Generalized Spectral Decomposition for Stochastic Non Linear Problems

International audienceWe present an extension of the Generalized Spectral Decomposition method for the resolution of non-linear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multiwavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional non-linear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

Magnetodynamic vector hysteresis models for steel laminations of rotating electrical machines

This thesis focuses on the modeling and prediction of iron losses in rotating electrical machines. The aim is to develop core loss models that are reasonably accurate and efficient for the numerical electromagnetic field analysis. The iron loss components, including hysteresis, classical eddy-current, and excess losses, are determined by modeling the dynamic hysteresis loops, whereby the incorporation of the core losses into the field solution is feasible and thus the influence of the core losses on the performance of the machine is investigated. The thesis presents a magnetodynamic vector hysteresis model that produces not only an accurate, overall prediction of the iron losses, but also explicitly models the magnetization behavior and the loop shapes. The model is found to be efficient, stable, and adequate for providing accurate predictions of the magnetization curves, and hence iron losses, under alternating and rotating flux excitations. It is demonstrated that the model satisfies the rotational loss property and reproduces the shapes of the experimental loops. In addition, a more simplified, efficient, and robust version of the magnetodynamic vector hysteresis model is introduced. The thesis also aims to analyze the convergence of the fixed-point method, examine the barriers behind the slow convergence, and show how to overcome them. The analysis has proved useful and provided sound techniques for speeding up the convergence of the fixed-point method. The magnetodynamic lamination models have been integrated into a two-dimensional finite-element analysis of rotating electrical machines. The core losses of two induction motors have been analyzed and the impact of core losses on the motor characteristics has been investigated. The simulations conducted reveal that the models are relatively efficient, accurate, and suitable for the design purposes of electrical machines.reviewe

Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation

Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo

Incompressible lagrangian fluid flow with thermal coupling

A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation. First of all, the Navier-Stokes equations of motion coupled with the Boussinesq approximation must be reformulated in the Lagrangian framework, whereas they have been mostly derived in an Eulerian context. Secondly, the Lagrangian formulation implies to follow the material particles during their motion, which means to convect the mesh in the case of the Finite Element Method (FEM), the spatial discretisation method chosen in this work. This provokes various difficulties for the mesh generation, mainly in three dimensions, whereas it eliminates the classical numerical difficulty to deal with the convective term, as much in the Navier-Stokes equations as in the energy equation. Even without the discretization of the convective term, an efficient iterative solver, which constitutes the only viable alternative for three dimensional problems, must be designed for the class of Generalized Stokes Problems (GSP), which could be able to behave well independently of the mesh Reynolds number, as it can vary greatly for coupled fluid-thermal analysis. Moreover, it offers a natural framework to treat free-surface problems like wave breaking and rough fluid-structure contact. On one hand, the convection of the mesh during one time step after the resolution of the non-linear system provides explicitly the locus of the domain to be considered. On the other hand, fluid-to-fluid and fluid-to-wall contact, as well as the update of the domain due to the remeshing, must be accurately and efficiently performed. Finally, the solidification of the fluid coupled with its motion through a variable viscosity is considered An efficient overall algorithm must be designed to bring the method effective, particularly in a three dimensional context, which is the ambition of this monograph. Various numerical examples are included to validate and highlight the potential of the method

Fast numerical methods for robust nonlinear optimal control under uncertainty

This thesis treats different aspects of nonlinear optimal control problems under uncertainty in which the uncertain parameters are modeled probabilistically. We apply the polynomial chaos expansion, a well known method for uncertainty quantification, to obtain deterministic surrogate optimal control problems. Their size and complexity pose a computational challenge for traditional optimal control methods. For nonlinear optimal control, this difficulty is increased because a high polynomial expansion order is necessary to derive meaningful statements about the nonlinear and asymmetric uncertainty propagation. To this end, we develop an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision. The algorithmic contribution is complemented by a convergence proof showing that the solutions of the optimal control problem after application of the polynomial chaos method approach the correct solution for increasing expansion orders. To obtain a further speed-up in solution time, we develop a structure-exploiting algorithm for the fast derivative generation. The algorithm makes use of the special structure induced by the spectral projection to reuse model derivatives and exploit sparsity information leading to a fast automatic sensitivity generation. This greatly reduces the computational effort of Newton-type methods for the solution of the resulting high-dimensional surrogate problem. Another challenging topic of this thesis are optimal control problems with chance constraints, which form a probabilistic robustification of the solution that is neither too conservative nor underestimates the risk. We develop an efficient method based on the polynomial chaos expansion to compute nonlinear propagations of the reachable sets of all uncertain states and show how it can be used to approximate individual and joint chance constraints. The strength of the obtained estimator in guaranteeing a satisfaction level is supported by providing an a-priori error estimate with exponential convergence in case of sufficiently smooth solutions. All methods developed in this thesis are readily implemented in state-of-the-art direct methods to optimal control. Their performance and suitability for optimal control problems is evaluated in a numerical case study on two nonlinear real-world problems using Monte Carlo simulations to illustrate the effects of the propagated uncertainty on the optimal control solution. As an industrial application, we solve a challenging optimal control problem modeling an adsorption refrigeration system under uncertainty

