arXiv subjects

Ulrich Langer

Publications and source records attributed to Ulrich Langer.

52 records · Page 3Linked to original sources

arXiv2015

Dual-Primal Isogeometric Tearing and Interconnecting Solvers for large-scale systems of multipatch continuous Galerkin IgA equations

The dual-primal isogeometric tearing and interconnecting (IETI-DP) method is the adaption of the dual-primal finite element tearing and interconnecting (FETI-DP) method to isogeometric analysis of scalar elliptic boundary value problems like, e.g., diffusion problems with heterogeneous diffusion coefficients. The purpose of this paper is to extent the already existing results on condition number estimates to multi-patch domains, which consist of different geometrical mappings for each patch. Another purpose is to prove a polylogarithmic condition number bound for the preconditioned system with stiffness scaling in case of $C^0$ smoothness across patch interfaces. Numerical experiments validate the presented theory.

math.NA↗

arXiv2015

Functional a posteriori error estimates for time-periodic parabolic optimal control problems

This paper is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds.

math.OC↗

arXiv2015

Discontinuous Galerkin Isogeometric Analysis of Elliptic Diffusion Problems on Segmentations with Gaps

We propose a new discontinuous Galerkin Isogeometric Analysis (IgA) technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with non-matching interfaces. Due to an incorrect segmentation procedure, it may happen that the interfaces of adjacent subdomains don't coincide. In this way, gap regions, which are not present in the original physical domain, are created. In this paper, the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains. We apply a multi-patch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem. The last formulation is further modified by deriving interface conditions without the presence of the solution in the gap. Finally, the solution of this modified problem is approximated by a special discontinuous Galerkin IgA technique. The ideas are illustrated on a model diffusion problem with discontinuous diffusion coefficients. We develop a rigorous theoretical framework for the proposed method clarifying the influence of the gap size onto the convergence rate of the method. The theoretical estimates are supported by numerical examples in two- and three-dimensional computational domains.

math.NA↗

arXiv2015

Mesh Grading in Isogeometric Analysis

This paper is concerned with the construction of graded meshes for approximating so-called singular solutions of elliptic boundary value problems by means of multipatch discontinuous Galerkin Isogeometric Analysis schemes. Such solutions appear, for instance, in domains with re-entrant corners on the boundary of the computational domain, in problems with changing boundary conditions, in interface problems, or in problems with singular source terms. Making use of the analytic behavior of the solution, we construct the graded meshes in the neighborhoods of such singular points following a multipatch approach. We prove that appropriately graded meshes lead to the same convergence rates as in the case of smooth solutions with approximately the same number of degrees of freedom. Representative numerical examples are studied in order to confirm the theoretical convergence rates and to demonstrate the efficiency of the mesh grading technology in Isogeometric Analysis.

math.NA↗

arXiv2015

Space-Time Isogeometric Analysis of Parabolic Evolution Equations

We present and analyze a new stable space-time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IgA spaces.

math.NA↗

arXiv2015

The Mid-Infrared Instrument for JWST, II: Design and Build

The Mid-InfraRed Instrument (MIRI) on the James Webb Space Telescope (JWST) provides measurements over the wavelength range 5 to 28.5 microns. MIRI has, within a single 'package', four key scientific functions: photometric imaging, coronagraphy, single-source low-spectral resolving power (R ~ 100) spectroscopy, and medium-resolving power (R ~ 1500 to 3500) integral field spectroscopy. An associated cooler system maintains MIRI at its operating temperature of < 6.7 K. This paper describes the driving principles behind the design of MIRI, the primary design parameters, and their realization in terms of the 'as-built' instrument. It also describes the test program that led to delivery of the tested and calibrated Flight Model to NASA in 2012, and the confirmation after delivery of the key interface requirements.

astro-ph.IM↗

arXiv2015

Numerical simulation of parabolic moving and growing interface problems using small mesh deformation

In this work, we develop a cutting method for solving problems with moving and growing interfaces in 3D. This new method is able to resolve large displacement or deformation of immersed objects by combining the Arbitrary Lagrangian-Eulerian method with only small local mesh deformation defined on the reference domain, that is decomposed into the macro-elements. The linear system of algebraic equations arising after the temporal and spatial discretizations of a model parabolic interface heat-conduction-like problem with vector-valued functions is solved by either an all-at-once or a segregated algebraic multigrid method.

math.NA↗

arXiv2015

Convection-adapted BEM-based FEM

We present a new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.

math.NA↗

arXiv2015

Shape optimization of an electric motor subject to nonlinear magnetostatics

The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

math.OC↗

arXiv2014

Functional a posteriori error estimates for parabolic time-periodic boundary value problems

