Finite Element Analysis for Transient Non-Newtonian Flow

(English) This thesis deals with finite element methods to solve compressible flow problems, an important branch of Computational Fluid Dynamics (CFD) whose applications are widespread in many areas of engineering and science. In spite of the increasing amount of computational resources made available for the scientific and engineering research communities, the numerical simulation of complex compressible phenomena in many practical applications is still a challenge. These type of flow problems are extremely demanding in what concerns numerical computations and memory requirements. In particular, in this thesis we investigate the possibility to solve the underlying algebraic systems in a decoupled manner, a technique usually called fractional step or segregation method. Although segregation techniques have been broadly studied and analyzed for the incompressible Navier-Stokes equations, allowing for the separate resolution of velocity and pressure unkonwns, much less has been explored for compressible problems. The interest on this type of technique not only comes from the fact that it permits a segregated calculation of the problem unkonwns (usually leading to better conditioned systems) but from the associated reduction of the computational cost. We study three different problems inside the compressible CFD research branch in separated chapters: the isentropic Navier-Stokes equations, the Navier-Stokes problem written in primitive variables (velocity, pressure and temperature), and the navier-Stokes problem written in the classical formulation with conservative variables (momentum, density, total energy). For each of these problems, first we propose a finite element stabilized formulation framed within the Variational MultiScale concept, which allows to use equal interpolation spaces for all the variables in play. Second, and once space and time discretizations are selected, we derive fractional step methods up to second order in time. Finally, all the schemes are implemented in a parallel multiphysics code and representative simulations are carried out in order to analyze the performance of the proposed techniques. (Español) Esta tesis se centra en la aplicación del método de elementos finitos a problemas de flujo compresible, unas de las ramas más importantes dentro de la mecánica de fluidos computacional (CFD, por sus siglas en inglés), y cuya aplicación es amplia en distintas áreas de ciencia e ingeniería. A pesar del aumento de los recursos computacionales que se ponen a disposición de la comunidad científica, la solución numérica de problemas complejos de flujo compresible para distintas aplicaciones prácticas es aún un reto. Este tipo de problemas de flujo son extremadamente exigentes en lo que se refiere al cálculo numérico y los requisitos de memoria computacionales. En particular, en esta tesis se estudia la posibilidad de resolver los sistemas algebraicos que subyacen tras la aplicación del método de elementos finitos de una forma desacoplada, una técnica que usualmente se denomina de paso fraccionado o método de segregación. Aunque las técnicas de segregación han sido ampliamente estudiadas y analizadas para el problema incompresible de Navier-Stokes, lo que permite un cálculo separado de las variables velocidad y presión, poco se ha explorado su aplicación a problemas de flujo compresible. El interés en este tipo de técnicas radica no solo en el hecho de que permite un cálculo segregado de las incógnitas del problema (generalmente dando lugar además a sistemas algebraicos mejor condicionados), sino también en la reducción de coste computacional que ofrecen. Se estudian tres problemas distintos dentro de la rama de investigación en CFD de flujos compresibles: las ecuaciones isentrópicas de Navier-Stokes, el problema compresible de Navier-Stokes escrito en variables primitivas (velocidad, presión y temperatura) y el problema compresible de Navier-Stokes planteado de manera clásica en variables conservativas (momento, densidad y energía total). Para cada uno de estos casos, primero se propone una formulación de elementos finitos, estabilizada bajo el marco de trabajo de las multiescalas variacionales (VMS, por sus siglas en inglés), lo que en la práctica permite utilizar la misma interpolación para las variables del problema. En segundo lugar, y una vez las discretizaciones en tiempo y espacio han sido seleccionadas, se proponen métodos de paso fraccionado para cada problema de hasta segundo orden en tiempo. Finalmente, todos los algoritmos se implementan en una plataforma de código paralelo para problemas multifísicos y se llevan a cabo simulaciones representativas con el fin de analizar el rendimiento de las técnicas propuestas.

