@misc{ErdmannLangMateraetal., author = {Erdmann, Bodo and Lang, Jens and Matera, Sebastian and Wilmanski, Krzysztof}, title = {Adaptive Linearly Implicit Methods for Linear Poroelastic Equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9312}, number = {06-37}, abstract = {Adaptive numerical methods in time and space are introduced and studied for linear poroelastic models in two and three space dimensions. We present equivalent models for linear poroelasticity and choose both the {\em displacement--pressure} and the {\em stress--pressure} formulation for our computations. Their discretizations are provided by means of linearly implicit schemes in time and linear finite elements in space. Our concept of adaptivity opens a way to a fast and reliable simulation of different loading cases defined by corresponding boundary conditions. We present some examples using our code {\sf Kardos} and show that the method works efficiently. In particular, it could be used in the simulation of some bone healing models.}, language = {en} } @misc{FranzoneDeuflhardErdmannetal., author = {Franzone, Piero Colli and Deuflhard, Peter and Erdmann, Bodo and Lang, Jens and Pavarino, Luca Franco}, title = {Adaptivity in Space and Time for Reaction-Diffusion Systems in Electrocardiology}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8649}, number = {05-30}, abstract = {Adaptive numerical methods in space and time are introduced and studied for multiscale cardiac reaction-diffusion models in three dimensions. The evolution of a complete heartbeat, from the excitation to the recovery phase, is simulated with both the anisotropic Bidomain and Monodomain models, coupled with either a variant of the simple FitzHugh-Nagumo model or the more complex phase-I Luo-Rudy ionic model. The simulations are performed with the {\sc kardos} library, that employs adaptive finite elements in space and adaptive linearly implicit methods in time. The numerical results show that this adaptive method successfully solves these complex cardiac reaction-diffusion models on three-dimensional domains of moderate sizes. By automatically adapting the spatial meshes and time steps to the proper scales in each phase of the heartbeat, the method accurately resolves the evolution of the intra- and extra-cellular potentials, gating variables and ion concentrations during the excitation, plateau and recovery phases.}, language = {en} } @misc{KoberErdmannLangetal., author = {Kober, Cornelia and Erdmann, Bodo and Lang, Jens and Sader, Robert and Zeilhofer, Hans-Florian}, title = {Adaptive Finite Element Simulation of the Human Mandible Using a New Physiological Model of the Masticatory Muscles}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7917}, number = {04-16}, abstract = {Structural mechanics simulation of bony organs is of general medical and biomechanical interest, because of the interdependence of the inner architecture of bone and its functional loading already stated by Wolff in 1892. This work is part of a detailed research project concerning the human mandible. By adaptive finite element techniques, stress/strain profiles occurring in the bony structure under biting were simulated. Estimates of the discretization errors, local grid refinement, and multilevel techniques guarantee the reliability and efficiency of the method. In general, our simulation requires a representation of the organ's geometry, an appropriate material description, and the load case due to teeth, muscle, or joint forces. In this paper, we want to focus on the influence of the masticatory system. Our goal is to capture the physiological situation as far as possible. By means of visualization techniques developed by the group, we are able to extract individual muscle fibres from computed tomography data. By a special algorithm, the fibres are expanded to fanlike (esp. for the musc. temporalis) coherent vector fields similar to the anatomical reality. The activity of the fibres can be adapted according to compartmentalisation of the muscles as measured by electromyological experiments. A refined sensitivity analysis proved remarkable impact of the presented approach on the simulation results.}, language = {en} } @misc{ErdmannLangRoitzsch, author = {Erdmann, Bodo and Lang, Jens and Roitzsch, Rainer}, title = {KARDOS - User"s Guide}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7096}, number = {02-42}, abstract = {The adaptive finite element code {\sc Kardos} solves nonlinear parabolic systems of partial differential equations. It is applied to a wide range of problems from physics, chemistry, and engineering in one, two, or three space dimensions. The implementation is based on the programming language C. Adaptive finite element techniques are employed to provide solvers of optimal complexity. This implies a posteriori error estimation, local mesh refinement, and preconditioning of linear systems. Linearely implicit time integrators of {\em Rosenbrock} type allow for controlling the time steps adaptively and for solving nonlinear problems without using {\em Newton's} iterations. The program has proved to be robust and reliable. The user's guide explains all details a user of {\sc Kardos} has to consider: the description of the partial differential equations with their boundary and initial conditions, the triangulation of the domain, and the setting of parameters controlling the numerical algorithm. A couple of examples makes familiar to problems which were treated with {\sc Kardos}. We are extending this guide continuously. The latest version is available by network: {\begin{rawhtml} Downloads. \end{rawhtml}}}, language = {en} } @misc{ErdmannKoberLangetal., author = {Erdmann, Bodo and Kober, Cornelia and Lang, Jens and Sader, Robert and Zeilhofer, Hans-Florian and Deuflhard, Peter}, title = {Efficient and Reliable Finite Element Methods for Simulation of the Human Mandible}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6403}, number = {01-14}, abstract = {By computed tomography data (CT), the individual geometry of the mandible is quite well reproduced, also the separation between cortical and trabecular bone. Using anatomical knowledge about the architecture and the functional potential of the masticatory muscles, realistic situations were approximated. The solution of the underlying partial differential equations describing linear elastic material behaviour is provided by an adaptive finite element method. Estimations of the discretization error, local grid refinement, and multilevel techniques guarantee the reliability and efficiency of the method.}, language = {en} } @misc{LangErdmann, author = {Lang, Jens and Erdmann, Bodo}, title = {Adaptive Linearly Implicit Methods for Heat and Mass Transfer Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5892}, number = {00-21}, abstract = {Dynamical process simulation of complex real-life problems often requires the use of modern algorithms, which automatically adapt both the time and space discretization in order to get error-controlled approximations of the solution. In this paper, a combination of linearly implicit time integrators of Rosenbrock type and adaptive multilevel finite elements based on a posteriori error estimates is presented. This approach has proven to work quite satisfactorily for a wide range of challenging practical problems. We show the performance of our adaptive method for two applications that arise in the study of flame balls and brine transport in porous media.}, language = {en} } @misc{LangMerz, author = {Lang, Jens and Merz, Wilhelm}, title = {Two-Dimensional Adaptive Simulation of Dopant Diffusion in Silicon}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5716}, number = {00-03}, abstract = {One important step in the fabrication of silicon-based integrated circuits is the creation of semiconducting areas by diffusion of dopant impurities into silicon. Complex models have been developed to investigate the redistribution of dopants and point defects. In general, numerical analysis of the resulting PDEs is the central tool to assess the modelling process. We present an adaptive approach which is able to judge the quality of the numerical approximation and which provides an automatic mesh improvement. Using linearly implicit methods in time and multilevel finite elements in space, we are able to integrate efficiently the arising reaction-drift-diffusion equations with high accuracy. Two different diffusion processes of practical interest are simulated.}, language = {en} } @phdthesis{Lang, author = {Lang, Jens}, title = {Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4089}, number = {SC-99-20}, abstract = {This monograph has been written to illustrate the interlocking of theory, algorithm, and application in developing solution techniques for complex PDE systems. A deep theoretical understanding is necessary to produce a powerful idea leading to a successful algorithm. Efficient and robust implementation is the key to make the algorithm perform satisfactorily. The extra insight obtained by solving real--life problems brings out the structure of the method more clearly and suggests often ways to improve the numerical algorithm. It is my intention to impart the beauty and complexity found in both the theoretical investigation of the adaptive algorithm proposed here, i.e., the coupling of Rosenbrock methods in time and multilevel finite elements in space, and its realization. I hope that this method will find many more interesting applications.}, language = {en} } @misc{LourencoRosaCastroetal., author = {Lourenco, Maria Jos{\´e} and Rosa, Samuel Costa S. and Castro, Carlos Alberto Nieto de and Albuquerque, C. and Erdmann, Bodo and Lang, Jens and Roitzsch, Rainer}, title = {Simulation of the Transient Heating in an Unsymmetrical Coated Hot--Strip Sensor with a Self--Adaptive Finite Element Method}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3656}, number = {SC-98-22}, abstract = {The transient heating in an unsymmetrical coated hot--strip sensor was simulated with a self--adaptive finite element method. The first tests of this model show that it can determine with a small error the thermal conductivity of liquids, from the transient temperature rise in the hot--strip, deposited in a substrate and coated by an alumina spray.}, language = {en} } @misc{Lang, author = {Lang, Jens}, title = {Adaptive Incompressible Flow Computations with Linearly Implicit Time Discretization and Stabilized Finite Elements}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3599}, number = {SC-98-16}, abstract = {Fully adaptive solutions of imcompressible flow problems employing the discretization sequence first in time then in space are presented. The time discretization is done by linearly implicit one--step methods possibly of high order with automatic step size control. A posteriori error estimates for the stabilized finite element discretization in space are obtained by solving local Dirichlet problems with higher accuracy. Once those estimates have been computed, we are able to control time and space grids with respect to required tolerances and necessary computational work. The devised method is applied to two benchmark problems in 2D.}, language = {en} }