Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

A Stabilized and Coupled Meshfree/Meshbased Method for the Incompressible Navier-Stokes Equations : Part I: Stabilization

A stabilized meshfree Galerkin method is employed for the approximation of the incompressible Navier-Stokes equations in Eulerian or arbitrary Lagrangian-Eulerian (ALE) formulation. Equal-order interpolations for velocities and pressure are used. It is well-known from the meshbased context, i.e. from finite volume and finite element methods, that in convection-dominated flow problems in Eulerian or ALE formulation, stabilization is a crucial requirement for reliable solutions. Also, stabilization is needed in order to enable equal-order interpolations of the incompressible Navier-Stokes equations. Standard stabilization techniques, developed in a meshbased context, are extended to meshfree methods. It is found that the same structure of the stabilization schemes may be used, however the aspect of the stabilization parameter, weighing the stabilization terms, has to be reconsidered. In part II of this work, the resulting stabilized meshfree Galerkin method is coupled with a stabilized finite element method. The resulting coupled method employs the comparatively costly meshfree Galerkin method only where it is needed---i.e. in areas of the domain, where a mesh is difficult to maintain---, and the efficient meshbased finite element method is used in the rest of the domain. The fluid solver resulting from this technique is able to solve complex flow problems, involving large deformations of the physical domain and/or moving and rotating obstacles

A nodal-based implementation of a stabilized Finite Element Method for Incompressible Flow Problems

The objective of this paper is twofold. First, a stabilized finite element method for the incompressible Navier-Stokes is presented, and several numerical experiments are conducted to check its performance. This method is able to deal with all the instabilities that the standard Galerkin method presents, namely, the pressure instability, the instability arising in convection dominated situations and also the less popular instabilities found when the Navier-Stokes equations have a dominant Coriolis force, or there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal-based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over element and thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this given rise to questions regarding the consistency and the conservation properties of the final scheme that are addressed in this paper

The emergence of biofilms:Computational and experimental studies

The response of biofilms to any external stimuli is a cumulative response aggregated from individual bacteria residing within the biofilm. The organizational complexity of biofilm can be studied effectively by understanding bacterial interactions at cell level. The overall aim of the thesis is to explore the complex evolutionary behaviour of bacterial biofilms. This thesis is divided into three major studies based on the type of perturbation analysed in the study. The first study analyses the physics behind the development of mushroom-shaped structures from the influence of nutrient cues in biofilms. Glazier-Graner-Hogeweg model is used to simulate the cell characteristics. From the study, it is observed that chemotaxis of bacterial cells towards nutrient source is one of the major precursors for formation of mushroom-shaped structures. The objective of the second study is to analyse the impact of environmental conditions on the inter-biofilm quorum sensing (QS) signalling. Using a hybrid convection-diffusion-reaction model, the simulations predict the diffusivity of QS molecules, the spatiotemporal variations of QS signal concentrations and the competition outcome between QS and quorum quenching mutant bacterial communities. The mechanical effects associated with the fluid-biofilm interaction is addressed in the third study. A novel fluid-structure interaction model based on fluid dynamics and structural energy minimization is developed in the study. Model simulations are used to analyse the detachment and surface effects of the fluid stresses on the biofilm. In addition to the mechanistic models described, a separate study is carried out to estimate the computational efficiency of the biofilm simulation models

A Stabilized and Coupled Meshfree/Meshbased Method for Fluid-Structure Interaction Problems

