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...

We present a new approach to enforcing local maximum principles in finite element schemes for the compressible Euler equations. In contrast to synchronized limiting techniques for systems of conservation laws, the density, momentum, and total energy are constrained in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum are adjusted to incorporate the irreversible effect of density cha...

This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system as well as existence of optimal solutions, admitting global-in-time solutions, to the optimization problem were shown in the the companion ...

In this thesis we propose a coordinate ascent method for a class of semidefinite programming problems arising in the reformulation of non-convex quadratic optimization problems where the variables are restricted to subsets of the integer numbers. It is known that non-convex quadratic integer problems are NP-hard for two reasons: the non-convexity of the objective function and the restrictions of integrality on the variables. Therefore no polynomial time algorithm is known for solving this cl...

The aim of present paper is to establish the detailed numerical results for bioconvection boundary-layer flow of two-phase dusty nanofluid. The dusty fluid contains gyrotactic microorganisms along an isothermally heated vertical wall. The physical mechanisms responsible for the slip velocity between the dusty fluid and nanoparticles, such as thermophoresis and Brownian motion, are included in this study. The influence of the dusty nanofluid on heat transfer and flow characteristics are i...

This paper addresses the problem of determining cost-minimal process designs for ideal multi-component distillation columns. The special case of binary distillation was considered in former work. Therein, a problem-specific bound-tightening strategy based on monotonic mole fraction profiles of single components was developed to solve the corresponding MINLPs, globally. In the multi-component setting, the mole fraction profiles of single components may not be monotonic, which is why the bound-...

In this paper numerical solutions of a two-phase natural convection dusty fluid flow are presented. The two-phase particulate suspension is investigated along a vertical cone by keeping variable viscosity and thermal conductivity of the carrier phase. Comprehensive flow formations of the gas and particle phases are given with the aim to predict the behavior of heat transport across the heated cone. The influence of i) air with particles, water with particles and oil with particles are shown o...

We present a unique approach for integrating research in High Performance Computing (HPC) as well as photovoltaic (PV) solar farming and battery technologies into a container-based compute center designed for a maximum of energy efficiency, performance and extensibility/scalability. We use NVIDIA Jetson TK1 boards to build a considerably dimensioned cluster of 60 low-power GPUs, attach a 7:5 kWp solar farm and a 8 kWh Lithium-Ion battery power supply and integrate everything into a single-co...