the energy needs of the world, using clean energy sources would not provide the sufficient answer to the world’s energy issues if it is not accompanied by developing energy storage systems that are capable of storing energy efficiently. Lithium-ion batteries are the main energy storage devices that are developed to satisfy the ever-growing energy needs of the modern world. However, there are still important features of Li-ion battery systems (such as the battery microstructural effects) that need to be studied to a broader extent. In this regard, some of the battery microstructural phenomena, such as the formation of solid electrolyte interface, is believed to be the main reason behind battery degradation and drop in performance. Previous studies have focused on the experimental and computational investigation of micro- and macro- structural features of the Li-ion battery; however, further study is needed to focus on incorporating the effects of microscale features of the Li-ion batteries into the total response of the battery system. In the present work, the details of developing a multiscale mathematical model for a Li-ion battery system is explained, and a multiscale model for the battery system is developed by employing variational multiscale modeling method. The developed model is capable of considering the effects of the battery microstructural features (e.g., the random shape of the active material particles) on the total battery performance. In the developed multiscale framework, the microstructural effects are accounted for in the governing equations of the battery macroscale with the help of Green’s function and variational formulation. This part of the present work provides a clear framework for understanding the details and process of developing a multiscale mathematical model for a Li-ion battery system. Learning mathematics is essential in engineering education and practice. With increasing number of students and emergence of online/distance learning programs, it is critical to look for new approaches in teaching mathematics that different in content development and design. Special consideration should be in place in designing an online program for teaching mathematics to ensure students’ success and satisfaction in the engineering curriculum. Previous investigations studied the effects of enrolling in online programs on students’ achievement. However, more implementations of such educational frameworks are needed to recognize their shortcomings and enhance the quality of online learning programs. In addition, the idea of the blended classroom should be put into practice to a further extent to ensure the high-quality development of online instructional content. In this work, an online learning program was provided for engineering students enrolled in an introductory engineering mechanics course. Online interactive instructional modules were developed and implemented in the targeted engineering course to cover prerequisite mathematical concepts of the course. Students with access to the developed online learning modules demonstrated improvement in their learning and recommended employing such modules to teach fundamental concepts in other courses. This part of the work improves the understanding of the development process of the online learning modules and their implementation in lecture-based classrooms