A method is presented which combines features of meshfree and meshbased methods in order to enable the simulation of complex flow problems involving large deformations of the domain or moving and rotating objects. Conventional meshbased methods like the finite element method have matured as standard tools for the simulation of fluid and structure problems. They offer efficient and reliable approximations, provided that a conforming mesh with sufficient quality can be maintained throughout the simulation. This, however, may not be guaranteed for complex fluid and fluid-structure interaction problems. Meshfree methods on the other hand approximate partial differential equations based on a set of nodes without the need for an additional mesh. Therefore, these methods are frequently used for problems where suitable meshes are prohibitively expensive to construct and maintain. This advantage of meshfree methods comes at the price of being considerably more time-consuming than their meshbased counterparts. A coupled meshfree/meshbased fluid solver is developed which combines the advantages of both methodologies. A meshfree method, closely related to the element-free Galerkin method, is used in small parts of the domain where a mesh is difficult to maintain, whereas the efficient meshbased finite element method is employed in the rest of the domain. Major steps in the development of the coupled flow solver are the extension of standard stabilization methods to meshfree approximations, and the realization of the coupling on the level of the shape functions, which are involved in the approximation of the governing equations. Concerning the stabilization, it is found that the same structure of the standard stabilization schemes may be used for meshfree approximations, however, the aspect of the stabilization parameter, weighing the stabilization terms, requires special care. For the coupling, these requirements for a reliable stabilization lead to modifications of the existing coupling approaches. The stabilized and coupled flow solver is verified and used for the simulation of geometrically complex fluid-structure interaction problems. Conventional meshbased methods are not suitable for the approximation of these test cases due to the prohibitively large deformations of the geometry.Es wird ein Verfahren vorgestellt, das Eigenschaften netzfreier und netzbasierter Methoden nutzt, um die Simulation komplexer Strömungsprobleme zu ermöglichen, bei denen große Verformungen des Gebietes oder sich bewegende und rotierende Objekte in der Strömung berücksichtigt werden können. Konventionelle netzbasierte Verfahren wie die Finite Element Methode haben sich zu Standardwerkzeugen bei der numerischen Analyse von Strömungen und Festkörpern entwickelt. Sie ermöglichen eine schnelle und zuverlässige Approximation, vorausgesetzt, daß ein konformes Netz mit geeigneter Qualität während der gesamten Simulation aufrecht erhalten werden kann. Dies kann jedoch bei komplexen Strömungs- und Fluid-Struktur-Interaktionsproblemen eine entscheidende Einschränkung in der Anwendbarkeit dieser Verfahren darstellen. Netzfreie Verfahren approximieren dagegen die zugrundeliegenden Modellgleichungen nur in Abhängigkeit einer gegebenen Knotenverteilung, ohne ein Netz zu erfordern, das die Konnektivität a priori festlegt. Deshalb werden diese Verfahren häufig dort eingesetzt, wo die Generierung oder Erhaltung geeigneter Netze nicht mit vertretbarem Aufwand möglich ist. Allerdings geht der Vorteil der Netzunabhängigkeit bei netzfreien Verfahren einher mit deutlich aufwendiger zu konstruierenden Ansatzfunktionen, was eine empfindliche Erhöhung des Rechenaufwandes bei der Integration der zugrundeliegenden Gleichungen erfordert. Hierin wird ein gekoppelter netzfreier/netzbasierter Strömungslöser entwickelt, der die Vorteile beider Verfahren kombiniert. Ein netzfreies Verfahren, das eng mit der "element-free" Galerkin Methode verwandt ist, wird nur in kleinen Teilen des Gebietes verwendet, wo ein Netz zu Schwierigkeiten führt, und im gesamten Rest des Gebiets wird die Finite Element Methode als effizientes netzbasiertes Standardverfahren eingesetzt. Wichtige Schritte bei der Entwicklung der gekoppelten Methode sind die Erweiterung von Standard-Stabilisierungsansätzen auf netzfreie Verfahren und die Umsetzung der Kopplung auf der Ebene der Ansatzfunktionen, die für die Approximation eingesetzt werden. Bezüglich der Stabilisierung wird gezeigt, daß die Struktur der verschiedenen Stabilisierungsmethoden direkt auf netzfreie Verfahren anwendbar ist, allerdings erfordert die Wahl geeigneter Stabilisierungsparameter, die den Stabilisierungseinfluß wichten, besondere Beachtung. Bei der Kopplung führen die Voraussetzungen, die für eine zuverlässige Stabilisierung gegeben sein müssen, zu Modifikationen der Standardansätze. Der gekoppelte Strömungslöser wird verifiziert und für die Simulation geometrisch komplexer Fluid-Struktur-Interaktionsprobleme eingesetzt. Die Fähigkeiten des gekoppelten Verfahrens werden dabei sichtbar, denn klassische netzbasierte Standardverfahren versagen bei den gezeigten Anwendungsbeispielen

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

Continuum and discrete approach in modeling biofilm development and structure: a review

The scientific community has recognized that almost 99% of the microbial life on earth is represented by biofilms. Considering the impacts of their sessile lifestyle on both natural and human activities, extensive experimental activity has been carried out to understand how biofilms grow and interact with the environment. Many mathematical models have also been developed to simulate and elucidate the main processes characterizing the biofilm growth. Two main mathematical approaches for biomass representation can be distinguished: continuum and discrete. This review is aimed at exploring the main characteristics of each approach. Continuum models can simulate the biofilm processes in a quantitative and deterministic way. However, they require a multidimensional formulation to take into account the biofilm spatial heterogeneity, which makes the models quite complicated, requiring significant computational effort. Discrete models are more recent and can represent the typical multidimensional structural heterogeneity of biofilm reflecting the experimental expectations, but they generate computational results including elements of randomness and introduce stochastic effects into the solutions