923 research outputs found

    Advancements in Fluid Simulation Through Enhanced Conservation Schemes

    Get PDF
    To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study new methods in solving the Navier-Stokes equations using models which have enhanced conservation qualities, in particular, the energy, momentum, and angular momentum conserving (EMAC) scheme. The EMAC scheme has gained popularity in the mathematics community over the past few years as a desirable method to model fluid flow. It has been proven to conserve energy, momentum, angular momentum, helicity, and others. EMAC has also been shown to perform better and maintain accuracy over long periods of time compared to other schemes. We investigate a fully discrete error analysis of EMAC and SKEW. We show that a problematic dependency on the Reynolds number is present in the analysis for SKEW, but not in EMAC under certain conditions. To further explore this concept, we include some numerical experiments designed to highlight these differences in the error analysis. Additionally, we include other projection methods to measure performance. Following this, we introduce a new EMAC variant which applies a differential spatial filter to the EMAC scheme, named EMAC-Reg. Standard models, including EMAC, require especially fine meshes with high Reynold\u27s numbers. This is problematic because the linear systems for 3D flows will be far too large and take an extraordinary amount of time to compute. EMAC-Reg not only performs better on a coarser mesh, but maintains conservation properties as well. Another topic in fluid flow computing that has been gaining recognition is reduced order models. This method uses experimental data to create new models of reduced computational complexity. We introduce the concept of consistency between a full order and a reduced order model, i.e., using the same numerical scheme for the full order and reduced order model. For inconsistency, one could use SKEW in the full order model and then EMAC for the reduced order model. We explore the repercussions of having inconsistency between these two models analytically and experimentally. To obtain a proper linear system from the Navier-Stokes equations, we must solve the nonlinear problem first. We will explore a method used to reduce iteration counts of nonlinear problems, known as Anderson acceleration. We will discuss how we implemented this using the finite element library deal.II \cite{dealII94}, measure the iteration counts and time, and compare against Newton and Picard iterations

    Cut finite element discretizations of cell-by-cell EMI electrophysiology models

    Full text link
    The EMI (Extracellular-Membrane-Intracellular) model describes electrical activity in excitable tissue, where the extracellular and intracellular spaces and cellular membrane are explicitly represented. The model couples a system of partial differential equations in the intracellular and extracellular spaces with a system of ordinary differential equations on the membrane. A key challenge for the EMI model is the generation of high-quality meshes conforming to the complex geometries of brain cells. To overcome this challenge, we propose a novel cut finite element method (CutFEM) where the membrane geometry can be represented independently of a structured and easy-to-generated background mesh for the remaining computational domain. Starting from a Godunov splitting scheme, the EMI model is split into separate PDE and ODE parts. The resulting PDE part is a non-standard elliptic interface problem, for which we devise two different CutFEM formulations: one single-dimensional formulation with the intra/extracellular electrical potentials as unknowns, and a multi-dimensional formulation that also introduces the electrical current over the membrane as an additional unknown leading to a penalized saddle point problem. Both formulations are augmented by suitably designed ghost penalties to ensure stability and convergence properties that are insensitive to how the membrane surface mesh cuts the background mesh. For the ODE part, we introduce a new unfitted discretization to solve the membrane bound ODEs on a membrane interface that is not aligned with the background mesh. Finally, we perform extensive numerical experiments to demonstrate that CutFEM is a promising approach to efficiently simulate electrical activity in geometrically resolved brain cells.Comment: 25 pages, 7 figure

    Nitsche method for Navier-Stokes equations with slip boundary conditions: Convergence analysis and VMS-LES stabilization

    Full text link
    In this paper, we analyze the Nitsche's method for the stationary Navier-Stokes equations on Lipschitz domains under minimal regularity assumptions. Our analysis provides a robust formulation for implementing slip (i.e. Navier) boundary conditions in arbitrarily complex boundaries. The well-posedness of the discrete problem is established using the Banach Ne\v{c}as Babu\v{s}ka and the Banach fixed point theorems under standard small data assumptions, and we also provide optimal convergence rates for the approximation error. Furthermore, we propose a VMS-LES stabilized formulation, which allows the simulation of incompressible fluids at high Reynolds numbers. We validate our theory through numerous numerical tests in well established benchmark problems

    Hybrid BEM-FEM for 2D and 3D dynamic soil-structure interaction considering arbitrary layered half-space and nonlinearities

    Get PDF
    Experiences and studies have shown that soil-structure interaction (SSI) effect has a vital role in the dynamic behaviour of a soil-structure system. Despite this, analyses involving dynamic SSI are still challenging for practicing engineers due to their complexity and accessibility. In this thesis, the hybrid BEM-FEM implementation is aimed at practicality by combining commercial software and an in-house code. The pre-processing task can be performed under one graphical environment, and it is enhanced with the capability to compute different types of dynamic sources and other improvements to increase its efficiency, accuracy, and modeling flexibility. Further, the underlying soil is commonly a layered profile with arbitrary geometries. Most existing solutions solve the problem through simplification of the geometry and pattern. One of the main contributions in this thesis is the development of layer-wise condensation method to solve these cases using hybrid BEM-FEM. The method significantly reduces the computational memory requirement. Another challenge in the dynamic SSI addressed in this work is the consideration of secondary nonlinearities. Existing solutions using the time domain BEM and iterative hybrid method are computationally costly, and implementation of such a hybrid method on commercial software is tedious. The solution to address this case using a sequential frequency-time domain procedure is presented. The relatively simple approach makes it possible to consider the nonlinearities in the simulation without using the time domain BEM and without requiring additional iterations. Case studies demonstrating the application of the enhanced hybrid method are presented including cases of bridges, containment structures, and a 3D multi-storey structure under point source and double-couple sources. These case studies illustrate the role of following critical factors such as the site effect, inhomogeneity, and nonlinearities

