Resources for and from Research, Teaching and Studying
Recent Submissions
Recently, we devised an approach to a posteriori error analysis, which clarifies the role of oscillation and where oscillation is bounded in terms of the current approximation error. Basing upon this approach, we derive plain convergence of adaptive linear finite elements approximating the Poisson problem. The result covers arbritray H^-1-data and characterizes convergent marking strategies.
The purpose of present study is to numerically investigate the natural convection flow of Ostwalde-de Waele type power law non-Newtonian fluid along the surface of rotating axi-symmetric round-nosed body. For computational purpose rotating hemisphere is used as a case study in order to examine the heat transfer mechanism near such transverse curvature geometries. The numerical scheme is applied after converting the dimensionless system of equations into primitive variable formulations. Implic...
In an unconventional approach to combining the very successful Finite Element Methods (FEM) for PDE-based simulation with techniques evolved from the domain of Machine Learning (ML) we employ approximate inverses of the system matrices generated by neural networks in the linear solver. We demonstrate the success of this solver technique on the basis of the Poisson equation which can be seen as a fundamental PDE for many practically relevant simulations [Turek 1999]. We use a basic Richardson ...
Based on the benchmark results in [1] for a 2D rising bubble, we present the extension towards 3D providing test cases with corresponding reference results, following the suggestions in [2]. Additionally, we include also an axisymmetric configuration which allows 2.5D simulations and which provides further possibilities for validation and evaluation of numerical multiphase flow components and software tools in 3D.
In this article we present a Radial Basis Function (RBF)-Finite Difference (FD) level set based method for numerical solution of partial differential equations (PDEs) of the reaction-diffusion-convection type on an evolving-in-time hypersurface Γ (t). In a series of numerical experiments we study the accuracy and robustness of the proposed scheme and demonstrate that the method is applicable to practical models.
An analysis is performed to study the two-phase natural convection flow of nano fluid along a vertical wavy surface. The model includes equations expressing conservation of total mass, momentum and thermal energy for two-phase nano fluid. Primitive variable formulations (PVF) are used to transform the dimensionless boundary layer equations into a convenient coordinate system and the resulting equations are integrated numerically via implicit finite difference iterative scheme. The effect ...
We revisit the MIT Benchmark 2001 and introduce a viscoelastic constitutive law into the fluid in motion. Our goal is to study the effect of viscoelasticity into the periodical behavior of the physical quantities of the corresponding benchmark. We use a robust numerical technique in simulating complex fluid flow problems based on higher order Finite Element discretization. While marching in time, an A-stable method of second order is favorable, i.e Crank-Nicolson scheme, to reproduce peri...
This dissertation presents a fully implicit, monolithic finite element solution scheme to effectively solve the governing set of differential algebraic equations of incompressible poroelastodynamics. Thereby, a two-dimensional, biphasic, saturated porous medium model with intrinsically coupled and incompressible solid and fluid constituents is considered. Our schemes are based on some well-accepted CFD techniques, originally developed for the efficient simulation of incompressible flow proble...
We study combinatorial problems with ellipsoidal uncertainty in the objective function concerning their theoretical and practical solvability. Ellipsoidal uncertainty is a natural model when the coefficients are normally distributed random variables. Robust versions of typical combinatorial problems can be very hard to solve compared to their linear versions. Complexity and approaches differ fundamentally depending on whether uncorrelated or correlated uncertainty occurs. We distinguish betw...
The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for some two-dimension...
We develop a general convergence theory for adaptive discontinu- ous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi interpolation into a newly developed limit space of the adaptively create...
In this paper, we present an entropy stable scheme for solving the compressible Navier-Stokes equations in two space dimensions. Our scheme uses entropy variables as degrees of freedom. It is an extension of an existing spacetime discontinuous Galerkin method for solving the compressible Euler equations. The physical diffusion terms are incorporated by means of the symmetric (SIPG) or nonsymmetric (NIPG) interior penalty method, resulting in the two versions ST-SDSC-SIPG and ST-SDSC-NIP...
This thesis is concerned with new numerical and algorithmic tools for flows with pressure and shear dependent viscosity together with the necessary background of the generalized Navier-Stokes equations. In general the viscosity of a material can be constant, e.g. water and this kind of fluid is called as Newtonian fluid. However the flow can be complicated for quasi-Newtonian fluid, where the viscosity can depend on some physical quantity. For example, the viscosity of Bingham fluid is a fun...
This thesis is concerned with the development of new approaches to redistancing and conservation of mass in finite element methods for the level set transport equation. The first proposed method is a PDE- and optimization-based redistancing scheme. In contrast to many other PDE-based redistancing techniques, the variational formulation derived from the minimization problem is elliptic and can be solved efficiently using a simple fixed-point iteration method. Artificial displacements are effe...
A robust mesh optimisation method is presented that directly enforces the resulting deformation to be orientation preserving. Motivated by aspects from mathematical elasticity, the energy functional of the mesh deformation can be related to a stored energy functional of a hyperelastic material. Formulating the functional in the principal invariants of the deformation gradient allows fine grained control over the resulting deformation. Solution techniques for the arising nonconvex and highl...
The aim of this paper is to present a boundary-layer analysis of two-phase dusty non-Newtonian fluid flow along a vertical surface by using a modified power-law viscosity model. This investigation particularly reports the flow behavior of spherical particles suspended in the non-Newtonian fluid. The governing equations are transformed into nonconserved form and then solved straightforwardly by implicit finite difference method. The numerical results of rate of heat transfer, rate of shea...
A viscous damage model including two damage variables - a local and a nonlocal one - coupled through a penalty term is investigated on three different levels: unique solvability, behaviour as the penalization parameter approaches ∞ and optimal control. Existence, uniqueness and regularity of the solutions are proven. In particular, we give an improved result regarding spacial regularity of the nonlocal damage. Lipschitz continuity as well as Fréchet-differentiability of the solution operato...
In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini’s problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton’s method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous...
In this paper, numerical solutions to thermally radiating Marangoni convection of dusty fluid flow along a vertical wavy surface are established. The results are obtained with the understanding that the dust particles are of uniform size and dispersed in optically thick fluid. The numerical solutions of the dimensionless transformed equations are obtained through straightforward implicit finite difference scheme. In order to analyze the influence of various controlling parameters, results are...