Polygonal finite element modeling of crack propagation via automatic adaptive mesh refinement

Summary An adaptive three-dimensional mesh refinement method based on the law of mass conservation has been introduced and tested using some analytical velocity fields as accurate in identifying singular point, asymptotic plane and drawing closed streamlines. This paper further investigates the adaptive mesh refinement method using a velocity field that is due to a uniform strain and a point vortex. Similar results have been obtained.

Adaptive mesh refinement in polygonal finite elements using clustering technique

A new modeling strategy is adopted to simulate the compression cycle of a small hermetic reciprocating compressor. The simulation model is based on a transient three-dimensional formulation to account for time and spatial variations of quantities in the computational domain. The discretization method automatically generates an orthogonal grid with adaptive mesh refinement (AMR) that locally adjusts the grid at each time step to yield accurate results. The time-dependent flow field and the valve dynamics were coupled and solved simultaneously, including the presence of suction and discharge mufflers. Numerical results were validated via comparison with experimental data and an analysis was carried out to identify the contributions of mufflers and valves to the thermodynamic losses.

The aim of this work is to provide a solver for viscoelastic multi-phase flows within the Bounded Support Spectral Solver (BoSSS) currently under development at the Chair of Fluid Dynamics at the Technical University of Darmstadt. The discretisation in BoSSS consists of a high-order Discontinuous Galerkin (DG) method for single-phase flow and a high-order eXtended Discontinuous Galerkin (XDG) method for the multi-phase purpose. The solver shall be used to investigate numerically the behaviour of viscoelastic droplets. The macroscopic Oldroyd B model which is used in a wide range of applications is chosen as the constitutive model. A detailed derivation of the system of equations including the modeling principles for the Oldroyd B model is presented. A DG discretisation of the system of equations including the Local Discontinuous Galerkin (LDG) method is presented after introducing the field of the DG method. The derivation of appropriate flux functions for the constitutive equations and the extra stress tensor are one of the key derivations of this scientific work. Difficulties arising in the numerical solution of viscoelastic flow problems for higher Weissenberg numbers for different discretisation methods are due to the convection dominated, mixed hyperbolic-elliptic-parabolic nature of the system of equations. Several strategies are presented which overcome these problems and are known from the literature. A key achievement of this scientific work is the application of the LDG method, originally developed for a hyperbolic system of equations for a Newtonian fluid, on the viscoelastic system of equations which renders methods for preserving ellipticity unnecessary. Furthermore, various strategiesn to enhance and to support convergence of the solution of the DG discretised system are presented. These are the Newton method with different approaches determining the Jacobian of the system, a homotopy continuation method based on the Weissenberg number for a better initial guess for the Newton method, and a troubled cell indicator combined with an artificial diffusion approach or an adaptive mesh refinement strategy, respectively. For the completeness of this work the XDG method is presented using a sharp interface approach with a signed distance level-set function as it is implemented in BoSSS. The single-phase solver is combined with these methods and appropriate flux functions for the interface are implemented to enable multi-phase applications for viscoelastic fluid. Several numerical experiments are conducted to verify and to validate the viscoelastic singlephase solver and to show the capability of the viscoelastic multi-phase solver to simulate viscoelastic droplets. Advantages and disadvantages of the implementation and an outlook for future research can be found in the conclusion.

Combining Shape Optimization (SO) with Adaptive Mesh Refinement (AMR) potentially offers a higher accuracy and higher computational efficiency, especially if the applied target error for AMR is reduced in the course of the optimization process. The disadvantage of that approach is that the rate of convergence of the corresponding optimization processes can be significantly lower as compared to processes which apply a fixed target error for AMR. In the present paper the so-called Multipoint Approximation Method (MAM) is used as a basis for SO in conjunction with AMR. Several techniques for improvement of the rates of convergence are presented and investigated. Firstly, alternative algorithms for determining the approximation functions using a weighted least squares method are investigated. The focus is on weights which depend on the discretization errors. Secondly, different strategies for moving and resizing the search sub-regions in the space of design variables are presented. The proposed methods are illustrated by means of several optimization problems in which the effect of AMR with changing discretization errors is modelled by artificially introduced numerical noise.

HiFlow3 Version 2.0 continues the path as a multi-purpose finite element software, which provides powerful tools for efficient and accurate solution of a wide range of problems modeled by partial differen- tial equations (PDEs). New features and functionalities, which allow to run numerical simulations with more advanced solution algorithms and discretizations in comparison to previous releases of HiFlow3, have been implemented. These comprise Uncertainty Quantification (UQ), energy-efficient multigrid techniques, Schur complement solvers for saddle-point problems, extended third-party library support, and adaptive local mesh refinement in a parallel computing environment. Furthermore, HiFlow3 has been successfully integrated into advanced and state-of-the-art simulation environments by means of the Medical Simulation Markup Language (MSML), for example.The presented new algorithms and features as well as general under-the-hood improvements have enabled excellent and relevant research activities in the fields of both medical engineering and meteorol- ogy and environmental sciences. The described show cases demonstrate the potential and advantages, which HiFlow3 can offer in performing a numerical simulation by means of finite element methods. Especially, the high performance computing capabilities of HiFlow3 – not only in the mentioned fields of applications, but also in general – have been significantly improved in Version 2.0.