    Error analysis for a Crouzeix-Raviart approximation of the variable exponent Dirichlet problem

    Full text link
    In the present paper, we examine a Crouzeix-Raviart approximation for non-linear partial differential equations having a (p(⋅),ή)(p(\cdot),\delta)-structure. We establish a medius error estimate, i.e., a best-approximation result, which holds for uniformly continuous exponents and implies a priori error estimates, which apply for H\"older continuous exponents and are optimal for Lipschitz continuous exponents. The theoretical findings are supported by numerical experiments.Comment: 28 pages, 4 tables, this article extends the methods in arXiv:2210.12116 to the variable exponent settin

    Design and analysis of a hybridized discontinuous Galerkin method for incompressible flows on meshes with quadrilateral cells

    Full text link
    We present and analyse a hybridized discontinuous Galerkin method for incompressible flow problems using non-affine cells, proving that it preserves a key invariance property that illudes most methods, namely that any irrotational component of the prescribed force is exactly balanced by the pressure gradient and does not influence the velocity field. This invariance property can be preserved in the discrete problem if the incompressibility constraint is satisfied in a sufficiently strong sense. We derive sufficient conditions to guarantee discretely divergence-free functions are exactly divergence-free, and give examples of divergence-free finite elements on meshes containing triangular, quadrilateral, tetrahedral, or hexahedral cells generated by a (possibly non-affine) map from their respective reference cells. In the case of quadrilateral cells, we prove an optimal error estimate for the velocity field that does not depend on the pressure approximation. Our theoretical analysis is supported by numerical results

    Fast boundary element methods for the simulation of wave phenomena

    Get PDF
    This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) fĂŒr ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nĂŒtzliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten fĂŒhrt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren EintrĂ€ge die Integration singulĂ€rer Kernfunktionen ĂŒber Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln fĂŒr die schwierigsten FĂ€lle. Dies ermöglicht die genaue Berechnung der MatrixeintrĂ€ge mit geringerem Rechenaufwand als konventionelle numerische Integration. Außerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation prĂ€sentiert, der die Matrizen komprimiert und die KomplexitĂ€t der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der rĂ€umlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die ĂŒberlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt

    Least-squares finite elements for distributed optimal control problems

    Full text link
    We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is inf⁥\inf--sup⁥\sup stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented

    Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions

    Full text link
    In this work, we present a phase-field model for tumour growth, where a diffuse interface separates a tumour from the surrounding host tissue. In our model, we consider transport processes by an internal, non-solenoidal velocity field. We include viscoelastic effects with the help of a general Oldroyd-B type description with relaxation and possible stress generation by growth. The elastic energy density is coupled to the phase-field variable which allows to model invasive growth towards areas with less mechanical resistance. The main analytical result is the existence of weak solutions in two and three space dimensions in the case of additional stress diffusion. The idea behind the proof is to use a numerical approximation with a fully-practical, stable and (subsequence) converging finite element scheme. The physical properties of the model are preserved with the help of a regularization technique, uniform estimates and a limit passage on the fully-discrete level. Finally, we illustrate the practicability of the discrete scheme with the help of numerical simulations in two and three dimensions

    Unstabilized hybrid high-order method for a class of degenerate convex minimization problems

    Get PDF
    The relaxation procedure in the calculus of variations leads to minimization problems with a quasi-convex energy density. In some problems of nonlinear elasticity, topology optimization, and multiphase models, the energy density is convex with some convexity control plus two-sided pp-growth. The minimizers may be non-unique in the primal variable, but define a unique stress variable σ\sigma. The approximation by hybrid high-order (HHO) methods utilizes a reconstruction of the gradients in the space of piecewise Raviart-Thomas finite element functions without stabilization on a regular triangulation into simplices. The application of the HHO methodology to this class of degenerate convex minimization problems allows for a unique H(Ă·)H(\div) conform stress approximation σh\sigma_h. The a priori estimates for the stress error σ−σh\sigma - \sigma_h in the Lebesgue norm are established for mixed boundary conditions without additional assumptions on the primal variable and lead to convergence rates for smooth solutions. The a posteriori analysis provides guaranteed error control, including a computable lower energy bound, and a convergent adaptive scheme. Numerical benchmarks display higher convergence rates for higher polynomial degrees and provide empirical evidence for the superlinear convergence of the lower energy bound. Although the focus is on the unstabilized HHO method, a detailed error analysis is provided for the stabilized version with a gradient reconstruction in the space of piecewise polynomials
    • 

    corecore