A hybridized discontinuous Galerkin method for nonlinear dispersive water waves

Simulation of water waves near the coast is an important problem in different branches of engineering and mathematics. For mathematical models to be valid in this region, they should include nonlinear and dispersive properties of the corresponding waves. Here, we study the numerical solution to three equations for modeling coastal water waves using the hybridized discontinuous Galerkin method (HDG). HDG is known to be a more efficient and in certain cases a more accurate alternative to some other discontinuous Galerkin methods, such as local DG. The first equation that we solve here is the Korteweg-de Vries equation. Similar to common HDG implementations, we first express the approximate variables and numerical fluxes in each element in terms of the approximate traces of the scalar variable, and its first derivative. These traces are assumed to be single-valued on each face. We next impose the conservation of the numerical fluxes via two sets of equations on the element boundaries. We solve this equation by Newton-Raphson method. We prove the stability of the proposed method for a proper choice of stabilization parameters. Through numerical examples, we observe that for a mesh with kth order elements, the computed variable and its first and second derivatives show optimal convergence at order k + 1 in both linear and nonlinear cases, which improves upon previously employed techniques. Next, we consider solving the fully nonlinear irrotational Green-Naghdi equation. This equation is often used to simulate water waves close to the shore, where there are significant dispersive and nonlinear effects involved. To solve this equation, we use an operator splitting method to decompose the problem into a dispersive part and a hyperbolic part. The dispersive part involves an implicit step, which has regularizing effects on the solution of the problem. On the other hand, for the hyperbolic sub-problem, we use an explicit hybridized DG method. Unlike the more common implicit version of the HDG, here we start by solving the flux conservation condition for the numerical traces. Afterwards, we use these traces in the original PDEs to obtain the internal unknowns. This process involves Newton iterations at each time step for computing the numerical traces. Next, we couple this solver with the dispersive solver to obtain the solution to the Green-Naghdi equation. We then solve a set of numerical examples to verify and validate the employed technique. In the first example we show the convergence properties of the numerical method. Next, we compare our results with a set of experimental data for nonlinear water waves in different situations. We observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data.Engineering Mechanic

Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations

Multigrid methods belong to the best-known methods for solving linear systems arising from the discretization of elliptic partial differential equations. The main attraction of multigrid methods is that they have an asymptotically meshindependent convergence behavior. Multigrid with Vanka (or local multilevel pressure Schur complement method) as smoother have been frequently used for the construction of very effcient coupled monolithic solvers for the solution of the stationary incompressible Navier-Stokes equations in 2D and 3D. However, due to its innate Gauß-Seidel/Jacobi character, Vanka has a strong influence of the underlying mesh, and therefore, coupled multigrid solvers with Vanka smoothing very frequently face convergence issues on meshes with high aspect ratios. Moreover, even on very nice regular grids, these solvers may fail when the anisotropies are introduced from the differential operator. In this thesis, we develop a new class of robust and efficient monolithic finite element multilevel Krylov subspace methods (MLKM) for the solution of the stationary incompressible Navier-Stokes equations as an alternative to the coupled multigrid-based solvers. Different from multigrid, the MLKM utilizes a Krylov method as the basis in the error reduction process. The solver is based on the multilevel projection-based method of Erlangga and Nabben, which accelerates the convergence of the Krylov subspace methods by shifting the small eigenvalues of the system matrix, responsible for the slow convergence of the Krylov iteration, to the largest eigenvalue. Before embarking on the Navier-Stokes equations, we first test our implementation of the MLKM solver by solving scalar model problems, namely the convection-diffusion problem and the anisotropic diffusion problem. We validate the method by solving several standard benchmark problems. Next, we present the numerical results for the solution of the incompressible Navier-Stokes equations in two dimensions. The results show that the MLKM solvers produce asymptotically mesh-size independent, as well as Reynolds number independent convergence rates, for a moderate range of Reynolds numbers. Moreover, numerical simulations also show that the coupled MLKM solvers can handle (both mesh and operator based) anisotropies better than the coupled multigrid solvers

