The programme will be updated as presentations are scheduled.
The Finite Element Method is a very effective and versatile tool to approximate solutions to complex mathematical models. It is especially well suited for applications from structural mechanics and engineering, but there are also numerous surprising applications from everyday life including biology and control. In this first introductory talk an elementary boundary value problem will be used to illustrate the Finite Element Method. The aim is to introduce the concepts needed in preparation for more advanced topics. It will be accessible to post-graduate students who have not encountered the method before.
In this second introductory talk to the Finite Element Method, different basis functions will be considered. Elementary examples will still be used to illustrate the main ideas. The aim of the introductory talks is to introduce the concepts needed in preparation for more advanced topics. It will be accessible to post-graduate students who have not encountered the method before (except for the first introductory talk).
This talk will showcase the modelling of blood flowing through a fistula using the finite element method. It will briefly look at various techniques employed in this application such a multidimensionality, non-linearity, mixed finite elements, and stable time integration. The talk will not go into detail; but the aim is rather to stimulate curiosity and leave clues that the intrigued student can hunt down to discover the riches available in this field.
Despite continuous advancements in medical technologies and imaging, cardiovascular disease forms the leading cause of mortality and death. Computational modeling provides a low-cost, non-invasive modality that complements animal testing and routine clinical care. Simulation-based diagnosis has demonstrated a growing impact in the clinic, ultimately leading to improved decision-making and patient outcomes. While this translation was successfully achieved in vascular flow applications, cardiac biomechanics (representing tissue mechanics and blood flow in the heart chambers) has remained distant, partly due to the significant cost and complexity involved in modeling the underlying function. These include multi-physics multi-scale interactions, feedback mechanisms, moving boundaries, and fluid-structure interaction effects. In this talk, I will present a robust and efficient framework to perform patient-specific modeling of ventricular hemodynamics with examples from single ventricle physiology. I will then briefly discuss the utility of the framework in embryonic cardiac flow modeling to understand shear-regulated mechanotransduction during cardiac development. Finally, I will present our novel approach to multiphysics modeling of heart function to address the above challenges and demonstrate examples of verification and validation.
Defects in a liquid crystal director field can arise due to external factors such as applied electric or magnetic fields, or the constraining geometry of the cell containing the liquid crystal material. Understanding the formation and dynamics of defects is important in the design and control of liquid crystal devices, and poses significant challenges for numerical modelling. In this talk we consider the numerical solution of a Q-tensor model of a nematic liquid crystal, where defects arise through rapid changes in the Q-tensor over a very small physical region in relation to the dimensions of the liquid crystal device. The efficient solution of the resulting six coupled partial differential equations is achieved using a finite element based adaptive moving mesh approach, where an unstructured triangular mesh is adapted towards high activity regions, including those around defects. Spatial convergence studies show the adaptive method to be optimally convergent using quadratic triangular finite elements. The full effectiveness of the method can be seen when solving a challenging two-dimensional dynamic Pi-cell problem involving the creation, movement, and annihilation of defects.
Discontinuous Galerkin (DG) methods constitute a generalization of the standard conforming finite element method in that they allow for discontinuities in the approximation across interior element boundaries. They have various advantages, for example, in representing complex geometries and irregular meshes with hanging nodes, so that they are well suited to adaptive strategies. The purpose of this presentation is, first, to provide an introduction to DG methods for elliptic problems. The second objective is to examine conditions under which DG methods are locking-free, that is, uniformly convergent, for problems with a large parameter such as occurs in situations of near-incompressibility in elasticity. It is shown that while standard DG approaches are indeed locking-free for piecewise-linear approximations on meshes of triangles or tetrahedra in two and three dimensions respectively, the situation for bilinear approximation on quadrilaterals (resp. trilinear on hexahedra) is more complex, and requires modifications of the standard approach to render these stable and convergent. These modifications are discussed and their numerical performance illustrated with various examples.
Nearly-incompressible elastic materials, and nearly-inextensible elastic materials, are two applications in which the standard finite element method yields poor approximations, due to the large material parameters involved in each. Discontinuous Galerkin (DG) methods, in which the interelement continuity requirement of the standard method is relaxed, circumvent the problem in each case, and provide uniformly convergent results with respect to both the compressibility and the extensibility parameters. The purpose of this presentation is to show the effectiveness of DG methods in these two contexts. Firstly, numerical results will be shown for problems in near-incompressibility with meshes of quadrilateral elements, with a focus on the effect of decreasing element shape regularity. Secondly, a DG method for transversely isotropic elasticity will be presented, with theoretical and numerical convergence results that highlight performance in the inextensible limit.