Chapter 11 - Mixed finite element–computational fluid dynamics method for nonlinear fluid–solid interactions

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

SummaryThe fractional step method (FSM) is an efficient solution technique for the particle finite element method, a Lagrangian‐based approach to simulate fluid–structure interaction (FSI). Despite various refinements, the applicability of the FSM has been limited to low viscosity flow and FSI simulations with a small number of equations along the fluid–structure interface. To overcome these limitations, while incorporating nonlinear response in the structural domain, an FSM that unifies structural and fluid response in the discrete governing equations is developed using the quasi‐incompressible formulation. With this approach, fluid and structural particles do not need to be treated separately, and both domains are unified in the same system of equations. Thus, the equations along the fluid–structure interface do not need to be segregated from the fluid and structural domains. Numerical examples compare the unified FSM with the non‐unified FSM and show that the computational cost of the proposed method overcomes the slow convergence of the non‐unified FSM for high values of viscosity. As opposed to the non‐unified FSM, the number of iterations required for convergence with the unified FSM becomes independent of viscosity and time step, and the simulation run time does not depend on the size of the FSI interface. Copyright © 2016 John Wiley & Sons, Ltd.

We study domain decomposition counterparts of the classical alternating direction implicit (ADI) and fractional step (FS) methods for solving the large linear systems arising from the implicit time stepping of parabolic equations. In the classical ADI and FS methods for parabolic equations, the elliptic operator is split along coordinate axes; they yield tridiagonal linear systems whenever a uniform grid is used and when mixed derivative terms are not present in the differential equation. Unlike coordinate-axes-based splittings, we employ domain decomposition splittings based on a partition of unity. Such splittings are applicable to problems on nonuniform meshes and even when mixed derivative terms are present in the differential equation and they require the solution of one problem on each subdomain per time step, without iteration. However, the truncation error in our proposed method deteriorates with smaller overlap amongst the subdomains unless a smaller time step is chosen. Estimates are presented for the asymptotic truncation error, along with computational results comparing the standard Crank-Nicolson method with the proposed method.

Computational fluid dynamics (CFD) is increasingly becoming the tool of choice for researchers and engineers facing complex fluid flow problems. One area of CFD that has attracted significant attention in recent years is the development of meshless or mesh-free methods[1]. These methods have so far been used in many applications like fracture mechanics, astrophysics and fluid flow. The key advantage of meshless methods over mesh-based methods like finite volume (FV) and finite element (FE) methods is its non-requirement of pre-specified connectivity between nodes in order to derive approximation and interpolation. This leads to the ability of finite difference (FD) methods to treat irregular domains and boundaries. A direct result of this is another advantage of mesh-free methods having the adaptability for problems involving large movement or wholesale motion of boundaries or embedded bodies. In this area, it has the edge over composite or overset grid methods in being able to use only one coordinate frame.A convecting meshless scheme for boundary driven fluid flow is described. This scheme can be applied to self-propulsion problems, or, in general, flows involving motion of rigid or deformable bodies embedded within the flow.The generalized finite difference method[2] (GFD) was chosen this scheme. A meshless cloud of nodes is embedded around the body surface along with the structured Cartesian background nodes for the entire flow field. By incorporating a subdomain for each node and its neighbours, the use of Taylor series expansion enables the derivatives to be found. Two forms of incompressible momentum equations are used: the standard Eulerian form in primitive variables for the structured nodes and the Arbitrary Lagrangian-Eulerian (ALE) form for the meshless cloud of nodes. GFD with moving least squares approximation is used for spatial discretization of the convecting meshless nodes. The projection method with Crank-Nicolson discretization is used to achieve second order accurate time discretization. Following prescribed body motion or fluid structure interaction, the meshless cloud acquire new locations in the next time step and the momentum equations in ALE form can then be solved.Test cases were performed to verify code usability and method accuracy and convergence. The simulation of the decaying vortex has been a popular benchmark for the gauging of accuracy. Basically the exact solutions of the time-dependent function values are known and both residues and absolute errors were calculated. Solving using different grid sizes proved the present method to be of second order spatial accuracy. The meshless method was then tried on the driven cavity flow at Reynolds number 1000. The results were well-compared with popular benchmark results[4]. In considering the effects of implementing the convecting meshless nodes across a normal flow field, the decaying vortex was again simulated, with a moving patch of nodes within the Cartesian field. The residues were well controlled and the errors were not significantly increased. Finally, a couple of external flows cases were simulated, basically on a flow past a stationary bluff body and a flapping ellipse inside an enclosed body of fluid.Future works in the pipeline include writing a parallel code for larger scale simulation and simulating oscillating cylinders and flapping of elliptic aerofoil(s).