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe

Accelerated Temporal Schemes for High-Order Unstructured Methods

The ability to discretize and solve time-dependent Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) remains of great importance to a variety of physical and engineering applications. Recent progress in supercomputing or high-performance computing has opened new opportunities for numerical simulation of the partial differential equations (PDEs) that appear in many transient physical phenomena, including the equations governing fluid flow. In addition, accurate and stable space-time discretization of the partial differential equations governing the dynamic behavior of complex physical phenomena, such as fluid flow, is still an outstanding challenge. Even though significant attention has been paid to high and low-order spatial schemes over the last several years, temporal schemes still rely on relatively inefficient approaches. Furthermore, academia and industry mostly rely on implicit time marching methods. These implicit schemes require significant memory once combined with high-order spatial discretizations. However, since the advent of high-performance general-purpose computing on GPUs (GPGPU), renewed interest has been focused on explicit methods. These explicit schemes are particularly appealing due to their low memory consumption and simplicity of implementation. This study proposes low and high-order optimal Runge-Kutta schemes for FR/DG high-order spatial discretizations with multi-dimensional element types. These optimal stability polynomials improve the stability of the numerical solution and speed up the simulation for high-order element types once compared to classical Runge-Kutta methods. We then develop third-order accurate Paired Explicit Runge-Kutta (P-ERK) schemes for locally stiff systems of equations. These third-order P-ERK schemes allow Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria, while seamlessly pairing at their interface. We then generate families of schemes optimized for the high-order flux reconstruction spatial discretization. Finally, We propose optimal explicit schemes for Ansys Fluent finite volume density-based solver, and we investigate the effect of updating and freezing reconstruction gradient in intermediate Runge-Kutta schemes. Moreover, we explore the impact of optimal schemes combined with the updated gradients in scale-resolving simulations with Fluent's finite volume solver. We then show that even though freezing the reconstruction gradients in intermediate Runge-Kutta stages can reduce computational cost per time step, it significantly increases the error and hampers stability by limiting the time-step size

Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent studies suggest that fractional differential and integral operators are well suited to model physical phenomena with intrinsic memory retention and anomalous behaviour. The global property of fractional operators presents difficulties in fnding either closed-form solutions or accurate numerical solutions to fractional differential equations. In rare cases, when analytical solutions are available, they often exist only in terms of complex integrals and special functions, or as infinite series. Similarly, obtaining an accurate numerical solution to arbitrary order differential equation is often computationally demanding. Fractional operators are non-local, and so it is practicable that when approximating fractional operators, non-local methods should be preferred. One such non-local method is the spectral method. In this thesis, we solve problems that arise in the ow of non-Newtonian fluids modelled with fractional differential operators. The recurrent theme in this thesis is the development, testing and presentation of tractable, accurate and computationally efficient numerical schemes for various classes of fractional differential equations. The numerical schemes are built around the pseudo{spectral collocation method and shifted Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral methods converge geometrically, are accurate and computationally efficient. The objective of this thesis is to show, among other results, that these features are true when the method is applied to a variety of fractional differential equations. A survey of the literature shows that many studies in which pseudo-spectral methods are used to numerically approximate the solutions of fractional differential equations often to do this by expanding the solution in terms of certain orthogonal polynomials and then simultaneously solving for the coefficients of expansion. In this study, however, the orthogonality condition of the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature are used to numerically find the coefficients of the series expansions. This approach is then applied to solve various fractional differential equations, which include, but are not limited to time{space fractional differential equations, two{sided fractional differential equations and distributed order differential equations. A theoretical framework is provided for the convergence of the numerical schemes of each of the aforementioned classes of fractional differential equations. The overall results, which include theoretical analysis and numerical simulations, demonstrate that the numerical method performs well in comparison to existing studies and is appropriate for any class of arbitrary order differential equations. The schemes are easy to implement and computationally efficient