In this visual talk, we will explore and exploit discretization errors in finite element based shape optimization. The aim is to understand the implications and limitations of certain discretization choices. Shape optimization is concerned with finding an optimal surface to perform a certain task subject to some restrictions. As the shape changes the underlying discretization changes. Smooth and continuous discretization changes result in smooth and continuous response changes, often realized using mesh movement strategies that preserve mesh topology. On the other hand, non-smooth and abrupt discretization changes result in discontinuous response changes, mostly the result of remeshing that changes mesh topology. The former is restrictive, while the latter apparently challenging. Join the discussion and equip yourself with understanding discontinuous responses and how to elegantly handle them to enable multi-fidelity models in finite element based shape optimization.
Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray-Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear.
In the derivation of a macroscopic model such as the deterministic Gray-Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g. fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations.
The randomness leads to a variate of new phenomena and may have a highly non-trivial impact on the behaviour of the solution. E.g. it has been shown by numerical modelling that the stochastic extension leads to a broader range of parameters with Turing patterns by a genetically engineered synthetic bacterial population in which the signalling molecules form a stochastic activator-inhibitor system. The stochastic extension may lead to multistability and noise-induced transitions between different states.
In the talk, we will introduce the Gray Scott system, which is a special case of an activator-inhibitor system. Then, we introduce its numerical modelling and highlight the proof of convergence.
References:
https://www.sciencedirect.com/science/article/abs/pii/S0377042719303322?via%3Dihub
Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations. J. Comput. Appl. Math. 364.
Shape optimization is a research area of high industrial relevance. Besides the canonical usage of FEM for simulation tasks, shape optimization can significantly profit from variational formulations and thus FEM. This pertains to the proper formulation of necessary optimality conditions, the proper implementation of shape metrics and also very recent approaches for mesh optimization within shape optimization algorithms leading to the so-called pre-shape calculus. This talks illustrates all these effects on the basis of current applications in science and industry.
We propose a class of adaptive enriched Galerkin-characteristics finite element methods for efficient numerical solution of the convection-diffusion equations and the incompressible Navier-Stokes equations in primitive variables. The proposed approach combines the modified method of characteristics to deal with convection terms, the finite element discretization to manage irregular geometries, a direct conjugate gradient algorithm to solve the Stokes problem, and an adaptive L2-projection using quadrature rules to improve the efficiency and accuracy of the method. In the present study, the gradient of the velocity field is used as an error indicator for adaptation of enrichments by increasing the number of quadrature points where it is needed without refining the mesh. Unlike other adaptive finite element methods for incompressible Navier-Stokes equations, linear systems in the proposed enriched Galerkin-characteristics finite element method preserve the same structure and size at each refinement in the adaptation procedure. We examine the performance of the proposed method for a coupled Burgers problem with known analytical solution and for the benchmark problem of flow past a circular cylinder. We also solve a transport problem in the Mediterranean Sea to demonstrate the ability of the method to resolve complex flow features in irregular geometries. Comparisons to the conventional Galerkin-characteristics finite element method are also carried out in the current work. The computed results support our expectations for an accurate and highly efficient enriched Galerkin-characteristics finite element method for incompressible Navier-Stokes equations.
In this talk I will start by motivating the stochastic wave equation through an application to underwater acoustics. The approach to wave equation will be through re-writing as a first order system and using the semi-group generated. I will introduce the basic ideas of space time noise before turning attention to the numerical approximation of the stochastic wave equation both in time and in space. In time we use a Verlet type integrator and in space we use a discontinuous finite element method DGFEM. I will present results on the convergence of the scheme. Finally we examine energy in the system before presenting some numerical experiments.
I will present an overview of our group's recent efforts at simulating two-phase flows with moving interfaces. The first part of the talk will describe a diffuse-interface methodology that we have developed, which uses finite elements with adaptive meshing for computing flow in complex geometries. The second part will be a case study focusing on a few
problems with distinctive features, including drop dynamics in viscoelastic fluids and self-assembly of drops in a liquid crystalline medium.
In the first part of this talk we will look at how error estimates for the semi-discrete and fully discrete Galerkin approximations of a general linear second order hyperbolic partial differential equation with general damping (which includes boundary damping) are derived. The results can be applied to a variety of cases e.g. vibrating systems of linked elastic bodies.
In the second part we will investigate the Mixed Finite Element method applied to a Timoshenko beam model, to determine the advantages of the method and to explain why the method yields improved results.
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