The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests confirm the efficiency of the a posteriori error bounds derived.

math.NA↗

arXiv2014

Multipatch Discontinuous Galerkin Isogeometric Analysis

Isogeometric analysis (IgA) uses the same class of basis functions for both, representing the geometry of the computational domain and approximating the solution. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This multi-patch representation corresponds to a decomposition of the computational domain into non-overlapping subdomains also called patches in the geometrical framework. We will present discontinuous Galerkin (dG) methods that allow for discontinuities across the subdomain (patch) boundaries. The required interface conditions are weakly imposed by the dG terms associated with the boundary of the sub-domains. The construction and the corresponding discretization error analysis of such dG multi-patch IgA schemes will be given for heterogeneous diffusion model problems in volumetric 2d and 3d domains as well as on open and closed surfaces. The theoretical results are confirmed by numerous numerical experiments which have been performed in G+SMO. The concept and the main features of the IgA library G+SMO are also described.

math.NA↗

arXiv2014

Numerical Simulation of Fluid-Structure Interaction Problems with Hyperelastic Models: A Monolithic Approach

In this paper, we consider a monolithic approach to handle coupled fluid-structure interaction problems with different hyperelastic models in an all-at-once manner. We apply Newton's method in the outer iteration dealing with nonlinearities of the coupled system. We discuss preconditioned Krylov sub-space, algebraic multigrid and algebraic multilevel methods for solving the linearized algebraic equations. Finally, we compare the results of the monolithic approach with those of the corresponding partitioned approach that we studied in our previous work.

math.NA↗

arXiv2014

Analysis of Multipatch Discontinuous Galerkin IgA Approximations to Elliptic Boundary Value Problems

In this work, we study the approximation properties of multi-patch dG-IgA methods, that apply the multipatch Isogeometric Analysis (IgA) discretization concept and the discontinuous Galerkin (dG) technique on the interfaces between the patches, for solving linear diffusion problems with diffusion coefficients that may be discontinuous across the patch interfaces. The computational domain is divided into non-overlapping sub-domains, called patches in IgA, where $B$-splines, or NURBS finite dimensional approximations spaces are constructed. The solution of the problem is approximated in every sub-domain without imposing any matching grid conditions and without any continuity requirements for the discrete solution across the interfaces. Numerical fluxes with interior penalty jump terms are applied in order to treat the discontinuities of the discrete solution on the interfaces. We provide a rigorous a priori discretization error analysis for problems set in 2d- and 3d- dimensional domains, with solutions belonging to $W^{l,p}, l\geq 2,{\ } p\in ({2d}/{(d+2(l-1))},2]$. In any case, we show optimal convergence rates of the discretization with respect to the dG - norm.

math.NA↗

arXiv2014

Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces

Isogeometric analysis uses the same class of basis functions for both, representing the geometry of the computational domain and approximating the solution. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This patch representation corresponds to a domain decomposition. In this paper, we present for the first time a Discontinuous Galerkin (DG) Method that allows for discontinuities only along the sub-domain (patch) boundaries. The required smoothness is obtained by the DG terms associated with the boundary of the sub-domains. The construction and corresponding discretization error analysis of such DG scheme is presented for Elliptic PDEs in a 2D as well as on open and closed surfaces. Furthermore, we present numerical results to confirm the theory presented.

math.NA↗

arXiv2013

Numerical Simulation of Fluid-Structure Interaction Problems with Hyperelastic Models I: A Partitioned Approach

In this work, we consider fluid-structure interaction simulation with nonlinear hyperelastic models in the solid part. We use a partitioned approach to deal with the coupled nonlinear fluid-structure interaction problems. We focus on handling the nonlinearity of the fluid and structure sub-problems, the near-incompressibility of materials, the stabilization of employed finite element discretization, and the robustness and efficiency of Krylov subspace and algebraic multigrid methods for the linearized algebraic equations.

math.NA↗

arXiv2013

Topology Optimization of Electric Machines based on Topological Sensitivity Analysis

Topological sensitivities are a very useful tool for determining optimal designs. The topological derivative of a domain-dependent functional represents the sensitivity with respect to the insertion of an infinitesimally small hole. In the gradient-based ON/OFF method, proposed by M. Ohtake, Y. Okamoto and N. Takahashi in 2005, sensitivities of the functional with respect to a local variation of the material coefficient are considered. We show that, in the case of a linear state equation, these two kinds of sensitivities coincide. For the sensitivities computed in the ON/OFF method, the generalization to the case of a nonlinear state equation is straightforward, whereas the computation of topological derivatives in the nonlinear case is ongoing work. We will show numerical results obtained by applying the ON/OFF method in the nonlinear case to the optimization of an electric motor.

math.OC↗