Adaptive Coupling of Finite Element Methods for Simulation of Hydrodynamics and Pollutant Transport in Lakes

Gegenstand dieser Arbeit ist die Entwicklung neuer numerischer Methoden zur Lösung von Problemen der Hydrodynamik in Seen. Für die Berechnung von Transportprozessen von Schadstoffen ist es wichtig Fronten scharf aufzulösen. Dies erfordert eine hohe Genauigkeit in bestimmten Bereichen des Gebiets. Um die erforderliche Genauigkeit zu erreichen und gleichzeitig die Kosten bei der Berechnung moderat zu halten, lösen wir das dreidimensioale Gebiet nicht überall komplett auf. In den Teilen des Gebiet, in denen nur geringe Genauigkeit gefordert wird, genügt eine zweidimensioale Lösung. Für die Bereiche in unserem Gebiet, in denen wir bessere Genauigkeit erzielen wollen, addieren wir zu der zweidimensionalen Lösung eine dreidimensionale Korrektur. Auf diese Weise erreichen wir in gewissen Teilen des Gebiets eine genauere, dreidimensionale Lösung bei moderatem Mehraufwand. Die Gleichungen, die durch diese Kopplung entstehen, werden hergeleitet. Für die Vorkonditionierung des gekoppelten Systems verwenden wir einen Block-Vorkonditionierer. Für die einzelnen Blöcke haben wir einen Mehrgitter-Vorkonditionierer für stetige Finite Elemente auf adaptiv verfeinerten Gittern entwickelt. Dabei geschieht die Glättung nur lokal. Anhand von numerischen Beispielen zeigen wir die Effizienz für Elemente höherer Ordnung

High Performance Computing Based Methods for Simulation and Optimisation of Flow Problems

The thesis is concerned with the study of methods in high-performance computing for simulation and optimisation of flow problems that occur in the framework of microflows. We consider the adequate use of techniques in parallel computing by means of finite element based solvers for partial differential equations and by means of sensitivity- and adjoint-based optimisation methods. The main focus is on three-dimensional, low Reynolds number flows described by the instationary Navier-Stokes equations

Parallel Algorithms for the Solution of Large-Scale Fluid-Structure Interaction Problems in Hemodynamics

This thesis addresses the development and implementation of efficient and parallel algorithms for the numerical simulation of Fluid-Structure Interaction (FSI) problems in hemodynamics. Indeed, hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the arterial wall. The simulation of fluid-structure interaction problems requires the approximation of a coupled system of Partial Differential Equations (PDEs) and the set up of efficient numerical solution strategies. Blood is modeled as an incompressible Newtonian fluid whose dynamics is governed by the Navier-Stokes equations. Different constituive models are used to describe the mechanical response of the arterial wall; specifically, we rely on hyperelastic isotropic and anistotropic material laws. The finite element method is used for the space discretization of both the fluid and structure problems. In particular, for the Navier-Stokes equations we consider a semi-discrete formulation based on the Variational Multiscale (VMS) method. Among a wide range of possible solution strategies for the FSI problem, here we focus on strongly coupled monolithic approaches wherein the nonlinearities are treated in a fully implicit mode. To cope with the high computational complexity of the three dimensional FSI problem, a parallel solution framework is often mandatory. To this end, we develop a new block parallel preconditioner for the coupled linearized FSI system obtained after space and time discretization. The proposed preconditioner, named FaCSI, exploits the factorized form of the FSI Jacobian matrix, the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations, and the use of a SIMPLE preconditioner for unsteady Navier-Stokes equations. In FSI problems, the different resolution requirements in the fluid and structure physical domains, as well as the presence of complex interface geometries make the use of matching fluid and structure meshes problematic. In such situations, it is much simpler to deal with discretizations that are nonconforming at the interface, provided however that the matching conditions at the interface are properly fulfilled. In this thesis we develop a novel interpolation-based method, named INTERNODES, for numerically solving partial differential equations by Galerkin methods on computational domains that are split into two (or several) subdomains featuring nonconforming interfaces. By this we mean that either a priori independent grids and/or local polynomial degrees are used to discretize each subdomain. INTERNODES can be regarded as an alternative to the mortar element method: it combines the accuracy of the latter with the easiness of implementation in a numerical code. The aforementioned techniques have been applied for the numerical simulation of large-scale fluid-structure interaction problems in the context of biomechanics. The parallel algorithms developed showed scalability up to thousands of cores utilized on high performance computing machines

