Monolithic Finite Element Method for the simulation of thixo-viscoplastic flows

[EN] This note is concerned with the application of Finite Element Method (FEM) and NewtonMultigrid solver to simulate thixo-viscoplastic flows. The thixo-viscoplastic stress dependent on material microstructure is incorporated via viscosity approach into generalized Navier-Stokes equations. The full system of equations is solved in a monolithic framework based on Newton-Multigrid FEM Solver. The developed solver is used to analyze the thixo-viscoplastic flow problem in a Lid-driven cavity configuration.The authors acknowledge the funding provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 446888252. Additionally, the authors acknowledge the financial grant provided by the Bundesministerium fr Wirtschaft und Energie aufgrund eines Beschlusses des Deutschen Bundestages through AiF-Forschungsvereinigung: Forschungs- Gesellschaft Verfahrens Technik e. V. - GVT under the IGF project number 20871 N. We would also like to gratefully acknowledge the support by LSIII and LiDO3 team at ITMC, TU Dortmund University, Germany.Begum, N.; Ouazzi, A.; Turek, S. (2022). Monolithic Finite Element Method for the simulation of thixo-viscoplastic flows. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 170-179. https://doi.org/10.4995/YIC2021.2021.12250OCS17017

Natural convection of incompressible viscoelastic fluid flow

We revisit the MIT Benchmark 2001 and introduce a viscoelastic constitutive law into the fluid in motion. Our goal is to study the effect of viscoelasticity into the periodical behavior of the physical quantities of the corresponding benchmark. We use a robust numerical technique in simulating complex fluid flow problems based on higher order Finite Element discretization. While marching in time, an A-stable method of second order is favorable, i.e Crank-Nicolson scheme, to reproduce periodical behaviors. We use a differential form of viscoelastic model, i.e Oldroyd-B type and find out that a small amount of viscoelasticity reduces the oscillatory behavior

Finite Element Methods for the simulation of thixotropic flow problems

This note is concerned with the application of Finite Element Methods (FEM) and Newton-Multigrid solvers for the simulation of thixotropic flow problems. The thixotropy phenomena are introduced into viscoplastic material by taking into account the internal material micro structure using a scalar structure parameter. Firstly, the viscoplastic stress is modified to include the thixotropic stress dependent on the structure parameter. Secondly, an evolution equation for the structure parameter is introduced to induce the time-dependent process of competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. Substantially, this is done simply by introducing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The modified thixotropic viscoplastic stress w.r.t. the structure parameter is integrated in quasi-Newtonian manner into the generalized Navier-Stokes equations and the evolution equation for the structure parameter constitutes the main core of full set of modeling equations, which are creditable as the privilege answer to incorporate thixotropy phenomena. A fully coupled monolithic finite element approach has been exercised which manages the material internal micro structure parameter, velocity, and pressure fields simultaneously. The nonlinearity of the corresponding problem, related to the dependency of the diffusive stress on the material parameters and the nonlinear structure parameter models on the other hand, is treated with generalized Newton's method w.r.t. the Jacobian's singularities having a global convergence property. The linearized systems inside the outer Newton loops form a typical saddle-point problem which is solved using a geometrical multigrid method with a Vanka-like smoother taking into account a stable FEM approximation pair for velocity and pressure with discontinuous linear pressure and biquadratic velocity spaces. We examine the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver throughout the solution of thixotropic viscoplastic flow problems in Couette device and in 4:1 contraction

Newton-Multigrid FEM Solver for the Simulation of Quasi-Newtonian Modeling of Thixotropic Flows

This paper is concerned with the application of Finite Element Methods (FEM) and NewtonMultigrid solvers to simulate thixotropic flows using quasi-Newtonian modeling. The thixotropy phenomena are introduced to yield stress material by taking into consideration the internal material microstructure using a structure parameter. Firstly, the viscoplastic stress is modified to include the thixotropy throughout the structure parameter. Secondly, an evolution equation for the structure parameter is introduced to induce the time-dependent process of competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. This is done simply by introducing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The nonlinearity, related to the dependency of the diffusive term on the material parameters, is treated with generalized Newton's method w.r.t. the Jacobian's singularities having a global convergence property. The linearized systems inside the outer Newton loops are solved using the geometrical multigrid with a Vanka-like linear smoother taking into account a stable FEM approximation pair for velocity and pressure with discontinuous pressure and biquadratic velocity spaces. We analyze the application of using the quasi-Newtonian modeling approach for thixotropic flows, and the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver throughout the solution of the thixotropic flows using manufactured solutions in a channel and the prototypical configuration of thixotropic flows in Couette device

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

Multilevel Domain Decomposition Algorithms for Monolithic Fluid-Structure Interaction Problems with Application to Haemodynamics

Finite element techniques for solving the problem of fluid-structure interaction of an elastic solid material in a laminar incompressible viscous flow are described. The mathematical problem consists of the Navier-Stokes equations in the Arbitrary Lagrangian-Eulerian formulation coupled with a non-linear structure model, considering the problem as one continuum. The coupling between the structure and the fluid is enforced inside a monolithic framework which computes simultaneously for the fluid and the structure unknowns within a unique solver. We used the well-known Crouzeix-Raviart finite element pair for discretization in space and the method of lines for discretization in time. A stability result using the Backward-Euler time-stepping scheme for both fluid and solid part and the finite element method for the space discretization has been proved. The resulting linear system has been solved by multilevel domain decomposition techniques. Our strategy is to solve several local subproblems over subdomain patches using the Schur-complement or GMRES smoother within a multigrid iterative solver. For validation and evaluation of the accuracy of the proposed methodology, we present corresponding results for a set of two FSI benchmark configurations which describe the self-induced elastic deformation of a beam attached to a cylinder in a laminar channel flow, allowing stationary as well as periodically oscillating deformations, and for a benchmark proposed by COMSOL multiphysics where a narrow vertical structure attached to the bottom wall of a channel bends under the force due to both viscous drag and pressure. Then, as an example of fluid-structure interaction in biomedical problems, we considered the academic numerical test which consists in simulating the pressure wave propagation through a straight compliant vessel. All the tests show the applicability and the numerical efficiency of our approach to both two-dimensional and three-dimensional problems

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

Implementation of linear and non-linear elastic biphasic porous media problems into FEATFLOW and comparison with PANDAS

This dissertation presents a fully implicit, monolithic finite element solution scheme to effectively solve the governing set of differential algebraic equations of incompressible poroelastodynamics. Thereby, a two-dimensional, biphasic, saturated porous medium model with intrinsically coupled and incompressible solid and fluid constituents is considered. Our schemes are based on some well-accepted CFD techniques, originally developed for the efficient simulation of incompressible flow problems, and characterized by the following aspects: (1) a special treatment of the algebraically coupled volume balance equation leading to a reduced form of the boundary conditions; (2) usage of a higher-order accurate mixed LBBstable finite element pair with piecewise discontinuous pressure for the spatial discretization; (3) application of the fully implicit 2nd-order Crank-Nicolson scheme for the time discretization; (4) use of a special fast multigrid solver of Vanka-type smoother available in FEATFLOW to solve the resulting discrete linear equation system. Furthermore, a new adaptive time stepping scheme combined with Picard iteration method is introduced to solve a non-linear elastic problem with special hyper-elastic model. For the purpose of validation and to expose themerits and benefits of our new solution strategies in comparison to other established approaches, canonical one- and two-dimensional wave propagation problems are solved and a large-scale, dynamic soil-structure interaction problem serves to reveal the efficiency of the special multigrid solver and to evaluate its performance for different formulations