This paper presents a new and effective method for re-initializing the Level-Set (LS) field within the framework of Chorin's projection method. The Navier-Stokes equations (NSE) are solved numerically using the Finite Element Method (FEM) in combination with Chorin's projection method. The proposed approach improves accuracy and efficiency, ensuring precise mass conservation of the LS field. The effectiveness and efficiency of the re-initialization method are validated through benchmark test cases. This study provides an efficient approach for solving time-dependent incompressible fluid flow problems by using Chorin's projection method to separate pressure and velocity fields. Additionally, it introduces an efficient technique for re-initializing the LS field. The findings demonstrate the accuracy, efficiency, and mass conservation capabilities of the method, making a valuable contribution to numerical analysis and computational fluid dynamics.

PurposeThe purpose of this paper is to numerically solve Eikonal and Hamilton‐Jacobi equations using the finite element method; to use both explicit Taylor Galerkin (TG) and implicit methods to obtain shortest wall distances; to demonstrate the implemented methods on some realistic problems; and to use iterative generalized minimal residual method (GMRES) method in the solution of the equations.Design/methodology/approachThe finite element method along with both the explicit and implicit time discretisations is employed. Two different forms of governing equations are also employed in the solution. The Eikonal equation in its original form is used in the explicit Taylor Galerkin discretisation to save computational time. For implicit method, however, the convection‐diffusion form in its conservation form is used to maintain spatial stability.FindingsThe finite element solution obtained is both accurate and smooth. As expected the implicit method is much faster than the explicit method. Though the proposed finite element solution procedures in serial is slower than the standard search procedure, they are suitable to be used in a parallel environment.Originality/valueThe finite element procedure for Eikonal and Hamilton‐Jacobi equations are attempted for the first time. Though the finite volume and finite difference‐based computational fluid dynamics (CFD) solvers have started employing differential equations for wall distance calculations, it is not common for finite element solvers to use such wall distance calculations. The results presented here clearly show that the proposed methods are suitable for unstructured meshes and finite element solvers.

A stabilized edge-based implicit incompressible flow formulation

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

The immersed boundary approach for the modeling of complex geometries in incompressible flows is examined critically from the perspective of satisfying boundary conditions and mass conservation. It is shown that the system of discretized equations for mass and momentum can be inconsistent, if the velocity is used in defining the force density to satisfy the boundary conditions. As a result, the velocity is generally not divergence free and the pressure at locations in the vicinity of the immersed boundary is not physical. However, the use of the pseudo‐velocities in defining the force density, as frequently done when the governing equations are solved using a fractional step or projection method, combined with the use of the specified velocity on the immersed boundary, is shown to result in a consistent set of equations which allows a divergence‐free velocity but, depending on the time step, is shown to have the undesirable effects of inaccurately satisfying the boundary conditions and allowing a significant permeability of the immersed boundary. If the time step is reduced sufficiently, the boundary conditions on the immersed boundary can be satisfied. However, this entails an unacceptable increase in computational expense. Two new methods that satisfy the boundary conditions and allow a divergence‐free velocity while avoiding the increased computational expense are presented and shown to be second‐order accurate in space. The first new method is based on local time step reduction. This method is suitable for problems where the immersed boundary does not move. For these problems, the first new method is shown to be closely related to the second new method. The second new method uses an optimization scheme to minimize the deviation from the interpolation stencil used to represent the immersed boundary while ensuring a divergence‐free velocity. This method performs well for all problems, including those where the immersed boundary moves relative to the grid. Additional results include showing that the force density that is added to satisfy the boundary conditions at the immersed boundary is unbounded as the time step is reduced and that the pressure in the vicinity of the immersed boundary is unphysical, being strongly a function of the time step. A method of computing the total force on an immersed boundary which takes into account the specifics of the numerical solver used in the iterative process and correctly computes the total force irrespective of the residual level is also presented. Copyright © 2007 John Wiley & Sons, Ltd.