An Ale Approach For The Numerical Simulation Of Insect Flight

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2014Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2014Bu çalışmada öncelikle büyük ölçekli (large-scale) hareketli yüzey problemlerinin tamamen birleşmiş (fully coupled) formda çözülmesi için kenar merkezli yapısal olmayan sonlu hacimler yöntemine dayalı Arbitrary Lagrangian-Eulerian (ALE) yöntemi geliştirilmiştir. Kenar merkezli sonlu hacim metoduna dayanan bu sayısal yöntemde hız vektör bileşenleri her bir elemanın yüzeylerinin orta noktasında tanımlanırken, basınç değerleri her bir elemanın merkezinde tanımlanmaktadır. Basınç ve hız değerlerinin mevcut şekilde düzenlenmesi kararlı bir sayısal şemaya yol açar ve böylece basınç noktalarının birbirleriyle etkileşmesi (pressure coupling) için ayrıca doğal olmayan bir değişikliğe ihtiyaç kalmaz. Süreklilik denklemi her bir eleman içerisinde tam olarak sağlanmakta ve bu süreklilik denklemlerinin toplamı hesaplama bölgesinin sınırlarında tanımlanan küresel süreklilik denklemini vermektedir. Geometrik korunum kanununun (GCL) ayrık biçimde (discrete formda) sağlanması için özel bir özen gösterilmiştir. Ağ deformasyonu her bir zaman adımında direkt olmayan radyal bazlı fonksiyon interpolasyonun çözülmesi ile elde edilmiş ve bu tekrar ağ oluşumunu gerektirmediğinden sayısal yöntemin performansını artırmıştır. Küçük zaman adımlı zamana bağlı akışların çözümü için projeksiyon metodunda olduğu gibi oluşan cebirsel denklemler üç ayrı matrise ayrıklaştırılmış ve bu matrislerin tersi önkoşullandırıcı olarak kullanılmıştır. Burada oluşan ayrık ölçekli Laplacian operatörünün tersi yerine iki adım HYPRE BoomerAMG önkoşullandırıcısı kullanılmıştır. Paralel önkoşullandırılmış iteratif yöntemlerin verimini artırmak için PETSc ve HYPRE kütüphanelerinden yararlanılmıştır. Hareketli ağlar üzerinde şu testler yapılmıştır: Azalan Taylor-Green Girdap akışı, kanal içindeki salınım hareketi yapan silindir etrafındaki akış, yere paralel salınım hareketi yapan küp içerisindeki küre etrafındaki akış.An arbitrary Lagrangian-Eulerian (ALE) approach has been developed in order to investigate the near wake structure of Drosophila flight. The numerical algorithm is based on side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face while the pressure is defined at the element centroid. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance pressure coupling. A special attention is also given to to satisfy the discrete global conservation law. An efficient and robust mesh-deformation algorithm based on the indirect radial basis function method is developed at each time level in order to enhance numerical robustness. For the algebraic solution of the resulting large-scale equations, a matrix factorization is introduced similar to that of the projection method for the whole coupled system and we use two-cycle of BoomerAMG solver for the scaled discrete Laplacian provided by the HYPRE library, which we access through the PETSc library. The present numerical algorithm is initially validated for the decaying Taylor-Green vortex flow, the flow past an oscillating circular cylinder in a channel and the flow induced by an oscillating sphere in a cubic cavity. Then the numerical method is applied to the numerical simulation of flow field around a pair of flapping Drosophila wings in hover flight. Finally, the numerical calculations with different wing kinematics are carried out to simulate the flow field around a pair of flapping Drosophila wings in hover.DoktoraPh

Multi space reduced basis preconditioners for parametrized partial differential equations

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs. In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation. The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system. We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency. We apply our strategy to a broad range of problems described by parametrized PDEs: (i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations. Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results. Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques. Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling

Parallel Overlapping Schwarz Preconditioners and Multiscale Discretizations with Applications to Fluid-Structure Interaction and Highly Heterogeneous Problems

Accurate simulations of transmural wall stresses in artherosclerotic coronary arteries may help to predict plaque rupture. Therefore, a robust and efficient numerical framework for Fluid-Structure Interaction (FSI) of the blood flow and the arterial wall has to be set up, and suitable material laws for the modeling of the fluid and the structural response have to be incorporated. In this thesis, monolithic coupling algorithms and corresponding monolithic preconditioners are used to simulate FSI using highly nonlinear anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models for the arterial wall. An MPI-parallel FSI software from the LifeV library is coupled to the software FEAP in order to enable access to the structural material models implemented in FEAP. To define a benchmark test for highly nonlinear material models in FSI, a simple geometry corresponding to a section of an idealized coronary artery, suitable boundary conditions, and material parameters adapted to experimental data are used. In particular, the geometry is chosen to be nonsymmetric to make effects due to the anisotropy of the structure visible. An initialization phase and several heartbeats are simulated, and systematical studies with meshes of increasing refinement and different space discretizations are carried out. The results indicate that, for the highly nonlinear material models, piecewise quadratic or F-bar element discretizations lead to significantly better results than piecewise linear shape functions. The results using piecewise linear shape functions are less accurate with respect to the displacements and, in particular, to the approximation of the stresses. To improve the performance of the FSI simulations, a more robust preconditioner for the highly nonlinear structural material models has to be used. Therefore, a parallel implementation of the GDSW (Generalized Dryja-Smith-Widlund) preconditioner, which is a geometric two-level overlapping Schwarz preconditioner with energy-minimizing coarse space, is presented. The implementation, which is based on the software library Trilinos, is held flexible to make further extensions of the preconditioner possible. Even though the dimension of its coarse space is comparably large, parallel scalability for two and three dimensional scalar elliptic and linear elastic problems for thousands of cores is demonstrated. Also for unstructured domain decompositions and for a hybrid version of the preconditioner, convincing scalability is presented. When used as a preconditioner for the structure block in FSI simulations, the GDSW preconditioner shows excellent performance as well: scalability for up to 512 cores and a significant reduction of the simulation time and of the number of iterations with respect to the previously used preconditioner, IFPACK, are observed. IFPACK is an algebraic one-level overlapping Schwarz preconditioner. Finally, highly heterogeneous (multiscale) problems are investigated. Since the GDSW coarse space is not robust for general problems of this type, spaces based on Approximate Component Mode Synthesis (ACMS) are considered. On the basis of the ACMS space, coarse spaces for overlapping Schwarz methods are constructed, and a parallel implementation of a special finite element method is presented. For the coarse spaces, preliminary results indicating numerical scalability and robustness are discussed. For the parallel implementation of the special finite element method, very good parallel weak scalability is observed with respect to the construction of the basis functions and to the solution of the resulting linear system using the FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) method

