Collision in a cross-shaped domain --- A steady 2D Navier--Stokes example demonstrating the importance of mass conservation in CFD

In the numerical simulation of the incompressible Navier-Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete LBB constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor-Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore we present a simple steady Navier-Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Finite elements with h -adaptation for momentum, heat and mass transport with application to environmental flow

Self-adaptive algorithms for 2 and 3-dimensional unstructured finite element grids are recent to the solution of partial differential equations, particularly those equations describing environmental transport. An h-adaptive grid embedding method is developed to solve the incompressible Navier-Stokes equations for fluid flow and scalar transport. An application to atmospheric mass transport is presented; This h-adaptive algorithm, in combination with the finite element method, has been designed to solve 2 and 3-dimensional problems on Pentium PC\u27s, including problems involving complex geometry on high end PC\u27s, workstations and mainframes. The Galerkin finite element solver is a 2 point Gauss-Legendre integration scheme which employs mass lumping, Cholesky skyline L-U decomposition, and Petrov-Galerkin upwinding; This dissertation introduces and explains the application of the Galerkin weighted residual finite element method. Development of the weak statements for the non-dimensional primitive variable Navier-Stokes equations is presented along with a Poisson formulation for resolving pressure. The semi-implicit solution process of this Poisson formulation is described in detail. Various adaptive methods are presented with emphasis on grid embedDing Finally the application of the adaptive process coupled with the finite element solver is applied to the solution of the Navier-Stokes equations along with the species transport equation; Adaptive methods are becoming common place in the solution of partial differential equations. In this thesis, an algorithm employing h-adaptation is developed for the solution of the non-linear Navier-Stokes equations for incompressible flow and its application to environmental fluid dynamics. Improvements in computational requirements are discussed including comparison with solutions on globally refined domains. Comparison of solutions is provided by using benchmark problems that provide a means for assuring the verification and validation of the computer code. Implementation of the algorithm for environmental species transport is an effective method to improve the accuracy of transport prediction

A stabilized discontinuous Galerkin method for variational embedding of physics-based data

A stabilized variational framework that admits overlapping as well as non overlapping coupling of domains for a variety of Partial Differential Equations (PDEs) is employed in this work. This method accommodates non-matching meshes across the interfaces between the subdomain boundaries and allows for sharp changes in mechanical material properties. Interface coupling operators that emanate via embedding of Discontinuous Galerkin ideas in the continuous Galerkin framework provide a unique avenue to embed physics-based data in the modeling and analysis of the system. Physics-based data, either in discrete or in distributed form can be embedded via the interface operators that are otherwise devised to enforce continuity of the fields across internal discontinuities. The least-squares form of the interface coupling operators is exploited for its inherent linear regression type structure, and it is shown that it helps improve the overall accuracy of the numerical solution. Method is applicable to multi-PDE class of problems wherein different PDEs are operational on adjacent domains across the common interface. The method also comes equipped with a residual based error estimation method which is shown to be applicable to test problems employed. Different test cases are employed to investigate the mathematical attributes of the method

Pore-scale Direct Numerical Simulation of Flow and Transport in Porous Media

This dissertation presents research on the pore-scale simulation of flow and transport in porous media and describes the application of a new numerical approach based on the discontinuous Galerkin (DG) finite elements to pore-scale modelling. In this approach, the partial differential equations governing the flow at the pore-scale are solved directly where the main advantage is that it does not require a body fitted grid and works on a structured partition of the domain. Furthermore this approach is locally mass conservative, a desirable property for transport simulation. This allows the investigation of pore-scale processes and their effect on macroscopic behaviour more efficiently. The Stokes flow in two and three dimensional disordered packing was solved and the flow field was used in a random-walk particle tracking model to simulate the transport through the packing. The permeabilities were computed and asymptotic behaviour of solute dispersion for a wide range of Péclet numbers was studied. The simulated results agree well with the data reported in the literature, which indicates that the approach chosen here is well suited for pore-scale simulation

Parallel computational strategies for modelling transient Stokes fluid flow.

The present work is centred on two main research areas; the development of finite element techniques for the modelling of transient Stokes flow and implementation of an effective parallel system on distributed memory platforms for solving realistic large-scale Lagrangian flow problems. The first part of the dissertation presents the space-time Galerkin / least-square finite element implicit formulation for solving incompressible or slightly compressible transient Stokes flow with moving boundaries. The formulation involves a time discontinuous Galerkin method and includes least-square terms in the variational formulation. Since the additional terms involve the residual of the Euler- Lagrangian equations evaluated over element interiors, it prevents numerical oscillation on the pressure field when equal lower order interpolation functions for velocity and pressure fields are used, without violating the Babuska-Brezzi stability condition. The space-time Galerkin / least-square formulation has been successfully extended into the finite element explicit analysis, in which the penalty based discrete element contact algorithm is adopted to simulate fiuid-structure or fluid-fluid particle contact. The second part of the dissertation focuses on the development of an effective parallel processing technique, using the natural algorithm concurrency of finite element formulations. A hybrid iterative direct parallel solver is implemented into the ELFEN/implicit commercial code. The solver is based on a non-overlapping domain decomposition and sub-structure approach. The modified Cholesky factorisation is used to eliminate the unknown variables of the internal nodes at each subdomain and the resulting interfacial equations are solved by a Krylov subspace iterative method. The parallelization of explicit fluid dynamics is based on overlapping domain decomposition and a Schwarz alternating procedure. Due to the dual nature of the overlapping domain decomposition a buffer zone between any two adjacent subdomains is introduced for handling the inter-processor communication. Both solvers are tested on a PC based interconnected network system and its performances are judged by the parallel speed-up and efficiency

A state of the art review of the Particle Finite Element Method (PFEM)

The particle finite element method (PFEM) is a powerful and robust numerical tool for the simulation of multi-physics problems in evolving domains. The PFEM exploits the Lagrangian framework to automatically identify and follow interfaces between different materials (e.g. fluid–fluid, fluid–solid or free surfaces). The method solves the governing equations with the standard finite element method and overcomes mesh distortion issues using a fast and efficient remeshing procedure. The flexibility and robustness of the method together with its capability for dealing with large topological variations of the computational domains, explain its success for solving a wide range of industrial and engineering problems. This paper provides an extended overview of the theory and applications of the method, giving the tools required to understand the PFEM from its basic ideas to the more advanced applications. Moreover, this work aims to confirm the flexibility and robustness of the PFEM for a broad range of engineering applications. Furthermore, presenting the advantages and disadvantages of the method, this overview can be the starting point for improvements of PFEM technology and for widening its application fields.Technology Innovation Program funded by the Ministry of Trade, Industry & Energy (MI, Korea), Grant/Award Number: 10053121; Universiti Teknologi PETRONAS (UTP) Internal Grant, Grant/Award Number: URIF 0153AAG24Peer ReviewedPostprint (published version