Molecular dynamics (MD) and finite-element (FE) simulations of a nano-crystalline diamond (NCD) grain system subjected to indentation were undertaken to understand how contact behavior pertaining to the surface roughness of NCD crystals can be spanned from the molecular to the continuum scale. It is shown that if the same surface roughness morphology, grain orientations, and elastic properties are used for both MD and FE simulations, there is agreement between contact pressures for relatively low indentation loads and shallow substrates. It should be emphasized that the strong correlation between MD and FE methods can also be due to the lack of defect nucleation associated with the elastic deformation of the NCD system. These predictions can be used for a hierarchical computational framework to harness the advantages of both computational approaches. Inherent to this approach is representation of surface roughness and crystal orientation that is physically consistent for both computational approaches. Using the techniques presented herein, MD simulations of fluids and fluid solid interactions may be able to be accurately performed using computational fluid dynamics (CFD) and FE simulations if an appropriate set of translation rules can be found for converting MD models to CFD models. An initial analysis of the requirements to convert MD models to CFD is undertaken and a subset of the necessary rules is postulated.

We present the mathematical foundation of a fluid animation method for unstructured meshes. Key contributions not previously treated are the extension to include diffusion forces and higher order terms of non-linear force approximations. In our discretization we apply a fractional step method to be able to handle advection in a numerically simple Lagrangian approach. Following this a finite element method is used for the remaining terms of the fractional step method. The key to deriving a discretization for the diffusion forces lies in restating the momentum equations in terms of a Newtonian stress tensor. Rather than applying a straightforward temporal finite difference method followed by a projection method to enforce incompressibility as done in the stable fluids method, the last step of the fractional step method is rewritten as an optimization problem to make it easy to incorporate non-linear force terms such as surface tension.

Hull girder collapse behavior is investigated in the case of a 5250 TEU containership using a coupling between Computational Fluid Dynamics (CFD) and Finite Element Method (FEM). The target collapse part is modeled with an elasto-plastic material with a fine mesh, while the remaining part is modeled with an elastic material. Still water load (SWL) has been applied first, which causes the containership to bend in a hogging direction in calm water condition. Thereafter, a design wave load (DWL) has been applied using CFD, which pushes the structure further towards its ultimate hogging strength. In this regard, one-way and two-way coupling between CFD and FEM has been chosen to analyze and understand the effect of direct coupling on the ultimate strength assessment of the structure under extreme wave conditions. An in-house coupling code is used to communicate between the structure and fluid platform. The collapse behaviors obtained from one-way and two-way couplings are discussed in comparison to a pure bending static test.

In many engineering problems, the meshfree methods (MMs) have been dynamically projected and increasingly advanced in order to overwhelm some hitches in the predictable numerical methods. Over the past three decades in many different application area, MMs have found their way ranging from solid mechanics analysis, fluid problems, vibration analysis, heat transfer and optimization to numerical solutions of all kinds of (partial) differential equations. As every technique has shortcomings, the meshfree method also has drawbacks like higher computational cost and imposition of boundary condition which can be overruled by coupling it with the finite element method (FEM). In the past two decades, coupled MMs and FEM have appeared into a new session of computational methods with significant achievement. In addition, a noteworthy amount of growth has been made in addressing the major deficiencies that were present in the conventional methods and MMs at the premature phases. The objective of the present work is to provide a comprehensive review of various coupling techniques used for interface elements of MMs and FEM and general discussion on shape function formulation of FE and element free Galerkin method (EFGM). Key contribution of coupling techniques for coupled EFGM and FEM to structure mechanics application as primary concern.