Monolithic multigrid methods for high-order discretizations of time-dependent PDEs

A currently growing interest is seen in developing solvers that couple high-fidelity and higher-order spatial discretization schemes with higher-order time stepping methods for various time-dependent fluid plasma models. These problems are famously known to be stiff, thus only implicit time-stepping schemes with certain stability properties can be used. Of the most powerful choices are the implicit Runge-Kutta methods (IRK). However, they are multi-stage, often producing a very large and nonsymmetric system of equations that needs to be solved at each time step. There have been recent efforts on developing efficient and robust solvers for these systems. We have accomplished this by using a Newton-Krylov-multigrid approach that applies a multigrid preconditioner monolithically, preserving the system couplings, and uses Newton’s method for linearization wherever necessary. We show robustness of our solver on the single-fluid magnetohydrodynamic (MHD) model, along with the (Navier-)Stokes and Maxwell’s equations. For all these, we couple IRK with higher-order (mixed) finiteelement (FEM) spatial discretizations. In the Navier-Stokes problem, we further explore achieving more higher-order approximations by using nonconforming mixed FEM spaces with added penalty terms for stability. While in the Maxwell problem, we focus on the rarely used E-B form, where both electric and magnetic fields are differentiated in time, and overcome the difficulty of using FEM on curved domains by using an elasticity solve on each level in the non-nested hierarchy of meshes in the multigrid method

Parallel Overlapping Schwarz Preconditioners for Incompressible Fluid Flow and Fluid-Structure Interaction Problems

Efficient methods for the approximation of solutions to incompressible fluid flow and fluid-structure interaction problems are presented. In particular, partial differential equations (PDEs) are derived from basic conservation principles. First, the incompressible Navier-Stokes equations for Newtonian fluids are introduced. This is followed by a consideration of solid mechanical problems. Both, the fluid equations and the equation for solid problems are then coupled and a fluid-structure interaction problem is constructed. Furthermore, a discretization by the finite element method for weak formulations of these problems is described. This spatial discretization of variables is followed by a discretization of the remaining time-dependent parts. An implementation of the discretizations and problems in a parallel C++ software environment is described. This implementation is based on the software package Trilinos. The parallel execution of a program is the essence of High Performance Computing (HPC). HPC clusters are, in general, machines with several tens of thousands of cores. The fastest current machine, as of the TOP500 list from November 2019, has over 2.4 million cores, while the largest machine possesses over 10 million cores. To achieve sufficient accuracy of the approximate solutions, a fine spatial discretization must be used. In particular, fine spatial discretizations lead to systems with large sparse matrices that have to be solved. Iterative preconditioned Krylov methods are among the most widely used and efficient solution strategies for these systems. Robust and efficient preconditioners which possess good scaling behavior for a parallel execution on several thousand cores are the main component. In this thesis, the focus is on parallel algebraic preconditioners for fluid and fluid-structure interaction problems. Therefore, monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and Navier-Stokes problems are presented. Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on block factorizations. In order to obtain numerically scalable algorithms, coarse spaces obtained from the Generalized Dryja-Smith-Widlund (GDSW) and the Reduced dimension GDSW (RGDSW) approach are used. These coarse spaces can be constructed in an essentially algebraic way. Numerical results of the parallel implementation are presented for various incompressible fluid flow problems. Good scalability for up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, were achieved. A comparison of these monolithic approaches and commonly used block preconditioners with respect to time-to-solution is made. Similarly, the most efficient construction of two-level overlapping Schwarz preconditioners with GDSW and RGDSW coarse spaces for solid problems is reported. These techniques are then combined to efficiently solve fully coupled monolithic fluid-strucuture interaction problems