This thesis investigates the flexibility associated to Discontinuous Galerkin (DG) discretization on very general meshes obtained by means of agglomeration techniques. The work begins with a brief overview of the main tools that have been extended or specifically developed to deal with arbitrarily shaped elements in the DG context. Then two different implementations of the BRMPS scheme introduced by Bassi, Rebay, Mariotti, Pedinotti and Savini in [16] for the DG discretization of the Laplace operator on arbitrarily shaped elements have been presented. The validation of the scheme on a Poisson problem shows that the discrete polynomial space preserves optimal convergence properties. The discretization of the second order differential operator has been directly extended to the Navier-Stokes equations and the Reynolds Averaged Navier-Stokes (RANS) equations coupled with the k-w turbulence model of Wilcox [54]. In this regard, an implicit time integration strategy has been considered and assessed on classical validation test cases for the compressible f uid dynamics. Then a simple alternative approach to high-order mesh generation is presented. Indeed, once a standard fine grid able to provide an accurate domain discretization has been produced by means of standard low-order grid generation tools, a computational mesh suitable for the desired accuracy and computationally affordable can be obtained via agglomeration while keeping the boundary resolution of the fine grid. The effectiveness of this approach in representing the geometry of the domain is numerically assessed both on a Poisson model problem and on challenging inviscid and viscous test cases. Finally, the freedom in simply defining the topology of agglomerated meshes leads to a nonstandard approach to h-adaptivity that exploits adaptive agglomeration coarsening of a properly fine underlying grid. The effectiveness of this approach has been assessed on test cases involving both error-based and fl ow feature-based simple estimators.

The aim of this paper is to utilize a numeri- cal model to compute bubble motion in quiescent Newto- nian and viscoelastic liquids. For our numerical method, we use a coupled level set and volume-of-fluid method with a second order treatment for the jump conditions related to surface tension. We investigate axisymmetric gas-liquid systems with large density and viscosity ratios as well as buoyancy-driven flows with complex changes in topology. We present comparisons to previous compu- tational results as well as experimental results. keyword: Level set, Volume-of-fluid, Viscoelastic, Projection method, Adaptive mesh refinement

The present dissertation deals with the application of an adaptive mesh hp-refinement method to the time-dependent compressible Euler equations describing the flow of an inviscid fluid. The physical simulation space is a two-dimensional channel with bump through which an ideal gas flows. A big advantage of the Euler equations is the applicability to different Mach regimes. Thus, smooth solutions and those containing discontinuities can be simulated. As the underlying numerical technique, a pseudo-spectral element method using least squares is used. The grid consists of rectangles that have to be transformed into curved deformed elements in the area of the bump. The h-refinement is realized by means of a quadtree and the p-refinement by means of a stepwise increase of the polynomial degree by 2, whereby hanging nodes can arise. Compared to previous research, the Euler equations are examined with respect to four different refinement criteria in combination with the hp-adaptive pseudo-spectral element method. The refinement criteria are the gradient criterion, two spectral criteria and an edge detection method. For a first determination of the suitability of the criteria for the h-refinement and for a validation of the pseudo-spectral element method, these are used for the solution of three steady state convection equations for which the exact solutions are known. Finally, the full method is applied to three test problems for the Euler equations. The flow through the channel is realized by a smooth subsonic flow, a transonic flow with a shock, and a supersonic flow that contains reflective shocks. Since no analytical solution is known, we compare our results to other works that consider stationary grids for these test cases. It turns out that the applied adaptive method refines the grid in the regions of the stagnation points and those that have discontinuities. Thus prior knowledge about the location of the problem areas is not necessary. A smaller number of elements and locally higher polynomial degrees reduce the sizes of the systems of equations to be solved.

High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows

Towards adaptive kinetic-fluid simulations of weakly ionized plasmas

In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.

High-order adaptive mesh refinement offers the potential for accurate and efficient resolution of problems in fluid dynamics and other fields where a wide range of length scales is present. A critical requirement for the interface closures used with these methods is stability in the context of hyperbolic systems of partial differential equations. In this study, a class of energy-stable high-order finite-difference interface closures is presented for grids with step resolution changes in one dimension. Asymptotic stability in time for these schemes is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface closures compatible with interior fourth- and sixth-order explicit, and fourth-order implicit centered schemes are presented. Validation tests include linear and nonlinear problems in one and in two dimensions with tensor-product grid refinement. A second class of stable high-order interface closures is presented for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Stability is achieved by again imposing a summation-by-parts condition, resulting in nondissipative closures, at the cost of accuracy at corner interfaces. Interface stencils for an explicit fourth-order finite-difference scheme are presented for each geometry. Validation tests confirm the stability and accuracy of these closures for linear and nonlinear problems. The Richtmyer-Meshkov instability is investigated using a novel first-order perturbation of the two-dimensional Navier-Stokes equations about a shock-resolved base flow. The computational domain is efficiently resolved using the one-dimensional fourth-order interface scheme. Results are compared to analytic models of the instability, showing agreement with predicted asymptotic growth rates in the inviscid range, while significant discrepancies are noted in the transient growth phase. Viscous effects are found to be poorly predicted by existing models.

Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic simulations in particular, this distinguishing characteristic translates into an enhanced capability for conformal and multi-scale modeling. The method also facilitates adaptive discretization refinements, which provides an important tool to decrease memory consumption and computation time. In this paper, a refinement strategy is introduced for RPIM. In the proposed node adaptation algorithm, the accuracy of a solution is increased iteratively based on an initial solution with a coarse discretization. In contrast to the commonly used residual-based adaptivity algorithms, this definition is extended by an error estimator based on the solution gradient. In the studied cases this strategy leads to increased convergence rates compared with the standard algorithm. Numerical examples are provided to illustrate the effectiveness of the algorithm.

Gegenstand dieser Arbeit ist die numerische Analysis von Optimierungsproblemen mit partiellen Differentialgleichungen (PDEs), deren Zustand oder Steuerung punktweisen Ungleichungsbeschränkungen unterliegt. Wir interessieren uns insbesondere für nichtkonvexe Probleme mit semilinearer Zustandsgleichung. Es liegt in der Natur derartiger Probleme, dass eine Lösung oftmals nur numerisch gefunden werden kann. Man interessiert sich deshalb für den Fehler zwischen einer (lokalen) Lösung des kontinuierlichen Problems und einer zugehörigen (lokalen) diskreten Lösung. Eine umfassende Diskussion des kontinuierliche Problems ist dabei eine Grundvoraussetzung. Insbesondere bei vorhanden punktweisen Zustandsbeschränkungen treten spezifische Schwierigkeiten sowohl analytischer als auch numerischer Art auf, denen man entweder direkt oder mit Hilfe von Regularisierungsansätzen begegnen kann. Mit dieser Dissertation leisten wir auf verschiedene Weise neue Beiträge zur Diskussion von Optimalsteuerungsproblemen mit punktweisen Zustandsschranken aber auch reinen Kontrollschranken. Nach kurzer Einführung in die Thematik und Bereitstellung gewisser Grundlagen beschäftigen wir uns in Kapitel 3 mit einem elliptischen semiinfiniten Optimalsteuerungsproblem. Bekannte Resultate zu notwendigen und hinreichenden Optimalitätsbedingungen, die vergleichsweise hohe Regularität von Lösungen elliptischer PDEs und die Endlichdimensionalität des Steuerungsraumes lassen eine direkte Diskussion von a priori Diskretisierungsfehlerabschätzungen für dieses Problem zu. Unser Hauptergebnis in diesem Kapitel ist eine a priori Fehlerschranke der Ordnung O(h^2|ln h|) für lokale Lösungen einer Finite-Elemente-Diskretisierung des Optimalsteuerungsproblems mit Gitterweite h in einem zweidimensionalen Ortsgebiet. Dazu stellen wir gewisse Annahmen an die Struktur der aktiven Menge. In Kapitel 4 betrachten wir ein parabolisches Optimalsteuerungsproblem mit punktweise beschräkten Steuerungsfunktionen, semilinearer Zustandsgleichung und punktweisen Zustandsbeschränkungen im gesamten Orts-Zeit-Gebiet. Im Gegensatz zu dem in Kapitel 3 diskutierten elliptischen Problem sind hier u.a. hinreichende Bedingungen zweiter Ordnung nur für eindimensionale Ortsgebiete verfügbar. Auch läßt die Verwendung von Steuerungsfunktionen an Stelle endlich vieler Parameter keine sinnvollen a priori Annahmen an die Struktur der aktiven Menge zu. Wir regularisieren daher das Problem mit der auf Meyer, Rösch und Tröltzsch zurückgehenden Lavrentievregularisierung, und können so unter anderem auf bekannte Resultate zurückgreifen, die eine höhere Regularität der Lagrangeschen Multiplikatoren sichern und eine tiefergehende Analysis ermöglichen. Wir beweisen ein Konvergenzresultat für lokale Lösungen des regularisierten Problems und weisen die lokale Eindeutigkeit regularisierter Lösungen nach. In Kapitel 5 untersuchen wir die Finite-Element-Diskretisierung eines kontrollbeschränkten parabolischen Optimalsteuerungsproblems mit semilinearer Zustandsgleichung. Wir beweisen Fehlerordnungen für diskrete lokale Lösungen in der L^2-Norm. Dabei erweitern wir Resultate, die für linear-quadratische Probleme bekannt sind, auf den nichtkonvexen Fall. Es müssen insbesondere Beschränktheitsresultate in der L^infty-Norm der semidiskreten und diskreten Zustände gezeigt werden, die unabhängig von den Diskretisierungsparametern gelten. Außerdem erfordert die Diskussion lokal optimaler Lösungen, dass Konvergenzresultate und quadratische Wachstumsbedingungen in denselben Normen betrachtet werden.